Chapter 9 Financial Mathematics (Concepts)

Perpetuity and Sinking Funds

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Financial Mathematics applies mathematical concepts and methods to solve problems in finance. It involves calculations related to interest, investments, loans, and valuation of financial instruments. Two specific concepts we explore here are Perpetuities and Sinking Funds, which deal with streams of cash flows over time.

Both concepts are related to annuities, which are series of equal payments made at regular intervals. The difference lies in the duration and purpose of the payments.

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Perpetuity

A Perpetuity is a type of annuity where the periodic payments are scheduled to continue indefinitely or forever. In other words, it's a stream of infinite cash flows that are equal in amount and occur at regular intervals. Perpetuities are theoretical constructs, but they are used in finance to model payments from certain types of investments (like some forms of preferred stock) or in valuation models where distant future cash flows are considered perpetual.

Since payments never end, the future value of a perpetuity with a positive interest rate is infinite. Therefore, the main focus when dealing with perpetuities is their Present Value (PV). The present value of a perpetuity is the lump sum amount today that is equivalent in value to the infinite stream of future payments, considering a specific interest rate.

Present Value of a Simple Perpetuity

A simple perpetuity assumes that the payments are equal and are made at the end of each period, and the interest is compounded at the end of each period.

Let:

• $PMT$ (or $P$) be the amount of the constant payment made at the end of each period.
• $r$ be the interest rate per period (expressed as a decimal). This is the discount rate used to bring future payments back to their present value.

The present value (PV) of the perpetuity is the sum of the present values of the payments occurring at the end of periods 1, 2, 3, and so on, infinitely. The present value of the payment at the end of period 1 is $PMT \times (1+r)^{-1} = \frac{PMT}{1+r}$. The present value of the payment at the end of period 2 is $PMT \times (1+r)^{-2} = \frac{PMT}{(1+r)^2}$. The present value of the payment at the end of period 3 is $PMT \times (1+r)^{-3} = \frac{PMT}{(1+r)^3}$, and so on.

The total present value is the sum of this infinite series:

This is an infinite geometric series with the first term $a = \frac{PMT}{1+r}$ and the common ratio $k = \frac{1}{1+r}$. The sum of an infinite geometric series $a + ak + ak^2 + \dots$ converges to $S = \frac{a}{1-k}$ if $|k| < 1$. Assuming the interest rate $r > 0$, we have $1+r > 1$, so $0 < \frac{1}{1+r} < 1$. Thus, the common ratio $k$ is less than 1, and the series converges.

Substituting the values of $a$ and $k$ into the sum formula:

Simplify the denominator: $1 - \frac{1}{1+r} = \frac{(1+r) - 1}{1+r} = \frac{r}{1+r}$.

Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator:

The formula for the present value of a simple perpetuity is:

where $PMT$ is the periodic payment and $r$ is the interest rate per period.

Present Value of a Growing Perpetuity

In some situations, the payments of a perpetuity may grow at a constant rate ($g$) each period. If the first payment is $PMT_1$ (at the end of period 1), the subsequent payments are $PMT_1(1+g)$, $PMT_1(1+g)^2$, and so on. The present value is $PV = \frac{PMT_1}{1+r} + \frac{PMT_1(1+g)}{(1+r)^2} + \frac{PMT_1(1+g)^2}{(1+r)^3} + \dots$. This is an infinite geometric series with $a = \frac{PMT_1}{1+r}$ and $k = \frac{1+g}{1+r}$. For convergence, we need $r > g$. The sum is:

where $PMT_1$ is the first payment (at the end of period 1), $r$ is the discount rate per period, and $g$ is the growth rate per period ($r > g$).

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Sinking Fund

A Sinking Fund is a financial strategy where an individual or a company makes periodic deposits into an account that earns interest, with the goal of accumulating a specific sum of money by a certain future date. Sinking funds are typically established to meet future obligations such as the repayment of a debt (like the principal of a bond issue), the replacement of an aging asset, or funding a future large expenditure.

The process of contributing to a sinking fund is an application of the concept of the future value of an annuity. The periodic payments into the sinking fund form an annuity, and the target amount to be accumulated is the future value of this annuity.

Calculating the required periodic payment for a sinking fund involves determining the equal amount that must be deposited at the end of each period into an interest-bearing account to reach a specified future value.

Let:

• $FV$ be the desired future value (the target amount) to be accumulated.
• $PMT$ be the fixed periodic payment made into the sinking fund.
• $r$ be the interest rate per period earned by the fund (as a decimal).
• $n$ be the total number of periods over which the payments are made.

Assuming payments are made at the end of each period (ordinary annuity), the future value of the series of payments is given by the formula:

This formula calculates the total value in the fund at time $n$, including the sum of all payments and the accumulated interest on them. To find the required periodic payment ($PMT$) for a sinking fund, we need to rearrange this formula to solve for $PMT$:

Divide both sides of Equation (3) by $\frac{(1+r)^n - 1}{r}$:

The formula for the required sinking fund payment is:

where $FV$ is the future amount needed, $r$ is the interest rate per period, and $n$ is the number of periods. The term $\frac{r}{(1+r)^n - 1}$ is sometimes called the sinking fund factor. This formula gives the equal payment required at the end of each period to accumulate $FV$ in $n$ periods at rate $r$.

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Examples

Answer:

Given:

To Find:

Solution:

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Answer:

Given:

To Find:

Solution:

Calculation of EMI

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One of the most common applications of financial mathematics in everyday life is the calculation of loan repayments, specifically the Equated Monthly Installment (EMI). An EMI is the fixed amount paid by a borrower to a lender on a specific date each month until the loan is fully repaid. It is a convenient way to manage loan repayments as the amount remains constant throughout the loan tenure. Each EMI payment consists of both the interest due for the outstanding loan amount and a portion of the principal amount.

Initially, in the early months of the loan, a larger portion of the EMI goes towards paying the interest, and a smaller portion is applied to reduce the principal. As the loan progresses, the interest component decreases (because the outstanding principal is reducing), and consequently, a larger portion of the fixed EMI is used to repay the principal.

The calculation of EMI requires the following key pieces of information:

• The Principal Loan Amount ($P$).
• The Interest Rate.
• The Loan Tenure (the total duration for repayment).

The calculation assumes that payments are made at the end of each period (typically monthly), making it an application of the present value of an ordinary annuity.

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EMI Calculation Formula

The formula for calculating the EMI is derived from the concept that the principal amount of the loan today is the present value of all the future EMI payments. The series of future equal EMI payments made at regular intervals (usually monthly) constitutes an ordinary annuity.

The formula for the Present Value (PV) of an ordinary annuity is given by:

where:

• $PV$ is the Present Value of the annuity.
• $PMT$ is the amount of each equal payment (the EMI in our case).
• $r$ is the interest rate per period.
• $n$ is the total number of periods (installments).

In the context of a loan, the Principal Loan Amount ($P$) is the present value received by the borrower. The EMI is the periodic payment ($PMT$). So, we can write:

To find the EMI amount, we rearrange this equation to solve for EMI. Multiply both sides by $r$:

Divide both sides by $[1 - (1+r)^{-n}]$:

The standard formula for calculating EMI is:

where:

• $\mathbf{EMI}$ is the Equated Monthly Installment.
• $\mathbf{P}$ is the Principal Loan Amount (the initial amount borrowed).
• $\mathbf{r}$ is the interest rate per installment period. It is crucial that this rate matches the frequency of the payments. If the annual interest rate is $R\%$ compounded monthly, the monthly interest rate $r = \frac{R/100}{12}$.
• $\mathbf{n}$ is the total number of installments. This is the loan tenure expressed in the same time unit as the payments. If the loan tenure is $T$ years and payments are monthly, then $n = T \times 12$.

Consistency in the units of $r$ and $n$ is essential for correct EMI calculation.

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Loan Amortization

An Amortization Schedule is a table that details the breakdown of each loan payment (EMI) into its interest and principal components and shows the remaining loan balance after each payment. It provides a clear picture of how the loan is being repaid over its tenure.

For each payment period:

• Interest Paid: This is calculated on the outstanding loan balance at the beginning of the period. Interest Paid = Starting Balance $\times r$.
• Principal Paid: This is the portion of the EMI that reduces the outstanding loan balance. Principal Paid = EMI - Interest Paid.
• Ending Balance: This is the remaining loan amount after the payment is made. Ending Balance = Starting Balance - Principal Paid.

The Ending Balance of one period becomes the Starting Balance for the next period.

Example Structure of an Amortization Table:

Period	Starting Balance	EMI Payment	Interest Paid	Principal Paid	Ending Balance
1	$P$	EMI	$P \times r$	EMI - ($P \times r$)	$P$ - (EMI - ($P \times r$))
2	Ending Balance of Period 1	EMI	Ending Balance of Period 1 $\times r$	EMI - (Interest Paid in Period 2)	Ending Balance of Period 1 - (Principal Paid in Period 2)
...	...	...	...	...	...
$n$	Starting Balance for last period (should be close to EMI - last period interest)	EMI	Starting Balance for last period $\times r$	EMI - (Interest Paid in last period)	Should be 0 (or very close to 0 due to rounding)

An amortization schedule shows how the interest component is higher at the beginning and the principal component is higher towards the end of the loan tenure.

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Example

Answer:

Given:

To Find:

Solution:

Calculation of Returns and Nominal Rate of Return

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In financial mathematics, evaluating the performance of an investment is essential for making informed decisions. The performance of an investment is typically measured by its return. Return represents the gain or loss on an investment over a specified period of time. It is usually expressed as a percentage. There are different ways to calculate and interpret returns, depending on the context and the period considered.

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Simple Return (Holding Period Return)

The most basic measure of return is the Simple Return, also known as the Holding Period Return (HPR). It measures the total return on an investment over the entire period it was held (the holding period), ignoring any compounding within that period. It is calculated as the percentage change in the value of the investment from the beginning to the end of the holding period.

If the investment value at the beginning of the period is $V_{start}$ and the value at the end of the period is $V_{end}$, the simple return is calculated as:

This formula gives the return as a decimal. To express it as a percentage, multiply by 100:

If the investment generated any income during the holding period, such as dividends from stocks or interest from bonds/deposits, this income must be added to the change in value to calculate the total return. Let $I$ be the total income received during the period.

This formula applies to any holding period, whether it is a day, a month, a year, or several years.

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Nominal Rate of Return

The term Nominal Rate of Return is often used to refer to a stated or contractual rate of return over a specific period, before taking into account certain factors that can affect the actual purchasing power of the return or the total return earned over a longer period due to compounding. It is the simple percentage increase in the amount of money invested.

Key aspects of the nominal rate:

• It does not account for the effect of inflation. The return adjusted for inflation is called the real rate of return.
• If the stated nominal rate is an annual rate but compounding occurs more frequently (e.g., monthly or quarterly), the nominal rate does not reflect the total return earned over the year due to the effect of compounding. The rate that accounts for this is called the effective annual rate.

For example, if a bank offers a fixed deposit with an interest rate of "8% per annum", this 8% is the nominal annual rate. If the interest is compounded half-yearly, you don't actually earn exactly 8% over the year on your initial deposit due to the compounding effect within the year.

Nominal Annual Rate vs. Effective Annual Rate (EAR)

The Effective Annual Rate (EAR) or Effective Interest Rate is the actual annual rate of return earned on an investment (or paid on a loan) when the effects of compounding more frequently than annually are taken into account.

If a nominal annual interest rate is $R_{nom}$ (expressed as a decimal) and interest is compounded $m$ times per year, the interest rate per compounding period is $\frac{R_{nom}}{m}$. Over one year, there are $m$ such compounding periods. The effective annual rate $R_{eff}$ (or EAR) is calculated using the formula:

Here, $R_{nom}$ and $R_{eff}$ are expressed as decimals. The term $\left(1 + \frac{R_{nom}}{m}\right)^m$ calculates the total accumulation factor over one year, and subtracting 1 gives the net effective rate of return for the year.

Example: Suppose a fixed deposit offers a nominal annual interest rate of 6%, compounded quarterly. What is the effective annual rate? Given: $R_{nom} = 6\% = 0.06$. Compounding frequency $m=4$ (quarterly). Using the formula (3):

Calculate $(1.015)^4$: $(1.015)^2 = 1.030225$ $(1.015)^4 = (1.030225)^2 \approx 1.0613635$

Expressed as a percentage, the effective annual rate is approximately $0.0613635 \times 100\% \approx 6.136\%$. This means that a nominal rate of 6% compounded quarterly is effectively equivalent to earning a single rate of approximately 6.136% annually. The EAR is higher than the nominal rate due to the benefit of compounding within the year. If compounding were only annual ($m=1$), then $R_{eff} = (1 + R_{nom}/1)^1 - 1 = R_{nom}$.

In the context of simply calculating "returns", the term "nominal rate of return" for a single period typically means the simple return calculated for that period, unadjusted for other factors like inflation.

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Examples

Answer:

Given:

To Find:

Solution:

Compound Annual Growth Rate (CAGR)

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When evaluating the performance of an investment or the growth of a metric (like revenue, users, etc.) over multiple periods, simply calculating the average of the simple returns for each period can be misleading. This is because simple averages do not account for the effect of compounding. The Compound Annual Growth Rate (CAGR) is a metric designed to provide a smoothed, annualized measure of growth that accounts for compounding over a period longer than one year.

The Compound Annual Growth Rate (CAGR) represents the average annual rate at which an investment or a value has grown over a specified number of years, assuming that the growth occurred at a constant rate each year and that any returns were reinvested. It is a geometric progression ratio that provides a single figure representing the yearly growth rate over a multi-year period.

CAGR is particularly useful for:

• Comparing the growth of different investments over the same or different time periods.
• Smoothing out year-to-year volatility to see the underlying growth trend.
• Making projections based on past performance.

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CAGR Calculation Formula

The formula for CAGR is derived from the fundamental concept of compound growth. If a value $V_{start}$ grows at a constant annual rate $r$ for $n$ years, its ending value $V_{end}$ would be given by the compound growth formula:

In the context of CAGR, we assume that the actual growth over the $n$ years can be represented by a constant annual growth rate, which we call CAGR (expressed as a decimal). So, if the value started at $V_{start}$ and ended at $V_{end}$ after $n$ years, the relationship is:

To derive the formula for CAGR, we need to solve this equation for CAGR. Divide both sides by $V_{start}$:

Take the $n$-th root of both sides. The $n$-th root of a number $X$ is $X$ raised to the power of $\frac{1}{n}$.

Finally, subtract 1 from both sides to isolate CAGR:

where:

• $\mathbf{V_{end}}$ is the ending value of the investment or metric.
• $\mathbf{V_{start}}$ is the starting value of the investment or metric.
• $\mathbf{n}$ is the number of years over the period. Ensure that $n$ represents the exact duration in years. For example, from the beginning of 2018 to the end of 2022 is exactly 5 full years.

CAGR is typically expressed as a percentage by multiplying the result by 100.

CAGR vs. Arithmetic Mean Return

The simple arithmetic average of annual returns can overstate the actual growth rate over multiple periods, especially when there is volatility in returns. CAGR provides a more accurate representation of the compounded growth.
Example: Investment value: Year 0: ₹100, Year 1: ₹150 (50% return), Year 2: ₹100 (-33.33% return). Arithmetic Mean Return = $(50\% + (-33.33\%))/2 \approx 8.33\%$. CAGR = $\left(\frac{100}{100}\right)^{\frac{1}{2}} - 1 = (1)^{0.5} - 1 = 1 - 1 = 0\%$. The CAGR of 0% correctly reflects that the investment ended at the same value it started, while the arithmetic mean return is misleading. CAGR is a better measure of growth over multiple periods due to compounding.

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Example

Answer:

Given:

To Find:

Solution:

Linear Method of Depreciation

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In accounting and finance, assets like machinery, vehicles, furniture, buildings, etc., lose their value over time due to various factors such as normal wear and tear, physical deterioration, obsolescence (becoming outdated), and usage. This loss of value of a tangible asset over its useful life is recognized as depreciation. Depreciation is an accounting method used to allocate the cost of a tangible asset over its estimated useful life. It is treated as a non-cash expense on the income statement and reduces the book value of the asset on the balance sheet.

The Linear Method of Depreciation, also widely known as the Straight-Line Method, is the simplest and most commonly used method for calculating depreciation. This method assumes that the asset provides equal economic benefits or services throughout its useful life. Consequently, the cost of the asset (less any estimated salvage value) is spread evenly over its useful life, resulting in the same amount of depreciation expense being recorded each year (or period).

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Straight-Line Depreciation Formula

The annual depreciation expense under the straight-line method is calculated by subtracting the estimated salvage value from the cost of the asset and then dividing the result by the estimated useful life of the asset.

Let:

• $C$ be the original Cost of the Asset. This includes the purchase price plus any costs necessary to get the asset ready for its intended use (e.g., transportation, installation, testing costs).
• $S$ be the estimated Salvage Value (or Scrap Value) of the asset. This is the expected residual value of the asset at the end of its useful life, which the company expects to recover through sale or disposal.
• $N$ be the estimated Useful Life of the Asset. This is the period (typically in years) over which the asset is expected to be available for use by the entity. Useful life can also be measured in units of production or service hours.

The amount of the asset's cost that is subject to depreciation is called the Depreciable Amount.

The annual depreciation expense is then calculated as:

Or, using the depreciable amount:

The annual depreciation expense remains constant each year throughout the useful life of the asset under this method.

Depreciation Rate

The straight-line depreciation can also be expressed as a percentage of the depreciable amount per year, known as the Annual Depreciation Rate. This rate is constant each year.

The annual depreciation rate is calculated based on the useful life:

Once the rate is determined, the annual depreciation can also be calculated as: Annual Depreciation = Depreciable Amount $\times$ Annual Depreciation Rate (as a decimal).

Book Value of the Asset

The Book Value (or carrying value) of an asset at any point in time is its original cost less the total accumulated depreciation charged up to that point. Accumulated depreciation is the sum of the depreciation expense recognized from the time the asset was put into use up to the current date.

Under the straight-line method, the accumulated depreciation after $t$ years is simply the annual depreciation amount multiplied by the number of years $t$.

So, the book value after $t$ years is:

At the end of the asset's useful life (i.e., when $t = N$), the accumulated depreciation will equal the depreciable amount (Cost - Salvage Value), and the book value should be equal to the estimated salvage value. Book Value at end of useful life = Cost - (Annual Depreciation $\times N$) $= \text{Cost} - \left(\frac{\text{Cost} - \text{Salvage Value}}{N}\right) \times N$ $= \text{Cost} - (\text{Cost} - \text{Salvage Value})$ $= \text{Cost} - \text{Cost} + \text{Salvage Value}$ $= \text{Salvage Value}$. This confirms the relationship holds true.

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Example

Answer:

Given:

To Find:

Solution: