Question

2. It is now date zero. Currently the yield on a one-year bond is 10%, on a two-year bond is 15%, and on a three-year bond is 17%. Furthermore, everyone expects inflation to run at 5% this year, 10% next year and at 15% the year after that. Assuming that the pure expectations model correctly describes the behavior of the term structure, calculate the market’s expectation of the following:

a. The nominal interest rate on a one-year bond originating date 1 and maturing date 2. (The one-year rate next year.)

b. The real interest rate on a one-year bond originating date 1 and maturing date 2. (The one-year real rate next year.)

c. The real interest rate on a one-year bond originating date 2 and maturing date 3. (The one-year real rate two years from now.)

3. Suppose the situation is as described in question 2. However, you are firmly convinced that next year’s one-year nominal interest rate will be 12% rather than the answer you found to 2.a. a. Describe the steps you would take today to put yourself in a position to profit if your belief is correct.

b. If next year it turns out you are right and the one-year interest rate is indeed 12%, describe the steps you would take at that time and demonstrate that a profit results.

Answer 1

This is question of bond forward rates

a
Nominal interest rate on a one-year bond originating date 1
(1+15%)^2 = (1+10%)^1 * (1+rf1)^1
(1+rf1)^1= (1+15%)^2 / (1+10%)^1
(1+rf1)^1=
rf1= 20.2273%

b
Real interest rate on a one-year bond originating date 1
(1+20.2273%) = (1+10%) * (1+rr2)
(1+rr2) = (1+20.2273%) / (1+10%)
(1+rr2) =
rr2= 9.2975%

c
Nominal interest rate on a one-year bond originating date 2
(1+17%)^3 = (1+15%)^2 * (1+rf2)^1
(1+rf2)^1= (1+17%)^3 / (1+15%)^2
(1+rf2)^1=
rf2= 21.205%

3.a, b
The interest rate found as per the current market rates is 20.23%. However, the expected 1 year interest rate 1 year from now is 12% which is lower than the calculation. The calculated rate is higher is driven by the higher spot rate of the 2-year bond. This rate is expected to decrease. When the rate decreases, the bond price increases. Hence, we should go long on this bond i.e. purchase this bond today and hold for 1 year. The price of this bond today will be 69.18 [excel function- =PRICE(4/2/19,4/2/21,0,20.23%,100,1,1)]. One year later when the 2 year bond approaches the 12% rate, the price will increase to 89.29 [excel function- =PRICE(4/2/19,4/2/21,0,20.23%,100,1,1)]. We sell the bond at this time. Hence, we can have a profit of 20.11 (=89.29-69.18) from this trade at the end of one year.