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CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
PROBLEM SETS.
1. In general, the forward rate can be viewed as the sum of the market’s expectation of
the future short rate plus a potential risk (or liquidity) premium. According to the
The liquidity preference theory, on the other hand, specifies that the liquidity
premium is positive so that the forward rate is greater than the market’s expectation
2. True. Under the expectations hypothesis, there are no risk premia built into bond
3. Uncertain. Expectations of lower inflation will usually lead to lower nominal
4. The liquidity theory holds that investors demand a premium to compensate them for
5. The pure expectations theory, also referred to as the unbiased expectations theory,
purports that forward rates are solely a function of expected future spot rates. Under
15-1
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
6. The yield curve slopes upward because short-term rates are lower than long-term
7. Maturity Price a. YTM b. Forward Rate
1 $943.40 6.00%
8. The expected price path of the 4-year zero coupon bond is shown below. (Note that
we discount the face value by the appropriate sequence of forward rates implied by
this year’s yield curve.)
Beginnin
g of Year a. Expected Price b. Expected Rate of Return
1 $792.16 ($839.69/$792.16) – 1 = 6.00%
2
69.839$
07.106.105.1
000,1$
($881.68/$839.69) – 1 = 5.00%
3
68.881$
07.106.1
000,1$
($934.58/$881.68) – 1 = 6.00%
4
58.934$
07.1
000,1$
($1,000.00/$934.58) – 1 = 7.00%
9. First calculate the short rates
1
2
2
3
3
4
4
( ) 5.00%
1.06
( ) 7.01%
1.05
1.065
( ) 7.51%
1.05 1.0701
1.07
( ) 8.51%
1.05 1.0701 1.0751
E r
E r
E r
E r
=
= =
= =
´
= =
´ ´
In one year, the market’s expectation of the yield curve is:
Bond Years to Maturity YTM(%)
B 1 7.01%
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
10. a. A 3-year zero coupon bond with face value $100 will sell today at a yield of
6% and a price of:
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, resulting in a
holding period return of 6%.
b. The forward rates based on today’s yield curve are as follows:
Year Forward Rate
Using the forward rates, the forecast for the yield curve next year is:
Maturit
y
YTM
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.
11. a.
2
$9 $109 $101.86
1.07 1.08
P= + =
b. The yield to maturity is the solution for y in the following equation:
2
$9 $109 $101.86
1 (1 )y y
+ =
+ +
c. The forward rate for next year, derived from the zero-coupon yield curve, is
the solution for f 2 in the following equation:
2
2
(1.08)
1 1.0901
1.07
f+ = =
15-3
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
forecast bond price is:
$109
P $99.99
1.0901
d. If the liquidity premium is 1% then the forecast interest rate is:
The forecast of the bond price is:
92.100$
0801.1
109$
15-4
b. If one year from now y = 8%, then the bond price will be:
13.
Year
Forward
Rate PV of $1 received at period end
b. To find the yield to maturity, solve for y in the following equation:
This can be solved using a financial calculator to show that y = 6.60%:
c.
Period
Payment Received
at End of Period:
Will Grow by
a Factor of:
To a Future
Value of:
CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
The holding period return is:
%88.50588.0
10.984$
)18.2$(60$
14. a. The return on the one-year zero-coupon bond will be 6.1%.
The price of the 4-year zero today is:
Next year, if the yield curve is unchanged, today’s 4-year zero coupon bond
b. Therefore, in this case, the longer-term bond is expected to provide the higher
return because its YTM is expected to decline during the holding period.
c. If you believe in the expectations hypothesis, you would not expect that the
yield curve next year will be the same as today’s curve. The upward slope in
15. The price of the coupon bond, based on its yield to maturity, is:
If the coupons were stripped and sold separately as zeros, then, based on the yield to
maturity of zeros with maturities of one and two years, respectively, the coupon
payments could be sold separately for:
08.111,1$
06.1
120,1$
05.1
120$
2
The arbitrage strategy is to buy zeros with face values of $120 and $1,120, and
respective maturities of one year and two years, and simultaneously sell the coupon
bond. The profit equals $2.91 on each bond.
16. a. The one-year zero-coupon bond has a yield to maturity of 6%, as shown below:
1
$100
$94.34 1y
=+
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CHAPTER 15: THE TERM STRUCTURE OF INTEREST RATES
The yield on the two-year zero is 8.472%, as shown below:
2
2
$100
$84.99 (1 )y
=+
y2 = 0.08472 = 8.472%
b. The price of the coupon bond is:
51.106$
)08472.1(
112$
06.1
12$
2
Therefore: yield to maturity for the coupon bond = 8.333%
[On a financial calculator, enter: n = 2; PV = –106.51; FV = 100; PMT = 12]
c.
2 2
2
2
1
(1 ) (1.08472)
1 1 0.1100 11.00%
1 1.06
y
fy
+
= - = - = =
+
d. Expected price
90.100$
11.1
112$
(Note that next year, the coupon bond will have one payment left.)
Expected holding period return =
%00.60600.0
51.106$
)51.106$90.100($12$
This holding period return is the same as the return on the one-year zero.
e. If there is a liquidity premium, then: E(r2) < f 2
E(Price) =
2
$112 $100.90
1 ( )E r >
+
E(HPR) > 6%
15-7
