Chapters 1 and 2 Exercises
1. Is there any risk when investing in treasury bonds, supposing there is no chance that
the US government will default?
Yes, interest rate risk, reinvestment risk, among others.
2. Ignore any complication, write down the formula for a 3-year fixed payment coupon
bond. Annual coupon rate is c, and coupons are semiannually paid. Face value is m,
interest rate/discount rate/yield to maturity is r.
You will be able to do this, won’t you?
3. Questions 9 chapter 1
(a) reference rate is the 1-month LIBOR. (b) Quoted margin is 220 basis points. (c) 5%
4. Question 10, chapter 1
bond issues whose coupon interest rate moves in the opposite direction from the change
in interest rates.
5. Questions 7, chapter 2
Find the present values of $2 million, $3 million, $5.4 million, and $5.8 million, sum
them up. We have total amount that should be invested today: $13,111,510.
6. Question 9, chapter 2
Use financial calculator, spreadsheet, formula or whatever, you should get the following
results:
A. 541.25;
B. New price is 509.09; and the percent of the price change is –5.94%;
C. 1077.95;
D. New price is 1000, and the percent of price change is -7.23%
E. Price volatility is higher in a low interest rate environment.
7. Question 11, chapter 2
Errors in V, X, Y.
Chapter 3 Exercises
(1) Problem 2 of chapter 3
EAY = (1+4.3%)**2-1=8.785%
(2) Problem 3 of chapter 3
Yield to maturity is the interest rate that will make the present value of the cash flows
equal to the price or the initial investment. At the time of bond purchase, it is equal to the
market interest rate.
(3) Problem 5b of chapter 3
a: bond W: 10%;
b: bond X: 9%;
c: bond Y: 10%;
d: bond Z: 8%
(4) Problem 12 of chapter 3
a: FV of the investment = 816*(1.045)40=4746.15
b: total coupon payment=nC=1400
c: FV of coupons and maturity value = FV of the bond = 4746.05. If there is no rounding
error, it should exactly equate the FV of the investment.
d: Interest on interest = FV of all coupon payments – nC = 3746.06-1400=2346.06;
e: It must equal 2346.06 since reinvestment rate of return = YTM.
(5) Problem 13 of chapter 3
There is no reinvestment risk related to zero coupon in this question, so return = YTM=
8%.
(6) Problem 16 of chapter 3
Step 1: calculating coupon interest + interest on interest = FV of coupon payment at 9.4%
reinvestment rate (4.7% for the semi-annual period) = 558.14.
Step 2: Determining the projected sale price at the end of 5 years.
N=4, YTM=5.6%, PMT=45, FV=1000  PV=961.53.
Step 3: Getting the total of steps 1 and 2
1519.67
Step 4: (1519.67/1000)1/10-1=4.2738%
Step 5: Double the rate=8.55%
Chapter 4 Exercises
(1) Question 10, Chapter 4
Percent change for bond A is –1.5%; percent change for bond B is –1.75%.
Thus investing in B is more volatile.
(2) Question 15, Chapter 15
a. Calculate the weighted average duration of the portfolio, we have portfolio duration
=8.96 years.
b. using the formula, you have percentage change in price = -4.48%
c. ignore this one.
Chapter 5 Exercises
(1) Question 10, chapter 5
The key is that each cash flow should be valued in today’s dollar using a discount rate
that reflects the required rate of return associated with that time period.
(2) Question 14, chapter 5
(a) Since YTM of the bond is 5.5%, coupon payment is $2.75 (for 19 periods)
We know the spot rate from period 1 through 18, so we do the following:
Summing up the present values for first payments. using Excel, you can easily find it
equals to 36.16.
The bond sold at par=100. So 36.16+102.75/(1+r)^19=100  r=2.534%
Annual rate = 2*r = 5.07% for year 9.5.
Applying the same logic, you r=4.80% for year 10.
(b). Now we have coupon rate 5%, semi-annual payment=2.5, face value=100.
Using the spot rates provided there, we can have present value for each cash flow.
Summing them up, we have the price of the bond $89.23.
Chapter 6 Exercises
(1) Question 2, chapter 6
c. (1) the dollar coupon interest at the end of the first six months is 151.50
c. (2) the principal at the end of the first six months is 1100
see the book for answers of other parts.
(2) Question 7, Chapter 6
a. 84437.5
don’t worry about other parts, I am not sure what they mean.
(3) Question 8, Chapter 6
STRIPS
Chapter 7 Exercises
(1) Question 13, Chapter 7
See book.
(2) Question 19, Chapter 7
Not really.
Chapter 8 Exercises
(1) Question 17, Chapter 8
(a) equivalent taxable rate = 12%
(b) other tax can impose problem on tax-exempt bonds.
Chapter 22
7. Answer the below questions.
(a) Compute the tracking error from the following information:
Month 2001
Portfolio A’s Return (%)
Lehman Aggregate Bond Index Return (%)
January
February
March
April
May
June
July
August
September
October
November
December
2.15
0.89
1.15
–0.47
1.71
0.10
1.04
2.70
0.66
2.15
–1.38
–0.59
1.65
–0.10
0.52
–0.60
0.65
0.33
2.31
1.10
1.23
2.02
–0.61
–1.20
The tracking error is the standard deviation of the active returns where an active return is the
portfolio A’s return minus the benchmark’s return for each month. The below exhibit has each
active return in the “Active Return” column. [Note that when subtracting a negative index return
from a portfolio return, the negative return is actually added to the portfolio return to get the
active return (i.e., for February, we have 0.89% – –0.10% = 0.89% + 0.10% = 0.99%).] To
compute the standard deviation of these active returns, we subtract the average (or mean) active
return from each active return, and then square each difference. Each difference squared value is
given in the exhibit below in the “Differences Squared” column. We then divided this sum by 12
– 1 = 11. We then multiply by 100 to convert to basis points. One basis point equals 0.0001 or
0.01%. We can then annualize the monthly basis points by multiplying by the square root of 12.
At the bottom of the below table we list details including the mean active return, variance,
standard deviation or tracking error (in terms of both percentage and basis points), and the
annualized tracking error (in terms of basis points).
Month 2001
Portfolio A’s
Return
Lehman Aggregate
Bond Index Return
Active Return
Differences
Squared
January
February
March
April
May
June
July
August
September
October
2.15%
0.89%
1.15%
–0.47%
1.71%
0.10%
1.04%
2.70%
0.66%
2.15%
1.65%
–0.10%
0.52%
–0.60%
0.65%
0.33%
2.31%
1.10%
1.23%
2.02%
0.50%
0.99%
0.63%
0.13%
1.06%
-0.23%
-1.27%
1.60%
-0.57%
0.13%
0.0707(%2)
0.5713(%2)
0.1567(%2)
0.0109(%2)
0.6820(%2)
0.2155(%2)
2.2625(%2)
1.8655(%2)
0.6467(%2)
0.0109(%2)
–1.38%
–0.59%
November
December
–0.61%
–1.20%
-0.77%
0.61%
Mean Active Return =
0.2342%
Variance (sum of differences squared / 11) =
Standard Deviation = Tracking Error =
Tracking error in basis points =
Tracking error in basis points annualized =
1.0084(%2)
0.1413(%2)
0.6947(%2)
0.8335%
83.35
288.74
(b) Is the tracking error computed in part (a) backward-looking or forward-looking tracking
error?
The tracking error just computed is based on the actual active returns observed for a portfolio.
Calculations computed for a portfolio based on a portfolio’s actual active returns reflect the
portfolio manager’s decisions during the observation period with respect to the factors that affect
tracking error. We call tracking error calculated from observed active returns for a portfolio
backward-looking tracking error. It is also called the ex-post tracking error and the actual
tracking error.
(c) Compare the tracking error found in part (a) to the tracking error found for Portfolios A and B
in Exhibit 19-1. What can you say about the investment management strategy pursued by this
portfolio manager?
The tracking error found for our problem is greater especially compared to Portfolio A in Exhibit
19-1. A greater tracking error means greater deviation from the benchmark. This is seen if we
compare active return values from our exhibit with the greater active return values found in
Exhibit 19-1. For our problem, it appears the manager may be employing a high-risk strategy to
enhance the indexed portfolio’s return. This strategy is commonly referred to as enhanced
indexing or indexing plus.
15. Below are two portfolios with a market value of $500 million. The bonds in both portfolios
are trading at par value. The dollar duration of the two portfolios is the same.
Bonds Included in Portfolio I
Issue
A
B
C
D
Bonds Included in Portfolio II
Years to Maturity
2.0
2.5
20.0
20.5
Par Value (in millions)
$120
$130
$150
$100
E
F
G
9.7
10.0
10.2
$200
$230
$ 70
(a) Which portfolio can be characterized as a bullet portfolio?
In a bullet strategy, the portfolio is constructed so that the maturities of the securities in the
portfolio are highly concentrated at one point on the yield curve. Thus, Portfolio II can be
characterized as a barbell portfolio because the maturities of its securities are concentrated around
one maturity (ten years).
(b) Which portfolio can be characterized as a barbell portfolio?
In a barbell strategy, the maturities of the securities included in the portfolio are concentrated at
two extreme maturities. Thus, Portfolio I can be characterized as a bullet portfolio because the
maturities of its securities are concentrated at two extreme maturities (two years and twenty
years).
(c) The two portfolios have the same dollar duration; explain whether their performance will be
the same if interest rates change.
For two portfolios with the same dollar duration, the portfolio with the greater the convexity will
perform better when yields change. What is necessary to understand is that the larger the dollar
convexity, the greater the dollar price change due to a portfolio’s convexity. Consider a barbell
portfolio and a bullet portfolio designed with the same duration as illustrated in the text. Although
both portfolios have the same dollar duration, the yield of the bullet portfolio is greater than the
yield of the barbell portfolio. This is because the dollar convexity of the barbell portfolio is
greater than that of the bullet portfolio. The difference in the two yields is sometimes referred to
as the cost of convexity (i.e., giving up yield to get better convexity).
(d) If they will not perform the same, how would you go about determining which would perform
best assuming that you have a six-month investment horizon?
To determine which portfolio would have the superior performance, we would want to look at the
total return for the six-month investment horizon given expectations about change in yields and
how the yield curve will shift. More details are given below.
The key point here is that looking at measures such as yield (yield to maturity or some type of
portfolio yield measure), duration, or convexity tells us little about performance over some
investment horizon, because performance depends on the magnitude of the change in yields and
how the yield curve shifts. When a manager wants to position a portfolio based on expectations as
to how (s)he might expect the yield curve to shift, it is imperative to perform total return analysis.
For example, in a steepening yield curve environment, it is often stated that a bullet portfolio
would be better than a barbell portfolio. As can be seen from Exhibit 19-8, it is not the case that a
bullet portfolio would outperform a barbell portfolio for this environment. Whether the bullet
portfolio outperforms the barbell depends on how much the yield curve steepens. An analysis
similar to that in Exhibit 19-8 based on total return for different degrees of steepening of the yield
curve can reveal to a manager whether a particular yield curve strategy will be superior to
another.