Final answer:
The bond's value is determined by calculating the present value of the semiannual coupons and face amount, using the given spot rates for each period and summing these values.
Step-by-step explanation:
The value of a bond based on zero coupon bond spot rates can be calculated using the present value (PV) of the bond's cash flows, which are the coupon payments and the face amount that will be received by the bondholder at maturity. In this scenario, a 10% bond with face amount 100 that matures in 3 years will pay semiannual coupon payments (5% of the face amount, which is $5) because the nominal annual rate is convertible semiannually. The present value of these cash flows must be calculated using the given term structures for zero coupon bond spot rates.
For the first term structure scenario, we would find the present value of each coupon payment and the face amount by discounting them at the corresponding spot rates for each period, and then sum these present values to find the total present value of the bond. The calculations would be as follows:
- PV of first coupon (6 months) = $5 / (1 + 0.75%/2)^1
- PV of second coupon (1 year) = $5 / (1 + 0.775%/2)^2
- PV of third coupon (1.5 years) = $5 / (1 + 0.08%/2)^3
- ...and so on, until the last coupon and face amount payment at 3 years.
For the second term structure, the process is similar, but with different spot rates. We would calculate the present value of each future cash flow by discounting them at the provided spot rates for each semiannual period:
- PV of first coupon (6 months) = $5 / (1 + 0.14%/2)^1
- PV of second coupon (1 year) = $5 / (1 + 0.1375%/2)^2
- PV of third coupon (1.5 years) = $5 / (1 + 0.135%/2)^3
- ...and so on, until the last coupon and face amount payment at 3 years.
After calculating all the present values, we sum them to obtain the total bond value.