Calculating Annual Loan Payment with Compound Interest

This problem asks us to find the equal annual payment required to repay a borrowed sum over a specific period at a given compound interest rate. This is a classic loan amortization problem, where the present value of all future equal payments must equal the initial loan amount.

Understanding the Concepts: Loan Amortization

When a loan is repaid with equal installments over time, each payment covers both the interest accrued on the outstanding balance and a portion of the principal amount. The sum of the present values of all these future payments, discounted at the compound interest rate, equals the initial principal amount of the loan.

Identifying the Variables

We are given the following information:

• Principal Loan Amount (P) = Rs. 16400
• Annual Compound Interest Rate (r) = 5% = 0.05
• Loan Tenure (n) = 2 years
• Equal Annual Payment = A (This is what we need to find)

Applying the Formula for Present Value of an Annuity

The formula that relates the principal amount (P) to the equal annual payment (A), interest rate (r), and number of periods (n) is the present value of an ordinary annuity formula. For a loan, the principal amount is the present value of the stream of future payments:

$$P = \frac{A}{(1+r)^1} + \frac{A}{(1+r)^2} + \dots + \frac{A}{(1+r)^n}$$

In this specific case, the loan is to be paid back in 2 years (n=2). So, the formula simplifies to:

$$P = \frac{A}{(1+r)} + \frac{A}{(1+r)^2}$$

Step-by-Step Calculation

Now, let's substitute the given values into the formula:

$$16400 = \frac{A}{(1+0.05)} + \frac{A}{(1+0.05)^2}$$

$$16400 = \frac{A}{1.05} + \frac{A}{(1.05)^2}$$

Calculate the terms involving the interest rate:

$$1.05^2 = 1.1025$$

Substitute this back into the equation:

$$16400 = \frac{A}{1.05} + \frac{A}{1.1025}$$

To solve for A, we can find a common denominator for the terms on the right side, which is 1.1025:

$$16400 = \frac{A \times (1.1025 / 1.05)}{1.1025} + \frac{A}{1.1025}$$

$$16400 = \frac{A \times 1.05}{1.1025} + \frac{A}{1.1025}$$

Combine the terms on the right side:

$$16400 = \frac{1.05A + A}{1.1025}$$

$$16400 = \frac{2.05A}{1.1025}$$

Now, isolate A by multiplying both sides by 1.1025 and dividing by 2.05:

$$A = \frac{16400 \times 1.1025}{2.05}$$

Calculate the numerator:

$$16400 \times 1.1025 = 18081$$

Now, divide by 2.05:

$$A = \frac{18081}{2.05}$$

$$A = 8820$$

Result

The calculated annual payment required to repay the loan of Rs. 16400 over 2 years at 5% compound interest is Rs. 8820.

Revision Table: Compound Interest Loan Repayment

• Principal Amount (P): The initial sum borrowed.
• Interest Rate (r): The rate at which interest is compounded annually.
• Loan Tenure (n): The total number of years for repayment.
• Annual Payment (A): The fixed amount paid each year.
• Formula Used: Present Value of Ordinary Annuity, $P = A \times \frac{1 - (1+r)^{-n}}{r}$ or $P = \sum_{t=1}^{n} \frac{A}{(1+r)^t}$. We used the sum form for clarity for n=2 years.

Additional Information: Loan Amortization Schedule

An amortization schedule shows how each annual payment is applied towards interest and principal, and the remaining loan balance over time. For this case:

• Year 1 Payment (Rs. 8820):
  • Interest for Year 1: $16400 \times 0.05 = Rs. 820$
  • Principal paid in Year 1: $8820 - 820 = Rs. 8000$
  • Remaining Balance after Year 1: $16400 - 8000 = Rs. 8400$
• Year 2 Payment (Rs. 8820):
  • Interest for Year 2: $8400 \times 0.05 = Rs. 420$
  • Principal paid in Year 2: $8820 - 420 = Rs. 8400$
  • Remaining Balance after Year 2: $8400 - 8400 = Rs. 0$

The total principal paid is $8000 + 8400 = Rs. 16400$, which matches the initial loan amount. The total interest paid is $820 + 420 = Rs. 1240$. The total paid back is $2 \times 8820 = Rs. 17640$. The difference $17640 - 16400 = 1240$ is the total interest.