Economics 0281
Homework #1: Answers
1.
Suppose you purchase a $1,000 face-value coupon bond with a
coupon rate of 10%, and a maturity of 3 years at a price of
$1,079. Assume that the inflation rate is zero.
a. Is the yield to maturity above or below 10%? Explain.
The yield to maturity is below 10% because the price
of the bond is above the par (face) value of the bond.
b. Calculate the present value (price) of this bond when
the interest rate is 8%.
$100
$100
$100 + $1,000
PV = ───── + ────── + ────────────── = $1,051.54
1.08
1.082
1.083
= $92.59 + $85.73 + $873.22 = $1,051.54
PV = AF(n=3,i=8%)($100) + PVF(n=3,i=8%)($1,000)
= 2.577($100) + (0.7938)($1,000) = $257.7 + $793.8
= $1,051.5
c. Will
Explain.
the
yield
to
maturity
be
above
or
below
8%?
Below 8% because a lower interest rate will produce a
higher PV which will be closer to the price of $1,079.
The yield to maturity should be 7% because this yields
a present value of $1,078.72.
2.
Do the following problems on the rate of return:
a. Suppose you purchase a consol with a yearly payment of
$100 on January 1 when its yield to maturity is 10% and
sell it on December 31 when its yield to maturity has
fallen to 5%. What is the rate of return on this consol?
rt,t+1 = (C + Pt+1 - Pt)/Pt
PV = $100/0.10 = $1,000 = Pt
PV = $100/0.05 = $2,000 = Pt+1
rt,t+1 = ($100 + $2,000 -$1,000)/$1,000
= $1,100/$1,000 = 1.1
= 110%
b. Suppose you purchase a 2 year 10% coupon bond for $1,000
(its par value) on January 1 and sell it on December 31
when its yield to maturity has fallen to 5%. What is the
rate of return on this coupon bond?
$100
$100 + $1,000
PV = ───── + ────────────── = $1,000.00 = Pt
1.10
1.102
PV = AF(n =2,i=10%)($100) + PVF(n=2,i=10%)($1,000)
= (1.736)($100) + (.8264)($1,000) = $173.6 + $826.4
= $1,000
$100 + $1,000
PV = ───────────── = $1,047.62 = Pt+1
1.05
rt,t+1 = (C + Pt+1 - Pt)/Pt
= ($100 + $1,047.62 - $1,000)/$1,000 = 0.148
= 14.8%
c. Which bond has the greater rate of return? Explain.
The consol has the greater rate of return for the same
fall in interest rates because the consol has the
longer maturity.
d. Suppose you purchase a 5 year 8% coupon bond on January
1, 2005 when the interest rate i = 6%. This bond has a par
value of $100,000 and a maturity date on December 31, 2007
when you purchase it. (NOTE: Coupon payments are made on
December 31.)
(1) What price do you pay for this bond?
Pt = ($8,000)AF(n=3,i=6%) + ($100,000)PVF(n=3,i=6%)
= $8,000(2.673) + $100,000(0.8396)
= $21,384 + $83,960 = $105,344
(2) Suppose you sell the bond on January 1, 2006 when
the interest rate i = 5%. What is the selling price of
the bond?
Pt+1 = ($8,000)AF(n=2,i=5%)] + ($100,000)PVF(n=2,i=5%)
= $8,000(1.859) + $100,000(0.9070)
= $14,872 + $90,700 = $105,572
(3) What is the rate of return on this bond?
rt,t+1 = (C + Pt+1 - Pt)/Pt
rt,t+1 = ($8,000 + $105,572 - $105,344)/$105,344
rt,t+1 = $8,228/$105,344 = 7.81%
3.
Suppose Alphonso has just bought a 5 year coupon bond with
a $10,000 face value and a 5% coupon rate. The interest
rate is 7%.
a. If the bond is not indexed and the inflation rate is 2%
in the first year, 3% in the second year, 4% in the third
year, 5% in the fourth year, and 6% in the fifth year,
calculate the (1) nominal value and (2) real value of this
bond.
Year
0
Price
Index 100
1
102.0
2
105.06
3
109.26
4
114.72
5
121.60
NOTE: The Price Index in Year 2 is 102(1.03) = 105.06
The Price index in Year 3 is 105.06(1.04) = 109.26
(1) The nominal value of the unindexed bond is
PV = $500[1/1.07 + 1/1.072 + 1/1.073 + 1/1.074 + 1/1.075]
+ $10,000/1.075
= ($500)AF(5,7%) + ($10,000)PVF(5,7%)
= $500[4.100] + $10,000[.7130] = $2,050 + $7,130
= $9,180
(2) The real value of the unindexed bond is
PV = $500[PV(1,7%)/(1.02) + PV(2,7%)/(1.02)(1.03) +
PV(3,7%)/(1.02)(1.03)(1.04) +
PV(4,7%)(1.02)(1.03)(1.04)(1.05)] +
$10,000[PV(4,7%)/(1.02)(1.03)(1.04)(1.05)
PV = $500[.9346/1.02 + .8734/1.0506 + .8163/1.0926
+ .7629/1.1472 + .7130/1.216] + $10,000(.7130)/1.216
= $500[.9163 + .8313 + .7471 + .6650 + .5863]
+ $10,000[.5863]
= $500[3.746] + $10,000[.5863] = $1,873 + $5,863
= $7,736
b. If the bond is indexed, calculate part a. again.
Year
Price
Index
0
1
100
102.0
Indexed
Coupon
Payment
$510
2
105.06
3
109.26
4
5
114.72
121.60
$525.30 $546.30 $573.60 $608.00
Indexed
Face Value
$12,160.00
(1) The nominal value of the indexed bond is
PV = ($510)PVF(1,7%) + ($525.30)PVF(2,7%) +
($546.30)PVF(3,7%) + ($573.60)PVF(4,7%) +
($608)PVF(5,7%) + ($12,160)PVF(5,7%)
PV = $510[.9346] + $525.30[.8734] + $546.30[.8163]
+ $573.60[.7629] + $608[.713] + $12,160[.713]
= $476.65 + $458.80 + $445.94 + $437.60 + $433.50
+ $8,670
= $10,922.49
(2) The real value of the indexed bond is
PV = $510[PV(1,7%)/(1.02)] + $525.3[PV(2,7%)/(1.02)(1.03)]
+ $546.3[PV(3,7%)/(1.02)(1.03)(1.04)] +
$573.6[PV(4,7%)(1.02)(1.03)(1.04)(1.05)] +
$12,160[PV(4,7%)(1.02)(1.03)(1.04)(1.05)]
PV = $510[.9346/1.02] + $525.3[.8734/1.0506]
+ $546.30[.8163/1.0926] + $573.60[.7629/1.1472]
+ $608[.713/1.216] + 12,160[.713/1.216]
= $510[.9163] + $525.30[.8313] + $546.30[.7471]
+ $573.60[.6650] + $608[.5863] + $12,160[.5863]
= $467.31 + $436.68 + $408.14 + $381.44 + $356.47
+ $7,129.41 = $9,179.45
Note that the real value of the indexed bond should be
equal to the nominal value of the unindexed bond.
4.
Use the concepts from the lecture on
interest rates to answer the following:
the
behavior
of
a. The Fed can decrease the money supply by selling
Treasury securities to the public/banks. Using the supply
and demand for bonds framework show what effect this Fed
policy has on interest rates.
The Fed action will increase the supply of bonds in the
bond market and will thereby drive the price of bonds down
and interest rates up.
Figure 1
P
i
BS0
BS1
P0
P1
i0
i1
BD
(a) Bond Market
B
Note: The arrows next to the axes indicate that prices
increase in an upward direction from the origin while
interest rates increase in a downward direction.
b. What effect will a sudden increase in the volatility of
gold prices have on interest rates. Show and explain.
This is increased riskiness of an alternative asset to
bonds so the relative riskiness of bonds decreases,
resulting in an increased demand for bonds and lower
interest rates.
Figure 2
P
i
BS
P1
P0
i1
i0
B D1
B D0
(a) Bond Market
B
c. Show and explain what effect an increase in the
riskiness of bonds has on interest rates. Use the supply
and demand for bonds in your answer.
Increased riskiness of bonds has the opposite effect to
part b. Namely, the demand for bonds will decrease
resulting in increased interest rates.
Figure 3
P
i
BS
P0
P1
B D0
B D1
(a) Bond Market
B
i0
i1
d. Predict what would happen to interest rates if the
public suddenly expects a large increase in stock prices.
Show and explain.
Large increases in
return relative to
return on bonds),
bonds which lowers
stock prices would reduce their rate of
bonds (or increase the relative rate of
resulting in an increase in demand for
interest rates.
Figure 4
P
i
BS
P1
P0
i1
i0
B D1
B D0
(a) Bond Market
B
e. Predict what would happen to the market for long-term
AT&T bonds if interest rates are expected to rise.
The expected rise in interest rates will reduce the rate of
return on AT&T bonds (as well as all other bonds) so the
demand for bonds will decrease which produces rising
interest rates.
Figure 5
P
i
BS
P0
P1
B D0
B D1
(a) Bond Market
B
i0
i1
5.
Use the concepts from the lecture on the term structure of
interest rates to do the following:
a. Assuming the expectations hypothesis is the correct
theory of the term structure, calculate the yields for
maturities of one to five years and plot the resulting
yield curves for the following series of one year interest
rates over the next five years:
(1) 3%, 5%, 7%, 9%, 11%
1 YR: it = 3%
2 YR: i2t = (it + iet+1)/2 = (3% + 5%)/2 = 4%
3 YR: i3t = (it + iet+1 + iet+2)/3
= (3% + 5% + 7%)/3 = 5%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4
= (3% + 5% + 7% + 9%)/4 = 6%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5
= (3% + 5% + 7% + 9% + 11%)/5 = 7%
Figure 6
Yield
(%)
7
6
5
4
3
1 2
3
4
5
Maturity
(in Years)
(2) 5%, 5%, 5%, 5%, 5%
1 YR: 5%
2 YR: i2t = (it + iet+1)/2
= (5% + 5%)/2 = 5%
3 YR: i3t = (it + iet+1 + iet+2)/3
= (5% + 5% + 5%)/3 = 5%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4
= (5% + 5% + 5% + 5%)/4 = 5%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5
= (5% + 5% + 5% + 5% + 5%)/5 = 5%
Figure 7
Yield
(%)
5
1
2 3 4
5
Maturity
(in Years)
(3) 7%, 6%, 5%, 4%, 3%
1 YR: 7%
2 YR: i2t = (it + iet+1)/2
= (7% + 6%)/2 = 6.5%
3 YR: i3t = (it + iet+1 + iet+2)/3
= (7% + 6% + 5%)/3 = 6%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4
= (7% + 6% + 5% + 4%)/4 = 5.5%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5
= (7% + 6% + 5% + 4% + 3%)/5 = 5%
Figure 8
Yield
(%)
7.0
6.5
6.0
5.5
5.0
1
2
3
4
5
Maturity
(in Years)
b. Now assume that investors prefer short term securities
to long-term securities with the term premium knt for one
through five year bonds being 1%, 2%, 3%, 4%, 5%. Redo part
a. and explain what you find.
(1) 3%, 5%, 7%, 9%, 11%
1 YR: 4%
2 YR: i2t = (it + iet+1)/2 + k2t
= (3% + 5%)/2 + 2% = 6%
3 YR: i3t = (it + iet+1 + iet+2)/3 + k3t
= (3% + 5% + 7%)/3 + 3% = 8%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4 + k4t
= (3% + 5% + 7% + 9%)/4 + 4% = 10%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5 + k5t
= (3% + 5% + 7% + 9% + 11%)/5 + 5% = 12%
Figure 9
Yield
(%)
12
10
8
6
4
1
2
3
4
5
Maturity
(in Years)
(2) 5%, 5%, 5%, 5%, 5%
1 YR: 6%
2 YR: i2t = (it + iet+1)/2 + k2t
= (5% + 5%)/2 + 2% = 7%
3 YR: i3t = (it + iet+1 + iet+2)/3 + k3t
= (5% + 5% + 5%)/3 + 3% = 8%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4 + k4t
= (5% + 5% + 5% + 5%)/4 + 4% = 9%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5 + k5t
= (5% + 5% + 5% + 5% + 5%)/5 + 5% = 10%
Figure 10
Yield
(%)
10
9
8
7
6
1 2
3
4
5
Maturity
(in Years)
(3) 7%, 6%, 5%, 4%, 3%
1 YR: 8%
2 YR: i2t = (it + iet+1)/2 + k2t
= (7% + 6%)/2 + 2% = 8.5%
3 YR: i3t = (it + iet+1 + iet+2)/3 + k3t
= (7% + 6% + 5%)/3 + 3% = 9%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4 + k4t
= (7% + 6% + 5% + 4%)/4 + 4% = 9.5%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5 + k5t
= (7% + 6% + 5% + 4% + 3%)/5 + 5% = 10%
Figure 11
Yield
(%)
10.0
9.5
9.0
8.5
8.0
1
2
3
4
5
Maturity
(in Years)
Note that all yield curves slope upwards: The slight
upwards slope of the curve indicates that future one
year interest rates will gently decline while a
regular upward slope indicates that future one year
interest rates will increase somewhat. In contrast, a
very steeply rising yield curve indicates sharply
rising one year interest rates.
c. Use the term premiums in part b and see if you can
construct a series of one year interest rates over five
years which results in an inverted yield curve.
knt for one through five year bonds are 1%, 2%, 3%, 4%, 5%
One year interest rates have to decline sharply: 12%, 9%,
6%, 3%, 0%
1 YR: 13.0%
2 YR: i2t = (it + iet+1)/2 + k2t
= (12% + 9%)/2 + 2% = 12.5%
3 YR: i3t = (it + iet+1 + iet+2)/3 + k3t
= (12% + 9% + 6%)/3 + 3% = 12.0%
4 YR: i4t = (it + iet+1 + iet+2 + iet+3)/4 + k4t
= (12% + 9% + 6% + 3%)/4 + 4% = 11.5%
5 YR: i5t = (it + iet+1 + iet+2 + iet+3 + iet+4)/5 + k5t
= (12% + 9% + 6% + 3% + 0%)/5 + 5% = 11.0%
d. Suppose that the expected inflation rate over the next
five years is rising: 1%, 1.5%, 2%, 2.5%, and 3%. Show and
explain how this affects the yield curve by using the
interest rates in a.(1) or a.(2). Then speculate on how a
decline in expected inflation should affect the yield
curve.
(1) 3%, 5%, 7%, 9%, 11%
1 YR: rt +Pet = 3% + 1% = 4%
2 YR: r2t = (rt +Pet + ret+1 + Pet+1)/2
= (3% + 1% + 5% + 1.5%)/2 = 5.25%
3 YR: r3t = (rt +Pet + ret+1 + Pet+1 + ret+2 + Pet+2)/3
= (3% + 1% + 5% + 1.5% + 7% +2%)/3 = 6.5%
4 YR: r4t = (rt + Pet + ret+1 + Pet+1 + ret+2 + Pet+2 +
ret+3 + Pet+3)/4
= (3% + 1% + 5% + 1.5% + 7% + 2% + 9% + 2.5%)/4
= 7.75%
5 YR: r5t = (rt + Pet + ret+1 + Pet+1 + ret+2 + Pet+2 +
ret+3 + Pet+3 + ret+4 + Pet+4)/5
= (3% + 1% + 5% + 1.5% + 7% + 2% + 9% + 2.5%
+ 11% + 3%)/5 = 9%
The result: A steeper yield curve (the yield spread
between 1 year and 5 year bonds is 9% - 4% = 5% as
opposed to 7% - 3% = 4%).
(2) 5%, 5%, 5%, 5%, 5%
1 YR: rt +Pet = 5% + 1% = 6%
2 YR: r2t = (rt +Pet + ret+1 + Pet+1)/2
= (5% + 1% + 5% + 1.5%) = 6.25%
3 YR: r3t = rt +Pet + ret+1 + Pet+1 + ret+2 + Pet+2)/3
= (5% + 1% + 5% + 1.5% + 5% + 2%)/3 = 6.5%
4 YR: i4t = (rt + Pet + ret+1 + Pet+1 + ret+2 + Pet+2 +
ret+3 + Pet+3)/4
= (5% + 1% + 5% + 1.5% + 5% + 2% + 5% + 2.5%)/4
= 6.75%
5 YR: r5t = (rt + Pet + ret+1 + Pet+1 + ret+2 + Pet+2 +
ret+3 + Pet+3 + ret+4 + Pet+4)/5
= (5% + 1% + 5% + 1.5% + 5% + 2% + 5% + 2.5% +
5% + 3%)/5 = 7%
Result: Steeper yield curve for the same reason.
Reversing our reasoning would imply that a fall in
inflationary expectations would produce an inverted
yield curve.