Understanding Loan Instalment Calculation with Compound Interest

This problem involves a loan that is paid back in equal yearly instalments. The key here is to understand how compound interest affects the value of future payments when calculating the original principal amount borrowed. The principal amount (P) is equal to the sum of the present values of each future instalment.

The formula for the present value (PV) of a single amount received in the future is:

\(PV = \frac{Future\ Value}{(1 + \frac{r}{100})^n}\)

Where:

• \(Future\ Value\) is the amount received in the future (the instalment).
• \(r\) is the annual interest rate.
• \(n\) is the number of years from the present.

In this specific problem:

• Each yearly instalment is Rs. 35,280.
• The interest rate is 5% per annum, compounded annually.
• There are two equal yearly instalments.

Step-by-Step Calculation of Principal Amount (P)

To find the principal amount (P), we need to calculate the present value of each instalment and sum them up.

Step 1: Calculate the present value of the first instalment.

The first instalment of Rs. 35,280 is paid at the end of the first year (n=1). The interest rate is 5%.

\(PV_{1st\ instalment} = \frac{35280}{(1 + \frac{5}{100})^1} = \frac{35280}{(1 + 0.05)^1} = \frac{35280}{1.05}\)

Calculating the value:

\(PV_{1st\ instalment} = \frac{35280}{1.05} = 33600\)

So, the present value of the first instalment is Rs. 33,600.

Step 2: Calculate the present value of the second instalment.

The second instalment of Rs. 35,280 is paid at the end of the second year (n=2). The interest rate is still 5%.

\(PV_{2nd\ instalment} = \frac{35280}{(1 + \frac{5}{100})^2} = \frac{35280}{(1 + 0.05)^2} = \frac{35280}{(1.05)^2}\)

First, calculate \((1.05)^2\):

\((1.05)^2 = 1.05 \times 1.05 = 1.1025\)

Now, calculate the present value:

\(PV_{2nd\ instalment} = \frac{35280}{1.1025} = 32000\)

So, the present value of the second instalment is Rs. 32,000.

Step 3: Calculate the total principal amount (P).

The total principal amount borrowed is the sum of the present values of all the instalments.

\(P = PV_{1st\ instalment} + PV_{2nd\ instalment}\)

\(P = 33600 + 32000\)

\(P = 65600\)

Therefore, the value of P, the principal amount borrowed, is Rs. 65,600.

Comparing this result with the given options, we find that it matches one of the options.

Revision Table: Key Concepts in Loan Instalments

Additional Information on Loan Repayments and Present Value

When a loan is repaid with equal instalments over time, each instalment includes a portion of the principal repayment and the interest accumulated since the last payment. Earlier instalments primarily cover interest, while later instalments pay down more principal. However, from the lender's perspective (or when calculating the original principal), the loan amount is the sum of the present values of all the future payments they expect to receive.

The concept of present value is fundamental in finance. It tells us how much a future amount is worth today, considering the time value of money and the earning potential (interest rate). A higher interest rate means a lower present value for a future sum because the discounting effect is stronger.

For a loan with equal instalments (an annuity), there is a general formula for the present value, but breaking it down by calculating the present value of each instalment separately, as done here, is also a valid and often clearer approach, especially for a small number of periods like two years.

In summary, calculating the principal amount of a loan repaid by equal instalments involves discounting each future instalment back to the present using the given interest rate compounded annually and summing these present values.