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
expectations theory of the term structure of interest rates, the liquidity premium is zero
2. True. Under the expectations hypothesis, there are no risk premia built into bond prices.
3. Uncertain. Expectations of lower inflation will usually lead to lower nominal interest
4.
Maturity
Price
YTM
Forward Rate
1
$943.40
6.00%
2
$898.47
5.50%
(1.0552/1.06) – 1 = 5.0%
3
$847.62
5.67%
(1.05673/1.0552) – 1 = 6.0%
4
$792.16
6.00%
(1.064/1.05673) – 1 = 7.0%
15-2
5. 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.)
Beginning
of Year
Expected Price
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%
6. a. A 3-year zero coupon bond with face value $100 will sell today at a yield of 6%
b. The forward rates based on today’s yield curve are as follows:
Year
Forward Rate
2
(1.052/1.04) – 1 = 6.01%
3
(1.063/1.052) – 1 = 8.03%
Maturity
YTM
1
6.01%
2
(1.0601 × 1.0803)1/2 – 1 = 7.02%
The market forecast is for a higher YTM on 2–year bonds than your forecast.
Thus, the market predicts a lower price and higher rate of return.
7. a.
86.101$
08.1
109$
07.1
9$
P2=+=
Chapter 15 – The Term Structure of Interest Rates
b. The yield to maturity is the solution for y in the following equation:
86.101$
)y1(
109$
y1
9$
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:
0901.1
07.1
)08.1(
f1 2
2==+
f 2 = 0.0901 = 9.01%.
Therefore, using an expected rate for next year of r2 = 9.01%, we find that the
forecast bond price is:
99.99$
0901.1
109$
P==
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:
92.100$
0801.1
109$ =
8. a. The current bond price is:
($85 0.94340) + ($85 0.87352) + ($1,085 0.81637) = $1,040.20
b. If one year from now y = 8%, then the bond price will be:
[$85 Annuity factor (8%, 2)] + [$1,000 PV factor (8%,2)] = $1,008.92
15-4
9.
Year
Forward
Rate
PV of $1 received at period end
1
5%
$1/1.05 = $0.9524
2
7%
$1/(1.051.07) = $0.8901
3
8%
$1/(1.051.071.08) = $0.8241
b. To find the yield to maturity, solve for y in the following equation:
c.
Period
Payment received
at end of period:
Will grow by
a factor of:
To a future
value of:
1
$60.00
1.07 1.08
$69.34
2
$60.00
1.08
$64.80
3
$1,060.00
1.00
$1,060.00
$1,194.14
$984.10 (1 + y realized)3 = $1,194.14
14.194,1$ 3/1
d. Next year, the price of the bond will be:
[$60 Annuity factor (7%, 2)] + [$1,000 PV factor (7%,2)] = $981.92
10. 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 will
have a 3-year maturity, a YTM of 6.3%, and therefore the price will be:
Chapter 15 – The Term Structure of Interest Rates
15-5
b. 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 today’s
curve would be evidence that expected short rates are rising and that the yield
11. The price of the coupon bond, based on its yield to maturity, is:
[$120 Annuity factor (5.8%, 2)] + [$1,000 PV factor (5.8%, 2)] = $1,113.99
If the coupons were stripped and sold separately as zeros, then, based on the yield to
maturity of zeros with maturities of one and two years, respectively, the coupon
payments could be sold separately for:
08.111,1$
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 $2.91 on each bond.
12. a. The one-year zero-coupon bond has a yield to maturity of 6%, as shown below:
1
y1
100$
34.94$ +
=
y1 = 0.06000 = 6.000%
The yield on the two-year zero is 8.472%, as shown below:
2
2)y1(
100$
99.84$ +
=
y2 = 0.08472 = 8.472%
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]
06.1
)08472.1(
y1
)y1(
1
2
2
2==−=−
+
+
Chapter 15 – The Term Structure of Interest Rates
15-6
c. 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.
d. If there is a liquidity premium, then: E(r2) < f 2
112$
13. a. We obtain forward rates from the following table:
Maturity
YTM
Forward Rate
Price (for parts c, d)
1 year
10%
$1,000/1.10 = $909.09
2 years
11%
(1.112/1.10) – 1 = 12.01%
$1,000/1.112 = $811.62
3 years
12%
(1.123/1.112) – 1 = 14.03%
$1,000/1.123 = $711.78
b. 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
1 year
$1,000/1.1201 = $892.78
12.01%
2 years
$1,000/(1.1201 × 1.1403) = $782.93
13.02%
c. Next year, the 2-year zero will be a 1-year zero, and will therefore sell at a price
of: $1,000/1.1201 = $892.78
Similarly, the current 3-year zero will be a 2-year zero and will sell for: $782.93
Expected total rate of return:
93.782$ =−=−
78.892$ =−=−
Chapter 15 – The Term Structure of Interest Rates
15-7
d. The current price of the bond should equal the value of each payment times the
present value of $1 to be received at the “maturity” of that payment. The present
value schedule can be taken directly from the prices of zero-coupon bonds
calculated above.
Similarly, the expected prices of zeros one year from now can be used to calculate
the expected bond value at that time:
Total expected rate of return =
%00.101000.0
68.003,1$
)68.003,1$02.984($120$==
−+
14. 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
$715.00
8.75%
9.50%
5
$650.00
9.00%
10.00%
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
3
-$1,000
The 3-year zero issued at time 0 matures;
the issuer pays out $1,000 face value
4
+$1,095
The 4-year zeros purchased at time 0 mature;
receive face value
This is a synthetic one-year loan originating at time 3. The rate on the synthetic
loan is 0.095 = 9.5%, precisely the forward rate for year 3.
Chapter 15 – The Term Structure of Interest Rates
15-8
c. For each 4-year zero issued today, use the proceeds to buy:
$715.00/$650.00 = 1.100 five-year zeros
Your cash flows are thus as follows:
Time
Cash Flow
0
$0
4
-$1,000
The 4-year zero issued at time 0 matures;
the issuer pays out $1,000 face value
5
+$1,100
The 5-year zeros purchased at time 0 mature;
receive face value
15. a. For each three-year zero you buy today, issue:
b. Your cash flows are thus as follows:
Time
Cash Flow
0
$0
3
+$1,000.00
The 3-year zero purchased at time 0 matures;
receive $1,000 face value
5
-$1,204.50
The 5-year zeros issued at time 0 mature;
issuer pays face value
c. The effective two-year interest rate on the forward loan is:
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:
%46.202046.01
085.1
09.1
1
)y1(
)y1(
)2(f 3
5
3
3
5
5
3==−=−
+
+
=
[Note: there is a slight discrepancy here due to rounding error in the YTM
calculations above.]
Chapter 15 – The Term Structure of Interest Rates
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
expected future short rates being higher than the current short rate. A downward-sloping
yield curve implies expected future short rates are lower than the current short rate. Thus
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
c. Under the expectations hypothesis, lower implied forward rates would indicate
lower expected future spot rates for the corresponding period. Since the lower
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:
030.1
02.1
025.1
f1 2==+
This means that the forward rate is 0.030 = 3.0% semiannually, or 6.0% annually.
15–11
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% per12 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
(semi-annual)
YTM
(semi-annual)
0
5.00%
0.00%
5.00%
2.500%
2.500%
6 months
4.50
0.05
4.55
2.275
2.387
12 months
4.00
0.10
4.10
2.050
2.275
18 months
4.00
0.15
4.15
2.075
2.225
24 months
5.00
0.20
5.20
2.600
2.300
30 months
5.00
0.25
5.25
2.625
2.354
9. a. Five-year Spot Rate:
5
5
4
4
3
3
2
2
1
1)y1(
070,1$
)y1(
70$
)y1(
70$
)y1(
70$
)y1(
70$
000,1$ +
+
+
+
+
+
+
+
+
=
5
5
432 )y1(
070,1$
)0716.1(
70$
)0605.1(
70$
)0521.1(
70$
)05.1(
70$
000,1$ +
++++=
5
5)y1(
070,1$
08.53$69.58$24.63$67.66$000,1$ +
++++=
5
5)y1(
070,1$
32.758$ +
=
32.758$
070,1$
)y1( 5
5=+
%13.71411.1y 5
5=−=
Five-year Forward Rate:
%01.710701.11
)0716.1(
)0713.1(
4
5=−=−
Chapter 15 – The Term Structure of Interest Rates
15–12
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
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 $1,000 to be received in 4 years. Using
annual compounding:
35.758$
)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:
%07.60607.01
11.1
09.1
1
)y1(
)y1(
)2(f
2/1
3
5
2/1
3
3
5
5
3==−
=−
+
+
=
b. Assuming a par value of $1,000, the bond price is calculated as follows:
10.987$
)09.1(
090,1$
)10.1(
90$
)11.1(
90$
)12.1(
90$
)13.1(
90$
)y1(
090,1$
)y1(
90$
)y1(
90$
)y1(
90$
)y1(
90$
P
54321
5
5
4
4
3
3
2
2
1
1
=++++=
+
+
+
+
+
+
+
+
+
=