Problem 4:This problem illustrates interest rate risk (the risk associated with future interest rates being uncertain) and how the magnitude of that risk depends upon horizon and on the characteristics of the cash flows associated with possible investment (in this case the characteristics of bonds).It also illustrates what we discussed in the last class (on 9/01) regarding the possible disconnection between YTM and the amount of wealth you accumulate over time when yield curves are not flat.

For the problem, you have $1,000,000.00 to invest in the following two bonds:

Bond A matures in two years and pays an annual coupon (once a year at the end of the year) of 1 percent.The face value is $1000.The market price of the bond is equal to the present value of the promised cashflows discounted using the appropriate rates of the yield curve for zero -coupon bonds (i.e., the zero yield curve) shown later in the problem.

Bond B matures in two years and pays and annual coupon (once a year at the end of the year) of 20 percent. The face value is $1000.The market price of this bond is also the present value of the promised cashflows using the zero yield curve.

A part of the zero yield curve (i.e., the yield curve based on zero coupon bonds) is given below:

TTM Zero YTM

1.03

2 .05

(TTM = time to maturity; YTM = yield to maturity)

Questions:

1.If the market is risk neutral, what does the market expect the 1-year zero YTM will be next year?That is, based on the current zero yield curve above, calculate the forward rate for the one-year zero YTM next year.

2.Given the price of Bond A is the present value of the promised cashflows using the current zero yield curve, what is the YTM for Bond A?

3.Similarly, what is the YTM for Bond B?

Think about:based on these YTMs, would you prefer to invest your $1,000,000 in Bond A over Bond B, or visa versa?The next set of questions explores what you will get under a variety of different scenarios.

4.First consider investing your $1,000,000 in just Bond A for two years.Note that since Bond A (and Bond B) pays a coupon at t = 1, you will want to take the cash paid then and invest it.Assume that you will invest it at the one-year rate that will exist next year.If you reinvest the cash you get from coupons at the 1-year zero rate you will get next year, how much money will you have at t = 2 (given the reinvested coupons plus the final payment of principal and coupon) if the 1-year rate next year equals the forward rate given the current yield curve (i.e., the rate equals your answer to question 1)?

5. Do the same think as 4 above, but invest in Bond B.How much will you have at t =2 if the one-year rate next year is the forward rate from the current yield curve?

6.By comparing your answers to 4 and 5 above, do you prefer (ex-post at t = 2) one bond over the other?

7.Now consider what would happen if your horizon was just one year. If you invest all of your $1,000,000 in Bond A, how much money will you have from that investment at t = 1 (i.e., after one year) if the one-year rate next year is the forward rate?Note that, because your horizon is only one year, you will have to sell the bond before it matures at t = 2, at a sales price that depends upon the interest rates that prevail at t = 1.Assume that the market prices of all bond are always equal to the present value of the remaining payments using the appropriate rates off the zero yield curve.Forward rate = 0.07y/t= 0.03 ; y/t 2 = 0.05

8.Next, similar to 7 above, how much would you have at t = 1 if you invest all of your $1,000,000 in Bond B?

9.By comparing your answers to 7 and 8, do you prefer one bond over the other?

10.How do answers to 6 and 9 make you feel about the YTM differences calculated in 2 and 3?f your calculations in 4 - 9, but assume that the one-year rate that actually occurs next year is the forward rate plus .005 (i.e., higher by a half a percentage point).

your calculation in 4 - 9, but assume that the one-year rate that actually occurs next year is the forward rate minus .005 (i.e., lower by half a percentage point).

13.Summary: You cannot perfectly predict the future one-year rate and, in fact, you face the uncertainty that the one-year rate next year will be the forward rate off the current yield curve plus or minus half a percentage point (where both outcomes are equally likely). How does that uncertainty effect the risk of each bond?Specifically, given all of the calculations and comparisons above, how does the magnitude of variation in what you get depend upon your horizon (either t = 1 or t = 2) and how does it depend upon the coupon rate (high or low)?