E6-1
The risk-free rate on T-bills recently was 1.23%. If the real rate of interest is estimated to be 0.80%,
what was the expected level of inflation?
Expected level of inflation:
r* = RF - IP
Estimated real rate of interest = r* = 0.80%
Risk free rate = 1.23%
0.80% = 1.23% - IP
IP = 1.23 – 0.08 = 0.43%
Expected level of inflation = 0.43%
E6-2
The yields for Treasuries with differing maturities on a recent day were as shown in the table on p.253.
a. Use the information to plot a yield curve for this date.
Yield curve:
b. If the expectation hypothesis is true, approximately what rate of return do investors expect a 5 year
Treasury not to pay 5 years from now.
Maturity------------------Yield
3 months -----------------1.41%
6 months------------------1.71%
2 years---------------------2.68%
3 years---------------------3.01%
5 years---------------------3.70%
10 years--------------------4.51%
30 years--------------------5.25%
Answer:
= {(4.51% * 10) (3.7% * 5)} / 5
= {45.1% - 18.5%} / 5
= 26.6% / 5 = 5.32%
The rate of return that investors expect is 5.32%
c. If the expectations hypothesis is true, approximately (ignoring compounding) what rate of return do
investors expect a 1 year Treasury security to pay starting 2 years from now?
Answer:
= (3.01% * 3) - (2.68% * 2)
= 9.03% - 5.36% = 3.67%
The rate of return that investors expect is 3.67%
d. Is it possible that even though the yield curve slopes up in this problem, investors do not expect
rising interest rates? Explain.
Answer:
Yield curve may slope up for several reasons beyond expectations of increasing interest rates. According
to the theory of liquidity preference, long-term interest rates to be more than short-term rates as long
term debt has less liquidity, more responsiveness to general interest rate movements, and borrower
willingness to pay a higher interest rate to lock in money for a long period. In addition to preference
theory and expectations theory, theory of market segementation allows for more interest rate raises
arising from either limited availability of funds or greater demand for funds at longer maturities.
E6-3
The yields for Treasuries with differing maturities, including an estimate of the real rate of interest, on
recent day were as shown in the following table:
Maturity-----------Yield-------------Real rate of interest
3 months----------1.41%---------------0.80%
6 months-----------1.71%--------------0.80
2 years--------------2.68%--------------0.80
3 years--------------3.01%--------------0.80
5 year---------------3.70%--------------0.80
10 year--------------4.51%--------------0.80
30 year--------------5.25%--------------0.80
Use the information in the proceeding table to calculate the inflation expectation for each maturity.
Inflation expectation for each maturity:
For a specific maturity, the inflation expectation is the difference between the yield and the real interest
rate at the maturity
Maturity
Yield
3 months
6 months
2 years
3 years
5 years
10 years
30 years
1.41%
1.71
2.68
3.01
3.70
4.51
5.25
Real Rate of Interest
0.80%
0.80
0.80
0.80
0.80
0.80
0.80
Inflation Expectation
0.61%
0.91
1.88
2.21
2.90
3.71
4.45
E6-4
Recently, the annual inflation rate measured by the Consumer Price Index (CPI) was forecast to be 3.3%.
How could it have had a zero real rate of return? What minimum rate of return must the T-bill have
earned to meet your requirement of a 2% real rate of return?
Answer:
A treasury bill can experience a negative real rate of return if its interest rate is less than the inflation
rate as measured by the CPI. The real rate of return would be 0 if the treasury bill rate was 3.3% that
macthing the CPI rate. To get a 2% real return, the reate of treasury bill have to be at least 5.3%
E6-5
Calculate the risk premium for each of the following rating classes of long-term securities, assuming
that the yield to maturity (YTM) for comparable Treasuries is 4.51%.
Rating class----------Normal interest rate
AAA--------------------5.12%
BBB---------------------5.78%
B--------------------------7.82%
Risk premium calculation:
To calculate other securities risk premium, subtract 4.51% risk free rate from each nominal interest rate:
Security
AAA
BBB
B
Nominal Interest Rate
5.12%
5.78
7.82
Risk Premium
5.12% - 4.51% = 0.61%
5.78% - 4.51% = 1.27%
7.82% - 4.51% = 3.31%
E6-6
You have two assets and must calculate their values today base on their different payment streams and
appropriate required returns. Asset 1 has a required return of 1.5% and will produce a stream of $500 at
the end of each year indefinitely. Asset 2 has a required return of 10% and will produce an end-of-year
cash flow of $1,200 in the first year, $1,500 in the second year, and $850 in its third and final year.
Values of two assets:
Finding the cash flow stream PV for the assets by discounting the expected cash flows with respective
required return:
Asset 1: PV = $500 / 0.15 = $3,333.33
Asset 2: [($1,200 / 1.10) + ($1,500 / (1.10)²) + ($850 / (1.10)³) = $2,969.20
E6-7
A bond with 5 year Treasury bond has a coupon rate of 4.5%.
a. Give examples of required rates of the return that would make the bond sell at a discount, at a
premium, and a par.
b. If the bond’s par value is $10,000, calculate the differing values for this bond given the required rates
you chose in part a.
Answer:
A bond PV is the PV of its future cash flows. In the 5 – year treasury bond case, the expected cash flows
are $1,200 for 5 years at the end of each year, plus the bond face value that will be received at the bond
maturity. With the help of financial calculator, the solution is as follows:
PV of interest:
PMT = -1,200
I = 8%/year
N = 5 periods
Solve for PV = $4,791.25
PV of the bond’s face value:
FV = $20,000
N = 5 periods
I = 8%/year
Solve for PV = $13,611.66
Bond PV = $4,791.25 + $13,611.66 = $18,402.91
The answer is consistent with the knowledge that when interest rate increase, the values of bonds that
are issue previously fall. The present value is a cash outflow, or cost of investor.