723G26/2012-10-10
Practice Set and Solutions #2
What to do with this practice set?
Practice sets are handed out to help students master the material of the course and
prepare for the final exam. These sets contain worked-out problems in corporate
finance and investments, with a bias toward quantitative problems. These sets are not
graded, and there is no need to hand-in the solutions. Students are strongly encouraged
to solve them, discuss the solutions with other course participants, and discuss any
problems with their teacher. Some questions in the final exam might resemble the
problems given here.
Question 1
The current one-year Treasury bill rate (1r1) is 5.2 percent, and the implicit forward one-year rate 1f2
12 months from now is 5.8 percent. According to the unbiased expectations theory, what should be
the current rate (1r2 ) for a two-year Treasury security?
Question 2
The Wall Street Journal reported interest rates of 6 percent, 6.35 percent, 6.65 percent, and 6.75
percent for three-year, four-year, five-year, and six-year Treasury notes, respectively. According to
the unbiased expectations theory, what are the expected one-year rates for years 4, 5, and 6?
Question 3
You note the following yield curve in The Wall Street Journal. According to the unbiased expectations
hypothesis, what is the implicit one-year forward rate for the period beginning two years from today,
2f1?
Maturity
Yield
One day
2%
One year
5.5%
Two years
6.5
Three years
9
Question 4
1
a) If both buyers (the public) and sellers of bonds (business) thinks that inflation will go up how
will that change the demand for bonds, interest rates and the slope of the yield curve.
b) If investors (demand for bonds) think that bad times are coming how will that change the
demand for bonds, interest rates and slope the yield curve?
Question 5
Based on economists’ = forecasts and analysis, one-year Treasury bill rates and liquidity premiums
for the next four years are expected to be as follows:
1R1
= 5.65%
E(2r1) = 6.75%
L2 = 0.05%
E(3r1) = 6.85%
L3 = 0.10%
E(4r1) = 7.15%
L4 = 0.12%
Using the liquidity premium hypothesis and plot the current yield curve. Make sure you label
the axes on the graph and identify the four annual rates on the curve both on the axes and on
the yield curve itself.
Question 6
A pension fund is attempting to balance one of the bond portfolios under its management. The fund
has identified three bonds which have five-year maturities and which trade at a yield to maturity
(YTM) of 9 percent, face value is $1,000. The bonds differ only in that the coupons are 7 percent, 9
percent, and 11 percent.
a. What is the duration for each bond?
b. What is the relationship between duration and the amount of coupon interest that is paid? Plot
the relationship.
Question 7
An investor can invest in a 5 year bond with annual coupon payments of 3%. The bond is trading at
an YTM of 4%. There also a zero coupon bond with a yield of 1.9%. Both bonds have a face value of
€1000.
a) What is the value of these bonds today?
b) If the investor have €10m to invest and wants to create a portfolio with a duration of 3 year how
should the investor allocate the portfolio?
2
Solutions
Solution 1
(1.052)(1.058) = (1 + 1r2)2 = 1.113016; (1 + 1r2) = 1.054996 ⇒ 1r2 = .0550 or 5.50 percent
Solution 2
[1 + E(ir1)] = (1 + 1Ri)i ÷ (1 + 1Ri-1)i-1
[1 + E(4r1)] = (1.0635)4 ÷ (1.06)3 = 1.0741 ⇒ E(4r1) = 7.41 percent for period 4
[1 + E(5r1)] = (1.0665)5 ÷ (1.0635)4 = 1.0786 ⇒ E(5r1) = 7.86 percent for period 5
[1 + E(6r1)] = (1.0675)6 ÷ (1.0665)5 = 1.0725 ⇒ E(6r1) = 7.25 percent for period 6
Solution 3
The implicit forward rates are
f2 = [(1 + 1r2)2/(1 + 1r1)] – 1
f2 = [(1.065)2/(1.055)] - 1 = 7.51%
Solution 4
a) If inflation is going up existing bonds, with given nominal values and coupons, do not offer a
yield that compensates for inflation (r = real rate + expected inflation). Thus, demand is
lowered and prices of bonds fall until they offer the required yield. In addition, if prices are
expected to go up it makes sense to consume today (incl. buying capital goods) before prices
increase. For business today’s bonds have a low real yield, why it is profitable to increase
the supply of bonds and invest more. They will increase the supply of long term bonds and
drive prices down. The outcome is a shift up in the yield curve, as the yield adjusts to higher
inflation expectations. We can make a case for a steeper slope, in particular if expected
inflation is associated with a boom in the economy.
b) Coming bad times means that you have to save more for the bad times going. Cut down on
current consumption and buy bonds that mature when the bad times are coming. Say that
you expect bad times 18 months from now, then you buy bonds with a maturity, which in
turn drives up the price of these bonds and brings down the yield. You could see the yield
curve becoming humped shaped with lower mid –term bonds selling at higher prices than
long-term bonds.
Solution 5
3
1r 1
= 5.65%
1r 2
= [(1 + .0565)(1 + .0675 + .0005)]½ - 1= 6.22%
1r 3
= [(1 + .0565)(1+.0675 + .0005)(1 + .0685 + .001)]1/3- 1 = 6.47%
1r 4
= [(1 + .0565)(1 + .0675 + .0005)(1+.0685 + .001)(1 + .0715 + .0012)]1/4- 1 = 6.67%
and the current yield curve will be upward sloping as shown:
YTM
6.67%
6.47%
6.22%
5.65%
_____________________________ Term to Maturity
0
1
2
3
4
(in years)
Solution 6
Five-year Bond: Par value = $1,000, Maturity = 5 years, R = 9%, Coupon rate = 7%
t
CF
PV of CF
PV of CF x t
1
$70
$64.22
$64.22
2
$70
$58.92
$117.84
3
$70
$54.05
$162.16
4
$70
$49.59
$198.36
5
$1,070
$695.43
$3,477.13
$922.21
$4,019.71
Duration = $4,019.71/$922.21 = 4.3588
R = 9%
t
Coupon rate = 9%
CF
PV of CF
PV of CF x t
1
$90
$82.57
$82.57
2
$90
$75.75
$151.50
3
$90
$69.50
$208.49
4
$90
$63.76
$255.03
5
$1,090
$708.43
$3,542.13
$1,000.00
$4,239.72
Duration = $4,239.72/$1,000.00 = 4.2397
R = 9%
Coupon rate = 11%
4
t
CF
PV of CF
PV of CF x t
1
$110
$100.92
$100.92
2
$110
$92.58
$185.17
3
$110
$84.94
$254.82
4
$110
$77.93
$311.71
5
$1,110
$721.42
$3,607.12
$1,077.79
$4,459.73
Duration = $4,459.73/$1,077.79 = 4.1378
b. What is the relationship between duration and the amount of coupon interest that is paid? Plot
the relationship.
Duration and Coupon Rates
Duration decreases as the amount of coupon interest
increases.
Years
4.3588
4.2397
Change in
4.1378
Duration
Coupon Duration
4.00
7%
9%
11%
Coupon Rates
4.3588
4.2397
7%
9%
-0.1191
4.1378
11%
-0.1019
Solution 7
a) The solution is given by the standard formulas for bond valuation (calculate the present value of
the cash flows, using YTM as the discount rate.
The duration of the five year bond is 4.71 years.
The duration of the one-year zero coupon bond is 1 year.
A portfolio with the duration 3 years is created by mixing the 5 and one year bonds such that the
combined portfolio has a duration of 3 years.
Le the weight in the portfolio of the one year bond be w1 and the weight of the five year bond is w2 =
(1-w1).
The solution is given by the weighted duration of the two bonds being equal to 3 (years),
3 = 1 x w1 + 4.71 x (1-w1)
The solution is
3 = w1 *1 + (1-w1)*4.71
3 = w1 – 4.71w1 + 4.71
5
-1.71 = (1-4.71) w1
w1 = -1.71/-3.71) = 0.46
and w1 = 0.46 and w2 = (1- w1) = 0.54.
In money terms the investor should allocate 0.46 x €10m = €4.6m in the one year bond, and the
€5.4m in the 5 year bond.
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