Term-structure of interest rates and Arbitrage
The 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? [5 Marks]
To find the current price of bond B, we need to calculate the present value of its cash flows, including both coupon payments and the face value at maturity. To do this, we can use the concept of the term structure of interest rates and apply the arbitrage-free pricing principle.
The term structure of interest rates represents the relationship between the interest rates and the time to maturity of bonds. In this case, we are given the spot interest rates for zero-coupon bonds with different maturities. These interest rates represent the rate of return an investor would require for investing in a risk-free bond over a specific time period.
The arbitrage-free pricing principle states that the price of a bond should be equal to the present value of its expected future cash flows, discounted at the appropriate interest rate for each time period.
In the case of bond B, it pays a coupon of 12% at regular annual intervals and has a face value of £100 at maturity. We need to discount each cash flow at the corresponding interest rate for each time period.
The present value of each cash flow can be calculated using the following formula:
PV = CF / (1 + r)^n
where PV is the present value of the cash flow, CF is the cash flow amount, r is the interest rate, and n is the time period.
Let's calculate the present value of each cash flow for bond B:
Year 1:
Coupon payment = 12% of face value = 0.12 * £100 = £12
PV = £12 / (1 + 8%)^1 = £12 / (1.08) = £11.11
Year 2:
Coupon payment = £12
PV = £12 / (1 + 10%)^2 = £12 / (1.10^2) = £9.92
Year 3:
Coupon payment = £12
PV = £12 / (1 + 11%)^3 = £12 / (1.11^3) = £9.22
Year 4:
Coupon payment = £12
Face value = £100
PV = (£12 + £100) / (1 + 12%)^4 = £112 / (1.12^4) = £81.68
To find the current price of bond B, we need to sum up the present values of all the cash flows:
Current price = £11.11 + £9.92 + £9.22 + £81.68 = £111.93
Therefore, the current price of bond B is £111.93.