MBA 566
Homework 2 (due 4/3/2013)
Individual Work: please submit your own work for this part
Topic
Assignments
Chapter 14
Problems (8), (11), (18), (19), (20); CFA Problem 2
Chapter 15
Problem (10); CFA Problems (4), (6), (9)
Chapter 16
Problem (20); CFA Problem (2)
Solution:
14.8.
The bond price will be lower. As time passes, the bond price, which is now above par
value, will approach par.
14.11. a. On a financial calculator, enter the following:
n = 40; FV = 1000; PV = –950; PMT = 40
You will find that the yield to maturity on a semi-annual basis is 4.26%. This implies a
bond equivalent yield to maturity equal to: 4.26% * 2 = 8.52%
Effective annual yield to maturity = (1.0426)2 – 1 = 0.0870 = 8.70%
b. Since the bond is selling at par, the yield to maturity on a semi-annual basis is the
same as the semi-annual coupon rate, i.e., 4%. The bond equivalent yield to maturity is
8%.
Effective annual yield to maturity = (1.04)2 – 1 = 0.0816 = 8.16%
c. Keeping other inputs unchanged but setting PV = –1050, we find a bond equivalent
yield to maturity of 7.52%, or 3.76% on a semi-annual basis.
Effective annual yield to maturity = (1.0376)2 – 1 = 0.0766 = 7.66%
14.18.
Time
0
1
2
3
Inflation
in year just
ended
2%
3%
1%
Par value
$1,000.00
$1,020.00
$1,050.60
$1,061.11
Coupon
Payment
Principal
Repayment
$40.80
$42.02
$42.44
$ 0.00
$ 0.00
$1,061.11
The nominal rate of return and real rate of return on the bond in each year are
computed as follows:
Nominal rate of return =
interest + price appreciation
initial price
Real rate of return =
1 + nominal return
1
1 + inflation
Second year
Third year
Nominal return
$42.02  $30.60
 0.071196
$1,020
$42.44  $10.51
 0.050400
$1,050.60
Real return
1.071196
 1  0.040  4.0%
1.03
1.050400
 1  0.040  4.0%
1.01
The real rate of return in each year is precisely the 4% real yield on the bond.
14.19. The price schedule is as follows:
Year
Remaining
Maturity (T)
Constant yield value
$1,000/(1.08)T
0 (now)
1
2
19
20
20 years
19
18
1
0
$214.55
$231.71
$250.25
$925.93
$1,000.00
Imputed interest
(Increase in constant
yield value)
$17.16
$18.54
$74.07
14.20. The bond is issued at a price of $800. Therefore, its yield to maturity is: 6.8245%
Therefore, using the constant yield method, we find that the price in one year (when
maturity falls to 9 years) will be (at an unchanged yield) $814.60, representing an increase
of $14.60. Total taxable income is: $40.00 + $14.60 = $54.60
14.2-CFA.
a. (i) Current yield = Coupon/Price = $70/$960 = 0.0729 = 7.29%
(ii) YTM = 3.993% semiannually or 7.986% annual bond equivalent yield.
On a financial calculator, enter: n = 10; PV = –960; FV = 1000; PMT = 35
Compute the interest rate.
(iii) Realized compound yield is 4.166% (semiannually), or 8.332% annual bond
equivalent yield. To obtain this value, first find the future value (FV) of reinvested
coupons and principal. There will be six payments of $35 each, reinvested semiannually
at 3% per period. On a financial calculator, enter:
PV = 0; PMT = 35; n = 6; i = 3%. Compute: FV = 226.39
Three years from now, the bond will be selling at the par value of $1,000 because the
yield to maturity is forecast to equal the coupon rate. Therefore, total proceeds in three
years will be: $226.39 + $1,000 =$1,226.39
Then find the rate (yrealized) that makes the FV of the purchase price equal to
$1,226.39:
$960 × (1 + yrealized)6 = $1,226.39  yrealized = 4.166% (semiannual)
b. Shortcomings of each measure:
(i) Current yield does not account for capital gains or losses on bonds bought at prices
other than par value. It also does not account for reinvestment income on coupon
payments.
(ii) Yield to maturity assumes the bond is held until maturity and that all coupon income
can be reinvested at a rate equal to the yield to maturity.
(iii) Realized compound yield is affected by the forecast of reinvestment rates, holding
period, and yield of the bond at the end of the investor's holding period.
10.
a.
A 3-year zero coupon bond with face value $100 will sell today at a yield of 6% and a
price of:
$100/1.063 =$83.96
Next year, the bond will have a two-year maturity, and therefore a yield of 6% (from
next year’s forecasted yield curve). The price will be $89.00, resulting in a holding
period return of 6%.
b.
The forward rates based on today’s yield curve are as follows:
Year
2
3
Forward Rate
(1.052/1.04) – 1 = 6.01%
(1.063/1.052) – 1 = 8.03%
Using the forward rates, the forecast for the yield curve next year is:
Maturity YTM
1
6.01%
2
(1.0601 × 1.0803)1/2 – 1 = 7.02%
The market forecast is for a higher YTM on 2–year bonds than your forecast. Thus, the market
predicts a lower price and higher rate of return.
15.11.
b.
a.
P
$9
$109

 $101.86
1.07
1.08 2
The yield to maturity is the solution for y in the following equation:
$9
$109

 $101.86
1  y (1  y) 2
[Using a financial calculator, enter n = 2; FV = 100; PMT = 9; PV = –101.86; Compute i]
YTM = 7.958%
c.
The forward rate for next year, derived from the zero-coupon yield curve, is the
solution for f 2 in the following equation:
1 f2 
(1.08) 2
 1.0901  f 2 = 0.0901 = 9.01%.
1.07
Therefore, using an expected rate for next year of r2 = 9.01%, we find that the
forecast bond price is:
P
$109
 $99.99
1.0901
d.
If the liquidity premium is 1% then the forecast interest rate is:
E(r2) = f2 – liquidity premium = 9.01% – 1.00% = 8.01%
The forecast of the bond price is:
$109
 $100.92
1.0801
15.4 (CFA).
The given rates are annual rates, but each period is a half-year. Therefore, the per
period spot rates are 2.5% on one-year bonds and 2% on six-month bonds. The semiannual
forward rate is obtained by solving for f in the following equation:
1 f 
1.025 2
 1.030
1.02
This means that the forward rate is 0.030 = 3.0% semiannually, or 6.0% annually.
15.6 (CFA).
a.
Based on the pure expectations theory, VanHusen’s conclusion is
incorrect. According to this theory, the expected return over any time horizon would be the same,
regardless of the maturity strategy employed.
b.
According to the liquidity preference theory, the shape of the yield curve implies that
short-term interest rates are expected to rise in the future. This theory asserts that
forward rates reflect expectations about future interest rates plus a liquidity premium
that increases with maturity. Given the shape of the yield curve and the liquidity
premium data provided, the yield curve would still be positively sloped (at least
through maturity of eight years) after subtracting the respective liquidity premiums:
2.90% – 0.55% = 2.35%
3.50% – 0.55% = 2.95%
3.80% – 0.65% = 3.15%
4.00% – 0.75% = 3.25%
4.15% – 0.90% = 3.25%
4.30% – 1.10% = 3.20%
4.45% – 1.20% = 3.25%
4.60% – 1.50% = 3.10%
4.70% – 1.60% = 3.10%
15.9 (CFA). a.
Five-year Spot Rate:
$1,000 
$70
$70
$70
$70
$1,070




1
2
3
4
(1  y1 )
(1  y 2 )
(1  y 3 )
(1  y 4 )
(1  y 5 ) 5
$1,000 
$70
$70
$70
$70
$1,070




2
3
4
(1.05) (1.0521)
(1.0605)
(1.0716)
(1  y 5 ) 5
$1,000  $66.67  $63.24  $58.69  $53.08 
$1,070
(1  y 5 ) 5
$758.32 
$1,070
(1  y 5 ) 5
(1  y 5 ) 5 
$1,070
 y 5  5 1.411  1  7.13%
$758.32
Five-year Forward Rate:
(1.0713) 5
 1  1.0701  1  7.01%
(1.0716) 4
b.
The yield to maturity is the single discount rate that equates the present value of a
series of cash flows to a current price. It is the internal rate of return.
The spot rate for a given period is the yield to maturity on a zero-coupon bond that matures at
the end of the period. A spot rate is the discount rate for each period. Spot rates are used to
discount each cash flow of a coupon bond in order to calculate a current price. Spot rates are
the rates appropriate for discounting future cash flows of different maturities.
A forward rate is the implicit rate that links any two spot rates. Forward rates are
directly related to spot rates, and therefore to yield to maturity. Some would argue (as
in the expectations hypothesis) that forward rates are the market expectations of
future interest rates. A forward rate represents a break-even rate that links two spot
rates. It is important to note that forward rates link spot rates, not yields to maturity.
Yield to maturity is not unique for any particular maturity. In other words, two bonds
with the same maturity but different coupon rates may have different yields to
maturity. In contrast, spot rates and forward rates for each date are unique.
c.
The 4-year spot rate is 7.16%. Therefore, 7.16% is the theoretical yield to maturity
for the zero-coupon U.S. Treasury note. The price of the zero-coupon note
discounted at 7.16% is the present value of $1,000 to be received in 4 years. Using
annual compounding:
PV 
$1,000
 $758.35
(1.0716) 4
16.20.a.
A. 8% coupon bond
Time
until
Cash
Period
Payment Flow
(Years)
1
0.5
$40
2
1.0
40
3
1.5
40
4
2.0
1,040
Sum:
B. Zero-coupon
1
2
0.5
1.0
$0
0
PV of CF
Discount rate =
6% per period
Weight
Years ×
Weight
$37.736
35.600
33.585
823.777
$930.698
0.0405
0.0383
0.0361
0.8851
1.0000
0.0203
0.0383
0.0541
1.7702
1.8829
$0.000
0.000
0.0000
0.0000
0.0000
0.0000
3
4
1.5
2.0
0
1,000
0.000
792.094
$792.094
Sum:
0.0000
1.0000
1.0000
0.0000
2.0000
2.0000
For the coupon bond, the weight on the last payment in the table above is less than it is in
Spreadsheet 16.1 because the discount rate is higher; the weights for the first three payments are
larger than those in Spreadsheet 16.1. Consequently, the duration of the bond falls. The zero
coupon bond, by contrast, has a fixed weight of 1.0 for the single payment at maturity.
b.
A. 8% coupon bond
Time
until
Cash
Period
Payment Flow
(Years)
1
0.5
$60
2
1.0
60
3
1.5
60
4
2.0
1,060
PV of CF
Discount rate =
5% per period
Weight
Years ×
Weight
$57.143
0.0552 0.0276
54.422
0.0526 0.0526
51.830
0.0501 0.0751
872.065
0.8422 1.6844
Sum:
$1,035.460
1.0000 1.8396
Since the coupon payments are larger in the above table, the weights on the earlier payments are
higher than in Spreadsheet 16.1, so duration decreases.
c.
The duration rule predicts a percentage price change of:
 D 
 7.515 

  0.01   
  0.01  0.0702  7.02%
 1.07 
 1.07 
This overstates the actual percentage decrease in price by 0.31%.
The price predicted by the duration rule is 7.02% less than face value, or 92.98% of
face value.
d.
The duration-with-convexity rule predicts a percentage price change of:


 7.515 

2
  1.07   0.01  0.5  64.933  0.01  0.0670  6.70%



The percentage error is 0.01%, which is substantially less than the error using the
duration rule.
The price predicted by the duration rule is 6.70% less than face value, or 93.30% of
face value.
16.2 (CFA). a. Bond price decreases by $80.00, calculated as follows:
b.
c.
d.
e.
f.
10  0.01  800 = 80.00
½  120  (0.015)2 = 0.0135 = 1.35%
9/1.10 = 8.18
(i)
(i)
(iii)