Chapter 9 Financial Mathematics (Q & A)

A perpetual annuity, or perpetuity, is a type of annuity that pays an infinite sequence of equal payments over a period of time. Unlike a regular annuity, payments for a perpetuity continue indefinitely, theoretically forever.

Derivation of the Present Value of a Perpetuity:

The present value (PV) of a perpetuity can be thought of as the sum of an infinite geometric series. If $A$ is the payment made at the end of each period and $r$ is the interest rate per period, the present value of each payment is:

• First payment (at end of period 1): $\frac{A}{(1+r)^1}$
• Second payment (at end of period 2): $\frac{A}{(1+r)^2}$
• Third payment (at end of period 3): $\frac{A}{(1+r)^3}$
• ... and so on indefinitely.

So, the present value of the perpetuity is the sum:

This is an infinite geometric series with:

• First term ($a$) = $\frac{A}{1+r}$
• Common ratio ($k$) = $\frac{1}{1+r}$

Since $r > 0$, we have $0 < \frac{1}{1+r} < 1$, which means the series converges.

The sum of an infinite geometric series is given by the formula $S = \frac{a}{1-k}$, provided $|k| < 1$.

Substituting the values of $a$ and $k$:

Simplify the denominator:

Now substitute this back into the PV formula:

Multiply the numerator by the reciprocal of the denominator:

Cancel out the $(1+r)$ terms:

Conclusion:

Based on the derivation, the present value of a perpetuity paying an amount $A$ at the end of each period, with an interest rate $r$ per period, is given by $PV = A / r$.

Comparing this with the given options:

• (A) $PV = A \times r$
• (B) $PV = A / r$
• (C) $PV = A \times (1+r)$
• (D) $PV = A / (1+r)$

Therefore, the correct option is (B).

The question asks to identify the term for a fund established to accumulate a specific sum of money in the future through regular payments. Let's analyze each option.

Explanation of Options:

(A) Annuity Due:

An annuity due is a series of equal payments made at the beginning of each period. While it involves regular payments, it primarily describes the timing of payments in an annuity, rather than the specific purpose of accumulating a future sum for a defined objective like repaying a debt or replacing an asset. It's a type of annuity, but not the specific fund type described.

(B) Perpetuity:

A perpetuity (or perpetual annuity) is an annuity that pays a fixed sum of money at regular intervals indefinitely, meaning the payments continue forever. This concept is about receiving a stream of income, not about accumulating a specific future sum by making payments into a fund.

(C) Sinking Fund:

A sinking fund is a fund established by a company or organization to accumulate money over time to repay a debt, replace a depreciating asset, or achieve some other specific future financial objective. Regular payments are made into this fund, which then accumulates with interest, to reach the target sum by a predetermined future date. This perfectly matches the description given in the question: "A fund created to accumulate a specific sum of money in the future by making regular payments."

(D) Endowment Fund:

An endowment fund is typically a donation of money or property to a non-profit organization or institution, where the principal amount is invested, and only the income generated from the investment is used for the institution's purposes. The goal is to provide a permanent source of income, and the principal itself is generally not spent. While it involves accumulating money, its primary purpose is not for making regular payments to reach a specific target sum for debt repayment or asset replacement in the way a sinking fund does. It's about providing long-term financial support through investment income.

Conclusion:

Based on the definitions, the term that accurately describes "a fund created to accumulate a specific sum of money in the future by making regular payments" is a Sinking Fund.

Therefore, the correct option is (C) Sinking Fund.

Explanation:

Let's break down why the other options are incorrect and why EMI is the right choice:

• Sinking Fund Payments: A sinking fund is a fund established by an organization to accumulate money over time to repay a debt or replace an asset. It's not directly related to the borrower's payments to the lender.
• Perpetuity Payments: A perpetuity is a stream of payments that continues forever. Loan repayments have a defined end date, so this doesn't fit.
• Compound Interest: Compound interest is a method of calculating interest where the interest earned on a principal is added to the principal, and then the next interest calculation is based on the new, higher principal. While compound interest is a component of loan calculations, it isn't the name for the periodic payments themselves.

Equated Monthly Instalments (EMI): EMIs are the most common way loans are repaid. They are fixed payments made at regular intervals (usually monthly) that cover both the principal amount and the interest charged on the loan. The EMI is calculated to ensure the loan is fully repaid over the agreed-upon term.

The correct answer is (C) Equated Monthly Instalments (EMI).

Explanation:

Simple return is calculated as the profit divided by the initial investment.

Profit = $\textsf{₹}12,000 - \textsf{₹}10,000 = \textsf{₹}2,000$

Simple Return = $\frac{\text{Profit}}{\text{Initial Investment}} = \frac{2000}{10000} = 0.2$

Converting to percentage: $0.2 \times 100 = 20\%$

The correct answer is (A) $20\%$.

Explanation:

• Simple interest rate: This is a fixed percentage of the principal, not considering compounding.
• Linear growth rate: This implies a constant amount of increase each period, not a compounded rate.
• Nominal interest rate: This is the stated interest rate without considering the effects of compounding or inflation.

CAGR specifically calculates the average annual growth rate over a period, assuming that profits are reinvested during the term of the investment. That is, it assumes compounding.

The correct answer is (C) Average annual growth rate over multiple periods, assuming compounding.

Explanation:

The linear method of depreciation, also known as the straight-line method, allocates an equal amount of depreciation expense to each period of an asset's useful life. The formula is:

Depreciation Expense = $\frac{\text{Cost of Asset - Salvage Value}}{\text{Useful Life}}$

Since the cost, salvage value, and useful life are fixed at the start, the depreciation expense remains the same each year.

The correct answer is (C) Constant.

Explanation:

The present value of a perpetuity is calculated using the formula:

PV = $\frac{\text{Payment}}{\text{Interest Rate}}$

In this case, the payment is $\textsf{₹}5000$ and the interest rate is $10\%$ (or 0.10).

PV = $\frac{5000}{0.10} = 50000$

The correct answer is (C) $\textsf{₹}50,000$.

Explanation:

The sinking fund payment can be calculated using the future value of an annuity formula:

FV = P $\times$ $\frac{(1+r)^n - 1}{r}$

Where:

• FV = Future Value ($\textsf{₹}10,00,000$)
• P = Annual Payment (what we want to find)
• r = Interest Rate (8% or 0.08)
• n = Number of years (5)

Rearranging the formula to solve for P:

P = $\frac{\text{FV} \times r}{(1+r)^n - 1}$

P = $\frac{10,00,000 \times 0.08}{(1+0.08)^5 - 1}$

P = $\frac{80,000}{(1.08)^5 - 1}$

P = $\frac{80,000}{1.4693 - 1}$

P = $\frac{80,000}{0.4693}$

P ≈ $1,70,461$

The correct answer is (A) $\textsf{₹}1,70,461$ (approx).

Explanation:

Since the interest is compounded monthly, the annual interest rate needs to be divided by the number of months in a year to find the monthly interest rate.

Monthly interest rate ($i$) = $\frac{\text{Annual Interest Rate}}{\text{Number of Months in a Year}}$

$i = \frac{12\%}{12} = 1\%$

The correct answer is (B) $1\%$.

Explanation:

The formula for CAGR is:

CAGR = $\left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{\frac{1}{\text{Number of Years}}} - 1$

In this case:

• Ending Value = $\textsf{₹}80,000$
• Beginning Value = $\textsf{₹}50,000$
• Number of Years = 4

So, the formula becomes:

CAGR = $\left(\frac{80000}{50000}\right)^{1/4} - 1$

The correct answer is (A) $\left(\frac{80000}{50000}\right)^{1/4} - 1$.

Explanation:

The linear method of depreciation calculates depreciation expense as:

Depreciation Expense = $\frac{\text{Cost of Asset - Salvage Value}}{\text{Useful Life}}$

In this case:

• Cost of Asset = $\textsf{₹}5,00,000$
• Salvage Value = $\textsf{₹}50,000$
• Useful Life = 10 years

So, Depreciation Expense = $\frac{5,00,000 - 50,000}{10} = \frac{4,50,000}{10} = \textsf{₹}45,000$

The correct answer is (D) $(\textsf{₹}5,00,000 - \textsf{₹}50,000) / 10 = \textsf{₹}45,000$.

Explanation:

EMI (Equated Monthly Installment) calculation is based on:

• Principal Loan Amount: The initial amount borrowed.
• Interest Rate: The rate at which interest is charged on the loan.
• Loan Tenure: The duration of the loan repayment period.

The borrower's age does not directly affect the EMI calculation itself. While it may be a factor in loan approval, it's not part of the EMI formula.

The correct answer is (D) Borrower's Age.

Explanation:

• Assertion (A): A perpetuity due *does* have a higher present value than an ordinary perpetuity. This is because payments are received at the *beginning* of each period instead of at the end.
• Reason (R): Receiving payments earlier *does* increase their present value due to the time value of money.

Since both the assertion and the reason are true, and the reason correctly explains the assertion, the answer is (A).

The correct answer is (A) Both A and R are true and R is the correct explanation of A.

Explanation:

A sinking fund is designed to accumulate a specific amount of money (future value) by making regular, periodic payments (an annuity). The future value of an annuity formula helps determine what that periodic payment needs to be to reach the desired future value.

• (A) is incorrect because we are looking for the *payment* amount, not the current value.
• (B) is incorrect because a sinking fund involves multiple payments, not a single present amount.
• (D) is incorrect because sinking funds are for accumulating savings, not repaying loans.

The correct answer is (C) We are calculating the periodic payment needed to reach a future target amount.

Explanation:

The standard formula for calculating EMI (Equated Monthly Installment) is:

$EMI = P \times \frac{i(1+i)^n}{(1+i)^n - 1}$

Where:

• $P$ = Principal Loan Amount
• $i$ = Interest Rate per period
• $n$ = Number of periods

The correct answer is (A) $EMI = P \times \frac{i(1+i)^n}{(1+i)^n - 1}$.

Explanation:

The nominal rate of return is the stated return on an investment without considering the impact of inflation. The real rate of return adjusts the nominal rate for inflation to reflect the true purchasing power of the investment's return.

The correct answer is (B) Inflation.

Explanation:

First, calculate the annual depreciation expense:

Total depreciation after 5 years = Purchase Price - Book Value = $\textsf{₹}10,00,000 - \textsf{₹}7,00,000 = \textsf{₹}3,00,000$

Annual depreciation expense = $\frac{\text{Total Depreciation}}{\text{Number of Years}} = \frac{3,00,000}{5} = \textsf{₹}60,000$

Next, calculate the total depreciation over the asset's useful life (10 years):

Total depreciation over 10 years = Annual Depreciation Expense $\times$ Useful Life = $\textsf{₹}60,000 \times 10 = \textsf{₹}6,00,000$

Finally, calculate the salvage value:

Salvage Value = Purchase Price - Total Depreciation = $\textsf{₹}10,00,000 - \textsf{₹}6,00,000 = \textsf{₹}4,00,000$

However, the options provided don't include $\textsf{₹}4,00,000$. Let's re-examine the approach.

Since the book value after 5 years is $\textsf{₹}7,00,000$, it means that in the remaining 5 years, the asset will depreciate by another $\textsf{₹}7,00,000 - \text{Salvage Value}$ to reach its salvage value.

The annual depreciation is $\textsf{₹}60,000$. So, over the next 5 years it will depreciate by $\textsf{₹}60,000 * 5 = \textsf{₹}3,00,000$.

Therefore, Salvage Value = $\textsf{₹}7,00,000 - \textsf{₹}3,00,000 = \textsf{₹}4,00,000$.

There appears to be a mistake in the provided options. If the question meant that asset was purchased for $\textsf{₹}10,00,000$ and the "Depreciable amount" is $\textsf{₹}7,00,000$, we would do:

$\textsf{₹}10,00,000$ - $\textsf{₹}7,00,000$ = $\textsf{₹}3,00,000$.

In this instance the answer might be closest to (C) $\textsf{₹}2,00,000$

The correct answer is none of the option is correct . The salvage value should be $\textsf{₹}4,00,000$.

Explanation:

The annual interest rate is $9\%$. Since the interest is compounded monthly, we need to divide the annual rate by 12 to get the monthly interest rate.

Monthly interest rate ($i$) = $\frac{\text{Annual Interest Rate}}{12} = \frac{9\%}{12} = 0.75\%$

The correct answer is (B) $0.75\%$.

Explanation:

The loan tenure is 20 years, and the payments are made monthly. To find the total number of monthly payments, we need to multiply the number of years by the number of months in a year.

Total number of monthly payments ($n$) = Loan Tenure (in years) $\times$ Number of Months in a Year = $20 \times 12 = 240$

The correct answer is (C) $20 \times 12 = 240$.

Explanation:

We are given the formula:

$EMI = P \times \frac{i(1+i)^n}{(1+i)^n - 1}$

Where:

• $P = 20,00,000$
• $i = 0.0075$
• $n = 240$

Plugging in the values:

$EMI = 20,00,000 \times \frac{0.0075(1+0.0075)^{240}}{(1+0.0075)^{240} - 1}$

$EMI = 20,00,000 \times \frac{0.0075(1.0075)^{240}}{(1.0075)^{240} - 1}$

First, we need to calculate $(1.0075)^{240}$. This is approximately 6.00915.

$EMI = 20,00,000 \times \frac{0.0075 \times 6.00915}{6.00915 - 1}$

$EMI = 20,00,000 \times \frac{0.045068625}{5.00915}$

$EMI = 20,00,000 \times 0.0090$ (approx)

$EMI = 18,000$ (approx)

The correct answer is (A) $\textsf{₹}18,000$.

Explanation:

• A perpetuity is a stream of payments that continues indefinitely (forever).
• "Due" indicates that the payments are made at the *beginning* of each period. An ordinary perpetuity has payments at the end of the period.

The correct answer is (B) A series of equal payments made at the beginning of each period indefinitely.

Explanation:

The purpose of a sinking fund is to accumulate enough money over time to meet a specific future obligation, such as repaying a debt or replacing an asset. Therefore, the goal is to reach the future value or replacement cost.

The correct answer is (B) Future value of the debt/replacement cost.

Explanation:

The interest component of the EMI is calculated on the outstanding loan balance.

Interest Component = Outstanding Balance $\times$ Monthly Interest Rate

Interest Component = $\textsf{₹}5,00,000 \times 0.01 = \textsf{₹}5000$

The correct answer is (B) $\textsf{₹}5,00,000 \times 0.01 = \textsf{₹}5000$.

Explanation:

The formula for CAGR is:

CAGR = $\left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{\frac{1}{\text{Number of Years}}} - 1$

In this case:

• Ending Value = $\textsf{₹}1,33,100$
• Beginning Value = $\textsf{₹}1,00,000$
• Number of Years = 3

CAGR = $\left(\frac{1,33,100}{1,00,000}\right)^{\frac{1}{3}} - 1$

CAGR = $\left(1.331\right)^{\frac{1}{3}} - 1$

CAGR = $1.1 - 1$

CAGR = $0.1$

Converting to percentage: $0.1 \times 100 = 10\%$

The correct answer is (A) $10\%$.

Explanation:

Under the linear (straight-line) depreciation method, an asset is depreciated evenly over its useful life until its book value equals its salvage value. The salvage value is the estimated value of the asset at the end of its useful life.

The correct answer is (C) Salvage Value.

Explanation:

A perpetuity due is the same as an ordinary perpetuity, except that the payments are made at the beginning of each period instead of at the end. This means that each payment is received one period earlier.

To find the present value of a perpetuity due, you can take the present value of an ordinary perpetuity and multiply it by $(1+r)$, where $r$ is the interest rate per period.

The correct answer is (B) $1+r$.

Explanation:

• Linear Method, Reducing Balance Method, and Sum of the Years' Digits Method are all standard depreciation methods.
• The Compound Annual Growth Rate (CAGR) method is a measure of investment growth, not a depreciation method.

The correct answer is (D) Compound Annual Growth Rate Method.

Explanation:

The Equated Monthly Installments (EMIs) represent a series of future payments. The loan amount represents the current value of those future payments, discounted back to the present. Therefore, the loan amount is the present value of the annuity of EMI payments.

The correct answer is (A) The loan amount is the present value of all future EMI payments.

Explanation:

For Year 2, the investment starts at $\textsf{₹}120$ and decreases to $\textsf{₹}110$.

Simple Return = $\frac{\text{Ending Value - Beginning Value}}{\text{Beginning Value}}$

Simple Return = $\frac{110 - 120}{120} = \frac{-10}{120} = -0.0833$

Converting to percentage: $-0.0833 \times 100 = -8.33\%$

The correct answer is (D) $-8.33\%$.

Explanation:

Let's match each term with its correct formula:

• (i) Perpetuity (Ordinary): The present value of an ordinary perpetuity is calculated as $A/r$, where $A$ is the periodic payment and $r$ is the interest rate. So, (i) matches with (c).
• (ii) Sinking Fund Payment: The sinking fund payment is calculated to determine the periodic payment needed to reach a future value (FV). The formula is $FV \times \frac{i}{(1+i)^n - 1}$. So, (ii) matches with (d).
• (iii) EMI: The EMI (Equated Monthly Installment) is calculated as $PV \times \frac{i(1+i)^n}{(1+i)^n - 1}$, where $PV$ is the present value (loan amount), $i$ is the interest rate per period, and $n$ is the number of periods. Note here that question uses $PV$ instead of $P$, where $P$ stands for principal amount. So, (iii) matches with (b).
• (iv) CAGR: The Compound Annual Growth Rate (CAGR) is calculated as $\left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{\frac{1}{\text{Number of Periods}}} - 1$. So, (iv) matches with (a).

Therefore, the correct matching is (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a).

The correct answer is (A) (i)-(c), (ii)-(d), (iii)-(b), (iv)-(a).

Explanation:

The present value of a perpetuity is calculated as:

PV = $\frac{\text{Annual Payment}}{\text{Interest Rate}}$

In this case, the annual payment is $\textsf{₹}1000$ and the interest rate is $8\%$ (or 0.08).

PV = $\frac{1000}{0.08} = \textsf{₹}12,500$

The correct answer is (D) $\textsf{₹}1000 / 0.08 = \textsf{₹}12,500$.

Explanation:

The sinking fund payment can be calculated using the future value of an annuity formula:

FV = P $\times$ $\frac{(1+r)^n - 1}{r}$

Where:

• FV = Future Value ($\textsf{₹}5,00,000$)
• P = Annual Payment (what we want to find)
• r = Interest Rate (6% or 0.06)
• n = Number of years (10)

Rearranging the formula to solve for P:

P = $\frac{\text{FV} \times r}{(1+r)^n - 1}$

P = $\frac{5,00,000 \times 0.06}{(1+0.06)^{10} - 1}$

P = $\frac{30,000}{(1.06)^{10} - 1}$

P = $\frac{30,000}{1.7908 - 1}$

P = $\frac{30,000}{0.7908}$

P ≈ $\textsf{₹}37,934$

The correct answer is (A) $\textsf{₹}37,934$.

Explanation:

EMI (Equated Monthly Installment) calculations are primarily used for loans where the interest is compounded (usually monthly). Simple interest loans typically have a different repayment structure, often involving a lump-sum payment of principal and accrued interest at the end of the loan term, or periodic payments of interest only, with the principal repaid at the end.

The correct answer is (B) Compound Interest.

Explanation:

The linear method of depreciation calculates depreciation expense as:

Depreciation Expense = $\frac{\text{Cost of Asset - Salvage Value}}{\text{Useful Life}}$

In this case:

• Cost of Asset = $\textsf{₹}2,00,000$
• Salvage Value = $\textsf{₹}20,000$
• Useful Life = 5 years

So, Depreciation Expense = $\frac{2,00,000 - 20,000}{5} = \frac{1,80,000}{5} = \textsf{₹}36,000$

The correct answer is (D) $(\textsf{₹}2,00,000 - \textsf{₹}20,000) / 5 = \textsf{₹}36,000$.

Explanation:

Book Value = Original Cost - Accumulated Depreciation

From Question 34, the annual depreciation is $\textsf{₹}36,000$.

Accumulated Depreciation after 3 years = $3 \times \textsf{₹}36,000 = \textsf{₹}1,08,000$

Book Value at the end of Year 3 = $\textsf{₹}2,00,000 - \textsf{₹}1,08,000 = \textsf{₹}92,000$

Option C can be described as Salvage Value + (Remaining useful life $\times$ annual depreciation)

However, the simplest way to calculate it is what is expressed in Option A.

The correct answer is (D) Both (A) and (C)

Explanation:

A positive return on an investment means that the investment has increased in value. This implies that the ending value is greater than the beginning value.

The correct answer is (C) Ending Value > Beginning Value.

Explanation:

The present value of a perpetuity is calculated as:

PV = $\frac{\text{Annual Payment}}{\text{Interest Rate}}$

We can rearrange the formula to solve for the annual payment:

Annual Payment = PV $\times$ Interest Rate

In this case, PV = $\textsf{₹}1,00,000$ and the interest rate is $5\%$ (or 0.05).

Annual Payment = $\textsf{₹}1,00,000 \times 0.05 = \textsf{₹}5000$

The correct answer is (A) $\textsf{₹}5000$.

Explanation:

The future value of an annuity (the sinking fund's target amount) is achieved through both the periodic payments and the interest earned on those payments. If the interest rate increases, the fund will earn more interest over time. Therefore, to reach the same target amount, the required annual payment will be lower.

The correct answer is (B) Decrease.

Explanation:

In an EMI (Equated Monthly Installment) structure, the EMI amount remains constant throughout the loan tenure. Initially, a larger portion of the EMI goes towards paying off the interest, and a smaller portion goes towards the principal. As the loan tenure progresses, the outstanding principal decreases, which means the interest charged on the outstanding principal decreases. Therefore, the portion of the EMI going towards interest *decreases*, while the portion going towards principal *increases*.

The correct answer is (B) Interest component.

Explanation:

The formula for CAGR is:

CAGR = $\left(\frac{\text{Ending Value}}{\text{Beginning Value}}\right)^{\frac{1}{\text{Number of Years}}} - 1$

In this case:

• Ending Value = $\textsf{₹}2,00,000$
• Beginning Value = $\textsf{₹}1,00,000$
• Number of Years = 7

CAGR = $\left(\frac{2,00,000}{1,00,000}\right)^{\frac{1}{7}} - 1$

CAGR = $(2)^{\frac{1}{7}} - 1$

The correct answer is (B) $(2)^{1/7} - 1$.

Explanation:

Under the linear depreciation method, Depreciation Expense = (Cost of Asset - Salvage Value) / Useful Life. If the salvage value is zero, then Depreciation Expense = Cost of Asset / Useful Life. Over the asset's entire useful life, the total depreciation will equal the original cost of the asset.

The correct answer is (A) Original Cost.

Explanation:

• Calculating EMI involves determining the fixed payment amount based on the loan's present value.
• Calculating amount needed for sinking fund involves calculating future value to repay a debt or replace an asset.
• Calculating the present value of a perpetuity directly involves determining the present value of an infinite stream of future cash flows.
• Calculating linear depreciation involves calculating an equal depreciation amount to each period of an assets useful life.

The correct answer is (C) Calculating the present value of a perpetuity.

Explanation:

• Assertion (A): A higher interest rate *decreases* the present value of a perpetuity. The present value represents the amount you would need to invest *today* to generate that stream of payments. A higher interest rate means you need to invest *less* today to achieve the same future payments. Therefore, A is *false*.
• Reason (R): The formula for the present value of a perpetuity *is* A/r. As r (the interest rate) increases, the value of A/r *decreases*. Therefore, R is *true*.

Since A is false and R is true, the answer is (D).

The correct answer is (D) A is false but R is true.

Explanation:

The total amount of depreciation is the difference between the original cost of the asset and its salvage value.

Total Depreciation = Original Cost - Salvage Value

Total Depreciation = $\textsf{₹}15,00,000 - \textsf{₹}3,00,000 = \textsf{₹}12,00,000$

The correct answer is (C) $\textsf{₹}15,00,000 - \textsf{₹}3,00,000 = \textsf{₹}12,00,000$.

Explanation:

In the linear method, the annual depreciation expense is the total depreciation divided by the useful life.

From Question 44, the total depreciation is $\textsf{₹}12,00,000$, and the useful life is 8 years.

Annual Depreciation Expense = $\frac{\text{Total Depreciation}}{\text{Useful Life}} = \frac{12,00,000}{8} = \textsf{₹}1,50,000$

The correct answer is (A) $\textsf{₹}12,00,000 / 8 = \textsf{₹}1,50,000$.

Option (A) calculates book value using: Initial Cost - (Years * Annual Depreciation).

$\textsf{₹}15,00,000 - 5 \times \textsf{₹}1,50,000 = \textsf{₹}7,50,000$. This is a standard calculation.

Option (C) calculates: $\textsf{₹}3,00,000 + (8-5) \times \textsf{₹}1,50,000 = \textsf{₹}7,50,000$. This implies initial cost is $\textsf{₹}15,00,000$, annual depreciation is $\textsf{₹}1,50,000$, useful life is 8 years, and salvage value is $\textsf{₹}3,00,000$. The formula seems to be Salvage Value + Depreciation for remaining years, which is not a direct book value calculation but coincidentally yields the correct result.

Since both (A) and (C) arrive at $\textsf{₹}7,50,000$, and (A) is a direct calculation, and (C) is also mathematically consistent if we infer the parameters, both are considered correct.

(D) Both (A) and (C)

The question asks to identify the measure calculated based on the ending value relative to the beginning value over multiple periods, assuming compounding.

Let's analyze the options:

(A) Simple Return: This is calculated as $\frac{Ending Value - Beginning Value}{Beginning Value}$ for a single period. It does not account for compounding over multiple periods.

(B) Nominal Return: This is the return on an investment before adjusting for inflation. While it can be calculated over multiple periods, the basic definition doesn't inherently assume compounding in its calculation method for the *rate* itself, but rather how the value grows. However, the question specifically asks about a calculation method that *assumes compounding* across periods.

(C) Real Return: This is the nominal return adjusted for inflation. Similar to nominal return, its core calculation doesn't specifically describe the compounding mechanism across periods.

(D) CAGR: The Compound Annual Growth Rate is specifically designed to measure the average annual growth of an investment over a period of time longer than one year, by assuming that the investment has been compounded at the end of each year. Its formula explicitly uses the beginning and ending values and the number of years to calculate a smoothed annual rate that reflects compounding.

The formula for CAGR is:

This formula clearly shows it's based on the ending value relative to the beginning value over multiple periods, with the exponent $\frac{1}{Number of Years}$ representing the effect of compounding.

Therefore, the correct option is (D) CAGR.

The nominal interest rate is the stated interest rate before taking inflation into account. When interest is compounded more frequently than annually (e.g., semi-annually, quarterly, monthly), the interest rate per period is calculated by dividing the nominal annual interest rate by the number of compounding periods in a year.

In this case:

Nominal annual interest rate = $10\%$ per annum

Compounding frequency = Semi-annually

Number of compounding periods per year = 2 (since semi-annually means twice a year)

To find the interest rate per period, we divide the nominal annual interest rate by the number of compounding periods:

The interest rate per period is $5\%$. Option (D) correctly shows the calculation and the result.

The question asks how the required payment for a sinking fund installment would change if payments are made at the beginning of each period (annuity due) instead of the end of each period (ordinary annuity), assuming the same future value goal.

Let's consider the mechanics of annuities:

Ordinary Annuity: Payments are made at the end of each period. Each payment earns interest for the duration of the remaining periods.

Annuity Due: Payments are made at the beginning of each period. Each payment earns interest for one additional period compared to an ordinary annuity.

Since each payment in an annuity due earns interest for an extra period, the total future value accumulated from an annuity due will be greater than that from an ordinary annuity, given the same payment amount, interest rate, and number of periods.

Conversely, if the objective is to reach a specific future value (as is the case with a sinking fund), and the payments are made as an annuity due, then each payment contributes more towards the goal due to the extra period of compounding. Consequently, a smaller payment amount is needed to reach the target future value compared to an ordinary annuity.

Therefore, if payments are made at the beginning of each period (annuity due), the required payment would be lower than the ordinary annuity payment to achieve the same sinking fund goal.

The correct option is (B) Lower than the ordinary annuity payment.

An Equated Monthly Installment (EMI) payment for a loan is structured such that the total payment remains constant throughout the loan tenure. However, the proportion of the EMI that goes towards paying the principal and the interest changes over time.

In the early periods of a loan:

• The outstanding loan principal is at its highest.
• The interest charged for the period is calculated on this high outstanding principal.
• Consequently, the interest component of the EMI is relatively high.

As the EMI is a fixed amount, and a larger portion of it is allocated to interest in the early stages, the remaining portion available to pay down the principal is smaller.

Therefore, in the early periods of a loan tenure, the principal component of an EMI payment is typically lower than the interest component.

As the loan progresses, the outstanding principal decreases. This leads to a lower interest amount being calculated for each subsequent period. As the EMI amount remains constant, a larger portion of the EMI is then allocated to the principal component, gradually reducing the outstanding loan balance.

The correct option is (B) Lower than the interest component.

The question asks which concept the formula for CAGR is derived from.

The given formula for CAGR is:

Where:

• $P_n$ is the final value of the investment.
• $P_0$ is the initial value (principal) of the investment.
• $n$ is the number of years.

Let's examine the options:

(A) Simple Interest: Simple interest is calculated only on the initial principal amount. The formula for the future value with simple interest is $FV = P(1+rt)$. This formula does not resemble the CAGR formula.

(B) Compound Interest: Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. The formula for the future value with compound interest is $FV = P(1+r)^n$. If we rearrange this formula to solve for the rate $r$, we get:

This derived formula for the rate $r$ is exactly the CAGR formula, with $FV$ being $P_n$ and $P$ being $P_0$. This indicates that CAGR is a measure of compound growth.

(C) Simple Return: A simple return is the total return over a period, not annualized or compounded. It's calculated as $\frac{Ending Value - Beginning Value}{Beginning Value}$. This does not involve compounding or an exponent of $1/n$.

(D) Depreciation: Depreciation is a decrease in value, whereas CAGR measures growth. While methods like reducing balance depreciation involve compounding principles, the CAGR formula is inherently about growth.

Therefore, the formula for CAGR is directly derived from the principles of Compound Interest.

The correct option is (B) Compound Interest.

Linear depreciation, also known as straight-line depreciation, is a method used to allocate the cost of an asset over its useful life. The basic formula for annual linear depreciation is:

The statement says that if the salvage value is zero, the annual linear depreciation amount is simply the original cost divided by the useful life.

Let's substitute a salvage value of zero into the formula:

This matches the statement given in the question. The statement is a direct application of the linear depreciation formula when the salvage value is zero.

The conditions in options (C) and (D) are irrelevant to the fundamental calculation of linear depreciation. The salvage value being zero is the only condition that simplifies the formula in this manner.

Therefore, the statement is true.

The correct option is (A) True.

A perpetuity is a stream of equal payments that continues indefinitely.

The present value (PV) of an ordinary perpetuity (where payments occur at the end of each period) is given by the formula:

Where:

$A$ = the amount of each payment

$r$ = the interest rate per period

A perpetuity due is a perpetuity where payments occur at the beginning of each period, starting immediately.

To find the present value of a perpetuity due, we can consider it as an immediate payment of $A$ plus the present value of an ordinary perpetuity that starts one period later. Alternatively, we can think of it as each payment in the ordinary perpetuity being discounted one period less.

So, the present value of a perpetuity due can be calculated as:

$PV_{due} = \text{First Payment} + PV \text{ of remaining ordinary perpetuity}$

$PV_{due} = A + \frac{A}{r}$

Another way to derive this is by noting that each payment in a perpetuity due is received one period earlier than in an ordinary perpetuity. Therefore, the present value of each payment is higher by a factor of $(1+r)$.

So, $PV_{due} = PV_{ordinary} \times (1+r)$ is incorrect for a perpetuity where the first payment is immediate. The relationship is more direct: the first payment is immediate (its PV is $A$), and the subsequent payments form an ordinary perpetuity that starts one period later.

Let's confirm with the standard financial formula for a perpetuity due:

$PV_{due} = A + A(1+r)^{-1} + A(1+r)^{-2} + \dots$

This is a geometric series with first term $a=A$ and common ratio $x = (1+r)^{-1}$. The sum of an infinite geometric series is $\frac{a}{1-x}$ provided $|x| < 1$.

In our case, the series for the payments *after* the first immediate one is $A(1+r)^{-1} + A(1+r)^{-2} + \dots$. This is an ordinary perpetuity starting one period later, with a payment of $A/(1+r)$ at time 1, $A/(1+r)^2$ at time 2, etc. The present value of this stream is $\frac{A/(1+r)}{r}$.

So, $PV_{due} = A + \frac{A/(1+r)}{r} = A + \frac{A}{r(1+r)}$ is also not fitting the options directly.

Let's use the relationship that a perpetuity due is simply an ordinary perpetuity where each cash flow is shifted one period earlier. So, the present value of a perpetuity due is the present value of an ordinary perpetuity compounded forward by one period.

The present value of an ordinary perpetuity is $\frac{A}{r}$.

If the first payment of $A$ occurs at time 0 (immediately), the next payment of $A$ occurs at time 1, the next at time 2, and so on.

$PV_{due} = A \times (1+r)^0 + A \times (1+r)^{-1} + A \times (1+r)^{-2} + \dots$

$PV_{due} = A + A \left[ (1+r)^{-1} + (1+r)^{-2} + \dots \right]$

The expression in the bracket is an infinite geometric series with the first term $a = (1+r)^{-1}$ and common ratio $x = (1+r)^{-1}$.

The sum of this geometric series is $\frac{a}{1-x} = \frac{(1+r)^{-1}}{1 - (1+r)^{-1}} = \frac{\frac{1}{1+r}}{1 - \frac{1}{1+r}} = \frac{\frac{1}{1+r}}{\frac{1+r-1}{1+r}} = \frac{\frac{1}{1+r}}{\frac{r}{1+r}} = \frac{1}{r}$.

Therefore, $PV_{due} = A + A \left[\frac{1}{r}\right] = A + \frac{A}{r}$.

Thus, the present value of a perpetuity due is $A + A/r$. This can also be written as $A(1 + 1/r)$.

The correct option is (B) $A + A/r$.

Let's analyze each statement regarding a sinking fund:

(A) It is used to provide for repayment of a future obligation.

This statement is TRUE. The primary purpose of a sinking fund is to set aside money systematically over time to meet a specific future financial obligation, such as retiring debt or replacing an asset.

(B) Payments are made into the fund periodically.

This statement is TRUE. Sinking funds are typically funded by making regular, periodic payments (e.g., monthly, quarterly, annually) into the fund.

(C) The fund earns interest.

This statement is generally TRUE. The money accumulated in a sinking fund is usually invested in interest-bearing securities or accounts. This interest earned helps the fund grow faster and reach its target amount more efficiently, reducing the size or number of payments required.

(D) The amount accumulated in the fund at the end must be exactly equal to the sum of all payments made.

This statement is FALSE. As mentioned in point (C), the funds in a sinking fund typically earn interest. This interest accrues over time and adds to the total amount accumulated. Therefore, the final accumulated amount in the fund will be the sum of all periodic payments plus the total interest earned on those payments. It is generally not just the sum of the payments.

Since the question asks for the INCORRECT statement, option (D) is the answer.

EMI stands for Equated Monthly Installment. It is the fixed amount that a borrower repays to a lender on a monthly basis. This fixed payment is designed to cover both the principal amount borrowed and the interest accrued on that principal over the loan tenure.

In the early stages of a loan, a larger portion of the EMI goes towards paying the interest, and a smaller portion goes towards reducing the principal. As the loan progresses, the outstanding principal decreases, leading to a lower interest component in each EMI, and consequently, a larger portion of the EMI goes towards repaying the principal.

Option (A) is incorrect because EMI also includes interest.

Option (B) is incorrect because EMI also includes principal repayment.

Option (D) is incorrect as the EMI is a recurring payment, not a one-time payment of the original loan amount.

Therefore, the EMI for a loan covers both the principal and interest components.

The correct option is (C) Both principal and interest components.

The question states that an investment yields a simple return of $15\%$ over 3 years.

Simple return is calculated as the total profit (or loss) as a percentage of the initial investment, without considering the time value of money or compounding effects. The formula for simple return is:

We are given that this total simple return over 3 years is $15\%$.

The annual simple return is the average rate of return per year, calculated by dividing the total simple return by the number of years.

Annual Simple Return = $\frac{\text{Total Simple Return}}{\text{Number of Years}}$

Annual Simple Return = $\frac{15\%}{3 \text{ years}}$

Annual Simple Return = $5\%$ per year.

Let's look at the options:

(A) $15\%$: This is the total return over 3 years, not the annual return.

(B) $15\% / 3 = 5\%$: This correctly calculates the annual simple return.

(C) $(1.15)^{1/3} - 1$: This formula is used to calculate the Compound Annual Growth Rate (CAGR), which assumes compounding, not simple return.

(D) $15\% \times 3 = 45\%$: This would be the total return if the rate were $15\%$ per year for 3 years under simple interest, but the problem states $15\%$ is the total return over 3 years.

Therefore, the annual simple return is $5\%$.

The correct option is (B) $15\% / 3 = 5\%$.

The linear depreciation method, also known as the straight-line method, is characterized by a constant depreciation expense recognized each accounting period over the asset's useful life.

The formula for annual linear depreciation is:

The book value of an asset at any point in time is calculated as:

Since the annual depreciation amount is constant under the linear method, the accumulated depreciation increases by the same constant amount each period. Consequently, the book value decreases by this same constant amount each period.

This process continues until the asset's book value equals its salvage value. At this point, depreciation stops, as the asset has been fully depreciated down to its residual or salvage value.

Let's examine the options:

(A) True: This aligns with the definition of linear depreciation.

(B) False: This contradicts the definition.

(C) Only if salvage value is zero: The constancy of the depreciation amount is inherent to the linear method, regardless of whether the salvage value is zero or a positive amount. The salvage value only affects the *amount* of depreciation, not the *method* of constant reduction.

(D) Only for the first few years: The linear method applies throughout the asset's useful life until the book value reaches the salvage value.

Therefore, the statement accurately describes the behavior of book value under linear depreciation.

The correct option is (A) True.

The question asks for the present value of the returns from Proposal A.

Proposal A offers a cash flow of $\textsf{₹}10,000$ annually forever, starting next year. This is an ordinary perpetuity.

The required rate of return is $8\%$, or $0.08$.

The formula for the present value of an ordinary perpetuity is:

Where $A$ is the periodic cash flow and $r$ is the discount rate.

Substituting the given values:

The present value of the returns from Proposal A is $\textsf{₹}1,25,000$.

Looking at the options:

(A) $\textsf{₹}10,000 \times 0.08 = \textsf{₹}800$

(B) $\textsf{₹}10,000 / 0.08 = \textsf{₹}1,25,000$

(C) $\textsf{₹}10,000$

(D) $\textsf{₹}1,00,000$

The correct option is (B) $\textsf{₹}10,000 / 0.08 = \textsf{₹}1,25,000$.

The Net Present Value (NPV) is calculated by taking the present value of all future cash inflows and subtracting the initial investment (cash outflow).

From the previous question (Question 58), we determined that the present value of the returns from Proposal A is $\textsf{₹}1,25,000$.

The initial investment for Proposal A is given as $\textsf{₹}1,00,000$.

Therefore, the NPV of Proposal A is calculated as:

Let's look at the options provided:

• (A) $\textsf{₹}1,25,000 - \textsf{₹}1,00,000 = \textsf{₹}25,000$: This matches our calculation.
• (B) $\textsf{₹}1,00,000 - \textsf{₹}1,25,000 = -\textsf{₹}25,000$: This incorrectly subtracts the present value of returns from the initial investment.
• (C) $\textsf{₹}1,25,000$: This is the present value of the returns, not the NPV.
• (D) $\textsf{₹}10,000$: This is the annual cash flow, not the NPV.

The correct option is (A) $\textsf{₹}1,25,000 - \textsf{₹}1,00,000 = \textsf{₹}25,000$.

The question asks for the simple return from Proposal B over 5 years. Simple return is calculated as the total profit divided by the initial investment.

For Proposal B:

• Initial Investment ($P_0$) = $\textsf{₹}5,00,000$
• Future Value after 5 years ($P_5$) = $\textsf{₹}8,00,000$

The total profit from the investment is the difference between the future value and the initial investment:

The simple return is this total profit as a percentage of the initial investment:

To express this as a percentage, we multiply by 100:

Now let's examine the options:

(A) $\textsf{₹}8,00,000 / \textsf{₹}5,00,000 = 1.6$: This calculates the ratio of the future value to the initial investment, not the simple return.

(B) $(\textsf{₹}8,00,000 - \textsf{₹}5,00,000) / \textsf{₹}5,00,000 = \textsf{₹}3,00,000 / \textsf{₹}5,00,000 = 0.6 = 60\%$: This correctly calculates the simple return.

(C) $60\% / 5 = 12\%$: This calculates the annual simple return, not the total simple return over 5 years.

(D) $(\textsf{₹}8,00,000 - \textsf{₹}5,00,000) / 5 = \textsf{₹}60,000$: This calculates the average annual profit, not the total simple return as a percentage.

Therefore, the simple return from Proposal B over the 5 years is $60\%$.

The correct option is (B) $(\textsf{₹}8,00,000 - \textsf{₹}5,00,000) / \textsf{₹}5,00,000 = \textsf{₹}3,00,000 / \textsf{₹}5,00,000 = 0.6 = 60\%$.

To determine which investment proposal is better, we need to analyze the results obtained for each from the previous questions and consider the appropriate decision-making criteria.

For Proposal A:

• Present Value of Returns = $\textsf{₹}1,25,000$
• Initial Investment = $\textsf{₹}1,00,000$
• Net Present Value (NPV) = $\textsf{₹}25,000$

A positive NPV indicates that the project is expected to generate more value than its cost, considering the time value of money and the required rate of return. Therefore, Proposal A appears to be a good investment.

For Proposal B:

• Initial Investment = $\textsf{₹}5,00,000$
• Future Value after 5 years = $\textsf{₹}8,00,000$
• Total Profit = $\textsf{₹}3,00,000$
• Simple Return over 5 years = $60\%$

A simple return of $60\%$ over 5 years is a raw measure of profitability. However, it does not account for the time value of money (i.e., when the returns are received) or the compounding of returns. To properly compare investments with different time horizons and cash flow patterns, a metric that considers the time value of money is necessary, such as NPV or Internal Rate of Return (IRR).

The question asks to compare based on NPV for A and simple return for B. While a positive NPV is a strong indicator of a good investment, relying solely on simple return for B without considering the timing of cash flows or its annualized equivalent is not a robust comparison method against NPV.

NPV is generally considered a superior method for investment appraisal because it directly measures the expected increase in wealth in today's terms. Simply having a high simple return does not guarantee that the investment is superior, especially if the returns are received far in the future or if the compounding effect is ignored.

Since Proposal A has a positive NPV of $\textsf{₹}25,000$, it is considered a financially sound investment. Proposal B's simple return of $60\%$ over 5 years needs to be converted to a comparable metric (like IRR or annualized return) to be properly compared with Proposal A's NPV. Without further information or calculation for Proposal B that accounts for the time value of money, directly comparing its simple return to Proposal A's NPV is problematic.

The question asks which is better based on these metrics. Proposal A is deemed good by NPV. Proposal B's simple return, while large, doesn't provide a directly comparable measure of value creation in present terms like NPV does.

Therefore, based on the fact that Proposal A has a positive NPV, indicating it is a value-creating investment, and the limitations of comparing simple return directly to NPV without further analysis for Proposal B, it's difficult to definitively say B is better. If we must choose based on the provided metrics, a positive NPV is a more direct indicator of value creation than a simple return over a period.

However, if we assume that the question implies that a positive NPV makes A better and a high simple return makes B better, and we need to pick one, we must acknowledge the incomparability. A positive NPV is a standard for accepting a project, while simple return is a less sophisticated metric.

Let's reconsider if there's an implied comparison. A positive NPV means the project adds value. A simple return of 60% over 5 years implies an average annual return of approximately 9.7% (using CAGR formula: $(1.6)^{1/5} - 1 \approx 0.0975$). If the required rate of return was significantly higher than this for Proposal B, then it might not be as good as Proposal A. Without knowing the required rate of return for Proposal B or its NPV, a direct comparison is indeed difficult.

Given the options, and the common understanding that NPV is a superior metric for comparing investment opportunities, the presence of a positive NPV for Proposal A makes it a strong contender. The simple return for B, while seemingly high, lacks the time value adjustment. Thus, a direct comparison is not ideal.

The most accurate statement considering the limitations of simple return for comparison purposes against NPV is that a direct comparison using these metrics is not ideal.

The correct option is (D) Cannot compare using these metrics.

In accounting, the value of an asset on a company's balance sheet is referred to as its book value. The book value is calculated by taking the original cost of the asset and subtracting the total depreciation that has been accumulated since the asset was put into use.

The formula for book value is:

Let's look at the other options to confirm why they are incorrect:

• (A) Salvage Value: This is the estimated resale value of an asset at the end of its useful life. It is used in the depreciation calculation but is not the difference between original cost and accumulated depreciation.
• (B) Depreciable Amount: This is the portion of an asset's cost that can be depreciated over its useful life. It is calculated as Original Cost - Salvage Value.
• (D) Useful Life: This is the period over which an asset is expected to be used productively by a company. It is a measure of time, not value.

Therefore, the difference between the original cost of an asset and its accumulated depreciation is its book value.

The correct option is (C) Book Value.

A sinking fund is a fund established by setting aside money regularly over time to accumulate a sufficient amount to meet a future financial obligation. This obligation could be the repayment of a debt, the replacement of an asset, or funding a specific project.

The target amount for a sinking fund is the total sum required to meet the future obligation. Since the money contributed to the sinking fund is typically invested and earns interest, the total accumulated amount will be greater than just the sum of the periodic payments made into the fund.

The target amount is the result of the periodic contributions (payments) and the compound interest earned on those contributions over the period until the target date.

Let's analyze the options:

• (A) The sum of all periodic payments only: This would be true if no interest were earned, which is generally not the case for a sinking fund.
• (B) The sum of all periodic payments plus the interest earned on them: This accurately describes how the target amount is reached, considering both contributions and investment growth.
• (C) Only the interest earned: This is incorrect, as the primary source of the target amount is the periodic payments themselves.
• (D) The principal amount of a loan: While a sinking fund might be used to repay a loan's principal, the target amount of the fund is not just the principal; it's the total amount needed, which includes the principal and potentially interest if the fund is for a specific repayment structure. However, the statement describes what the fund *includes*, and this option is too specific and not universally true for all sinking funds.

Therefore, the target amount of a sinking fund is the sum of all periodic payments made into it, plus all the interest earned on those payments over time.

The correct option is (B) The sum of all periodic payments plus the interest earned on them.

The calculation of EMI (Equated Monthly Installment) requires that the interest rate ($i$) and the number of periods ($n$) are consistent with the payment frequency. In this case, the payments are made monthly.

1. Interest Rate per Period ($i$):

The loan has an annual interest rate of $12\%$. Since the interest is compounded monthly, we need to find the monthly interest rate.

Annual Interest Rate = $12\% = 0.12$

Number of compounding periods per year = 12 (since it's compounded monthly)

Monthly Interest Rate ($i$) = $\frac{\text{Annual Interest Rate}}{\text{Number of months in a year}}$

So, the interest rate per period ($i$) is $0.01$ (or $1\%$).

2. Number of Periods ($n$):

The loan tenure is 5 years. Since the payments are made monthly, we need to convert the loan tenure into the total number of monthly periods.

Loan Tenure = 5 years

Number of payments per year = 12

Total Number of Periods ($n$) = Loan Tenure in Years $\times$ Number of payments per year

So, the total number of periods ($n$) is 60.

Therefore, the values to be used in the EMI formula are $i=0.01$ and $n=60$.

Let's examine the given options:

• (A) $i=0.12, n=5$: Incorrect, as $i$ should be monthly rate and $n$ should be total months.
• (B) $i=0.01, n=5$: Incorrect, as $n$ should be the total number of months.
• (C) $i=0.12, n=60$: Incorrect, as $i$ should be the monthly rate.
• (D) $i=0.01, n=60$: This correctly represents the monthly interest rate and the total number of monthly periods.

The correct option is (D) $i=0.01, n=60$.

The Compound Annual Growth Rate (CAGR) measures the average annual growth rate of an investment over a specified period of time, assuming that the profits are reinvested.

The formula for CAGR is:

In this problem:

• Beginning Value = $\textsf{₹}X$
• Ending Value = $\textsf{₹}2X$
• Number of Years = 5

Substitute these values into the CAGR formula:

The $\textsf{₹}X$ terms cancel out:

Now, we need to calculate $(2)^{1/5}$. This means finding the fifth root of 2.

$(2)^{1/5} \approx 1.148698$

So, the CAGR is approximately:

Converting this to a percentage:

Now let's evaluate the options:

• (A) $20\%$: This would be the simple annual return if the growth was linear ($(\textsf{₹}2X - \textsf{₹}X) / \textsf{₹}X / 5 = 1/5 = 20\%$), but CAGR accounts for compounding.
• (B) $14.87\%$: This is the calculated CAGR.
• (C) $(2)^{1/5} - 1 \approx 1.1487 - 1 = 0.1487 = 14.87\%$: This option shows the correct calculation and the resulting percentage, which is the same as option (B).
• (D) Both (B) and (C): Since options (B) and (C) express the same value and calculation, this is the most comprehensive correct answer.

The correct option is (D) Both (B) and (C).

First, let's determine the annual depreciation amount using the linear method.

The formula for annual linear depreciation is:

Given:

• Original Cost = $\textsf{₹}8,00,000$
• Salvage Value = $\textsf{₹}1,00,000$
• Useful Life = 7 years

The depreciable amount is $\textsf{₹}8,00,000 - \textsf{₹}1,00,000 = \textsf{₹}7,00,000$.

Annual Depreciation = $\frac{\textsf{₹}7,00,000}{7 \text{ years}} = \textsf{₹}1,00,000$ per year.

Next, we need to find the book value at the end of Year 4.

The book value is calculated as:

Accumulated Depreciation at the end of Year 4 = Annual Depreciation $\times$ Number of Years

Accumulated Depreciation = $\textsf{₹}1,00,000/\text{year} \times 4 \text{ years} = \textsf{₹}4,00,000$

Book Value at the end of Year 4 = $\textsf{₹}8,00,000 - \textsf{₹}4,00,000 = \textsf{₹}4,00,000$.

Now let's examine the given options to see which one correctly represents this calculation:

(A) $\textsf{₹}8,00,000 - 4 \times (\textsf{₹}7,00,000/7) = \textsf{₹}8,00,000 - \textsf{₹}4,00,000 = \textsf{₹}4,00,000$

This option correctly calculates the book value. It uses the original cost, subtracts the accumulated depreciation, where the annual depreciation is calculated as (Original Cost - Salvage Value) / Useful Life, and then multiplies by the number of years (4).

(B) $\textsf{₹}1,00,000 + (7-4) \times (\textsf{₹}7,00,000/7) = \textsf{₹}1,00,000 + 3 \times \textsf{₹}1,00,000 = \textsf{₹}4,00,000$

This option uses an alternative way to calculate the book value. It starts with the salvage value and adds the remaining depreciable amount. The remaining depreciable amount is the (remaining useful life $\times$ annual depreciation). Remaining useful life = Total useful life - Years passed = $7 - 4 = 3$ years. So, Salvage Value + (Remaining Useful Life $\times$ Annual Depreciation) = $\textsf{₹}1,00,000 + 3 \times (\textsf{₹}7,00,000/7) = \textsf{₹}1,00,000 + 3 \times \textsf{₹}1,00,000 = \textsf{₹}1,00,000 + \textsf{₹}3,00,000 = \textsf{₹}4,00,000$. This option also correctly calculates the book value.

(C) $\textsf{₹}7,00,000$: This is the depreciable amount (Original Cost - Salvage Value), not the book value at the end of Year 4.

(D) Both (A) and (B): Since both options (A) and (B) correctly arrive at the book value of $\textsf{₹}4,00,000$ using valid methods for calculating book value with linear depreciation, this option is correct.

The correct option is (D) Both (A) and (B).

The formula $PV = A/r$ is used to calculate the present value of an ordinary perpetuity.

An ordinary perpetuity is a series of equal cash payments that continue indefinitely, with each payment occurring at the end of a period.

The formula $PV = A/r$ is derived by considering that the present value of an infinite stream of payments of $A$ occurring at the end of each period (periods 1, 2, 3, ...) discounted at rate $r$ is $A/(1+r) + A/(1+r)^2 + A/(1+r)^3 + \dots$, which sums to $A/r$.

In contrast, a perpetuity due has payments occurring at the beginning of each period. If the first payment occurs immediately (at time 0), the formula for its present value is $A + A/(1+r) + A/(1+r)^2 + \dots$, which sums to $A + A/r$.

Therefore, the formula $PV = A/r$ specifically applies when the first payment occurs at the end of the first period.

Let's look at the options:

• (A) Immediately: This describes a perpetuity due's first payment, not an ordinary perpetuity.
• (B) At the end of the first period: This is the defining characteristic of an ordinary perpetuity for which $PV = A/r$ is used.
• (C) At the beginning of the first period: This also describes a perpetuity due.
• (D) At any time in the future: This is too vague; the timing of payments is crucial for present value calculations.

The correct option is (B) At the end of the first period.

A sinking fund is established to accumulate a specific target amount by making periodic payments over time. The growth of the sinking fund depends on two main factors: the amount of the periodic payments and the interest earned on those payments.

The formula for the future value of an ordinary annuity (which is often used to calculate sinking fund payments) is:

Where:

• $FV$ = Future Value (the target amount)
• $P$ = Periodic Payment
• $r$ = Interest rate per period
• $n$ = Number of periods

If the target amount ($FV$) and the number of periods ($n$) are fixed, we can rearrange the formula to solve for the periodic payment ($P$):

Now, consider what happens if the interest rate ($r$) increases, while $FV$ and $n$ remain constant. Looking at the formula for $P$, we see that $P$ is directly proportional to $r$ (in the numerator) and inversely related to the term $\frac{(1+r)^n - 1}{r}$ (in the denominator). As $r$ increases, the term $(1+r)^n$ grows faster, and the term $\frac{(1+r)^n - 1}{r}$ also increases.

A simpler way to think about it is that if the fund earns a higher rate of interest, the accumulated amount will grow faster. Since the target amount remains the same, and the fund is growing faster due to higher interest, fewer or smaller periodic payments are needed to reach that target.

Therefore, if a sinking fund earns a higher rate of interest than initially planned, the required periodic payment to meet the future target will decrease.

The correct option is (B) Decrease.

When you take out a loan, you borrow a certain amount, which is the principal or the original loan amount. Over the tenure of the loan, you repay this principal along with interest.

The EMI (Equated Monthly Installment) is the fixed amount paid each month. The total amount paid over the entire tenure of the loan is the EMI multiplied by the total number of payments (periods).

Total Amount Paid = EMI $\times$ Number of Periods

This total amount paid consists of two parts: the repayment of the principal amount borrowed and the total interest charged over the life of the loan.

Total Amount Paid = Principal Repaid + Total Interest Paid

Therefore, the total interest paid is the difference between the total amount paid and the principal amount borrowed (the original loan amount).

Total Interest Paid = Total Amount Paid - Original Loan Amount

Let's analyze the options:

• (A) First EMI amount: This is just one of the payments, not the total principal.
• (B) Last EMI amount: This is also just one payment, not the total principal.
• (C) Original Loan Amount: This is the principal amount borrowed, which is the part of the total payments that is not interest.
• (D) Interest rate: This is a percentage used to calculate interest, not a monetary amount representing the principal.

The correct option is (C) Original Loan Amount.

The Compound Annual Growth Rate (CAGR) formula is:

Given:

• Beginning Value = $\textsf{₹}500$
• Ending Value = $\textsf{₹}750$
• Number of Years = 2

Calculation:

Option (B) shows the correct formula structure but has a numerical approximation error, stating the result as $14.87\%$. However, it's the only option presenting the correct methodology.

The correct option is (B) $(750/500)^{1/2} - 1 = (1.5)^{0.5} - 1 \approx 1.2247 - 1 = 0.1487 = 14.87\%$.

The linear depreciation method, also known as the straight-line method, allocates the depreciable cost of an asset evenly over its useful life. The formula for annual depreciation is:

From this formula, we can see the components needed:

• (A) Original Cost: This is the initial purchase price of the asset. It is required.
• (B) Useful Life: This is the estimated period the asset is expected to be used. It is required as the denominator in the formula.
• (C) Salvage Value: This is the estimated residual value of the asset at the end of its useful life. It is required to calculate the depreciable amount (Original Cost - Salvage Value).

Let's consider option (D):

• (D) Market Value in each year: The market value of an asset fluctuates and is not directly used in the calculation of depreciation using the linear method. Depreciation is based on the asset's cost and its expected useful life and salvage value, not its current market worth. While market value is important for other financial analyses, it's not a component of the linear depreciation calculation.

Therefore, the market value in each year is NOT needed to calculate annual depreciation using the linear method.

The correct option is (D) Market Value in each year.

A perpetuity is a stream of equal cash payments that continue indefinitely. This means the payments never stop.

Let's analyze the given options:

(A) A short-term loan repayment: Short-term loan repayments have a defined end date, so they are not perpetual.

(B) The principal repayment component of an EMI: While an EMI includes principal repayment, it does so over a finite loan term, and the principal component itself changes over time. It's not a constant payment forever.

(C) A stream of cash flows from a permanent endowment: A permanent endowment is typically set up to provide a perpetual stream of income for a specific purpose (e.g., scholarships, research). The cash flows from such an endowment are intended to continue indefinitely, making it an ideal candidate for modeling with a perpetuity.

(D) Depreciation of an asset over its finite life: Depreciation is an expense recognized over the useful life of an asset, which is finite. It is not a cash flow stream, and it definitely does not continue indefinitely.

Therefore, a perpetuity is most suitable for modeling a stream of cash flows that is expected to continue forever, such as those from a permanent endowment.

The correct option is (C) A stream of cash flows from a permanent endowment.

Let's analyze the Assertion and Reason separately.

Assertion (A): A sinking fund helps a company manage the repayment of a large future liability.

This statement is TRUE. The purpose of a sinking fund is precisely to accumulate funds over time to meet a specific future financial obligation, such as repaying a large debt or replacing an asset. It breaks down a large, potentially daunting liability into smaller, manageable periodic payments.

Reason (R): By making regular, smaller payments into a fund that earns interest, the company can accumulate the required lump sum by the maturity date.

This statement is also TRUE. The core mechanism of a sinking fund is to make periodic contributions (payments) into an investment vehicle that earns interest. The compounding of interest on these payments helps the fund grow, allowing the company to reach the target lump sum amount by the time the liability is due, often with less financial strain than trying to gather the entire amount at once.

Now, let's determine if Reason (R) correctly explains Assertion (A).

Reason (R) describes *how* a sinking fund works (regular payments + interest accumulation) and *what it achieves* (accumulating a lump sum by maturity). This process directly explains *how* a sinking fund helps manage a large future liability, as stated in Assertion (A). The regular, smaller payments and the benefit of interest earned are the very reasons why it's an effective way to manage such liabilities.

Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

The correct option is (A) Both A and R are true and R is the correct explanation of A.

An Equated Monthly Installment (EMI) is a fixed payment made by a borrower to a lender at a specified frequency, usually monthly. Each EMI payment consists of two components: a principal repayment and an interest payment.

The calculation of EMI is structured such that the total payment is constant, but the allocation between principal and interest changes over the loan tenure.

In the early stages of a loan:

• The outstanding principal balance is at its highest.
• The interest charged for the period is calculated on this high principal amount.
• Therefore, a larger portion of the EMI goes towards paying the interest component.

As the loan progresses and payments are made:

• The outstanding principal balance gradually decreases.
• The interest calculated for each subsequent period also decreases because it's based on a lower principal.
• Since the total EMI amount remains constant, the portion of the EMI that goes towards principal repayment increases as the interest portion decreases.

In summary, as the loan tenure progresses, the proportion of the EMI allocated to principal repayment increases, while the proportion allocated to interest decreases.

Therefore, if your EMI remains constant throughout the loan tenure, the proportion of the EMI that goes towards principal repayment increases over time.

The correct option is (B) Increases over time.

The Compound Annual Growth Rate (CAGR) is a measure of the average annual growth rate of an investment over a specified period of time longer than one year. It represents the constant rate at which an investment would have grown each year if the growth had been steady.

The calculation of CAGR effectively smooths out the fluctuations in returns that might occur from one year to the next. It looks only at the beginning and ending values of the investment and the total time period. It does not account for the interim performance or the volatility of the investment's returns during that period.

For example, an investment might have returned $20\%$ in Year 1, $-5\%$ in Year 2, and $30\%$ in Year 3. CAGR would provide a single, smoothed annual rate that represents the equivalent steady growth over those three years, effectively ignoring the specific ups and downs of Year 1 and Year 2.

This smoothing aspect is a key characteristic of CAGR, making it useful for comparing the long-term performance of different investments, but it also means it doesn't reflect the actual year-to-year volatility experienced by the investor.

The statement is true for both positive and negative returns, as CAGR can be calculated for investments that have grown or shrunk in value over the period.

Therefore, the statement that CAGR provides a smoothed annual return figure, ignoring year-to-year volatility, is true.

The correct option is (A) True.

In the linear depreciation method, the annual depreciation is calculated as:

The term "Original Cost - Salvage Value" is known as the Depreciable Amount. Let's denote it as $D_{amount}$.

So, $Annual \ Depreciation = \frac{D_{amount}}{Useful \ Life}$.

The useful life is given as $N$ years.

$Annual \ Depreciation = \frac{D_{amount}}{N}$.

The accumulated depreciation at the end of Year $N$ (which is the end of the asset's useful life) is the sum of all annual depreciation charges over $N$ years.

$Accumulated \ Depreciation \ at \ end \ of \ Year \ N = Annual \ Depreciation \times N$

$Accumulated \ Depreciation = \left(\frac{D_{amount}}{N}\right) \times N$

$Accumulated \ Depreciation = D_{amount}$

Since $D_{amount}$ is the depreciable amount (Original Cost - Salvage Value), the accumulated depreciation at the end of Year $N$ will be equal to the depreciable amount.

Let's check the options:

• (A) Original Cost: The accumulated depreciation will equal the original cost only if the salvage value is zero. This is not always the case.
• (B) Salvage Value: The accumulated depreciation is subtracted from the original cost to arrive at the salvage value (book value at the end of useful life). So, accumulated depreciation is not equal to salvage value, unless the original cost was twice the salvage value and depreciation was calculated in a specific way.
• (C) Depreciable Amount (Cost - Salvage): This matches our calculation. The total depreciation over the asset's life is the amount that can be depreciated, which is the original cost minus the salvage value.
• (D) Annual Depreciation: This is the depreciation for just one year, not the total accumulated depreciation over $N$ years.

The correct option is (C) Depreciable Amount (Cost - Salvage).

A perpetuity is a financial concept referring to a stream of cash payments that are expected to continue indefinitely. This means that the payments are assumed to go on forever.

Let's examine the given options:

(A) For a fixed number of years: This describes an annuity, not a perpetuity. Annuities have a defined end date.

(B) Until the principal is repaid: This also describes a finite stream of payments, like a loan repayment schedule, where the principal is eventually paid off. A perpetuity does not have a principal to repay in this sense; it's an ongoing series of payments.

(C) Forever: This accurately describes the defining characteristic of a perpetuity – the payments continue without end.

(D) Until the interest rate changes: While interest rates are crucial for valuing perpetuities, the concept of a perpetuity itself does not depend on the interest rate changing or not changing for its continuation. The payments are assumed to continue regardless of interest rate fluctuations.

Therefore, the core assumption of a perpetuity is that the stream of payments continues forever.

The correct option is (C) Forever.

The statement describes a sinking fund where periodic payments are made, and the fund earns interest. The goal is to find the accumulated amount at the end of the period.

Let's break down the components of the statement:

• Sinking fund: A fund established by setting aside money regularly over time to accumulate a specific sum.
• Periodic payments: $\textsf{₹}20,000$ annually.
• Time period: 5 years.
• Interest rate: $7\%$ compounded annually.
• Goal: Accumulated amount at the end of 5 years.
• Calculation method mentioned: Formula for FV of an ordinary annuity.
• Parameters mentioned for the formula: periodic payment = $\textsf{₹}20,000$, $i=0.07$, and $n=5$.

The future value (FV) of an ordinary annuity is indeed the correct formula to calculate the accumulated amount when regular payments are made at the end of each period, and interest is compounded at the end of each period.

The parameters provided ($P = \textsf{₹}20,000$, $i = 0.07$ per year, $n = 5$ years) are consistent with the formula for the FV of an ordinary annuity when payments are annual and interest is compounded annually.

The formula for the FV of an ordinary annuity is:

The statement correctly identifies that this formula is used and provides the correct parameters ($P=\textsf{₹}20,000$, $i=0.07$, $n=5$) if the payments are made at the end of each year and interest is compounded annually.

Let's consider the conditions mentioned in options (C) and (D):

• (C) Only if the payments are made at the beginning of the year: If payments were made at the beginning of the year, it would be an annuity due, and a different FV formula would be used (FV of annuity due = FV of ordinary annuity $\times (1+i)$). The statement implicitly assumes end-of-period payments by referring to the FV of an *ordinary* annuity.
• (D) Only if the interest is compounded monthly: The statement explicitly says interest is compounded *annually*, so this condition is not met.

Since the problem specifies annual deposits, annual interest, and a 5-year period, and correctly states that the FV of an *ordinary* annuity formula with the given parameters ($P=\textsf{₹}20,000$, $i=0.07$, $n=5$) is used, the statement is accurate.

The correct option is (A) True.

When comparing investment performance, especially across different assets and different time periods, it is essential to use a metric that standardizes the returns to a comparable basis.

Let's analyze the suitability of each option:

• (A) Total Simple Return: This is the total profit as a percentage of the initial investment over the entire period. It's not suitable for comparing investments over different time periods because it doesn't account for the length of time the investment was held. An investment with a higher total simple return might have achieved it over a much longer period than another investment with a lower total simple return.
• (B) Annual Simple Return: This is the total simple return divided by the number of years. While this is better than total simple return as it annualizes the return, it still ignores the effect of compounding, which is crucial for understanding investment growth over time.
• (C) CAGR (Compound Annual Growth Rate): CAGR measures the average annual rate of return of an investment over a specified period of time, assuming that profits are reinvested at the end of each period. It effectively smooths out volatility and provides a standardized annual growth rate that accounts for compounding. This makes it ideal for comparing investments with different time periods and different compounding frequencies.
• (D) Nominal Return: Nominal return is the stated return before adjusting for inflation. While it's a basic measure, it doesn't account for the time value of money or compounding, nor does it standardize for different time periods, making it less suitable for direct comparison of performance over different durations.

CAGR is specifically designed to provide a standardized, annualized measure of an investment's growth over time, making it the most suitable metric for comparing investments with different time horizons and compounding effects.

The correct option is (C) CAGR.

The book value of an asset represents its value on a company's balance sheet. This value is determined by accounting principles, which involve allocating the asset's cost over its useful life through depreciation.

The formula for book value is:

This formula shows that the book value is directly reduced by the accumulated depreciation charged against the asset.

Let's consider the other options:

• (A) Market fluctuations: Book value is based on historical cost and depreciation, not the asset's current market value, which can fluctuate.
• (C) Inflation: Accounting practices typically use historical cost, and while inflation affects purchasing power, it's not directly accounted for in the book value calculation unless specific inflation accounting methods are used (which is uncommon for general book value).
• (D) Expected future profits: While expected future profits are crucial for investment decisions and valuation (like discounted cash flow analysis), they are not part of the book value calculation itself. Book value reflects the unallocated cost of the asset on the balance sheet.

Therefore, the book value of an asset is its worth as recorded in the accounting records after accounting for depreciation.

The correct option is (B) Depreciation.

A sinking fund is a financial strategy used to systematically accumulate funds over time to meet a specific future financial obligation. It involves making regular, periodic contributions to a fund that earns interest.

Let's examine why each option is or isn't suited for a sinking fund:

(A) Small and frequent: For liabilities that are small and frequent, managing them as they arise or through regular operating cash flow is usually more efficient. A sinking fund is designed for larger, less frequent obligations where advance planning and accumulation are necessary.

(B) Large and occur as a lump sum in the future: This is the primary use case for a sinking fund. When a significant liability (like a bond maturity or a large capital expenditure) is expected in the future, a sinking fund allows an entity to save incrementally over time, benefiting from compounding interest, to ensure the lump sum is available when needed. This makes the repayment manageable and less disruptive to cash flow.

(C) Unpredictable in amount and timing: Sinking funds require predictability in the target amount and the maturity date to be effective. If the liability is unpredictable, it's difficult to set a target and a saving plan. Other financial management strategies might be more appropriate for uncertain liabilities.

(D) Already repaid: A sinking fund is a planning tool for future liabilities. If a liability has already been repaid, there is no need for a sinking fund to plan for it.

Therefore, a sinking fund is most useful for planning the repayment of liabilities that are large and occur as a lump sum in the future, allowing for systematic saving and growth through interest.

The correct option is (B) Large and occur as a lump sum in the future.

The calculation of EMI (Equated Monthly Installment) relies on time value of money principles, specifically the present value of an annuity formula. For these formulas to be applied correctly, the interest rate per period and the number of periods must be consistent.

If payments are made monthly, the interest rate used in the calculation must be the monthly interest rate. This monthly interest rate is derived from the annual interest rate by dividing it by the number of months in a year (12), assuming the annual rate is compounded monthly.

Similarly, if payments were made quarterly, the interest rate used would need to be the quarterly rate (annual rate divided by 4), and the number of periods would be the total number of quarters.

This consistency ensures that the interest accrual (compounding) aligns with when the payments are made and when the interest is calculated within the loan structure.

Therefore, the frequency of compounding of the interest rate *must* match the frequency of the payments for the EMI calculation to be accurate.

Let's look at the options:

• (A) True: This aligns with the principle of consistent time periods in financial calculations.
• (B) False: This would lead to incorrect EMI calculations.
• (C) Only for simple interest loans: EMI calculations are for amortizing loans, which use compound interest. Simple interest is not typically used for EMI calculations.
• (D) Only for loans with balloon payments: Balloon payments are a different loan feature and do not alter the fundamental requirement for matching compounding and payment frequencies for EMI calculations.

The correct option is (A) True.

Simple return measures the total percentage gain or loss on an investment over its entire holding period. The formula is:

While it correctly uses the initial and final investment amounts, its major limitation when comparing investments over different time horizons is its failure to account for how returns are reinvested or compounded over time.

Let's consider why:

• (A) The initial investment amount: Simple return does account for the initial investment amount as the denominator in its calculation.
• (B) The final investment amount: Simple return directly uses the final investment amount in its calculation.
• (C) The effect of compounding: Simple return assumes that any returns generated are not reinvested. In reality, most investments allow returns to be reinvested, leading to compound growth where returns earn further returns. Simple return ignores this crucial aspect.
• (D) The time value of money (beyond a single period): Because simple return does not account for compounding, it doesn't fully capture the time value of money for periods longer than one, as it treats each period's return as independent and not contributing to future growth. While it uses the total time to give a "total" return, it doesn't annualize it in a way that reflects compounding growth.

The effect of compounding is a key aspect of the time value of money, particularly for investments held over multiple periods. CAGR, for instance, explicitly accounts for compounding. Simple return, by not compounding, can be misleading when comparing investments held for different lengths of time, as it doesn't reflect the potential for returns to generate further returns.

Both (C) and (D) are closely related and are indeed the primary disadvantages. However, the "effect of compounding" is the most direct consequence of not accounting for reinvestment, which is central to the time value of money over multiple periods.

In many contexts, the failure to account for compounding is considered the most significant disadvantage for comparing multi-period investments. This failure leads to an incomplete picture of the time value of money's impact on investment growth.

Let's consider which is the *primary* disadvantage. The reason simple return is less useful for comparing over different time horizons is precisely because compounding makes returns received earlier more valuable. Simple return doesn't capture this. So, the effect of compounding is the root cause.

The time value of money (beyond a single period) is a broader concept that is *affected* by compounding. So, the failure to account for compounding is the more specific and primary disadvantage in this context.

The correct option is (C) The effect of compounding.

Let's analyze each statement about linear depreciation:

(A) The annual depreciation amount is constant.

This statement is TRUE. Linear depreciation (or straight-line depreciation) is defined by a constant depreciation expense recognized each period throughout the asset's useful life.

(B) The book value decreases linearly over time.

This statement is TRUE. Since the annual depreciation amount (which is subtracted from the original cost to get book value) is constant, the book value decreases by the same amount each period, resulting in a linear decrease.

(C) It results in higher depreciation expense in the early years compared to some other methods.

This statement is FALSE. Linear depreciation recognizes depreciation evenly over the asset's life. Other methods, such as accelerated depreciation methods (like the double-declining balance or sum-of-the-years'-digits method), recognize higher depreciation expense in the early years of an asset's life and lower expense in later years. Linear depreciation is the least aggressive in terms of early-year depreciation among common methods.

(D) The total depreciation equals the depreciable amount.

This statement is TRUE. The depreciable amount is calculated as the Original Cost minus the Salvage Value. The sum of all the annual depreciation charges over the asset's useful life equals this depreciable amount, reducing the asset's book value from its original cost down to its salvage value.

Since the question asks for the INCORRECT statement, the answer is (C).

The correct option is (C) It results in higher depreciation expense in the early years compared to some other methods.

The present value (PV) of a perpetuity is calculated using the formula:

Where:

• $A$ is the constant cash flow per period.
• $r$ is the interest rate (or discount rate) per period.

This formula shows that the present value ($PV$) is in the numerator, and the interest rate ($r$) is in the denominator.

This relationship indicates an inverse relationship between the present value and the interest rate. As the interest rate ($r$) increases, the denominator gets larger, causing the present value ($PV$) to decrease. Conversely, as the interest rate ($r$) decreases, the denominator gets smaller, causing the present value ($PV$) to increase.

This inverse relationship holds true for both ordinary perpetuities ($PV = A/r$) and perpetuities due ($PV = A + A/r$). In both cases, the term $A/r$ is present, making the present value inversely related to $r$.

Therefore, the statement that the present value of a perpetuity is inversely related to the interest rate is true.

The correct option is (A) True.

When purchasing an asset like a car, the total cost of the asset is the sum of the down payment and the loan amount. The loan amount is what needs to be financed through EMI payments.

The total cost of the car is $\textsf{₹}8,00,000$.

The down payment made by the buyer is $\textsf{₹}2,00,000$.

The down payment is the portion of the total cost paid upfront, reducing the amount that needs to be borrowed.

The principal amount for the loan is the total cost of the asset minus the down payment:

This $\textsf{₹}6,00,000$ is the amount that will be financed through the loan, and therefore, it is the principal amount on which the EMI will be calculated.

Let's check the options:

• (A) $\textsf{₹}8,00,000$: This is the total cost of the car, not the loan principal.
• (B) $\textsf{₹}2,00,000$: This is the down payment, not the loan principal.
• (C) $\textsf{₹}8,00,000 - \textsf{₹}2,00,000 = \textsf{₹}6,00,000$: This correctly calculates the loan principal.
• (D) $\textsf{₹}8,00,000 + \textsf{₹}2,00,000 = \textsf{₹}10,00,000$: This incorrectly adds the down payment to the total cost.

The correct option is (C) $\textsf{₹}8,00,000 - \textsf{₹}2,00,000 = \textsf{₹}6,00,000$.

Given:

Future Value (FV) = $\textsf{₹}$5,00,000

Number of years (n) = 5 years

Interest rate (r) = 8% per annum = 0.08

Compounding frequency = Annually

To Find:

Annual contribution (A) required for the sinking fund.

Formula for Future Value of an Ordinary Annuity (Sinking Fund):

The formula for the future value of an ordinary annuity is used to calculate the periodic payment (annual contribution) needed to reach a future sum:

Where:

• FV = Future Value
• A = Annual Contribution
• r = Interest rate per period
• n = Number of periods

Calculation:

We need to rearrange the formula to solve for A:

Substitute the given values into the formula:

First, calculate $(1+0.08)^5$:

$(1.08)^5 \approx 1.469328$

Now, substitute this value back into the equation for A:

Answer:

The annual contribution required for the sinking fund is approximately $\textsf{₹}$85,228.02.

Definition of Perpetuity:

A perpetuity is a type of annuity that continues indefinitely, meaning it pays out a fixed amount of money at regular intervals forever. In simpler terms, it's a stream of cash flows that never ends.

Types of Perpetuities:

There are two main types of perpetuities, distinguished by the timing of their payments:

1. Ordinary Perpetuity:

In an ordinary perpetuity, the payments are made at the end of each period. For example, receiving a fixed amount of money at the end of every year forever.

The formula for the present value (PV) of an ordinary perpetuity is:

$PV = \frac{C}{r}$

…(i)

Where:

• PV = Present Value
• C = Cash flow per period (the fixed amount paid)
• r = Interest rate per period
2. Perpetuity Due:

In a perpetuity due, the payments are made at the beginning of each period. For instance, receiving a fixed amount of money at the start of every year forever.

Since each payment in a perpetuity due occurs one period earlier than in an ordinary perpetuity, its present value will be higher. The formula for the present value of a perpetuity due is:

$PV = \frac{C}{r} \times (1+r)$

…(ii)

Alternatively, it can be seen as the present value of an ordinary perpetuity plus the first payment, as the first payment is received immediately.

Difference between Ordinary Perpetuity and Perpetuity Due:

The fundamental difference lies in the timing of the cash flows:

• Timing of Payments: In an ordinary perpetuity, payments occur at the end of each period, while in a perpetuity due, payments occur at the beginning of each period.
• Present Value: For the same cash flow amount (C) and interest rate (r), a perpetuity due will always have a higher present value than an ordinary perpetuity because its payments are received earlier, allowing them to earn interest for an additional period.
• Formula: The present value formulas reflect this difference, with the perpetuity due formula including an extra $(1+r)$ multiplier to account for the earlier timing of payments.

Given:

Cash flow per year (C) = $\textsf{₹}$10,000

Discount rate (r) = 12% per annum = 0.12

The first payment is made at the end of the first year, indicating it is an ordinary perpetuity.

To Find:

The present value (PV) of the perpetuity.

Formula for Present Value of an Ordinary Perpetuity:

The present value of an ordinary perpetuity is calculated using the formula:

Where:

• PV = Present Value
• C = Cash flow per period
• r = Discount rate per period

Calculation:

Substitute the given values into the formula:

Now, perform the division:

Answer:

The present value of the perpetuity is $\textsf{₹}$83,333.33.

Given:

Loan Principal (P) = $\textsf{₹}$6,00,000

Annual Interest Rate = 10%

Loan Tenure = 3 years

Compounding Frequency = Monthly

Calculations:

First, we need to determine the monthly interest rate and the total number of monthly payments.

1. Monthly Interest Rate (r):

The annual interest rate is 10%. Since it is compounded monthly, we divide the annual rate by 12:

2. Total Number of Payments (n):

The loan tenure is 3 years, and payments are made monthly. So, the total number of payments is:

Formula for EMI:

The Equated Monthly Installment (EMI) can be calculated using the following formula:

Where:

• P = Loan Principal
• r = Monthly Interest Rate
• n = Total Number of Payments

Substituting the values:

Let's calculate $(1+r)^n = (1.008333)^{36}$:

$(1.008333)^{36} \approx 1.348185$

Now, plug this back into the EMI formula:

Answer:

The EMI for the loan is approximately $\textsf{₹}$19,360.20.

Given:

Principal Investment (P) = $\textsf{₹}$50,000

Final Amount Received (A) = $\textsf{₹}$55,000

Time Period (t) = 1 year

To Find:

Simple Return on Investment.

Formula for Simple Return:

The simple return on an investment is the profit made on the investment, expressed as a percentage of the initial investment. It is calculated as:

The profit is the difference between the final amount received and the principal investment:

Calculation:

First, calculate the profit:

Now, calculate the simple return using the formula:

Answer:

The simple return on this investment is 10%.

Given:

Cost of the Asset (C) = $\textsf{₹}$8,00,000

Salvage Value (S) = $\textsf{₹}$80,000

Useful Life of the Asset (n) = 10 years

To Find:

Annual depreciation amount using the linear method.

Formula for Linear Depreciation:

The linear depreciation method (also known as the straight-line method) allocates the cost of an asset evenly over its useful life. The annual depreciation amount is calculated as:

This can also be written as:

Calculation:

Substitute the given values into the formula:

Calculate the depreciable amount:

Now, calculate the annual depreciation:

Answer:

The annual depreciation amount using the linear method is $\textsf{₹}$72,000.

Concept of Compound Annual Growth Rate (CAGR):

The Compound Annual Growth Rate (CAGR) is a measure of the average annual growth of an investment or a metric over a specified period of time longer than one year. It represents the rate at which an investment would have grown if it had grown at a steady rate each year.

CAGR smooths out volatility and provides a single, representative growth rate over time, assuming that the growth was compounded annually. It is a widely used metric in finance to evaluate the performance of investments, businesses, or economic indicators.

Formula for CAGR:

The formula for CAGR is:

Where:

• Ending Value: The value of the investment at the end of the period.
• Beginning Value: The value of the investment at the start of the period.
• Number of Years: The total number of years in the period.

Why CAGR is Preferred Over Simple Average Growth Rate:

The Compound Annual Growth Rate (CAGR) is generally preferred over the simple average growth rate for several key reasons:

1. Accounts for Compounding: The primary advantage of CAGR is that it considers the effect of compounding. In financial markets, returns often compound over time, meaning that earnings in one period generate further earnings in subsequent periods. The simple average growth rate ignores this crucial aspect.
2. Smooths Out Volatility: Investments rarely grow at a steady, linear rate. There are usually fluctuations year-to-year. The simple average growth rate can be misleading if there are significant swings in growth. For example, a large positive growth in one year can disproportionately inflate the simple average, even if other years had poor or negative growth. CAGR provides a more realistic and smoother representation of the growth trend.
3. Provides a More Realistic Growth Trajectory: CAGR represents the constant rate at which an investment would have grown each year to reach its final value from its starting value. This gives a clearer picture of the overall growth performance over the entire period.
4. Comparability: CAGR makes it easier to compare the performance of different investments or assets over the same time horizon, even if their year-on-year growth patterns are different.

Example to illustrate the difference:

Consider an investment that grows as follows:

• Year 0: $\textsf{₹}$100
• Year 1: $\textsf{₹}$200 (Growth of 100%)
• Year 2: $\textsf{₹}$100 (Growth of -50%)

Simple Average Growth Rate:

Average Growth = $\frac{100\% + (-50\%)}{2} = \frac{50\%}{2} = 25\%$

CAGR:

Ending Value = $\textsf{₹}$100

Beginning Value = $\textsf{₹}$100

Number of Years = 2

In this example, the simple average growth rate suggests an average growth of 25%, which is highly misleading. The CAGR of 0% accurately reflects that the investment ended up at its starting value, despite the significant volatility in between.

Given:

Beginning Value (BV) = $\textsf{₹}$1,00,000

Ending Value (EV) = $\textsf{₹}$1,40,000

Number of Years (n) = 4 years

To Find:

Compound Annual Growth Rate (CAGR).

Formula for CAGR:

The formula for CAGR is:

Calculation:

Substitute the given values into the formula:

Simplify the fraction:

Calculate $(1.4)^{\frac{1}{4}}$ (the fourth root of 1.4):

$(1.4)^{\frac{1}{4}} \approx 1.087757$

Now, complete the calculation:

To express this as a percentage, multiply by 100:

Answer:

The Compound Annual Growth Rate (CAGR) is approximately 8.78%.

Definition of a Sinking Fund:

A sinking fund is a fund established by an organization or individual to set aside a certain amount of money over time for a specific future financial obligation. These funds are typically accumulated through regular contributions, and the money within the fund earns interest, helping it grow towards the target amount.

The primary purpose of a sinking fund is to ensure that sufficient funds are available to meet a future liability, thereby avoiding the need to borrow heavily at the last minute or to face financial distress.

Key Characteristics of a Sinking Fund:

• Specific Purpose: Sinking funds are created for a defined objective, such as repaying a debt, replacing an asset, or funding a project.
• Regular Contributions: Funds are usually deposited into the sinking fund at regular intervals (e.g., monthly, annually).
• Interest Earning: The money in the fund is invested and earns interest, which accelerates the accumulation process.
• Time Horizon: Sinking funds are typically established with a specific time horizon in mind, aligned with the due date of the obligation.

Example of its Application:

One of the most common applications of a sinking fund is by companies to retire their bonds.

Scenario: A company issues bonds with a face value of $\textsf{₹}$10,00,000 that mature in 10 years. Instead of raising the entire $\textsf{₹}$10,00,000 from borrowing or cash reserves at the maturity date, the company can set up a sinking fund.

Application:

• The company makes regular periodic payments into the sinking fund.
• The money in the sinking fund is invested, typically in low-risk, interest-bearing securities.
• As the fund grows with contributions and interest, it accumulates towards the target amount of $\textsf{₹}$10,00,000.
• By the time the bonds mature in 10 years, the sinking fund should have accumulated enough money to repay the bondholders.

This method helps the company manage its cash flow more effectively and ensures it can meet its financial obligations without a sudden, large cash outflow. Other examples include funds for replacing machinery, retiring preferred stock, or building up reserves for future capital expenditures.

Given:

Cost of the Machine (C) = $\textsf{₹}$5,00,000

Annual Depreciation (AD) = $\textsf{₹}$40,000

Estimated Useful Life (n) = 10 years

To Find:

Estimated Salvage Value (S).

Formula for Linear Depreciation:

The linear depreciation method is calculated as:

Rearranging this formula to solve for Salvage Value (S):

Calculation:

Substitute the given values into the rearranged formula:

Calculate the total depreciation over the useful life:

Now, find the salvage value:

Answer:

The estimated salvage value of the machine is $\textsf{₹}$1,00,000.

Given:

Cash flow per quarter (C) = $\textsf{₹}$5,000

Annual Discount Rate = 8%

Compounding Frequency = Quarterly

The payments are made at the beginning of each quarter, indicating a perpetuity due.

To Find:

The present value (PV) of the perpetuity due.

Calculations:

First, we need to determine the quarterly interest rate and confirm the cash flow is per quarter.

1. Quarterly Interest Rate (r):

The annual interest rate is 8%. Since it is compounded quarterly, we divide the annual rate by 4:

2. Cash Flow per Period:

The cash flow is given as $\textsf{₹}$5,000 per quarter, which is the correct period for our interest rate.

Formula for Present Value of a Perpetuity Due:

The present value of a perpetuity due is calculated as:

Alternatively, it can be expressed as:

Where:

• PV = Present Value
• C = Cash flow per period
• r = Discount rate per period

Calculation:

Substitute the given values into the formula:

Calculate $\frac{1}{0.02}$:

Now, complete the calculation:

Answer:

The present value of the perpetuity due is $\textsf{₹}$2,55,000.

Understanding the terms related to an asset's value is crucial in accounting and financial reporting. Here's an explanation of the differences between original cost, accumulated depreciation, and book value:

1. Original Cost (or Historical Cost):

The original cost of an asset is the total amount paid to acquire it and bring it into working condition. This includes the purchase price plus any directly attributable costs, such as:

• Purchase price of the asset
• Transportation or delivery costs
• Installation costs
• Taxes and duties related to the acquisition
• Any costs incurred to make the asset ready for its intended use.

This value remains constant on the balance sheet throughout the asset's life, unless there are subsequent capital expenditures that significantly enhance its capacity or efficiency.

2. Accumulated Depreciation:

Depreciation is the systematic allocation of the cost of a tangible asset over its useful life. It reflects the gradual reduction in the asset's value due to wear and tear, obsolescence, or usage.

Accumulated depreciation is the total amount of depreciation expense that has been recognized for an asset since it was put into use. It is a contra-asset account, meaning it reduces the carrying value of the asset on the balance sheet. It represents the cumulative portion of the asset's cost that has been expensed over time.

For example, if an asset has an annual depreciation of $\textsf{₹}$10,000 and has been in use for 3 years, its accumulated depreciation would be $\textsf{₹}$30,000.

3. Book Value (or Carrying Value):

The book value of an asset is its net value as recorded on the company's balance sheet. It is calculated by subtracting the accumulated depreciation from the original cost of the asset.

The formula is:

Book value represents the asset's value on the company's books at a specific point in time. It is not necessarily the same as the asset's market value or its resale value.

Summary of Differences:

• Original Cost: The initial purchase price plus all costs to get the asset ready for use. It's a historical figure.
• Accumulated Depreciation: The total depreciation charged to the asset to date. It represents the expensed portion of the asset's cost.
• Book Value: The asset's value on the balance sheet after accounting for depreciation (Original Cost minus Accumulated Depreciation). It reflects the unexpensed portion of the asset's cost.

Example:

Suppose a company buys a machine for $\textsf{₹}$1,00,000 (Original Cost). It depreciates the machine by $\textsf{₹}$10,000 per year for 5 years.

• Original Cost = $\textsf{₹}$1,00,000
• After 5 years, Accumulated Depreciation = 5 years $\times$ $\textsf{₹}$10,000/year = $\textsf{₹}$50,000
• Book Value after 5 years = Original Cost - Accumulated Depreciation = $\textsf{₹}$1,00,000 - $\textsf{₹}$50,000 = $\textsf{₹}$50,000

Given:

Purchase Price per Share (P) = $\textsf{₹}$150

Selling Price per Share (S) = $\textsf{₹}$180

Time Period of Investment = 15 months

No dividends were received.

To Find:

Annualised Return (Simple Annual Return).

Formula for Simple Return:

The simple return (or holding period return) for the entire period is calculated as:

To annualise this return, we need to adjust it for the time period over which it was earned.

Calculation:

First, calculate the profit per share:

Now, calculate the simple return for the 15-month period:

This means the investment generated a 20% return over 15 months.

Next, annualise this return:

To express this as a percentage:

Answer:

The annualised return (simple annual return) on this investment is 16%.

Given:

Loan Principal (P) = $\textsf{₹}$10,00,000

Annual Interest Rate = 9%

Loan Tenure = 5 years

Compounding Frequency = Monthly

To Set Up:

The formula to calculate the Equated Monthly Installment (EMI).

Formula for EMI:

The formula for calculating EMI is:

Where:

• P = Loan Principal
• r = Monthly Interest Rate
• n = Total Number of Payments

Deriving the values for 'r' and 'n':

1. Monthly Interest Rate (r):

The annual interest rate is 9%. Since it is compounded monthly, we divide the annual rate by 12:

2. Total Number of Payments (n):

The loan tenure is 5 years, and payments are made monthly. So, the total number of payments is:

Set up Formula:

Substituting the derived values into the EMI formula, we get:

This is the formula to calculate the EMI, with the values for principal, monthly interest rate, and total number of payments plugged in.

Given:

Book Value after 3 years (BV$_3$) = $\textsf{₹}$4,00,000

Annual Depreciation (AD) = $\textsf{₹}$60,000

Number of Years (n) = 3 years

To Find:

Original Cost of the machine (C).

Formula for Book Value:

The book value of an asset at a certain point in time is calculated as:

Accumulated Depreciation is the total depreciation charged over the years. For the linear method:

Rearranging the book value formula to solve for Original Cost (C):

Calculation:

First, calculate the accumulated depreciation after 3 years:

Now, calculate the original cost:

Answer:

The original cost of the machine was $\textsf{₹}$5,80,000.

Given:

Beginning Value (BV) = $\textsf{₹}$2,50,000

Ending Value (EV) = $\textsf{₹}$4,00,000

Number of Years (n) = 5 years

To Find:

Compound Annual Growth Rate (CAGR).

Formula for CAGR:

The formula for CAGR is:

Calculation:

Substitute the given values into the formula:

Simplify the fraction:

Calculate $(1.6)^{\frac{1}{5}}$ (the fifth root of 1.6):

$(1.6)^{\frac{1}{5}} \approx 1.098563$

Now, complete the calculation:

To express this as a percentage, multiply by 100:

Answer:

The Compound Annual Growth Rate (CAGR) is approximately 9.86%.

Calculating the present value of a perpetuity relies on several fundamental assumptions. These assumptions are crucial for the validity and applicability of the perpetuity valuation model:

Key Assumptions for Present Value of Perpetuity:

1. Perpetual Cash Flows: The most fundamental assumption is that the cash flows will continue indefinitely, without end. This is the defining characteristic of a perpetuity.
2. Constant Cash Flow Amount: It is assumed that the amount of cash flow received in each period remains constant. Whether it's $\textsf{₹}$1,000 per year or $\textsf{₹}$5,000 per quarter, this amount does not change over time.
3. Constant Discount Rate: A constant discount rate (or interest rate) is assumed to be applied to all future cash flows. This rate reflects the time value of money and the risk associated with receiving the cash flows. In reality, discount rates can fluctuate over time due to changes in market conditions, inflation, or perceived risk.
4. Timing of Cash Flows:
   • Ordinary Perpetuity: For an ordinary perpetuity, it is assumed that the first cash flow is received at the end of the first period.
   • Perpetuity Due: For a perpetuity due, it is assumed that the first cash flow is received at the beginning of the first period.
5. No Growth or Variable Growth: The standard perpetuity formula assumes no growth in the cash flows. If the cash flows are expected to grow at a constant rate (known as a growing perpetuity), a different formula is used, but the core concept of perpetual cash flows remains. The basic perpetuity model does not account for variable or uneven growth.
6. Discount Rate Exceeds Growth Rate (for Growing Perpetuities): If a growing perpetuity is considered, a critical assumption is that the discount rate ($r$) must be greater than the growth rate ($g$). If $r \le g$, the present value would either be infinite or negative, which is not a realistic financial scenario.

These assumptions simplify the complex reality of financial markets and cash flow streams. While the standard perpetuity model is a useful theoretical tool, it's important to be aware of these assumptions when applying it to real-world situations, as deviations from these assumptions can significantly impact the calculated present value.

Given:

Annual Deposit (A) = $\textsf{₹}$1,50,000

Interest Rate (r) = 7% per annum = 0.07

Number of Years (n) = 4 years

Compounding Frequency = Annually

Assuming the deposit is made at the end of each year (ordinary annuity).

To Find:

The accumulated amount in the sinking fund after 4 years.

Formula for Future Value of an Ordinary Annuity:

The accumulated amount (Future Value, FV) of a sinking fund (which is an ordinary annuity) is calculated using the formula:

Where:

• FV = Future Value (Accumulated Amount)
• A = Annual Deposit (Annuity Payment)
• r = Interest rate per period
• n = Number of periods

Calculation:

Substitute the given values into the formula:

First, calculate $(1+0.07)^4$:

$(1.07)^4 \approx 1.310796$

Now, substitute this value back into the FV formula:

Answer:

The accumulated amount in the sinking fund after 4 years will be approximately $\textsf{₹}$6,65,991.45.

An Equated Monthly Installment (EMI) is a fixed payment amount made by a borrower to a lender at a specified frequency (usually monthly) over the loan tenure. Each EMI payment comprises two main components:

1. Principal Repayment
2. Interest Component

1. Principal Repayment:

The principal component of an EMI is the portion that directly reduces the outstanding loan amount. When you make an EMI payment, a part of that payment goes towards paying off the actual amount you borrowed.

Behavior over the Loan Tenure:

• Early Stages: In the initial EMIs, the principal repayment portion is relatively small. This is because the outstanding loan amount is still high, and a larger portion of the EMI is allocated to covering the interest accrued on that high principal.
• Later Stages: As the loan tenure progresses and the outstanding principal decreases, the proportion of the EMI allocated to principal repayment gradually increases. More of your payment starts going towards reducing the actual debt.

2. Interest Component:

The interest component of an EMI is the cost of borrowing money. It is calculated based on the outstanding principal amount at the time of the payment and the applicable interest rate.

Behavior over the Loan Tenure:

• Early Stages: In the early EMIs, the interest component is significantly larger. This is because the interest is calculated on the full or near-full principal amount, which is at its highest.
• Later Stages: As the principal amount reduces with each EMI payment, the interest component also decreases. In the final EMIs, the interest portion becomes very small.

How the Components Work Together:

Each EMI payment is fixed in amount. However, the distribution between principal and interest changes with each payment:

• EMI = Principal Repayment + Interest Component

The sum of the principal repayment and interest component for each EMI remains constant, ensuring that the total loan amount, along with all accrued interest, is paid off by the end of the loan tenure. The amortization schedule for a loan details how each EMI is split between principal and interest over the life of the loan.

Example:

Consider a loan where an EMI of $\textsf{₹}$10,000 is paid. In the first EMI, it might be split as $\textsf{₹}$2,000 principal and $\textsf{₹}$8,000 interest. By the last EMI, it might be split as $\textsf{₹}$9,500 principal and $\textsf{₹}$500 interest, but the total EMI remains $\textsf{₹}$10,000.

Given:

Purchase Price (P) = $\textsf{₹}$950

Annual Coupon Payment (C) = $\textsf{₹}$80

Selling Price (S) = $\textsf{₹}$980

Time Period = 1 year

To Find:

Total Return on the investment for that year.

Formula for Total Return:

The total return on an investment like a bond over a period is the sum of the income generated (coupon payments) and any capital gain or loss from the change in the bond's price.

Total Return = (Coupon Payment + Capital Gain/Loss)

To express this as a percentage of the initial investment:

Where:

Capital Gain/Loss = Selling Price - Purchase Price

Calculation:

First, calculate the capital gain:

Now, calculate the total return in absolute terms:

Finally, calculate the total return as a percentage of the purchase price:

Answer:

The total return on the investment for that year is approximately 11.58%.

Given:

Original Purchase Price (P) = $\textsf{₹}$7,00,000

Book Value after 5 years (BV$_5$) = $\textsf{₹}$4,50,000

Salvage Value (S) = $\textsf{₹}$1,50,000

Time elapsed (n) = 5 years

Method used = Linear Depreciation Method

To Find:

Useful life of the vehicle (Useful life in years).

Solution:

The linear depreciation method (also known as the straight-line method) calculates depreciation as a constant amount each year. The formula for annual depreciation is:

We can also express the book value after 'n' years as:

First, let's calculate the annual depreciation from the given information:

We know that the book value after 5 years is $\textsf{₹}$4,50,000. Using the formula for book value:

Now, we need to solve for D:

Subtract 7,00,000 from both sides:

Divide both sides by -5:

So, the annual depreciation is $\textsf{₹}$50,000.

Now, we can use the annual depreciation to find the useful life of the vehicle. We know the original cost, salvage value, and the annual depreciation. Let the useful life be 'L' years.

Using the formula for annual depreciation:

Substitute the known values:

Now, solve for L:

Multiply both sides by L:

Divide both sides by 50,000:

Therefore, the useful life of the vehicle is 11 years.

The main purpose of calculating the Compound Annual Growth Rate (CAGR) is to measure the average annual growth of an investment over a specified period longer than one year, assuming that the profits were reinvested at the end of each year.

CAGR provides a smoothed-out rate of return that shows how an investment would have grown if it had grown at a steady rate each year. It helps to understand the performance of an investment or a business over time in a consistent and comparable manner, abstracting away from the volatility of year-to-year fluctuations.

CAGR is commonly used in several areas:

1. Investment Analysis:

It is widely used by investors to evaluate the historical performance of stocks, mutual funds, bonds, and other investment vehicles. By comparing the CAGR of different investments, investors can make more informed decisions about where to allocate their capital.

2. Business Performance Measurement:

Businesses use CAGR to track their growth in key metrics such as revenue, profit, customer base, or market share over multiple years. It helps management assess the effectiveness of their strategies and set future growth targets.

3. Financial Planning and Forecasting:

CAGR is used in financial planning to project future values of investments or business metrics based on historical growth rates. It helps in budgeting, setting financial goals, and understanding potential future outcomes.

4. Industry Analysis:

Analysts and researchers use CAGR to understand the growth trends within specific industries, compare the growth rates of different companies in the same sector, and identify emerging markets or declining segments.

5. Valuation:

In business valuation, CAGR is often used to estimate future earnings or cash flows, which are then discounted back to present values to determine the worth of a company.

In essence, CAGR is a valuable tool for understanding and comparing growth over time, providing a more stable and insightful perspective than simple year-over-year growth rates.

Given:

Periodic Payment (A) = $\textsf{₹}$12,000 per year

Annual Interest Rate (r) = 6% per annum = 0.06

Type of Cash Flow = Perpetuity

To Find:

Present Value (PV) of the perpetuity.

Solution:

A perpetuity is a series of equal payments that continue indefinitely. The formula for calculating the present value (PV) of a perpetuity is:

Where:

PV = Present Value

A = Periodic Payment

r = Periodic Interest Rate

In this case, the periodic payment is annual, and the interest rate is also annual. Therefore, we can directly use the given values in the formula.

Substitute the given values into the formula:

Now, calculate the present value:

Therefore, the present value of the perpetuity is $\textsf{₹}$200,000.

Given:

Loan Principal (P) = $\textsf{₹}$3,00,000

Annual Interest Rate = 12%

Interest Rate per month (r) = $\frac{12\%}{12} = 1\%$ per month = 0.01

EMI (Equated Monthly Installment) = $\textsf{₹}$10,000

To Find:

Number of months (n) required to repay the loan.

Equation Setup:

The formula for calculating the EMI of a loan is:

Where:

$P$ = Principal Loan Amount

$r$ = Monthly Interest Rate

$n$ = Number of months

We need to set up the equation to find 'n' by substituting the given values:

Simplifying the terms:

To isolate the term with 'n', we can divide both sides by 3,00,000:

The equation to find the number of months (n) is:

Alternatively, we can rearrange the original EMI formula to solve for n:

To solve for 'n', one would typically use logarithms:

The equation that needs to be solved for 'n' is:

Given:

Book Value after 4 years (BV$_4$) = $\textsf{₹}$6,00,000

Annual Depreciation (D) = $\textsf{₹}$75,000

Time elapsed (n) = 4 years

Method used = Linear Depreciation Method

To Find:

Original Cost of the asset (P).

Solution:

The linear depreciation method calculates depreciation as a constant amount each year. The formula for the book value after 'n' years is:

Where:

$BV_n$ = Book Value after 'n' years

$P$ = Original Cost

$n$ = Number of years

$D$ = Annual Depreciation

We are given $BV_4$, $n$, and $D$, and we need to find $P$.

Substitute the given values into the formula:

Calculate the total depreciation over 4 years:

Now, substitute this back into the equation:

To find the original cost (P), add 3,00,000 to both sides of the equation:

Therefore, the original cost of the asset is $\textsf{₹}$9,00,000.

Given:

Beginning Value (BV) = $\textsf{₹}$5,00,000

Ending Value (EV) = $\textsf{₹}$7,50,000

Number of Years (n) = 6 years

To Find:

Compound Annual Growth Rate (CAGR).

Solution:

The formula for calculating the Compound Annual Growth Rate (CAGR) is:

Where:

$EV$ = Ending Value

$BV$ = Beginning Value

$n$ = Number of years

Substitute the given values into the formula:

Simplify the ratio of the ending value to the beginning value:

Now, the formula becomes:

To calculate $(1.5)^{\frac{1}{6}}$, we take the sixth root of 1.5:

Now, subtract 1:

To express this as a percentage, multiply by 100:

Therefore, the Compound Annual Growth Rate (CAGR) of the investment is approximately 6.99%.

Given:

Future Value (FV) = $\textsf{₹}$10,00,000

Number of Years (n) = 8 years

Interest Rate (r) = 9% per annum = 0.09

Compounding Frequency = Annually

To Find:

Annual Deposit (A).

Solution:

This problem involves calculating the periodic payment (annual deposit) required to reach a future value, which is characteristic of an annuity. Specifically, this is a future value of an ordinary annuity problem, as the deposits are made at the end of each period (annually).

The formula for the Future Value of an Ordinary Annuity is:

Where:

$FV$ = Future Value

$A$ = Periodic Payment (Annual Deposit)

$r$ = Interest Rate per period

$n$ = Number of periods

We need to rearrange the formula to solve for the Annual Deposit (A):

Now, substitute the given values into the formula:

Calculate $(1.09)^8$:

Now, substitute this value back into the formula for A:

Calculate the fraction:

Finally, calculate the annual deposit:

Therefore, the annual deposit required is approximately $\textsf{₹}$90,673.44.

The Equated Monthly Installment (EMI) is the fixed amount that a borrower pays to a lender on a specified date each month. Several factors influence the EMI amount for a given loan. These are:

1. Principal Loan Amount (P):

This is the total amount of money borrowed from the lender. A higher principal amount will naturally result in a higher EMI, as more money needs to be repaid over the loan tenure. The relationship between the principal amount and EMI is directly proportional.

2. Interest Rate (r):

The interest rate is the cost of borrowing money. A higher interest rate means the borrower has to pay more interest over the life of the loan, leading to a higher EMI. Conversely, a lower interest rate will result in a lower EMI. The interest rate is a significant driver of the EMI amount. It's important to consider the rate per compounding period (e.g., monthly interest rate if compounded monthly).

3. Loan Tenure (n):

The loan tenure is the duration over which the loan is to be repaid. A longer loan tenure means that the principal amount and interest are spread over a greater number of installments, resulting in a lower EMI. Conversely, a shorter loan tenure means the installments are higher because the repayment is compressed into a shorter period. While a longer tenure reduces the EMI, it also means that more total interest will be paid over the life of the loan.

4. Compounding Frequency:

Although often implied by the interest rate (e.g., "per annum compounded monthly"), the frequency of compounding also affects the EMI. If interest is compounded more frequently (e.g., monthly vs. annually), the effective interest paid will be slightly higher, leading to a slightly higher EMI. For EMI calculations, the interest rate is typically converted to the rate for the payment period (e.g., monthly rate if EMI is monthly).

The EMI formula encapsulates these relationships:

Where:

$P$ = Principal Loan Amount

$r$ = Interest Rate per installment period

$n$ = Total number of installments

As seen from the formula, increasing P or r will increase EMI, while increasing n will decrease EMI.

Given:

Buying Price (BP) = $\textsf{₹}$1050

Selling Price (SP) = $\textsf{₹}$1080

Time Period = 9 months

To Find:

Simple Annualised Return.

Solution:

First, calculate the profit made from the transaction:

Next, calculate the simple return for the 9-month period:

Simplify the fraction:

To annualise this return, we need to find out what the return would be over 12 months (1 year). Since the return for 9 months is $\frac{1}{35}$, we can calculate the annual return using the following proportion:

Or more directly, divide the return for the period by the fraction of the year it represents:

To express this as a percentage, multiply by 100:

Therefore, the simple annualised return is approximately 3.81%.

Given:

Cost of the machine (Original Cost) = $\textsf{₹}$12,00,000

Useful Life of the machine = 15 years

Salvage Value of the machine = $\textsf{₹}$1,50,000

Depreciation Method = Linear Method

To Find:

Annual Depreciation.

Solution:

The linear depreciation method, also known as the straight-line method, calculates depreciation as an equal amount each year over the asset's useful life.

The formula for calculating annual depreciation using the linear method is:

Substitute the given values into the formula:

First, calculate the depreciable amount (Original Cost - Salvage Value):

Now, divide the depreciable amount by the useful life:

Calculate the annual depreciation:

Therefore, the annual depreciation of the machine using the linear method is $\textsf{₹}$70,000.

The difference between the nominal rate of return and the real rate of return lies in their consideration of inflation.

Nominal Rate of Return:

The nominal rate of return is the percentage gain on an investment before accounting for inflation. It represents the actual monetary return earned on an investment over a period. For example, if you invest $\textsf{₹}$100 and it grows to $\textsf{₹}$110 in a year, your nominal return is 10%, regardless of whether prices have increased or decreased.

It is the stated interest rate or return on investment without any adjustments for changes in the purchasing power of money.

Real Rate of Return:

The real rate of return, also known as the inflation-adjusted rate of return, measures the gain on an investment after accounting for the effects of inflation. Inflation erodes the purchasing power of money, meaning that even if your nominal return is positive, your actual ability to buy goods and services might not have increased, or could even have decreased, if inflation is higher than your nominal return.

The real rate of return indicates the actual increase in purchasing power derived from an investment.

Relationship between Nominal and Real Rate of Return:

The relationship between the nominal rate of return, the real rate of return, and the rate of inflation can be approximated by the following Fisher Equation:

A more precise formula, known as the Fisher Equation, is:

From this, the real rate of return can be calculated as:

Example:

Suppose an investment yields a nominal return of 8% in a year when the inflation rate is 5%.

Nominal Rate of Return = 8%

Inflation Rate = 5%

Using the approximation:

Real Rate $\approx$ 8% - 5% = 3%

Using the precise formula:

Real Rate = $\frac{1 + 0.08}{1 + 0.05} - 1 = \frac{1.08}{1.05} - 1 \approx 1.02857 - 1 \approx 0.02857$, which is approximately 2.86%.

In summary:

• Nominal Rate: The stated return, not adjusted for inflation.
• Real Rate: The return adjusted for inflation, reflecting the actual change in purchasing power.

When evaluating the true performance of an investment and its impact on your wealth, the real rate of return is a more meaningful measure.

Given:

Principal Investment (P) = $\textsf{₹}$80,000

Final Amount (A) = $\textsf{₹}$92,000

Time Period (t) = 3 years

To Find:

Simple Annual Return (Rate of Interest, R).

Solution:

First, calculate the total simple interest earned over the 3 years:

Now, use the formula for simple interest to find the annual rate of return:

Where:

$SI$ = Simple Interest

$P$ = Principal Investment

$R$ = Annual Rate of Interest (in %)

$t$ = Time Period (in years)

Rearrange the formula to solve for R:

Substitute the calculated values into the formula:

Calculate the denominator:

Now, calculate R:

So, the simple annual return is 5%.

EMI (Equated Monthly Installment) calculation serves distinct but complementary purposes for both the borrower and the lender in a loan transaction.

From the Borrower's Perspective:

1. Budgeting and Financial Planning:

The primary purpose of EMI calculation for a borrower is to understand the fixed monthly outflow required to repay the loan. This predictability allows the borrower to budget their finances effectively, ensuring they can comfortably meet their repayment obligations without straining their monthly income. Knowing the EMI amount helps in assessing affordability and planning other expenses.

2. Affordability Assessment:

Before taking a loan, borrowers use EMI calculators to determine if they can afford the monthly payments for a loan of a certain principal amount, interest rate, and tenure. This helps them decide on the loan amount they can realistically manage.

3. Understanding Total Repayment Cost:

While EMI focuses on the monthly payment, the calculation implicitly reveals the total interest paid over the loan's life. By extending the tenure, borrowers can lower their EMI, but they will end up paying more interest overall. Conversely, a shorter tenure means higher EMIs but less total interest paid. This understanding helps borrowers make informed decisions about loan terms.

4. Loan Comparison:

Borrowers can use EMI calculations to compare loan offers from different lenders. By inputting the same loan amount, interest rate, and tenure into various calculators, they can identify the most cost-effective loan option.

From the Lender's Perspective:

1. Revenue Generation:

For a lender, the EMI is the mechanism through which they earn revenue from the interest charged on the loan. The calculated EMI ensures that the lender receives the principal amount back, along with the agreed-upon interest, over the loan tenure.

2. Risk Management:

EMI calculation is a crucial part of risk management for lenders. By setting a fixed monthly payment that covers both principal and interest, lenders ensure a steady stream of income. The EMI is structured so that early payments are heavily weighted towards interest, and later payments contribute more towards the principal. This ensures that the lender recovers a significant portion of the interest even if the loan is repaid earlier than expected, and it also helps in managing the loan portfolio's risk.

3. Cash Flow Management:

Lenders rely on the predictable cash flow generated by EMIs to manage their own financial obligations and to fund new loans. The consistent inflow of payments from multiple borrowers allows banks and financial institutions to operate efficiently.

4. Standardization and Efficiency:

The use of a standardized EMI calculation method makes the lending process more efficient and less prone to errors. It allows for automated loan processing and servicing, reducing administrative costs.

5. Monitoring Loan Performance:

The EMI structure helps lenders track the repayment progress of each loan. Deviations from the expected EMI payment schedule (e.g., late payments or defaults) can be quickly identified and addressed.

In essence, EMI calculation is a fundamental process that balances the borrower's need for manageable repayment with the lender's objective of earning a profit and managing financial risk.

Given:

Original Cost of the machine = $\textsf{₹}$6,50,000

Book Value after 6 years = $\textsf{₹}$3,80,000

Time elapsed (n) = 6 years

Depreciation Method = Linear Method

To Find:

Annual Depreciation amount.

Solution:

The linear depreciation method involves calculating a constant depreciation amount each year. The total depreciation over the years is the difference between the original cost and the book value after those years.

First, calculate the total depreciation over the 6 years:

Now, to find the annual depreciation amount, divide the total depreciation by the number of years:

Calculate the annual depreciation:

Therefore, the annual depreciation amount is $\textsf{₹}$45,000.

In the context of a standard loan, which is essentially an annuity where the borrower receives a lump sum (present value) and repays it through a series of equal periodic payments (EMI) over a specific time period at a given interest rate, the relationship is governed by the formula for the present value of an ordinary annuity.

The formula that links these elements is:

Where:

• $PV$ is the Present Value of the loan, which is the principal amount borrowed.
• $PMT$ is the Periodic Payment, commonly known as EMI (Equated Monthly Installment).
• $r$ is the Interest Rate per period. If the loan has an annual interest rate (R) compounded m times per year, then $r = R/m$. For monthly compounding, $r = R/12$.
• $n$ is the Total Number of Periods (installments). If the loan has a tenure of T years and is compounded m times per year, then $n = T \times m$. For monthly compounding over T years, $n = T \times 12$.

Explanation of the Relationship:

Present Value (PV): This is the initial amount borrowed. It represents the value today of all the future EMIs, discounted back at the loan's interest rate.

Periodic Payment (PMT / EMI): This is the fixed amount paid at regular intervals (usually monthly) to repay the loan. The EMI includes both a portion of the principal and the interest accrued during that period.

Interest Rate (r): This is the cost of borrowing money. It determines how much interest accrues on the outstanding loan balance in each period. A higher interest rate means more of each EMI will go towards interest, and less towards the principal, which affects the total repayment period and the total interest paid.

Time Period (n): This is the total duration over which the loan is repaid. The number of periods directly influences the EMI amount. A longer time period means more installments, which generally results in a lower EMI but a higher total interest paid over the life of the loan. Conversely, a shorter time period means fewer installments, leading to higher EMIs but less total interest paid.

How they are interconnected:

• If you increase the Present Value (PV), holding other factors constant, the Periodic Payment (EMI) will increase.
• If you increase the Interest Rate (r), holding other factors constant, the Periodic Payment (EMI) will increase.
• If you increase the Time Period (n), holding other factors constant, the Periodic Payment (EMI) will decrease.

The formula shows that the present value of the loan is directly proportional to the EMI and inversely related to the interest rate and the number of periods in a complex way due to the exponentiation.

Given:

Beginning Value (BV) = $\textsf{₹}$2,00,000

Ending Value (EV) = $\textsf{₹}$3,10,000

Number of Years (n) = 7 years

To Find:

Compound Annual Growth Rate (CAGR).

Solution:

The formula for calculating the Compound Annual Growth Rate (CAGR) is:

Where:

$EV$ = Ending Value

$BV$ = Beginning Value

$n$ = Number of years

Substitute the given values into the formula:

Simplify the ratio of the ending value to the beginning value:

Now, the formula becomes:

To calculate $(1.55)^{\frac{1}{7}}$, we take the seventh root of 1.55:

Now, subtract 1:

To express this as a percentage, multiply by 100:

Therefore, the Compound Annual Growth Rate (CAGR) of the investment is approximately 6.55%.

Given:

Perpetual Periodic Payment (A) = $\textsf{₹}$50,000 per year

Annual Interest Rate (r) = 7.5% per annum = 0.075

The first payment is made after one year, which means it's a standard perpetuity.

To Find:

Initial amount needed to fund the scholarship (Present Value of the perpetuity, PV).

Solution:

A perpetuity is a series of equal payments that continue indefinitely. The formula for calculating the present value (PV) of a perpetuity is:

Where:

$PV$ = Present Value (the initial amount needed)

$A$ = Periodic Payment (the annual scholarship amount)

$r$ = Periodic Interest Rate (the annual interest rate)

Substitute the given values into the formula:

Now, calculate the present value:

Therefore, the initial amount needed to fund the perpetual scholarship is approximately $\textsf{₹}$666,666.67.

Given:

Accumulated Amount (Future Value, FV) = $\textsf{₹}7,00,000$

Annual Contribution (Periodic Payment, P) = $\textsf{₹}1,00,000$

Number of years (n) = 6

To Find:

Effective Interest Rate (i) per annum.

Formula:

The accumulated amount (Future Value) of an ordinary annuity is given by the formula:

$FV = P \times \frac{(1+i)^n - 1}{i}$

where:

FV = Future Value of the annuity

P = Periodic Payment

i = Interest rate per period

n = Number of periods

Solution:

Substitute the given values into the formula:

$7,00,000 = 1,00,000 \times \frac{(1+i)^6 - 1}{i}$

Divide both sides of the equation by $1,00,000$:

$\frac{7,00,000}{1,00,000} = \frac{(1+i)^6 - 1}{i}$

$7 = \frac{(1+i)^6 - 1}{i}$

Let $S_{n|i} = \frac{(1+i)^n - 1}{i}$ be the future value interest factor for an annuity. We need to find the value of $i$ such that $S_{6|i} = 7$. This equation is a polynomial in $i$ and cannot be easily solved directly using algebraic methods. The effective interest rate $i$ can be approximated using methods like trial and error, interpolation, or by using financial calculators or software.

Using trial and error, we evaluate $S_{6|i}$ for different values of $i$ to find which value gives a result close to 7.

Trial 1: Assume an interest rate $i = 5\% = 0.05$

$S_{6|0.05} = \frac{(1+0.05)^6 - 1}{0.05} = \frac{(1.05)^6 - 1}{0.05}$

$(1.05)^6 \approx 1.340096$

$S_{6|0.05} \approx \frac{1.340096 - 1}{0.05} = \frac{0.340096}{0.05} \approx 6.8019$

Since $6.8019 < 7$, the actual interest rate is higher than 5%.

Trial 2: Assume an interest rate $i = 6\% = 0.06$

$S_{6|0.06} = \frac{(1+0.06)^6 - 1}{0.06} = \frac{(1.06)^6 - 1}{0.06}$

$(1.06)^6 \approx 1.41852$

$S_{6|0.06} \approx \frac{1.41852 - 1}{0.06} = \frac{0.41852}{0.06} \approx 6.9753$

Since $6.9753 < 7$, the actual interest rate is higher than 6%.

Trial 3: Assume an interest rate $i = 6.3\% = 0.063$

$S_{6|0.063} = \frac{(1+0.063)^6 - 1}{0.063} = \frac{(1.063)^6 - 1}{0.063}$

$(1.063)^6 \approx 1.44091$

$S_{6|0.063} \approx \frac{1.44091 - 1}{0.063} = \frac{0.44091}{0.063} \approx 6.99857$

Since $6.99857$ is very close to 7 but slightly less, the rate is slightly higher than 6.3%.

Trial 4: Assume an interest rate $i = 6.32\% = 0.0632$

$S_{6|0.0632} = \frac{(1+0.0632)^6 - 1}{0.0632} = \frac{(1.0632)^6 - 1}{0.0632}$

$(1.0632)^6 \approx 1.44240$

$S_{6|0.0632} \approx \frac{1.44240 - 1}{0.0632} = \frac{0.44240}{0.0632} \approx 7.00006$

Since $7.00006$ is very close to 7 and slightly more, the actual rate is between 6.3% and 6.32%.

Based on the trial and error, the effective interest rate is approximately 6.32% per annum.

In accounting and finance, particularly in the context of calculating depreciation, 'salvage value' refers to the estimated residual value of a tangible asset at the end of its useful life.

It is the amount that a business expects to realize from selling the asset after it has been fully used and depreciated. This value is also known by other names such as scrap value or residual value.

Role in Depreciation Calculation

The primary purpose of determining salvage value is to calculate an asset's depreciable base. Depreciation is the process of allocating the cost of an asset over its useful life, and you cannot depreciate an asset below its salvage value.

The depreciable base is calculated using the following formula:

$\text{Depreciable Base} = \text{Original Cost of Asset} - \text{Salvage Value}$

This depreciable base is then spread over the asset's useful life to determine the annual depreciation expense.

For example, using the straight-line depreciation method:

$\text{Annual Depreciation Expense} = \frac{(\text{Original Cost} - \text{Salvage Value})}{\text{Useful Life}}$

Example:

Suppose a company buys a machine for $50,000. It estimates that the machine will have a useful life of 10 years and a salvage value of $5,000 at the end of those 10 years.

• Original Cost: $50,000
• Salvage Value: $5,000
• Useful Life: 10 years

The depreciable base would be:

$50,000 - $5,000 = $45,000$

The annual depreciation expense would be:

$\frac{$45,000}{10 \text{ years}} = $4,500 \text{ per year}$

Key Characteristics

• It is an Estimate: Salvage value is determined at the time an asset is acquired. Since it is a projection of future value, it is always an estimate and may need to be revised during the asset's life if expectations change significantly.
• Impact on Financial Statements: A higher salvage value results in a lower depreciable base and, consequently, lower annual depreciation expense. This leads to higher reported net income during the asset's life. Conversely, a lower salvage value increases depreciation expense and reduces net income.
• Zero Salvage Value: If an asset is expected to have no or negligible value at the end of its useful life, its salvage value is considered to be zero. In this case, the entire original cost of the asset is depreciated.

Initial Investment = $\textsf{₹}1,00,000$

Final Value after one year = $\textsf{₹}1,12,000$

Dividend Received during the year = $\textsf{₹}5,000$

1. Capital Appreciation

2. Dividend Yield

1. Calculation of Capital Appreciation

Capital appreciation is the increase in the market price of the investment, separate from any income (like dividends) it generates.

First, we need to find the value of the mutual fund units at the end of the year, excluding the dividend received. The final value of $\textsf{₹}1,12,000$ includes both the growth in the investment's price and the dividend payout.

Value of the investment at year-end (before dividend) = Final Value - Dividend Received

$= \textsf{₹}1,12,000 - \textsf{₹}5,000$

$= \textsf{₹}1,07,000$

Now, capital appreciation is the difference between this year-end value and the initial investment.

Capital Appreciation = Year-end Value of Investment - Initial Investment

$= \textsf{₹}1,07,000 - \textsf{₹}1,00,000$

$= \textsf{₹}7,000$

The capital appreciation is $\textsf{₹}7,000$.

2. Calculation of Dividend Yield

Dividend yield measures the return from dividends as a percentage of the original investment.

The formula for dividend yield is:

$\text{Dividend Yield} = \left( \frac{\text{Dividend Received}}{\text{Initial Investment}} \right) \times 100\%$

Substituting the given values:

$\text{Dividend Yield} = \left( \frac{\textsf{₹}5,000}{\textsf{₹}1,00,000} \right) \times 100\%$

$\text{Dividend Yield} = 0.05 \times 100\%$

$\text{Dividend Yield} = 5\%$

The dividend yield is 5%.

The capital appreciation on the investment is $\textsf{₹}7,000$.

The dividend yield is 5%.

Given:

EMI = $\textsf{₹}$25,000

Time period = 4 years = 48 months

Rate of interest (compounded monthly) = 10% per annum = $\frac{10}{12}\%$ per month = $\frac{10}{1200} = 0.0083\overline{3}$ per month

To Find:

The principal amount of the loan (P)

Formula used:

EMI is given by the formula:

$EMI = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Where:

• $P$ = principal
• $r$ = monthly interest rate
• $n$ = number of months

Substituting the known values:

$25,000 = P \cdot \frac{0.00833 \times (1 + 0.00833)^{48}}{(1 + 0.00833)^{48} - 1}$

First, compute $(1 + r)^{48}$

$(1 + 0.00833)^{48} \approx (1.00833)^{48} \approx 1.4896$

So,

$25,000 = P \cdot \frac{0.00833 \times 1.4896}{1.4896 - 1}$

$25,000 = P \cdot \frac{0.0124}{0.4896}$

$25,000 = P \cdot 0.0253$

Now solving for $P$:

$P = \frac{25,000}{0.0253} \approx 988,142.29$

∴ Principal amount of the loan = $\textsf{₹}$988,142.29

Given:

Cost of machine = $\textsf{₹}$9,00,000

Estimated life = 12 years

Book value after 8 years = $\textsf{₹}$4,00,000

To Find:

Estimated salvage value (S)

Using Linear Depreciation Method:

The formula for book value after $n$ years using linear depreciation:

$B = C - \frac{(C - S) \cdot n}{N}$

Where:

• $B$ = Book value after $n$ years
• $C$ = Original cost
• $S$ = Salvage value (to be found)
• $n$ = Number of years after which book value is calculated
• $N$ = Total life of the asset

Substituting the known values:

$4,00,000 = 9,00,000 - \frac{(9,00,000 - S) \cdot 8}{12}$

Solving the equation:

Bring constant term to one side:

$5,00,000 = \frac{8}{12}(9,00,000 - S)$

Multiply both sides by 12:

$60,00,000 = 8(9,00,000 - S)$

Divide both sides by 8:

$7,50,000 = 9,00,000 - S$

Now solve for $S$:

$S = 9,00,000 - 7,50,000 = 1,50,000$

∴ Estimated salvage value = $\textsf{₹}$1,50,000

Given:

Initial investment = $\textsf{₹}$3,00,000

Time period = 5 years

CAGR = 15% = 0.15

To Find:

Final value of the investment after 5 years

Formula used:

Final Value (FV) is calculated using the CAGR formula:

$FV = PV \cdot (1 + r)^n$

Where:

• $PV$ = Initial investment
• $r$ = CAGR rate
• $n$ = Number of years

Substituting the values:

$FV = 3,00,000 \cdot (1 + 0.15)^5$

$= 3,00,000 \cdot (1.15)^5$

$= 3,00,000 \cdot 2.011357$

$= 6,03,407.10$

∴ Final value of the investment after 5 years = $\textsf{₹}$6,03,407.10

Book Value:

The book value of an asset is its value as recorded in the company's books (accounts) after accounting for depreciation.

In the context of depreciation:

When an asset is purchased, it is recorded at its original cost. Over time, due to usage, wear and tear, or obsolescence, the asset loses value. This loss in value is called depreciation.

Each year, a portion of the asset’s cost is deducted as depreciation, and the remaining value is called the book value.

Mathematically,

$ \text{Book Value} = \text{Original Cost} - \text{Accumulated Depreciation} $

Key Points:

• Book value reflects the net worth of an asset at any point in time.
• It helps in financial reporting and decision-making regarding asset disposal or replacement.
• The method of depreciation (e.g., straight-line, reducing balance) affects the book value calculation.

Example:

If a machine is purchased for $\textsf{₹}$10,00,000 and depreciated at $\textsf{₹}$1,00,000 per year, then after 3 years, the book value will be:

$\textsf{₹}$10,00,000 - ($\textsf{₹}$1,00,000 × 3) = $\textsf{₹}$7,00,000

∴ Book value is the asset's current value in the accounts after accounting for depreciation.

Given:

Perpetual payment (A) = $\textsf{₹}$20,000 (every 6 months)

Rate of return = 6% per annum compounded semi-annually = $\frac{6}{2} = 3\%$ per half-year = 0.03

To Find:

Present value (PV) of the perpetuity

Formula used:

The present value of a perpetuity is given by:

$PV = \frac{A}{r}$

Where:

• $A$ = Periodic payment
• $r$ = Periodic rate of interest

Substituting the values:

$PV = \frac{20,000}{0.03}$

$= 6,66,666.67$

∴ Present value of the perpetuity = $\textsf{₹}$6,66,666.67

Return:

The return refers to the total gain or loss made on an investment over a period of time. It includes all sources of income, such as interest, dividends, and capital appreciation.

It is generally expressed as a percentage of the original investment amount.

Formula:

$\text{Return} = \frac{\text{Final Value} - \text{Initial Investment}}{\text{Initial Investment}} \times 100\%$

Nominal Rate of Return:

The nominal rate of return is the rate of return on an investment without adjusting for inflation. It shows the percentage increase in money received from an investment, but does not indicate how much of that return is real purchasing power.

Example:

If an investment earns 8% in a year but the inflation rate is 3%, then:

• Nominal return = 8%
• Real return = 8% − 3% = 5%

Key Differences:

∴ The return gives an overall measure of profit/loss, whereas the nominal rate of return specifically indicates the unadjusted percentage gain without considering inflation.

Given:

Annual deposit = $\textsf{₹}$80,000

Time period = 5 years

Rate of interest = 6% per annum = 0.06

To Find:

Accumulated amount in the fund at the end of 5 years

Formula used:

The accumulated amount in a sinking fund is calculated using the future value of an ordinary annuity formula:

$A = R \cdot \frac{(1 + r)^n - 1}{r}$

Where:

• $A$ = Accumulated amount
• $R$ = Annual deposit
• $r$ = Annual interest rate
• $n$ = Number of years

Substituting the values:

$A = 80,000 \cdot \frac{(1 + 0.06)^5 - 1}{0.06}$

$= 80,000 \cdot \frac{(1.3382 - 1)}{0.06}$

$= 80,000 \cdot \frac{0.3382}{0.06}$

$= 80,000 \cdot 5.6367$

$= 4,50,936$

∴ Accumulated amount in the sinking fund = $\textsf{₹}$4,50,936

Given:

Cost of the asset = $\textsf{₹}$5,00,000

Salvage value = $\textsf{₹}$50,000

Useful life = 8 years

To Find:

Annual depreciation rate using the linear method

Formula used:

Annual depreciation amount:

$D = \frac{\text{Cost} - \text{Salvage Value}}{\text{Useful Life}}$

Depreciation rate:

$\text{Depreciation Rate} = \left( \frac{D}{\text{Cost}} \right) \times 100\%$

Step 1: Calculate Annual Depreciation Amount

$D = \frac{5,00,000 - 50,000}{8} = \frac{4,50,000}{8} = 56,250$

Step 2: Calculate Depreciation Rate

$\text{Depreciation Rate} = \left( \frac{56,250}{5,00,000} \right) \times 100\% = 11.25\%$

∴ Annual depreciation rate = 11.25%

Given:

Final Value (FV) = $\textsf{₹}$10,00,000

CAGR = 8% = 0.08

Time period = 10 years

To Find:

Initial investment (PV)

Formula used:

$FV = PV \cdot (1 + r)^n$

$\Rightarrow PV = \frac{FV}{(1 + r)^n}$

Where:

• $FV$ = Final value
• $PV$ = Initial investment
• $r$ = CAGR
• $n$ = Number of years

Substituting the values:

$PV = \frac{10,00,000}{(1 + 0.08)^{10}} = \frac{10,00,000}{(1.08)^{10}}$

$(1.08)^{10} \approx 2.1589$

$PV = \frac{10,00,000}{2.1589} \approx 4,63,229.59$

∴ Initial investment amount = $\textsf{₹}$4,63,229.59

Given:

Principal amount (P) = $\textsf{₹}$4,00,000

Time period = 2 years = 24 months

Annual interest rate = 15% per annum compounded monthly

Monthly interest rate ($r$) = $\frac{15}{12} = 1.25\% = 0.0125$

To Find:

EMI (Equated Monthly Installment)

Formula used:

$EMI = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Where:

• $P$ = Loan amount
• $r$ = Monthly interest rate
• $n$ = Number of monthly installments

Substituting the values:

$EMI = 4,00,000 \cdot \frac{0.0125(1 + 0.0125)^{24}}{(1 + 0.0125)^{24} - 1}$

First, calculate $(1 + 0.0125)^{24}$

$(1.0125)^{24} \approx 1.34935$

So,

$EMI = 4,00,000 \cdot \frac{0.0125 \times 1.34935}{1.34935 - 1}$

$= 4,00,000 \cdot \frac{0.016867}{0.34935}$

$= 4,00,000 \cdot 0.04829$

$= 19,316$ (approx)

∴ EMI = $\textsf{₹}$19,316 (approximately)

Given:

Principal loan amount ($P$) = $\textsf{₹}$50,00,000

Annual interest rate = 8% per annum compounded monthly

Monthly interest rate ($r$) = $\frac{8}{12} = 0.6667\% = 0.006667$

Loan tenure = 20 years = 240 months

To Find:

1. Monthly EMI
2. Total interest paid over 20 years

Formula used for EMI:

$EMI = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Where:

• $P$ = Principal
• $r$ = Monthly interest rate
• $n$ = Number of monthly installments

Substituting the values:

$EMI = 50,00,000 \cdot \frac{0.006667(1 + 0.006667)^{240}}{(1 + 0.006667)^{240} - 1}$

First, calculate $(1 + 0.006667)^{240}$

$(1.006667)^{240} \approx 4.9268$

So,

$EMI = 50,00,000 \cdot \frac{0.006667 \cdot 4.9268}{4.9268 - 1}$

$= 50,00,000 \cdot \frac{0.03285}{3.9268}$

$= 50,00,000 \cdot 0.00837$

$= 41,850$ (approx)

Monthly EMI = $\textsf{₹}$41,850

Total amount paid over 20 years:

$= 41,850 \times 240 = \textsf{₹}1,00,44,000$

Total interest paid:

$= \textsf{₹}1,00,44,000 - \textsf{₹}50,00,000 = \textsf{₹}50,44,000$

∴ Mr. Sharma has to pay an EMI of $\textsf{₹}$41,850 per month.

∴ The total interest paid over 20 years is $\textsf{₹}$50,44,000.

Given:

Replacement cost = $\textsf{₹}$15,00,000

Salvage value = $\textsf{₹}$1,00,000

Effective amount required = $\textsf{₹}$15,00,000 − $\textsf{₹}$1,00,000 = $\textsf{₹}$14,00,000

Time period = 7 years

Rate of interest = 7% per annum compounded annually = $r = 0.07$

To Find:

1. Annual deposit required
2. Sinking fund schedule for the first 3 years

Formula used for sinking fund deposit:

$R = \frac{A \cdot r}{(1 + r)^n - 1}$

Where:

• $A$ = Target amount = $\textsf{₹}$14,00,000
• $r$ = Interest rate = 0.07
• $n$ = Number of years = 7

Substituting the values:

$R = \frac{14,00,000 \cdot 0.07}{(1.07)^7 - 1}$

$= \frac{98,000}{(1.6058 - 1)} = \frac{98,000}{0.6058}$

$= 1,61,741.47$

∴ Annual deposit required = $\textsf{₹}$1,61,741.47

Sinking Fund Schedule (First 3 Years)

Note: Interest = 7% of opening balance each year

∴ The company must deposit $\textsf{₹}$1,61,741.47 annually into the sinking fund, and the above table shows the schedule for the first 3 years.

Part (a): Present Value of Perpetuity

Given:

Annual payment ($A$) = $\textsf{₹}$30,000

Rate of return ($r$) = 9% = 0.09

Formula:

$PV = \frac{A}{r}$

Substituting the values:

$PV = \frac{30,000}{0.09} = \textsf{₹}$3,33,333.33

Part (b): CAGR for Bond Investment

Given:

Initial investment = $\textsf{₹}$3,50,000

Final amount = $\textsf{₹}$7,00,000

Time period = 8 years

Formula:

$CAGR = \left( \frac{FV}{PV} \right)^{\frac{1}{n}} - 1$

Substituting the values:

$CAGR = \left( \frac{7,00,000}{3,50,000} \right)^{\frac{1}{8}} - 1 = (2)^{\frac{1}{8}} - 1$

$= 1.0905 - 1 = 0.0905 = 9.05\%$

Conclusion:

• Present Value of perpetuity = $\textsf{₹}$3,33,333.33
• CAGR of bond = 9.05%

Since both options offer nearly the same rate (perpetuity = 9%, bond CAGR ≈ 9.05%) but:

• Perpetuity gives a steady income forever
• Bond doubles the money in 8 years but is a one-time payout

∴ Financially, the bond gives slightly better returns due to a higher CAGR, but the better choice depends on whether the investor prefers steady income (perpetuity) or capital growth (bond).

Given:

Cost of the machine = $\textsf{₹}$20,00,000

Salvage value = $\textsf{₹}$2,00,000

Estimated useful life = 10 years

(a) Annual Depreciation Amount:

$D = \frac{\text{Cost} - \text{Salvage Value}}{\text{Useful Life}}$

$D = \frac{20,00,000 - 2,00,000}{10} = \frac{18,00,000}{10} = \textsf{₹}1,80,000$

(b) Depreciation Schedule for 5 Years:

(c) Book Value on 31st March 2025:

2023–2024 depreciation = $\textsf{₹}$1,80,000 → Book Value = $11,00,000 - 1,80,000 = \textsf{₹}9,20,000$

2024–2025 depreciation = $\textsf{₹}$1,80,000 → Book Value = $9,20,000 - 1,80,000 = \textsf{₹}7,40,000$

∴ Book value of the machine on 31st March 2025 = $\textsf{₹}$7,40,000

Given:

Principal loan amount ($P$) = $\textsf{₹}$8,00,000

Annual interest rate = 10%

Monthly interest rate ($r$) = $\frac{10}{12} = 0.8333\% = 0.008333$

Loan period = 5 years = 60 months

(a) Monthly EMI:

Formula:

$EMI = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Substitute:

$EMI = 8,00,000 \cdot \frac{0.008333(1 + 0.008333)^{60}}{(1 + 0.008333)^{60} - 1}$

$(1.008333)^{60} \approx 1.647$ (approx)

$EMI = 8,00,000 \cdot \frac{0.008333 \cdot 1.647}{0.647} = 8,00,000 \cdot \frac{0.01373}{0.647}$

$EMI = 8,00,000 \cdot 0.02122 = \textsf{₹}16,976$ (approx)

∴ Monthly EMI = $\textsf{₹}$16,976

(b) Outstanding Principal After 24 Payments:

Formula:

$OP = EMI \cdot \frac{(1 + r)^n - (1 + r)^p}{r}$

Where:

• $n = 60$, $p = 24$, $r = 0.008333$
• $EMI = 16,976$

Compute:

$(1.008333)^{60} \approx 1.647$, $(1.008333)^{24} \approx 1.2314$

$OP = 16,976 \cdot \frac{1.647 - 1.2314}{0.008333}$

$= 16,976 \cdot \frac{0.4156}{0.008333} = 16,976 \cdot 49.87$

$= \textsf{₹}8,46,371$ (approx)

(Note: There may be rounding differences; exact EMI can slightly adjust this value.)

∴ Outstanding principal after 24 payments = $\textsf{₹}$8,46,371

(c) Total amount paid if Mr. Iyer prepays after 2 years:

• EMI paid for 24 months = $16,976 \times 24 = \textsf{₹}4,07,424$
• Final payment = Outstanding Principal = $\textsf{₹}$8,46,371

$\text{Total paid} = 4,07,424 + 8,46,371 = \textsf{₹}12,53,795$

∴ Total payment if prepaid after 2 years = $\textsf{₹}$12,53,795

Case 1: First Donation Made Immediately

Given:

Annual donation = $\textsf{₹}$75,000

Interest rate = 6% per annum

This is a case of a perpetuity due (first payment starts immediately)

Formula for Present Value of Perpetuity Due:

$PV = A + \frac{A}{r}$

Where:

• $A = 75,000$
• $r = 0.06$

Substitute:

$PV = 75,000 + \frac{75,000}{0.06} = 75,000 + 12,50,000 = \textsf{₹}13,25,000$

Case 2: First Donation Made at the End of the First Year

This is a case of an ordinary perpetuity.

Formula:

$PV = \frac{A}{r} = \frac{75,000}{0.06} = \textsf{₹}12,50,000$

Conclusion:

● If the first donation is made immediately, the company needs to deposit $\textsf{₹}$13,25,000

● If the first donation is made at the end of the first year, the company needs to deposit $\textsf{₹}$12,50,000

Annual Return Calculation:

Formula: $R = \frac{V_{\text{end}} - V_{\text{start}}}{V_{\text{start}}} \times 100\%$

Simple Average Annual Return:

$=\frac{12.5 + 6.67 + 10.42 + 3.77 + 12.73}{5} = \frac{46.09}{5} = 9.22\%$

Compound Annual Growth Rate (CAGR):

Formula: $CAGR = \left( \frac{V_{\text{final}}}{V_{\text{initial}}} \right)^{\frac{1}{n}} - 1$

$CAGR = \left( \frac{6,20,000}{4,00,000} \right)^{\frac{1}{5}} - 1 = (1.55)^{0.2} - 1$

$= 1.0923 - 1 = 0.0923 = 9.23\%$

Conclusion:

• Simple Average Annual Return = 9.22%
• Compound Annual Growth Rate (CAGR) = 9.23%

∴ CAGR is a more appropriate measure for representing the investment’s growth over the entire 5-year period because it accounts for the compounding effect and gives a single consistent annual rate.

Given:

Purchase cost = $\textsf{₹}$25,00,000

Salvage value = $\textsf{₹}$3,00,000

Useful life = 8 years

(a) Annual Depreciation Expense:

Using linear method:

$D = \frac{\text{Cost} - \text{Salvage Value}}{\text{Useful Life}}$

$D = \frac{25,00,000 - 3,00,000}{8} = \frac{22,00,000}{8} = \textsf{₹}2,75,000$

∴ Annual Depreciation = $\textsf{₹}$2,75,000

(b) Book Value at End of Year 4 and Year 7:

• After 4 years:

Book Value = $25,00,000 - (4 \times 2,75,000) = 25,00,000 - 11,00,000 = \textsf{₹}14,00,000$

• After 7 years:

Book Value = $25,00,000 - (7 \times 2,75,000) = 25,00,000 - 19,25,000 = \textsf{₹}5,75,000$

(c) Total Accumulated Depreciation at End of Year 6:

$AD = 6 \times 2,75,000 = \textsf{₹}16,50,000$

∴

• Book Value at end of Year 4 = $\textsf{₹}$14,00,000
• Book Value at end of Year 7 = $\textsf{₹}$5,75,000
• Total Accumulated Depreciation at end of Year 6 = $\textsf{₹}$16,50,000

Option A: Loan at 12% p.a. compounded monthly

Given:

• Principal ($P$) = $\textsf{₹}$10,00,000
• Annual interest rate = 12% → Monthly rate ($r$) = 1% = 0.01
• Time = 5 years = 60 months

EMI Formula:

$EMI = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Substitute:

$EMI = 10,00,000 \cdot \frac{0.01(1.01)^{60}}{(1.01)^{60} - 1}$

$(1.01)^{60} \approx 1.8194$

$EMI = 10,00,000 \cdot \frac{0.018194}{0.8194} = 10,00,000 \cdot 0.0222$

$EMI = \textsf{₹}22,200$ (approx)

Total payment over 5 years:

$22,200 \times 60 = \textsf{₹}13,32,000$

Total interest = $13,32,000 - 10,00,000 = \textsf{₹}3,32,000$

Option B: Loan at 13% p.a. compounded annually

Given:

• Principal ($P$) = $\textsf{₹}$10,00,000
• Annual rate = 13% = 0.13
• Time = 5 years

Installment Formula (Annual Instalment of Loan):

$A = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Substitute:

$A = 10,00,000 \cdot \frac{0.13(1.13)^5}{(1.13)^5 - 1}$

$(1.13)^5 \approx 1.8427$

$A = 10,00,000 \cdot \frac{0.2396}{0.8427} = 10,00,000 \cdot 0.2843$

$A = \textsf{₹}28,4300$ (approx)

Total payment over 5 years:

$28,4300 \times 5 = \textsf{₹}14,21,500$

Total interest = $14,21,500 - 10,00,000 = \textsf{₹}4,21,500$

Comparison:

Conclusion:

Option A results in a lower total interest payment of $\textsf{₹}$3,32,000 compared to Option B's $\textsf{₹}$4,21,500.

∴ Option A is financially better for the business.

Given:

Future Value (FV) = $\textsf{₹}$50,00,000

Time ($n$) = 10 years

Interest rate ($r$) = 6.5% = 0.065 (per annum)

(a) Required Annual Deposit (A):

Using the Future Value of Ordinary Annuity formula:

$FV = A \cdot \frac{(1 + r)^n - 1}{r}$

Rearranging to find $A$:

$A = \frac{FV \cdot r}{(1 + r)^n - 1}$

$A = \frac{50,00,000 \cdot 0.065}{(1.065)^{10} - 1}$

$(1.065)^{10} \approx 1.877$ → $1.877 - 1 = 0.877$

$A = \frac{50,00,000 \cdot 0.065}{0.877} = \frac{3,25,000}{0.877} \approx \textsf{₹}3,70,012.54$

∴ Required Annual Deposit = $\textsf{₹}$3,70,012.54 (approx)

(b) Total Interest Earned:

Total payment made = $3,70,012.54 \times 10 = \textsf{₹}37,00,125.40$

Interest earned = $50,00,000 - 37,00,125.40 = \textsf{₹}12,99,874.60$

∴ Total Interest Earned = $\textsf{₹}$12,99,874.60 (approx)

(c) Required Interest Rate if Annual Deposit = $\textsf{₹}$3,00,000

We use the same formula:

$FV = A \cdot \frac{(1 + r)^n - 1}{r}$

Where:

• $FV = 50,00,000$
• $A = 3,00,000$
• $n = 10$

Trial and error or numerical solution is required. Trying with $r = 10.6\% = 0.106$:

$\frac{(1.106)^{10} - 1}{0.106} \approx \frac{1.741 - 1}{0.106} \approx \frac{0.741}{0.106} \approx 6.99$

$FV \approx 3,00,000 \times 6.99 \approx 20,97,000$ → too low

Trying with $r = 0.16$ (16%):

$(1.16)^{10} \approx 4.41 \Rightarrow \frac{4.41 - 1}{0.16} = \frac{3.41}{0.16} \approx 21.31$

$3,00,000 \times 21.31 \approx 63,93,000$ → too high

Trying $r \approx 0.13$ (13%):

$(1.13)^{10} \approx 3.395 \Rightarrow \frac{3.395 - 1}{0.13} = \frac{2.395}{0.13} \approx 18.42$

$3,00,000 \times 18.42 = \textsf{₹}55,26,000$ → slightly high

Trying $r = 12.1\%$:

$(1.121)^{10} \approx 3.136 \Rightarrow \frac{2.136}{0.121} \approx 17.65$

$3,00,000 \times 17.65 = \textsf{₹}52,95,000$ → close

∴ Required interest rate ≈ 11.8% p.a. compounded annually (approx)

Final Answers:

• (a) Annual Deposit = $\textsf{₹}$3,70,012.54
• (b) Total Interest Earned = $\textsf{₹}$12,99,874.60
• (c) Required Interest Rate = Approximately 11.8%

Step 1: Calculate the amount after 3 years in savings account

Principal = $\textsf{₹}$1,50,000

Rate = 5% p.a.

Time = 3 years

Using compound interest formula:

$A = P(1 + r)^n = 1,50,000(1.05)^3$

$A = 1,50,000 \times 1.157625 = \textsf{₹}1,73,643.75$

Step 2: Invested in stock for 4 years and grew to $\textsf{₹}$2,20,000

Initial investment = $\textsf{₹}$1,73,643.75

Final value = $\textsf{₹}$2,20,000

Duration = 4 years

Step 3: Calculate total CAGR over 7-year period

Initial value = $\textsf{₹}$1,50,000

Final value = $\textsf{₹}$2,20,000

Time = 7 years

CAGR formula:

$CAGR = \left(\frac{\text{Final Value}}{\text{Initial Value}}\right)^{\frac{1}{n}} - 1$

$CAGR = \left(\frac{2,20,000}{1,50,000}\right)^{\frac{1}{7}} - 1$

$CAGR = (1.4667)^{1/7} - 1 \approx 1.0555 - 1 = 0.0555$

$CAGR = 5.55\%$ (approx)

Final Answer:

∴ The Compound Annual Growth Rate (CAGR) over the 7-year period is approximately $5.55\%$

Given:

Cost of building = $\textsf{₹}$80,00,000

Salvage value = $\textsf{₹}$10,00,000

Useful life = 40 years

(a) Annual Depreciation:

Annual Depreciation = $\frac{\text{Cost} - \text{Salvage value}}{\text{Useful life}} = \frac{80,00,000 - 10,00,000}{40}$

$= \frac{70,00,000}{40} = \textsf{₹}1,75,000$

(b) Accumulated Depreciation at the end of year 25:

$= 1,75,000 \times 25 = \textsf{₹}43,75,000$

(c) Book Value at the end of year 30:

Book Value = Cost – Accumulated Depreciation

$= 80,00,000 - (1,75,000 \times 30) = 80,00,000 - 52,50,000$

$= \textsf{₹}27,50,000$

(d) Year when book value ≤ $\textsf{₹}$30,00,000:

Let $n$ be the number of years:

Book Value = $80,00,000 - 1,75,000 \times n \leq 30,00,000$

$1,75,000 \times n \geq 50,00,000 \Rightarrow n \geq \frac{50,00,000}{1,75,000}$

$n \geq 28.57$

∴ In the 29^th year, book value will fall below or equal to $\textsf{₹}$30,00,000.

Final Answers:

• (a) Annual Depreciation = $\textsf{₹}$1,75,000
• (b) Accumulated Depreciation after 25 years = $\textsf{₹}$43,75,000
• (c) Book Value after 30 years = $\textsf{₹}$27,50,000
• (d) Book value will be ≤ $\textsf{₹}$30,00,000 in year 29

Given:

Monthly pension = $\textsf{₹}$15,000

Annual interest rate = 7.2% compounded monthly $\Rightarrow$ Monthly interest rate = $\frac{7.2\%}{12} = 0.6\% = 0.006$

Case 1: First payment at the end of the first month

In this case, the required corpus is calculated using the formula for the present value of a perpetuity (ordinary annuity):

$P = \frac{A}{r}$

$P = \frac{15,000}{0.006} = \textsf{₹}25,00,000$

∴ Required initial corpus = $\textsf{₹}$25,00,000

Case 2: First payment made immediately (annuity due)

For annuity due, the formula is adjusted as:

$P_{\text{due}} = P_{\text{ordinary}} \times (1 + r)$

$P_{\text{due}} = 25,00,000 \times (1 + 0.006) = 25,00,000 \times 1.006 = \textsf{₹}25,15,000$

∴ Required corpus if first payment is immediate = $\textsf{₹}$25,15,000

Difference in amount = $25,15,000 - 25,00,000 = \textsf{₹}$15,000

Final Answers:

• Required initial corpus (first payment after one month) = $\textsf{₹}$25,00,000
• Required initial corpus (first payment immediate) = $\textsf{₹}$25,15,000
• Extra amount required for immediate payment = $\textsf{₹}$15,000

Step 1: Determine the amount to be accumulated

Cost of new machine = $\textsf{₹}$12,00,000

Salvage value of old machine = $\textsf{₹}$1,50,000

Required amount to accumulate = $12,00,000 - 1,50,000 = \textsf{₹}10,50,000$

Step 2: Use sinking fund formula

The formula for annual sinking fund deposit is:

$A = \frac{F \times r}{(1 + r)^n - 1}$

Where:

• $F = \textsf{₹}10,50,000$ (Future Value)
• $r = 8\% = 0.08$
• $n = 8$ years

$A = \frac{10,50,000 \times 0.08}{(1.08)^8 - 1}$

$= \frac{84,000}{1.85093 - 1} = \frac{84,000}{0.85093} \approx \textsf{₹}98,682.66$

Final Answer:

∴ The required annual contribution to the sinking fund is approximately $\textsf{₹}$98,682.66

Concept:

Time Weighted Return (TWR) measures the compound growth rate of an investment by eliminating the impact of cash flows. It involves breaking the investment period into sub-periods defined by cash flows and computing the return in each, then compounding the returns.

Sub-Period 1: 01/01/2018 to 01/01/2019

Start value = $\textsf{₹}$5,00,000

Value just before withdrawal = $\textsf{₹}$5,50,000

Return$_1$ = $\frac{5,50,000}{5,00,000} = 1.10$

Sub-Period 2: 01/01/2019 to 01/01/2020

Start value after withdrawal = $\textsf{₹}$4,50,000

Value before next investment = $\textsf{₹}$5,00,000

Return$_2$ = $\frac{5,00,000}{4,50,000} = 1.\overline{11} \approx 1.1111$

Sub-Period 3: 01/01/2020 to 01/01/2023

Start value after investment = $\textsf{₹}$7,00,000

End value = $\textsf{₹}$10,00,000

Return$_3$ = $\frac{10,00,000}{7,00,000} = 1.4286$

Step 4: Compound the sub-period returns

TWR = $(1.10) \times (1.1111) \times (1.4286)$

TWR = $1.10 \times 1.1111 = 1.2222$

TWR = $1.2222 \times 1.4286 \approx 1.746$

TWR over 5 years = $1.746$

TWR annualized = $1.746^{1/5} - 1 \approx 1.1175 - 1 = 0.1175$

∴ TWR = 11.75% per annum (approx)

Final Answer:

∴ The Time Weighted Return (TWR) for the investment is approximately $11.75\%$ per annum

Given:

Cost of machinery = $\textsf{₹}$30,00,000

Salvage value = $\textsf{₹}$6,00,000

Useful life = 12 years

(a) Annual Depreciation (Linear Method):

$\text{Annual Depreciation} = \frac{\text{Cost} - \text{Salvage Value}}{\text{Useful Life}} = \frac{30,00,000 - 6,00,000}{12} = \frac{24,00,000}{12} = \textsf{₹}2,00,000$

(b) Depreciation Schedule for First 4 Years:

(c) Book Value at the End of Year 9:

Book value = $\text{Cost} - 9 \times \text{Annual Depreciation}$

Book value = $30,00,000 - 9 \times 2,00,000 = 30,00,000 - 18,00,000 = \textsf{₹}12,00,000$

Final Answers:

• Annual Depreciation = $\textsf{₹}$2,00,000
• Book Value at end of Year 9 = $\textsf{₹}$12,00,000

Concept:

Present value of a perpetuity is given by:

$PV = \frac{R}{r}$, where $R$ = annual payment, and $r$ = rate of return (in decimal form)

Case 1: When return = 8% p.a.

$PV = \frac{60,000}{0.08} = \textsf{₹}7,50,000$

Case 2: When return = 10% p.a.

$PV = \frac{60,000}{0.10} = \textsf{₹}6,00,000$

Difference in Initial Amount Required:

$\textsf{₹}7,50,000 - \textsf{₹}6,00,000 = \textsf{₹}1,50,000$

Final Answers:

• Present value at 8% = $\textsf{₹}$7,50,000
• Present value at 10% = $\textsf{₹}$6,00,000
• ∴ Reduction in required initial amount = $\textsf{₹}$1,50,000

Given:

Loan Amount = $\textsf{₹}$15,00,000

Annual Interest Rate = 14% p.a. compounded monthly → Monthly rate = $\frac{14}{12} = 1.167\% = 0.01167$

Tenure = 7 years = 84 months

(a) Monthly EMI:

EMI is given by the formula:

$EMI = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}$

$P = 15,00,000$, $r = 0.01167$, $n = 84$

$EMI = 15,00,000 \times \frac{0.01167(1 + 0.01167)^{84}}{(1 + 0.01167)^{84} - 1}$

$EMI \approx \textsf{₹}29,126.24$

(b) Amortization Schedule for First 3 Months:

Note: Interest = Opening Balance × Monthly Rate

Final Answers:

• EMI = $\textsf{₹}$29,126.24
• Amortization schedule prepared for first 3 months as shown above

Concept:

Compound Annual Growth Rate (CAGR) is calculated using the formula:

$CAGR = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} - 1$

Where:
$PV$ = Initial Investment
$FV$ = Final Value (including all cash flows and sale proceeds if not reinvested)
$n$ = Number of years

Given:

Initial Investment ($PV$) = $\textsf{₹}$10,00,000
Cash inflows (not reinvested):
- End of 2015: $\textsf{₹}$1,50,000
- End of 2016: $\textsf{₹}$2,00,000
- End of 2017: $\textsf{₹}$2,50,000
Final Sale Value on 31 Dec 2018: $\textsf{₹}$9,00,000

Total Value Received = $\textsf{₹}$1,50,000 + $\textsf{₹}$2,00,000 + $\textsf{₹}$2,50,000 + $\textsf{₹}$9,00,000 = $\textsf{₹}$15,00,000

Applying CAGR Formula:

$PV = 10,00,000$, $FV = 15,00,000$, $n = 4$ years

$CAGR = \left(\frac{15,00,000}{10,00,000}\right)^{\frac{1}{4}} - 1 = (1.5)^{\frac{1}{4}} - 1$

$CAGR \approx 1.1067 - 1 = 0.1067 = 10.67\%$

Final Answer:

The CAGR over the 4-year period is approximately $10.67\%$.

Given:

Original Cost of Machinery = $\textsf{₹}$40,00,000

Estimated Salvage Value = $\textsf{₹}$4,00,000

Estimated Useful Life = 20 years

(a) Annual Depreciation Expense:

Using the linear method:

$\text{Depreciation per year} = \frac{\text{Cost} - \text{Salvage Value}}{\text{Useful Life}}$

$= \frac{40,00,000 - 4,00,000}{20} = \frac{36,00,000}{20} = \textsf{₹}1,80,000$

(b) Book Value at the end of Year 15:

$\text{Total depreciation in 15 years} = 15 \times \textsf{₹}1,80,000 = \textsf{₹}27,00,000$

$\text{Book Value} = \text{Cost} - \text{Accumulated Depreciation}$

$= \textsf{₹}40,00,000 - \textsf{₹}27,00,000 = \textsf{₹}13,00,000$

(c) Profit or Loss on Sale:

Sale Price = $\textsf{₹}$15,00,000

Book Value at end of Year 15 = $\textsf{₹}$13,00,000

Since Sale Price > Book Value → Profit

$\text{Profit} = \textsf{₹}15,00,000 - \textsf{₹}13,00,000 = \textsf{₹}2,00,000$

Final Answers:

• (a) Annual Depreciation = $\textsf{₹}1,80,000$
• (b) Book Value at end of Year 15 = $\textsf{₹}13,00,000$
• (c) Profit on sale = $\textsf{₹}2,00,000$

Concept:

To calculate the present value of a perpetuity where the first payment starts after a delay, we first compute the present value of the perpetuity at the time just before the first payment starts, and then discount it back to the present.

The formula for perpetuity (starting one year from the valuation date) is:

$\text{PV}_{\text{deferred}} = \frac{A}{r}$ where $A$ is annual payment and $r$ is the rate of return.

Given:

Annual Payment ($A$) = $\textsf{₹}$1,20,000
Rate of Return ($r$) = 7% = 0.07
First payment starts 3 years from today

Step 1: PV at end of year 2 (just before first payment)

$\text{PV}_{\text{at year 2}} = \frac{1,20,000}{0.07} = \textsf{₹}17,14,285.71$

Step 2: Discount back to present (i.e., year 0)

$\text{PV}_{\text{today}} = \frac{17,14,285.71}{(1+0.07)^2} = \frac{17,14,285.71}{1.1449} \approx \textsf{₹}14,96,000$

Final Answer:

The initial amount required today for the endowment is approximately $\textsf{₹}14,96,000$.

Given:

Future Value to accumulate = $\textsf{₹}25,00,000$

Number of years ($n$) = 12

Rate of interest ($r$) = 5.5% = 0.055

(a) Required Annual Deposit:

Use the formula for Future Value of an Ordinary Annuity:

$FV = A \left(\frac{(1 + r)^n - 1}{r}\right)$

Substituting values:

$25,00,000 = A \left(\frac{(1 + 0.055)^{12} - 1}{0.055}\right)$

$25,00,000 = A \left(\frac{1.898298 - 1}{0.055}\right) = A \times 16.3327$

$A = \frac{25,00,000}{16.3327} \approx \textsf{₹}1,53,064.50$

(b) Interest Earned in the Last Year (Year 12):

At the beginning of year 12, 11 deposits have been made.

Amount accumulated after 11 years using FV formula:

$FV_{11} = 1,53,064.50 \left(\frac{(1.055)^{11} - 1}{0.055}\right) \approx 1,53,064.50 \times 14.8134 \approx \textsf{₹}22,67,106.26$

Interest in 12th year = 5.5% of $\textsf{₹}22,67,106.26$

$= \frac{5.5}{100} \times 22,67,106.26 = \textsf{₹}1,24,690.84$

(c) Total Interest Earned Over 12 Years:

Total amount accumulated = $\textsf{₹}25,00,000$

Total deposits made = $1,53,064.50 \times 12 = \textsf{₹}18,36,774$

Total interest = $\textsf{₹}25,00,000 - \textsf{₹}18,36,774 = \textsf{₹}6,63,226$

Final Answers:

• (a) Required Annual Deposit = $\textsf{₹}1,53,064.50$
• (b) Interest Earned in Year 12 = $\textsf{₹}1,24,690.84$
• (c) Total Interest over 12 Years = $\textsf{₹}6,63,226$

Given:

Principal $P = \textsf{₹}20,00,000$

Annual interest rate = 9% $\Rightarrow$ Monthly interest rate $r = \frac{9}{12} = 0.75\% = 0.0075$

Tenure = 10 years = $n = 10 \times 12 = 120$ months

(a) Monthly EMI Calculation:

EMI is calculated using the formula:

$EMI = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$

Substituting the values:

$EMI = 20,00,000 \cdot \frac{0.0075(1 + 0.0075)^{120}}{(1 + 0.0075)^{120} - 1}$

Calculating $(1 + 0.0075)^{120} \approx 2.348850$

So,

$EMI = 20,00,000 \cdot \frac{0.0075 \times 2.348850}{2.348850 - 1}$

$EMI = 20,00,000 \cdot \frac{0.0176164}{1.348850} \approx 20,00,000 \cdot 0.01306 \approx \textsf{₹}26,120.73$

(b) Total Amount Paid Over Loan Tenure:

Total number of payments = 120

Total amount paid = $26,120.73 \times 120 = \textsf{₹}31,34,487.60$

(c) Total Interest Paid Over Loan Tenure:

Total interest = Total amount paid $-$ Principal

Total interest = $\textsf{₹}31,34,487.60 - \textsf{₹}20,00,000 = \textsf{₹}11,34,487.60$

Final Answers:

• (a) Monthly EMI = $\textsf{₹}26,120.73$
• (b) Total Amount Paid = $\textsf{₹}31,34,487.60$
• (c) Total Interest Paid = $\textsf{₹}11,34,487.60$

Step 1: Calculate CAGR

The formula for Compound Annual Growth Rate (CAGR) is:

$ CAGR = \left( \frac{FV}{PV} \right)^{\frac{1}{n}} - 1 $

Where:

• $FV$ = Final value = $\textsf{₹}1,20,00,000$
• $PV$ = Initial value = $\textsf{₹}50,00,000$
• $n$ = Number of years = 7

$ CAGR = \left( \frac{1,20,00,000}{50,00,000} \right)^{\frac{1}{7}} - 1 = \left(2.4\right)^{\frac{1}{7}} - 1 $

$ CAGR \approx 1.1351 - 1 = 0.1351 = 13.51\% $

Step 2: Future Value of $\textsf{₹}5,00,000$ after 12 years

We are given that an investor started with $\textsf{₹}5,00,000$ on 1st April 2016 and wants to know the value on 31st March 2028 (i.e., after 12 years).

Using the compound interest formula:

$ FV = PV \cdot (1 + CAGR)^n $

$ FV = 5,00,000 \cdot (1 + 0.1351)^{12} = 5,00,000 \cdot (1.1351)^{12} $

$ FV \approx 5,00,000 \cdot 4.5562 \approx \textsf{₹}22,78,100 $

Final Answers:

• CAGR = 13.51%
• Value of $\textsf{₹}5,00,000$ investment on 31st March 2028 = $\textsf{₹}22,78,100$

Given:

Cost of patent = $\textsf{₹}10,00,000$

Estimated useful life = 15 years

(a) Annual Amortization Expense

Using the linear method:

$ \text{Annual Amortization} = \frac{\text{Cost of Patent}}{\text{Useful Life}} = \frac{10,00,000}{15} = \textsf{₹}66,666.67 $

(b) Book Value at the End of Year 7

$ \text{Total amortization till year 7} = 66,666.67 \times 7 = \textsf{₹}4,66,666.69 $

$ \text{Book Value} = 10,00,000 - 4,66,666.69 = \textsf{₹}5,33,333.31 $

(c) Accumulated Amortization at the End of Year 10

$ \text{Accumulated Amortization} = 66,666.67 \times 10 = \textsf{₹}6,66,666.70 $

(d) Profit or Loss on Sale at the End of Year 10

$ \text{Book Value at End of Year 10} = 10,00,000 - 6,66,666.70 = \textsf{₹}3,33,333.30 $

$ \text{Sale Price} = \textsf{₹}3,00,000 $

$ \text{Loss on Sale} = 3,33,333.30 - 3,00,000 = \textsf{₹}33,333.30 $

Final Answers:

• (a) Annual amortization expense = $\textsf{₹}66,666.67$
• (b) Book value at end of year 7 = $\textsf{₹}5,33,333.31$
• (c) Accumulated amortization at end of year 10 = $\textsf{₹}6,66,666.70$
• (d) Loss on sale = $\textsf{₹}33,333.30$