Prob Set 2.3 4. (3 points) A 3-year, $1,000 face value, zero-coupon Treasury bon

Question

Prob Set 2.3

4. (3 points) A 3-year, $1,000 face value, zero-coupon Treasury bond sells today for $882.616, when the Consumer Price Index (CPI) is 196. Assume the expected real rate of interest (r) is 1.50%. Use this Fisher equation:

                        R = r + ?e(Expected Inflation) or R = r + ?a(Actual Inflation) and show all work.  

a. Calculate the 3-year annual YTM, which is the 3-year nominal interest rate (R) for the bond (set calculator so that it will round to 2 decimal places for this problem).

b. Using the nominal rate (R) and the real rate (r), calculate and report the expected average, annual inflation rate over the next 3 years.  

c. If the CPI in three years is 218.8872 calculate the actual average, annual compounded rate of actual inflation over the 3 years (?a), based on the formula: FV = PV (1 + i)n. (Using the calculator you can enter the first CPI value as PV - make it negative - and enter the second CPI value as FV, and solve for I/YR). Then calculate the actual, realized real rate of interest earned on the bond based on actual inflation, compare it to the expected real rate of 1.50%.

d. If the CPI in three years is 204.3478, calculate the actual real rate of interest earned on the bond following the same procedure as in part c, and compare it to the expected real rate of 1.50%.

e. In a full essay, refer to your numerical results above, and explain how the actual, realized real rate of interest changes when the actual rate of inflation (?a) is different from the expected rate of inflation (?e).

Explanation / Answer

Part a. Based upon the given information we have : 882.616 * (1+R%)3 = 1000; where R is the nominal interest rate (3 year YTM). Solving for R we get R = 4.25%

Part b. Since R = 4.25% and real rate r is given to be 1.5%, using the Fisher equation, we get

R = r + expected inflation (e); plugging in the values, we get: 4.25% = 1.5% + e or e = 2.75%

Part c. We given that the CPI increases from 196 to 218.8872 in 3 years, hence the actual inflation (a) will be given by 218.8872 = 196 * (1+a)3 ; solving for a, we get a = 3.75%. Hence the actual inflation during the 3 year period is 3.75%. At this rate of inflation, the real return on the bond will be 4.25%-3.75% = 0.5% which is lower than the 1.5% expected at the time of investment in the bond.

Part d. If the CPI after 3 years is 204.3478, then the actual inflation will be : 204.3478 = 196 * (1+a)3; solving for a, we get a = 1.40%. At this level of actual inflation , the real return on the bond will be (4.25%-1.40%) = 2.85% which is much higher than the expected 1.5% real return at the time of investment.

Part e. The actual real return is the return earned adjusted for inflation and is the true appreciation in the purchasing power of the money invested. The investors look to be compensated in real terms and hence inflation expectation is built into interest rates, as stated in the Fisher equation where the nominal rate are sum of real return and inflation. Since these are exante calculations, the actual real returns may be different than exante returns due to actual inflation being different from expected inflation. This uncertainity increases even more for longer tenure bonds which is one of the reasons for longer term yields being generally higher. As we can see in the above case that at the time investment based upon the expected inflation and nominal rates the real rate of return was expected to be 1.50% bu after 3 years if the actual inflation turns out to be 3.75%, then the real return is only 0.5% and if the actual inflation turns out to be lower than exante expected inflation at 1.40% then the actual real return is higher at 2.85%

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