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CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
17. a. We obtain forward rates from the following table:
Maturit
y
YTM Forward Rate Price (for parts c, d)
1 year 10% $1,000/1.10 = $909.09
b., c. We obtain next year’s prices and yields by discounting each zero’s face value
at the forward rates for next year that we derived in part (a):
Maturity Price YTM
Note that this year’s upward sloping yield curve implies, according to the
expectations hypothesis, a shift upward in next year’s curve.
15-1
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
Expected total rate of return:
2-year bond:
%00.1011000.11
62.811$
78.892$
c. The current 3-year zero will be a 2-year zero and will sell for: $782.93
3-year bond:
%00.1011000.11
78.711$
93.782$
d. The current price of the bond should equal the value of each payment times
the present value of $1 to be received at the “maturity” of that payment. The
present value schedule can be taken directly from the prices of zero-coupon
bonds calculated above.
Current price = ($120 × 0.90909) + ($120 × 0.81162) + ($1,120 × 0.71178)
= $109.0908 + $97.3944 + $797.1936 = $1,003.68
e. The expected prices of zeros one year from now can be used to calculate the
expected bond value at that time:
Total expected rate of return =
%00.101000.0
68.003,1$
)68.003,1$02.984($120$
18. a.
Maturity
(years) Price YTM
Forward
Rate
1 $925.93 8.00%
2 853.39 8.25 8.50%
3 782.92 8.50 9.00
4 15.00 8.75 9.50
5 650.00 9.00 10.00
15-2
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
b. For each 3-year zero issued today, use the proceeds to buy:
Your cash flows are thus as follows:
Time Cash Flow
0 $ 0
c. For each 4-year zero issued today, use the proceeds to buy:
Your cash flows are thus as follows:
Time Cash Flow
This is a synthetic one-year loan originating at time 4. The rate on the
19. a. For each three-year zero you buy today, issue:
The time-0 cash flow equals zero.
b. Your cash flows are thus as follows:
Time Cash Flow
0 $ 0
This is a synthetic two-year loan originating at time 3.
c. The effective two-year interest rate on the forward loan is:
15-3
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
d. The one-year forward rates for years 4 and 5 are 9.5% and 10%, respectively.
Notice that:
The 5-year YTM is 9.0%. The 3-year YTM is 8.5%. Therefore, another way to
derive the 2-year forward rate for a loan starting at time 3 is:
55
5
33 3
3
(1 ) 1.09
(2) 1 1 0.2046 20.46%
(1 ) 1.085
y
fy
[Note: slight discrepancies here from rounding errors in YTM calculations]
CFA PROBLEMS
1. Expectations hypothesis: The yields on long-term bonds are geometric averages of
present and expected future short rates. An upward sloping curve is explained by
Liquidity preference hypothesis: Yields on long-term bonds are greater than the
expected return from rolling over short-term bonds in order to compensate investors
in long-term bonds for bearing interest rate risk. Thus bonds of different maturities
2. d. Investors bid up the price of short term securities and force yields to be relatively
3. a. (1+y4 )4 = (1+ y3 )3 (1 + f 4 )
b. The conditions would be those that underlie the expectations theory of the
term structure: risk neutral market participants who are willing to substitute
15-4
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
c. Under the expectations hypothesis, lower implied forward rates would
indicate lower expected future spot rates for the corresponding period. Since
4. 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:
2
1.025
1 1.030
1.02
f
5. The present value of each bond’s payments can be derived by discounting each cash
flow by the appropriate rate from the spot interest rate (i.e., the pure yield) curve:
Bond A:
53.98$
11.1
110$
08.1
10$
05.1
10$
PV
32
Bond B:
36.88$
11.1
106$
08.1
6$
05.1
6$
PV
32
Bond A sells for $0.13 (i.e., 0.13% of par value) less than the present value of its
stripped payments. Bond B sells for $0.02 less than the present value of its stripped
payments. Bond A is more attractively priced.
6. a. Based on the pure expectations theory, VanHusen’s conclusion is incorrect.
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
15-5
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
7. The coupon bonds can be viewed as portfolios of stripped zeros: each coupon can
stand alone as an independent zero-coupon bond. Therefore, yields on coupon
8. The following table shows the expected short-term interest rate based on the
projections of Federal Reserve rate cuts, the term premium (which increases at a
rate of 0.10% per 12 months), the forward rate (which is the sum of the expected
rate and term premium), and the YTM, which is the geometric average of the
forward rates.
Time
Expected
Short Rate
Term
Premium
Forward
Rate
(annual)
Forward Rate
(semiannual)
YTM
(semiannual)
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
2 3 4 5
5
$70 $70 $70 $70 $1,070
$1,000 (1.05) (1.0521) (1.0605) (1.0716) (1 )y
= + + + + +
5
5
$1,070
$1, 000 $66.67 $63.24 $58.69 $53.08 (1 )y
= + + + + +
5
5
$1,070
$758.32 (1 )y
5
5
$1,070
(1 ) $758.32
y
5
5
1.411 1 7.13%y= - =
Five-year forward rate:
%01.710701.11
)0716.1(
)0713.1(
4
5
b. The yield to maturity is the single discount rate that equates the present value
of a series of cash flows to a current price. It is the internal rate of return.
The spot rate for a given period is the yield to maturity on a zero-coupon bond
that matures at the end of the period. A spot rate is the discount rate for each
A forward rate is the implicit rate that links any two spot rates. Forward rates
are directly related to spot rates, and therefore to yield to maturity. Some
Yield to maturity is not unique for any particular maturity. In other words, two
c. The four-year spot rate is 7.16%. Therefore, 7.16% is the theoretical yield to
15-7
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
35.758$
)0716.1(
000,1$
PV
4
10. a. The two-year implied annually compounded forward rate for a deferred loan
beginning in 3 years is calculated as follows:
1/2 1/2
55
5
33 3
3
(1 ) 1.09
(2) 1 1 0.0607 6.07%
(1 ) 1.11
y
fy
é ù é ù
+
= - = - = =
ê ú ê ú
+ë û
ë û
b. Assuming a par value of $1,000, the bond price is calculated as follows:
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
$90 $90 $90 $90 $1, 090
(1 ) (1 ) (1 ) (1 ) (1 )
$90 $90 $90 $90 $1,090 $987.10
(1.13) (1.12) (1.11) (1.10) (1.09)
Py y y y y
= + + + +
+ + + + +
= + + + + =
