he current term-structure of spot interest rates for safe zero-coupon bonds is as follows:
Maturity, in years
Interest rate(r)
1 8%
2 10%
3 11%
4 12%
5 13%
There is a safe bond B which has 4 years before maturity and pays a coupon
of 12% at regular annual intervals and a face value of $100 at maturity.
(a) What will be the current price of bond B?
To calculate the current price of bond B, we need to determine the present value of the bond's future cash flows. The cash flows of the bond consist of periodic coupon payments and the final maturity payment.
Given:
Coupon rate: 12% per year
Face value: $100
Maturity: 4 years
Step 1: Calculate the present value of each coupon payment.
The bond pays a coupon of 12% annually. So, the coupon payment each year will be:
Coupon payment = Coupon rate * Face value = 0.12 * $100 = $12
To calculate the present value of each coupon payment, we need to discount it to its present value using the corresponding spot interest rate.
Current Spot Interest Rates:
Year 1: 8% (r1)
Year 2: 10% (r2)
Year 3: 11% (r3)
Year 4: 12% (r4)
Year 5: 13% (r5)
Present Value of Year 1 Coupon Payment:
PV1 = Coupon payment / (1 + r1) = $12 / (1 + 0.08) = $11.11
Present Value of Year 2 Coupon Payment:
PV2 = Coupon payment / (1 + r2)^2 = $12 / (1 + 0.10)^2 = $9.92
Present Value of Year 3 Coupon Payment:
PV3 = Coupon payment / (1 + r3)^3 = $12 / (1 + 0.11)^3 = $8.51
Present Value of Year 4 Coupon Payment:
PV4 = Coupon payment / (1 + r4)^4 = $12 / (1 + 0.12)^4 = $7.25
Step 2: Calculate the present value of the final maturity payment.
The face value of the bond is $100, which will be received at maturity. We need to discount it to its present value using the spot interest rate for the maturity year.
Present Value of Maturity Payment:
PVMaturity = Face value / (1 + r4)^4 = $100 / (1 + 0.12)^4 = $67.76
Step 3: Calculate the current price of bond B.
The current price of bond B is the sum of the present values of all the cash flows:
Current Price of bond B = PV1 + PV2 + PV3 + PV4 + PVMaturity
= $11.11 + $9.92 + $8.51 + $7.25 + $67.76
= $104.55
Therefore, the current price of bond B is $104.55.