Mr. Bansal is offered a choice: either Rs. 1,60,000 in cash now, or Rs. 7,500 at the end of every six months for ever. If money is worth 9% per annum compounded half-yearly, which offer is better for him, and by how much in present value?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → The half-yearly payments, by Rs. 6,666.67
Two offers are compared by their present values.
The half-yearly payments form a perpetuity paid at the end of each period. The rate per half-year is
\( \displaystyle i = \frac{0.09}{2} = 0.045 \)
Its present value is
\( \displaystyle P = \frac{7500}{0.045} = 166666.67 \)
This is more than the cash offer of Rs. 1,60,000 by
\(166666.67 - 160000 = 6666.67\)
So the half-yearly payments are better by Rs. 6,666.67.
Why the other options are incorrect
Option (2) The cash now, by Rs. 76,666.67 — Incorrect. This uses 7500 ÷ 0.09 = 83,333.33, the yearly rate with a half-yearly payment.
Option (3) The half-yearly payments, by Rs. 14,166.67 — Incorrect. This adds one extra payment of Rs. 7,500, as if the first payment were made today.
Option (4) The cash now, by Rs. 6,666.67 — Incorrect. The difference is right, but the perpetuity is worth more than the cash, not less.