3-3 INFLATION AND INTEREST RATES
In late 1980 the US Commerce Department released new figures which showed that inflation was running at an annual rate of close to 15 percent. However, many investors expected the new Reagan administration to be more effective incontrolling inflation than the Carter administration had been. At the time, the prime rate of interest was 21 percent, a record high. However many observers believed that the extremely high interest rates and generally tight credit, which resulted from the Federal Reserve System’s attempts to curb the inflation rate, would shortly bring back a recession, which in turn would lead to a decline in the inflation rate and also in the rate of interest. Assume that at the beginning of 1981 the expecte rate of inflation for 1981 was 13 percent; for 1982, 9 percent; for 1983, 7 percent ; and for 1984 and thereafter, 6 percent.

a) What was the average expected inflation rate over the 5 year period 1981- 1985? (use the arithmetic average).
SOLUTION:
IP5 = (13 + 9 + 7 + 6 + 6)/5 = 41 / 5 = 8.2%

b) What average nominal interest rate would, over the five year period, be expected to produce a 2 percent real risk free rate of return on 5-year Treasury securities?

SOLUTION:
r* = 2%
r = ?
r = r* + IP5 = 2% + 8.2% = 10.2%

c) Assuming a real risk-free rate of 2 percent and a maturity risk premium which starts at 0.1 percent and increases by 0.1 percent each year, estimate the interest rate in January 1981 on bonds that mature in 1, 2, 5, 10 and 20 years and draw a yield curve based on these data.

SOLUTION:
r* = 2%
MRP = 0.1% increase every year
r1, r2,r5,r10,r20 = ?
r = r* + IP + MRP + LP + DRP

r1=2% + 13% = 15%
r2=2% + 11% + 0.1% = 13.1% IP2=(13+9)/2=11
r5=2% + 8.2% +4(0.1%) = 10.6% IP5=8.2%
r10=2% + 7.1% + 9(0.1%) = 10% IP10= (13+9+7+6+6+6+6+66+6)/10=7.1
r20=2% + 6.55% + 19(0.1) = 10.45% IP20= (13+9+7+6+6+6+6+66+6+6+6+6+6+6+6+6+6+6+6)/20=6.55

d) Describe the general economic conditions that could be expected to produce an upward sloping yield curve.
SOLUTION:
If in the future we expect that the inflation will be higher than the current one, then we will obtain an upward sloping yield curve.

e) If the consensus among investors in early 1981 had been that the expected rate of inflation for every future year was 10 percent ( that is It = I t+1 = 10% for t = 1 to 8), what do you think the yield curve would have looked like? Consider all the factors that are likely to affect the curve. Does your answer here make you question the yield curve you drew in part c?
SOLUTION:
If the expected inflation rate for every future year was 10 percent, we would expect to have a flat yield curve.

3-7 EXPECTED RATE OF INTEREST
Suppose the annual yield on a 2 – year Treasury Bond is 11.5 percent, while that on a 1-year bond is 10 percent. k* is 3 percent and the maturity risk premium is zero.
a) Using the expectations theory, forecast the interest rate on a 1-year bond during the second year. (hint: under the expectations theory, the yield on a 2-year bond is equal to the average yield on 1-year bonds in Years 1 and 2.)

SOLUTION:

r 2year treasury bond = 11.5%
r 1year bond = 10%
r* = 3%
MRP=0

r second = ?
According to the expectation theory:
11.5% = (r 1year bond + r second) / 2
11.5 = (10 + r second) / 2
23 = 10 + r second
r second = 13%

b) What is the expected inflation rate in Year 1? Year 2?

SOLUTION:

I1 = ?
IP2 = ?

Since, r = r* + IP, we derive IP = r – r*, therefore we calculate the following:

I1 = r – r* = 10 – 3 = 7%
IP2 = 13 – 3 = 10%

3-8 EXPECTED RATE OF INTEREST
Assume that the real risk free rate is 4 percent and that the maturity risk premium is zero. If the nominal rate of interest on 1-year bonds is 11 percent and that on comparable risk2-year bonds is 13 percent, what is the 1-year interest rate that is expected for Year 2? What inflation rate is expected during year 2? Comment on why the average interest rate during the 2 year period differs from the 1-year interest rate expected for year 2.

SOLUTION:
r* = 4%
MRP = 0
r 1year bond = 11%
r 2 year bond = 13 %
r second = ?
From the expectation theory we know that r 2 year bond = (r 1 year bond + r second)/2
We substitute the variables with numbers:
13 = (11 + r second) / 2
26 = 11 + r second
r second = 26 – 11
r second = 15

To calculate the IP2 we use the formula:
IP2 = r second – r* = 15 – 4 = 11%

3-11 MATURITY RISK PREMIUM
Assume that the real risk-free rate , k* is 3 percent and that inflation is expected to be 8 percent in year 1, 5 percent in year 2, and 4 percent thereafter. Assume also that all treasury bonds are highly liquid and free of default risk. If 2-year and 5-year treasury bonds both yield 10 percent, what is the difference in the maturity risk premiums (MRPs) on the two bonds, that is, what is MRP5 minus MRP2?

SOLUTION:
r* = 3%
r2 = r5 = 10%
MRP5 – MRP2 = ?

Year
Inflation Rate
1
8
2
5
3
4
4
4
5
4

The general formula is r = r* + IP + MRP

For calculating MRP2 we must first calculate IP2 = (8+5) / 2 = 6.5 and now put it in the general formula

r 2= r* + IP2 + MRP2
10 = 3 + 6.5 + MRP2
MRP2 = 0.5

For calculating MRP5 we must frist find IP5 = (8+5+4+4+4)/5 = 5 and putin the general formula

r 5= r* + IP5 + MRP5

10 = 3 + 5 + MRP5
MRP5 = 2

So the MRP5 – MRP2 = 2 – 0.5 = 1.5%

3-13 INTEREST RATES
Due to the recession, the rate of inflation expected for the coming year is only 3 percent. However the rate of inflation in year 2 and thereafter is expected to be constant at some level above 3 percent. Assume that the real risk-free rate , k*, is 2 percent for all maturities and that the expectations theory fully explains the yield curve, so there are no maturity premiums. If 3-year treasury bonds yield 2 percentage points more than 1- year bonds, what rate of inflation is expected after year 1?

SOLUTION:
I 1= 3%
I 2 = 3 + x
r* = 2%
I2 = ?

r3 = r1 +2%

r1 = r* +IP = 2+3 = 5%

Knowing r1 we can now calculate r3 = 5% + 2% = 7%

r3 = r* + IP3
7 = 2 + [3 + ( 3+x) + (3+x)] / 3
21 = 6 + 3 + 2(3+x)
21 = 9 + 2(3+x)
12 = 2(3+x)
2(3+x) = 12 / :2
3+x = 6
x = 3

Now we can find the I2 = 3 + x = 3 + 3 = 6%