Calculating Equal Loan Installments with Compound Interest

This solution explains how to calculate the amount of each equal annual installment for a loan, considering the effect of compound interest. We need to determine the fixed amount that should be paid each year for 4 years to fully repay a loan of Rs 15000, given a 10% annual compound interest rate.

Understanding the Present Value of Annuity Formula

When a loan is repaid in equal installments over a period, it forms an annuity. The sum of the present values of all these future installments must equal the initial loan amount (the principal). The formula used to relate the loan amount (Present Value, PV) to the equal installment amount (Payment, Pmt) is the Present Value of an Ordinary Annuity formula:

$ PV = Pmt \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] $

Where:

• $PV$ is the Present Value of the loan (the principal amount).
• $Pmt$ is the amount of each equal installment (what we need to find).
• $r$ is the annual interest rate (as a decimal).
• $n$ is the number of installments.

Applying the Formula to the Loan Repayment Problem

From the question, we have the following information:

• Loan Amount ($PV$) = Rs 15000
• Number of Installments ($n$) = 4
• Compound Interest Rate ($r$) = 10% per annum = 0.10

We need to find the installment amount ($Pmt$). We can rearrange the annuity formula to solve for $Pmt$:

$ Pmt = \frac{PV \times r}{1 - (1 + r)^{-n}} $

Step-by-Step Calculation:

1. Substitute the known values into the rearranged formula:

   $ Pmt = \frac{15000 \times 0.10}{1 - (1 + 0.10)^{-4}} $

2. Simplify the numerator:

   $ 15000 \times 0.10 = 1500 $

3. Calculate the term $(1 + r)^-n$:

   First, find $(1.10)^4$:

   $ (1.10)^1 = 1.10 $

   $ (1.10)^2 = 1.21 $

   $ (1.10)^3 = 1.331 $

   $ (1.10)^4 = 1.4641 $

   Now, calculate the inverse:

   $ (1.10)^{-4} = \frac{1}{(1.10)^4} = \frac{1}{1.4641} \approx 0.68301 $

4. Calculate the denominator:

   $ 1 - (1.10)^{-4} = 1 - 0.68301 = 0.31699 $

5. Calculate the installment amount ($Pmt$):

   $ Pmt = \frac{1500}{0.31699} $

   $ Pmt \approx 4731.94 $

Determining the Approximate Installment Amount

The calculated amount for each annual installment is approximately Rs 4731.94. We need to find the closest option among the choices provided.

• Option 1: Rs. 4500
• Option 2: Rs. 4731
• Option 3: Rs. 4900
• Option 4: Rs. 5100

Comparing our calculated value (Rs 4731.94) with the options, the closest value is Rs 4731.

Conclusion

To repay a loan of Rs 15000 in 4 equal annual installments at a 10% compound interest rate, each installment should be approximately Rs 4731.