Question

You observe that the current three-year discount factor for default-risk free cash flows is 0.68. Remember, the t-year discount factor is the present value of $1 paid at time t, i.e. D=(1+r)^-t, where r(t) is the t-year spot interest rate (annual compounding). Assume all bonds have a face value of $100 and that all securities are default-risk free. All cash flows occur at the end of the year to which they relate.

c) you observe the following: a 2-year coupon bond paying 10% annual coupons with a market price of $97, and two annuities that are trading at the same market price as each other. The first annuity matures in 3 years and pays annual cash flows of $20, while the second annuity pays annual cash flows of $28 and matures in 2 years. Using this information:

i. Complete the term structure of interest rates, i.e. determine the one- and two-year discount factors, d1 and d2, respectively.

ii. Determine the price of the annuities.

Answer 1

i). Let the 1-year spot rate be a and the 2-year spot rate be b.

Then, we have the following equation for the price of the 2-year bond:

Bond price = coupon 1/(1+a) + (coupon 2 + par value)/(1+b)^2

97 = 10/(1+a) + (10+100)/(1+b)^2

97 = 10/(1+a) + 110/(1+b)^2 ---- Equation (1)

Similarly, for the two annuities, we have the equations:

Price of 1st annuity = 20/(1+a) + 20/(1+b)^2 + 20/(1+13.72%)^3

Price of 2nd annuity = 28/(1+a) + 28/(1+b)^2

Since both have the same price, we have

20/(1+a) + 20/(1+b)^2 + 20/(1+13.72%)^3 = 28/(1+a) + 28/(1+b)^2 or

8/(1+a) + 8/(1+b)^2 = 13.60 ---- Equation (2)

We have two quadratic equations with two variables a & b.

Solving for a & b, we get a = 11.12% and b = 11.80% (Note: The equations can be solved either by hand, or using Solver or an online quadratic solver.)

Term structure of interest rates: s1 = 11.12%; s2 = 11.80%; s3 = 13.72%

Discount factor d1 = 1/(1+s1) = 1/(1+11.12%) = 0.8999

Discount factor d2 = 1/(1+s2)^2 = 1/(1+11.80%)^2 = 0.8000

ii). Price of the annuity = 20/(1+11.12%) + 20/(1+11.12%)^2 + 13.60 = 47.60

Since both annuities have the same price, both are priced at 47.60.