Loan Amortization (Compound Interest) — ₹ 5,100 is to be repaid in two equal yearly installments at 4% p.a. compounded annually. Find each installment.
Correct Answer: ₹ 2,704
Introduction / Context:For equal annual installments under compound interest, the present value of all installments at the loan rate must equal the loan amount.
Given Data / Assumptions:
- Principal (loan) = ₹ 5,100.
- Rate i = 4% p.a. (compounded yearly).
- Installments: two, paid at ends of years 1 and 2.
Concept / Approach:Let each installment be A. Present value: PV = A/(1 + i) + A/(1 + i)^2. Set PV = 5,100 and solve for A. Equivalently, PV = A * [1 − (1 + i)^(−2)] / i.
Step-by-Step Solution:5,100 = A * [1 − (1.04)^(−2)] / 0.04[1 − (1.04)^(−2)] = 1 − 1/1.0816 = 0.075444…A = 5,100 * 0.04 / 0.075444… ≈ 5,100 * 0.53023… ≈ ₹ 2,704
Verification / Alternative check:PV check: 2,704/1.04 + 2,704/1.0816 ≈ 2,600 + 2,500 = 5,100 (rounded).
Why Other Options Are Wrong:They do not equate PV to ₹ 5,100 at 4% with two payments.
Common Pitfalls:Summing installments without discounting or using simple interest instead of compounding for PV.
Final Answer:₹ 2,704