Time Value of Money

Concept:Future value of an annuity of semi-annual deposits compounded semi-annually.Explanation:• Semi-annual interest rate = $\frac{10\%}{2}=5\% = 0.05$ per period.• Number of deposits: from son's age 6 months to 8 years (inclusive) at 6-month intervals = $\frac{8-0.5}{0.5}+1 = 16$ deposits. Each deposit = Rs. 1000.• Withdrawal at age 10 years. Time from first deposit (0.5 years) to withdrawal = 9.5 years = 19 semi-annual periods. Time from last deposit (8 years) to withdrawal = 2 years = 4 periods.• Future value of deposits at withdrawal = sum of each deposit compounded forward: $1000[(1.05)^{19}+(1.05)^{18}+\dots+(1.05)^{4}]$.• Rewrite as geometric series: $a = (1.05)^4$, $r=1.05$, $n=16$ terms. Sum = $a\frac{r^n-1}{r-1} = (1.05)^4 \times \frac{(1.05)^{16}-1}{0.05}$.• Given $(1.05)^{16}=2.1829$, and $(1.05)^4 = (1.1025)^2 \approx 1.21550625$.• Then sum $\approx 1.21550625 \times \frac{2.1829-1}{0.05} = 1.21550625 \times \frac{1.1829}{0.05} = 1.21550625 \times 23.658 \approx 28.756$.• Future value = $1000 \times 28.756 = Rs. 28756$.Answer:The son received Rs. 28756, which corresponds to option B.