Question1.18 What should the purchase price of a 2-year zero-coupon bond be, if the 2-year zero coupon bond with face value of $1,000 is purchased at the beginning of year 2, given the information in the table below? [table]
Year | 1-year forward rate 1 | 1f2 = 3% 2 | 2f3 = 4% 3 | 3f4 = 5% 4 | 4f5 = 6% 5 | 5f6 = 7%
[/table]Select one alternative: 898.47 915.75 933.53 881.68 924.55 ResetMaximum marks: 2.5 Flag question undefined

Question

BlackTom AI · Accepted Answer

The price at the beginning of year 2 (time t = 1) equals the present value at t = 1 of $1,000 paid two years later (at t = 3), discounted for the two successive one-year periods using the one‑year forward rates 1f2 and 2f3.

1) Relevant concepts
- One‑year forward rate i_jk (e.g., 1f2, 2f3) is the one‑year interest rate applicable in the specified future year. Under no‑arbitrage, to value cash flows spanning successive one‑year periods you multiply the one‑period gross discount factors (1 + forward rate) for each year.
- A 2‑year zero bought at the beginning of year 2 pays $1,000 at the end of two years (t = 3). Its price at t = 1 is 1000 discounted over the two one‑year forward rates relevant for years 2 and 3.

2) Step‑by‑step reasoning and calculations
- Forward rates given: 1f2 = 3% = 0.03, and 2f3 = 4% = 0.04.
- Gross discount factor over the two one‑year periods = (1 + 1f2) × (1 + 2f3) = (1.03) × (1.04).
- Compute product: 1.03 × 1.04 = 1.0712.
- Price at t = 1 = 1000 / 1.0712 = 933.525... → round to two decimals = 933.53.

3) Why this answer is correct
Discounting year‑3 cash flow back to t = 1 requires applying the forward one‑year rates for year 2 and year 3 sequentially. Using 3% then 4% yields the exact present value 1000/(1.03×1.04) = 933.53, so this matches the correct no‑arbitrage valuation.

4) Why the other options are incorrect (brief)
- 898.47, 915.75, 881.68, 924.55 all imply different two‑year effective discount factors or yields inconsistent with the stated 3% and 4% one‑year forward rates. They come from mis‑discounting (e.g., using wrong rates, compounding incorrectly, or using a single wrong spot rate for both years). None equals 1000/(1.03×1.04).

Conclusion: The correct purchase price is $933.53, which matches the provided answer.

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