SOLUTIONS TO END-OF-CHAPTER PROBLEMS
4-1
k* = 3%; I1 = 2%; I2 = 4%; I3 = 4%; MRP = 0; kT2 = ?; kT3 = ?
k = k* + IP + DRP + LP + MRP.
Since these are Treasury securities, DRP = LP = 0.
kT2 = k* + IP2.
IP2 = (2% + 4%)/2 = 3%.
kT2 = 3% + 3% = 6%.
kT3 = k* + IP3.
IP3 = (2% + 4% + 4%)/3 = 3.33%.
kT3 = 3% + 3.33% = 6.33%.
4-2
kT10 = 6%; kC10 = 8%; LP = 0.5%; DRP = ?
k = k* + IP + DRP + LP + MRP.
kT10 = 6% = k* + IP + MRP; DRP = LP = 0.
kC10 = 8% = k* + IP + DRP + 0.5% + MRP.
Because both bonds are 10-year bonds the inflation premium and maturity
risk premium on both bonds are equal. The only difference between them
is the liquidity and default risk premiums.
kC10 = 8% = k* + IP + MRP + 0.5% + DRP.
IP + MRP = 6%; therefore,
kC10 = 8% = 6% + 0.5% + DRP
1.5% = DRP.
4-3
kT1 = 5%;
kT2 =
4-4
1kT1
= 6%; kT2 = ?
5% + 6%
= 5.5%.
2
k* = 3%; IP = 3%; kT2 = 6.2%; MRP2 = ?
kT2 = k* + IP + MRP = 6.2%
kT2 = 3% + 3% + MRP = 6.2%
MRP = 0.2%.
But we know from above that k* +
4-5
Let x equal the yield on 2-year securities 4 years from now:
7.5% = [(4)(7%) + 2x]/6
0.45 = 0.28 + 2x
x = 0.085 or 8.5%.
4-6
k = k* + IP + MRP + DRP + LP.
k* = 0.03.
IP = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035.
MRP = 0.0005(6) = 0.003.
DRP = 0.
LP = 0.
kT7 = 0.03 + 0.035 + 0.003 = 0.068 = 6.8%.
4-7
a. k1 = 3%, and
k2 =
3% + 1k1
2
= 4.5%,
Solving for k1 in Year 2, 1k1, we obtain
1k1
= (4.5% × 2) - 3% = 6%.
b. For riskless bonds under the expectations theory, the interest rate
for a bond of any maturity is kn = k* + average inflation over n
years. If k* = 1%, we can solve for IPn:
Year 1: k1 = 1% + I1 = 3%;
I1 = expected inflation = 3% - 1% = 2%.
Year 2: k1 = 1% + I2 = 6%;
I2 = expected inflation = 6% - 1% = 5%.
Note also that the average inflation rate is (2% + 5%)/2 = 3.5%,
which, when added to k* = 1%, produces the yield on a 2-year bond,
4.5 percent. Therefore, all of our results are consistent.
Alternative solution:
Year 2 first:
kRF = k* + IP.
Solve for the inflation rates in Year 1 and
Year 1:
3% = 1% + IP1; IP1 = 2%, thus I1 = 2%.
Year 2:
4.5% = 1% + IP2; IP2 = 3.5%.
IP2 = (I1 + I2)/2
3.5% = (2% + I2)/2
I2 = 5%.
Then solve for the yield on the one-year bond in the second year:
Year 2: k1 = 1% + 5% = 6%.
4-8
k* = 2%; MRP = 0%; k1 = 5%; k2 = 7%;
1k1
1k1
= ?
represents the one-year rate on a bond one year from now (Year 2).
k1 + 1k1
2
5% + 1k 1
7% =
2
9% = 1k1.
k2 =
1k1
= k* + I2
9% = 2% + I2
7% = I2.
The average interest rate during the 2-year period differs from the 1year interest rate expected for Year 2 because of the inflation rate
reflected in the two interest rates.
The inflation rate reflected in
the interest rate on any security is the average rate of inflation
expected over the security’s life.
4-9
Basic relevant equations:
kt = k* + IPt + DRPt + MRPt + LPt.
But here IP is the only premium, so kt = k* + IPt.
IPt = Avg. inflation = (I1 + I2 + ...)/N.
We know that I1 = IP1 = 3% and k* = 2%.
kT1 = 2% + 3% = 5%.
Therefore,
kT3 = k1 + 2% = 5% + 2% = 7%.
kT3 = k* + IP3 = 2% + IP3 = 7%, so
IP3 = 7% - 2% = 5%.
We also know that It = Constant after t = 1.
We can set up this table:
But,
k*
2
2
2
1
2
3
I
3
I
I
Avg. I = IPt
3%/1 = 3%
(3% + I)/2 = IP2
(3% + I + I)/3 = IP3
k = k* + IPt
5%
k3 = 7%, so IP3 = 7% - 2% = 5%.
IP3 = (3% + 2I)/3 = 5%
2I = 12%
I = 6%.
4-10
kC8
8.3%
8.3%
8.3%
DRP8
=
=
=
=
=
k* + IP8 + MRP8 + DRP8 + LP8
2.5% + (2.8%  4 + 3.75%  4)/8 + 0.0% + DRP8 + 0.75%
2.5% + 3.275% + 0.0% + DRP8 + 0.75%
6.525% + DRP8
1.775%.
4-11
T-bill rate = k* + IP
5.5% = k* + 3.25%
k* = 2.25%.
4-12
We’re given all the components to determine the yield on the Cartwright
bonds except the default risk premium (DRP) and MRP. Calculate the MRP
as 0.1%(5 - 1) = 0.4%. Now, we can solve for the DRP as follows:
7.75% = 2.3% + 2.5% + 0.4% + 1.0% + DRP, or DRP = 1.55%.
4-13
First, calculate the inflation premiums for the next three and five
years, respectively. They are IP3 = (2.5% + 3.2% + 3.6%)/3 = 3.1% and
IP5 = (2.5% + 3.2% + 3.6% + 3.6% + 3.6%)/5 = 3.3%. The real risk-free
rate is given as 2.75%.
Since the default and liquidity premiums are
zero on Treasury bonds, we can now solve for the default risk premium.
Thus, 6.25% = 2.75% + 3.1% + MRP3, or MRP3 = 0.4%.
Similarly, 6.8% =
2.75% + 3.3% + MRP5, or MRP5 = 0.75%. Thus, MRP5 – MRP3 = 0.75% - 0.40% =
0.35%.
4-14
a.
Years to
Maturity
1
2
3
4
5
10
20
Real
Risk-Free
Rate (k*)
2%
2
2
2
2
2
2
IP**
7.00%
6.00
5.00
4.50
4.20
3.60
3.30
MRP
0.2%
0.4
0.6
0.8
1.0
1.0
1.0
kT = k* + IP + MRP
9.20%
8.40
7.60
7.30
7.20
6.60
6.30
**The computation of the inflation premium is as follows:
Expected
Inflation
7%
5
3
3
3
3
3
Year
1
2
3
4
5
10
20
Average
Expected Inflation
7.00%
6.00
5.00
4.50
4.20
3.60
3.30
For example, the calculation for 3 years is as follows:
7% + 5% + 3%
= 5.00%.
3
Thus, the yield curve would be as follows:
Interest
(%)
11.
10.
10.
9.
9.
8.
Exelo
8.
7.
Exxon
7.
6.
T0
2
4
6
8
10
12
14
16 18
Years
20
to
b. The interest rate on the Exxon Mobil bonds has the same components
as the Treasury securities, except that the Exxon Mobil bonds have
default risk, so a default risk premium must be included.
Therefore,
kExxon
Mobil
= k* + IP + MRP + DRP.
For a strong company such as Exxon Mobil, the default risk premium is
virtually
zero
for
short-term
bonds.
However,
as
time
to
maturity
increases, the probability of default, although still small, is sufficient
to warrant a default premium.
Thus, the yield risk curve for the Exxon
Mobil bonds will rise above the yield curve for the Treasury securities.
In the graph, the default risk premium was assumed to be 1.0 percentage
point on the 20-year Exxon Mobil bonds.
The return should equal 6.3% + 1%
= 7.3%.
c. Exelon bonds would have significantly more default risk than either
Treasury securities or Exxon Mobil bonds, and the risk of default
would increase over time due to possible financial deterioration.
In this example, the default risk premium was assumed to be 1.0
percentage point on the 1-year Exelon bonds and 2.0 percentage
points
on
the
20-year bonds. The 20-year return should equal 6.3% + 2% = 8.3%.
4-15
6
1
2
3
4
5
10
20
30
Term
months
year
years
years
years
years
years
years
years
Rate
5.1%
5.5
5.6
5.7
5.8
6.0
6.1
6.5
6.3
Interest Rate
(%)
10
8
6
4
2
0
0
5
10
15
20
25
30
Years to Maturity
4-16
a. The average rate of inflation for the 5-year period is calculated
as:
Average
inflation = (0.13 + 0.09 + 0.07 + 0.06 + 0.06)/5 = 8.20%.
rate
b. k = k* + IPAvg. = 2% + 8.2% = 10.20%.
c. Here is the general situation:
Year
1
2
3
5
.
.
1-Year
Expected
Inflation
13%
9
7
6
.
.
Arithmetic
Average
Expected
Inflation
13.0%
11.0
9.7
8.2
.
.
k*
2%
2
2
2
.
.
Maturity
Risk
Premium
0.1%
0.2
0.3
0.5
.
.
Estimated
Interest
Rates
15.1%
13.2
12.0
10.7
.
.
Interest Rate
(%)
15.0
12.5
10.0
7.5
5.0
2.5
0
.
10
20
.
6
6
2
4
.
7.1
6.6
6
8
10
.
2
2
12
14 16 18 20
Years to Maturity
.
1.0
2.0
.
10.1
10.6
d. The “normal” yield curve is upward sloping because, in “normal”
times, inflation is not expected to trend either up or down, so IP
is the same for debt of all maturities, but the MRP increases with
years, so the yield curve slopes up. During a recession, the yield
curve typically slopes up especially steeply, because inflation and
consequently short-term interest rates are currently low, yet people
expect inflation and interest rates to rise as the economy comes out
of the recession.
e. If inflation rates are expected to be constant, then the
expectations theory holds that the yield curve should be horizontal.
However, in this event it is likely that maturity risk premiums
would be applied to long-term bonds because of the greater risks of
holding long-term rather than short-term bonds:
Percen
(%
Actual yield curve
Maturit
ris
premiu
Pure expectations yield curve
Years to Maturity
If maturity risk premiums were added to the yield curve in Part e
above, then the yield curve would be more nearly normal; that is, the
long-term end of the curve would be raised. (The yield curve shown in
this answer is upward sloping; the yield curve shown in Part c is
downward sloping.)