Compound Interest

Compound Interest: Definition and Concept of Compounding

Definition

Compound Interest (CI) is a method of calculating interest where the interest earned during each period is added to the principal amount before calculating the interest for the next period. This fundamental concept is often summarized as earning "interest on interest".

Unlike simple interest, the principal amount on which interest is calculated does not remain fixed. Instead, it grows over time as the earned interest is periodically added to it. This process accelerates the growth of the investment or debt over time.

Concept of Compounding

The core mechanism of compound interest is the process of compounding. Compounding is the act of reinvesting the interest earned back into the principal. This creates a new, larger principal for the subsequent interest calculation period.

The compounding frequency is a crucial factor. It specifies how often the interest is calculated and added to the principal within a standard period, usually a year. The more frequently interest is compounded, the faster the principal grows, because interest starts earning interest sooner. Common compounding frequencies include:

• Annually: once a year.
• Semi-annually (Half-yearly): twice a year (every 6 months).
• Quarterly: four times a year (every 3 months).
• Monthly: twelve times a year.
• Daily: typically 365 times a year (ignoring leap years), or sometimes 360 days in commercial calculations.
• Continuously: A theoretical limit where compounding happens infinitely often.

The concept is best understood by comparing it directly with simple interest.

Illustration: Simple vs. Compound Interest Comparison

Let's illustrate the concept using an example: investing $\textsf{₹}\$ 1,000$ for 3 years at an annual interest rate of 10%. We will compare the outcomes under simple interest and compound interest (compounded annually).

Scenario A: Simple Interest

Principal (P) = $\textsf{₹}\$ 1,000$

Annual Rate (r) = 10% or 0.10

• End of Year 1:

  Interest for Year 1 = $P \times r \times 1 = 1000 \times 0.10 \times 1 = \textsf{₹}\$ 100$.

  Amount at end of Year 1 = $1000 + 100 = \textsf{₹}\$ 1100$.

  Principal for Year 2 remains the original $\textsf{₹}\$ 1000$.

• End of Year 2:

  Interest for Year 2 = $P \times r \times 1 = 1000 \times 0.10 \times 1 = \textsf{₹}\$ 100$.

  Amount at end of Year 2 = $1100 + 100 = \textsf{₹}\$ 1200$.

  Principal for Year 3 remains the original $\textsf{₹}\$ 1000$.

• End of Year 3:

  Interest for Year 3 = $P \times r \times 1 = 1000 \times 0.10 \times 1 = \textsf{₹}\$ 100$.

  Amount at end of Year 3 = $1200 + 100 = \textsf{₹}\$ 1300$.

Total Simple Interest after 3 years = $\textsf{₹}\$ 100 + \textsf{₹}\$ 100 + \textsf{₹}\$ 100 = \textsf{₹}\$ 300$.

Final Amount after 3 years = $\textsf{₹}\$ 1300$.

Scenario B: Compound Interest (Compounded Annually)

Principal (P) = $\textsf{₹}\$ 1,000$

Annual Rate (r) = 10% or 0.10

Compounding Frequency = Annually (interest is added once per year)

• End of Year 1 (1st Compounding Period):

  Interest for Year 1 ($I_1$) = $Principal_{Start} \times Annual\$ Rate = 1000 \times 0.10 = \textsf{₹}\$ 100$.

  Amount at end of Year 1 ($A_1$) = $Principal_{Start} + I_1 = 1000 + 100 = \textsf{₹}\$ 1100$.

  This amount $\textsf{₹}\$ 1100$ becomes the principal for the next period.

• End of Year 2 (2nd Compounding Period):

  Principal for Year 2 = $A_1 = \textsf{₹}\$ 1100$.

  Interest for Year 2 ($I_2$) = $Principal_{Year2} \times Annual\$ Rate = 1100 \times 0.10 = \textsf{₹}\$ 110$.

  Amount at end of Year 2 ($A_2$) = $Principal_{Year2} + I_2 = 1100 + 110 = \textsf{₹}\$ 1210$.

  This amount $\textsf{₹}\$ 1210$ becomes the principal for the next period.

• End of Year 3 (3rd Compounding Period):

  Principal for Year 3 = $A_2 = \textsf{₹}\$ 1210$.

  Interest for Year 3 ($I_3$) = $Principal_{Year3} \times Annual\$ Rate = 1210 \times 0.10 = \textsf{₹}\$ 121$.

  Amount at end of Year 3 ($A_3$) = $Principal_{Year3} + I_3 = 1210 + 121 = \textsf{₹}\$ 1331$.

Total Compound Interest after 3 years = $I_1 + I_2 + I_3 = \textsf{₹}\$ 100 + \textsf{₹}\$ 110 + \textsf{₹}\$ 121 = \textsf{₹}\$ 331$.

Final Amount after 3 years = $\textsf{₹}\$ 1331$.

Comparison Table

Here is a side-by-side comparison of the growth under simple vs. compound interest:

This illustration clearly shows that:

• In simple interest, the interest earned each year is constant ($\textsf{₹}\$ 100$). The growth is linear.
• In compound interest, the principal amount for calculation increases each year ($\textsf{₹}\$ 1000 \to \textsf{₹}\$ 1100 \to \textsf{₹}\$ 1210$), leading to increasing interest amounts each year ($\textsf{₹}\$ 100 \to \textsf{₹}\$ 110 \to \textsf{₹}\$ 121$). The growth is exponential.
• For periods longer than one year, compound interest results in a higher total interest and a larger accumulated amount than simple interest at the same nominal annual rate.

Compound interest is the standard method used in modern financial calculations for savings, investments, and loans because it accurately reflects the benefit of reinvesting earnings.

Compound Interest Formula (Annual Compounding)

Derivation of the Formula

Let's derive the general formula for calculating the Accumulated Amount (A) or Future Value (FV) when interest is compounded annually. We start with an initial Principal (P) and an annual interest rate, denoted by $r$ (expressed as a decimal).

• At the beginning (Time = 0):

  The initial investment or loan amount is the Principal, $P$.

  Amount at Time 0 = $P$.

• End of Year 1 (After 1st Compounding Period):

  Interest earned in Year 1 ($I_1$) = $Principal_{Beginning\_of\_Year\_1} \times Annual\$ Rate = P \times r$.

  The Amount at the end of Year 1 ($A_1$) is the sum of the initial principal and the interest earned:

  $A_1 = P + I_1 = P + Pr$.

  By factoring out the common term P, we get:

  $A_1 = P(1 + r)$

  This $A_1$ now becomes the principal for the next year.

• End of Year 2 (After 2nd Compounding Period):

  The principal for Year 2 is the amount from the end of Year 1, which is $A_1 = P(1+r)$.

  Interest earned in Year 2 ($I_2$) = $Principal_{Beginning\_of\_Year\_2} \times Annual\$ Rate = A_1 \times r = P(1+r) \times r$.

  The Amount at the end of Year 2 ($A_2$) is the principal at the start of Year 2 ($A_1$) plus the interest earned in Year 2 ($I_2$):

  $A_2 = A_1 + I_2 = A_1 + A_1 r$.

  Factoring out $A_1$:

  $A_2 = A_1 (1 + r)$.

  Now, substitute the expression for $A_1$ from the previous step:

  $A_2 = [P(1 + r)] (1 + r) = P(1 + r)^2$

  This $A_2$ becomes the principal for the next year.

• End of Year 3 (After 3rd Compounding Period):

  The principal for Year 3 is $A_2 = P(1+r)^2$.

  Interest earned in Year 3 ($I_3$) = $Principal_{Beginning\_of\_Year\_3} \times Annual\$ Rate = A_2 \times r = P(1+r)^2 \times r$.

  The Amount at the end of Year 3 ($A_3$) is the principal at the start of Year 3 ($A_2$) plus the interest earned in Year 3 ($I_3$):

  $A_3 = A_2 + I_3 = A_2 + A_2 r$.

  Factoring out $A_2$:

  $A_3 = A_2 (1 + r)$.

  Substitute the expression for $A_2$:

  $A_3 = [P(1 + r)^2] (1 + r) = P(1 + r)^3$

Observing the pattern that emerges:

• After 1 year: $A_1 = P(1+r)^1$
• After 2 years: $A_2 = P(1+r)^2$
• After 3 years: $A_3 = P(1+r)^3$

It is clear that the exponent matches the number of years (which is also the number of compounding periods in this case of annual compounding).

Generalizing this pattern, the Amount (A) accumulated after $t$ years, when the interest is compounded annually at an annual rate $r$ (as a decimal), is given by the formula:

$\mathbf{A = P(1 + r)^t}$

Where:

• A = The final Amount or Accumulated Value (Future Value) at the end of $t$ years.
• P = The original Principal amount.
• r = The annual interest rate, expressed as a decimal (e.g., an annual rate of $5\% = 0.05$). Remember to divide the percentage rate by 100.
• t = The time period in years. For this annual compounding formula, $t$ represents the number of full years the money is invested or borrowed. If the time is not a whole number of years, this specific formula might need adjustment or periodic compounding formula ($A = P(1+i)^n$) is more appropriate. However, for typical 'annual compounding' problems, $t$ is usually an integer or a fraction where compounding is assumed to occur annually for $t$ years.

The term $(1+r)^t$ is known as the Accumulation Factor or Growth Factor for compound interest compounded annually. Multiplying the principal by this factor calculates its value at the end of the period, including all the accumulated interest.

Compound Interest (CI) Calculation

The Compound Interest (CI) itself is the total interest earned or paid over the entire period. It is simply the difference between the final Accumulated Amount (A) and the original Principal (P).

$\mathbf{CI = Amount - Principal}$

$\mathbf{CI = A - P}$

By substituting the formula for A that we just derived, we can also get a formula to calculate the Compound Interest directly:

$\mathbf{CI = P(1 + r)^t - P}$

We can factor out the common term P from this expression:

$\mathbf{CI = P [ (1 + r)^t - 1 ]}$

This formula calculates the total compound interest earned over $t$ years when compounded annually. The term $(1+r)^t - 1$ represents the effective interest rate for the entire $t$-year period.

Worked Example

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Accumulation with Compound Interest (Calculating Amount)

Concept of Accumulation

As established, compound interest leads to the accumulation of wealth over time because earned interest is added back to the principal, forming a larger base for subsequent interest calculations. The final value achieved at the end of the investment or loan period is referred to as the Accumulated Value or the Future Value (FV). This is precisely the same as the Amount (A).

Calculating this Accumulated Amount is a primary objective in compound interest problems, as it represents the total worth of the initial principal after a period of growth fuelled by compounding. It includes both the original principal and all the "interest on interest" accumulated over time.

Formula for Accumulated Amount (Compounded Annually)

When interest is compounded annually, the growth occurs once per year. The formula derived in the previous section for the Amount (A) after $t$ years, given an initial Principal (P) and an annual interest rate $r$ (as a decimal), is:

$\mathbf{A = P(1 + r)^t}$

Where:

• A = The final Accumulated Amount or Future Value.
• P = The original Principal (the initial investment or loan amount).
• r = The annual interest rate, expressed as a decimal. Remember to convert the given percentage rate ($R\%$) into a decimal by dividing by 100: $r = R/100$.
• t = The time period in years. In this formula, $t$ represents the number of annual compounding periods. For annual compounding, if the time is given in years, $t$ will be that number. For example, if the time is 5 years, $t=5$. If the time is $2.5$ years, $t=2.5$, assuming compounding applies proportionally for the partial year (though often periodic compounding formula is clearer for non-integer years).

The term $(1+r)$ is the growth factor per year. Raising it to the power of $t$ accounts for the cumulative effect of this growth over $t$ years. The entire term $(1+r)^t$ is sometimes called the accumulation factor, as it is the factor by which the initial principal grows.

Worked Examples

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Compounding Frequency (Half-yearly, Quarterly, Monthly, etc.) and Formula Adjustments

Concept of Compounding Frequency

While the basic compound interest formula ($A = P(1+r)^t$) assumes interest is compounded annually (once per year), in many real-world financial products, interest is compounded more frequently. This means the earned interest is added to the principal and starts earning interest itself more often than just annually.

Common compounding frequencies include:

• Semi-annually (or Half-yearly): Interest is compounded twice a year, usually at the end of every 6-month period.
• Quarterly: Interest is compounded four times a year, typically at the end of every 3-month period.
• Monthly: Interest is compounded twelve times a year, at the end of every month. This is very common for savings accounts and loan EMIs.
• Daily: Interest is compounded every day (usually assuming 365 days in a year). This is often used for savings accounts where balances change daily.

The effect of more frequent compounding is that the total accumulated amount at the end of the overall term will be higher than if the interest was compounded less frequently (like annually), given the same nominal annual rate. This is because interest starts earning interest sooner and more often.

Formula Adjustments for Different Frequencies

To calculate the accumulated amount when interest is compounded more than once a year, we need to modify the basic formula $A = P(1+r)^t$. The modifications are necessary to reflect the rate applied during each compounding period and the total number of such periods over the entire term.

Let:

• P = The original Principal.
• r = The nominal annual rate of interest (expressed as a decimal). This is the stated yearly rate, *before* considering compounding frequency.
• t = The total time period of the investment or loan in years.
• m = The number of times interest is compounded per year (the compounding frequency).

The two adjustments needed are:

1. Determine the Periodic Interest Rate ($i$): This is the interest rate that applies during each compounding period. Since the nominal annual rate $r$ is spread over $m$ periods in a year, the rate for each period is the annual rate divided by the number of periods per year.

   $i = \frac{\text{Nominal Annual Rate (decimal)}}{\text{Number of Compounding Periods per Year}} = \frac{r}{m}$

   This rate $i$ is used for calculations within each shorter compounding period.

2. Determine the Total Number of Compounding Periods ($n$): The total duration of the investment or loan is $t$ years, and compounding happens $m$ times each year. So, the total number of times interest is compounded over the entire term is the number of years multiplied by the compounding frequency per year.

   $n = \text{Time in Years} \times \text{Number of Compounding Periods per Year} = t \times m$

   This $n$ represents the total number of times the periodic rate $i$ is applied to the growing principal.

Now, replace $r$ with the periodic rate $i$ and $t$ with the total number of periods $n$ in the basic compound interest formula $A = P(1+r)^t$.

$A = P(1 + i)^n$

Substituting the expressions for $i$ and $n$, we get the general formula for the Amount (A) with any compounding frequency:

$\mathbf{A = P\left(1 + \frac{r}{m}\right)^{mt}}$

Where:

• A = The final Accumulated Amount or Future Value.
• P = The original Principal.
• r = The nominal annual interest rate (as a decimal).
• m = The number of times interest is compounded per year.
• t = The total time period in years.

This is the most versatile formula for compound interest accumulation.

Common Compounding Frequencies and Formula Parameters

Based on the general formula $A = P\left(1 + \frac{r}{m}\right)^{mt}$, here are the specific values for $m$, $i$, and $n$ for common frequencies:

The Compound Interest (CI) for any frequency is still the difference between the final Amount and the Principal:

$\mathbf{CI = A - P = P\left(1 + \frac{r}{m}\right)^{mt} - P = P \left[ \left(1 + \frac{r}{m}\right)^{mt} - 1 \right]}$

Continuous Compounding

As the compounding frequency ($m$) increases infinitely, the accumulation approaches a limit. This theoretical scenario is called continuous compounding. Using calculus (specifically the limit definition of $e$), the formula for the amount under continuous compounding is:

$A = P e^{rt}$

where:

• A = Accumulated Amount.
• P = Principal.
• r = Nominal annual interest rate (as a decimal).
• t = Time in years.
• e = The base of the natural logarithm, an irrational number approximately equal to 2.71828.

Continuous compounding represents the maximum possible accumulation for a given nominal annual rate and time period.

Worked Example

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Problems based on Compound Interest

Mastering compound interest requires not only the ability to calculate the amount or interest given the principal, rate, and time, but also the skill to determine any one of these variables when the others are known. This section presents various problems where the formulas for compound interest need to be applied or rearranged to find an unknown value.

Core Formulas for Compound Interest

Recall the fundamental formulas for compound interest where $P$ is the Principal, $r$ is the nominal annual rate (as a decimal), $t$ is the time in years, $m$ is the number of compounding periods per year, $i = r/m$ is the periodic rate, and $n = mt$ is the total number of periods:

The formula for the Accumulated Amount (A) or Future Value is:

$\mathbf{A = P\left(1 + \frac{r}{m}\right)^{mt}}$

This is often more conveniently written in terms of the periodic rate ($i$) and total periods ($n$):

$\mathbf{A = P(1 + i)^n}$

The Compound Interest (CI) is the difference between the Amount and the Principal:

$\mathbf{CI = A - P}$

Rearranging Formulas to Find Unknowns

The amount formula $A = P(1 + i)^n$ is an equation relating A, P, i, and n. If any three of these are known, the fourth can be found by rearranging the formula:

• To find Principal (P): If you know A, i, and n, divide both sides by $(1+i)^n$:

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

• To find Periodic Rate (i): If you know A, P, and n, rearrange the formula:

  $\frac{A}{P} = (1 + i)^n$

  Take the $n$-th root of both sides: $\left(\frac{A}{P}\right)^{1/n} = 1 + i$

  Subtract 1: $i = \left(\frac{A}{P}\right)^{1/n} - 1$

  Once the periodic rate $i$ is found, the nominal annual rate $r$ can be found using $r = i \times m$. The annual percentage rate $R\%$ is $r \times 100\%$.

• To find Total Number of Periods (n): If you know A, P, and i, rearrange the formula:

  $\frac{A}{P} = (1 + i)^n$

  To solve for the exponent $n$, logarithms are typically required:

  $\log\left(\frac{A}{P}\right) = \log((1 + i)^n)$

  Using the logarithm property $\log(a^b) = b \log(a)$:

  $\log\left(\frac{A}{P}\right) = n \log(1 + i)$

  Divide by $\log(1+i)$:

  $n = \frac{\log(A/P)}{\log(1 + i)}$

  Once $n$ is found, the time in years $t$ can be found using $t = n/m$. (For simpler problems in exams, the numbers might be chosen such that $A/P$ is a clear power of $(1+i)$, avoiding the need for explicit log calculations).

Let's work through some examples covering these different scenarios.

Worked Examples

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