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%
4. Do the
following problems:
a. You win $10M in the Pennsylvania lottery.
You have a choice of being paid $500,000 in 20 yearly installments or $5M in
one lump sum right now. Which alternative do you choose? Assume the market
interest rate is 10%.
Use the annuity formula:
PV = $500,000(8.514) = $4,257,000
You should choose the lump sum payment.
b. Suppose that you can purchase a $15,000
car in two ways: (1) A $3,000 rebate and 10% financing or (2) No rebate and 4%
financing. Assume there are 5 annual payments.
(1) $12,000 = C[3.791] and C = $3,165.39
(2) $15,000 = C[4.452] and C = $3,369.27
The rebate is the better choice.
c. What rebate will make these two
alternatives equal?
$15,000 - X = $3,369.27[3.791] = $12,772.90
X = $15,000 - $12,772.90 = $2,227.10
5. Given that the yield to maturity on Treasury
bills is 10%:
a. Compute the price of the following:
(1) A one year Treasury bill with a face
value (FV) of $1,000
PV(nTB) = FV/(1 + i)n where n = 1,
2, 3
PV(1TB) = $1,000/(1.1) = $909.09 for n = 1
(2) A two year Treasury bill with a face
value of $1,000
PV(2TB) = $1,000/(1.1)2 = $826.45
for n = 2
(3) A three year Treasury bill with a face
value of $1,000
PV(3TB) = $1,000/(1.1)3 = $751.31
for n = 3
(4) A three year Treasury bill with a face
value of $10,000
PV(3TB) = $10,000/(1.1)3 =
$7,513.15 for n = 3
b. Given that the yield to maturity (i) of
Treasury coupon bonds is 10%, compute the price of a three year Treasury coupon
bond which has a face value (FV) of $10,000 and an annual coupon rate of 10%.
PV(3TBND) = C1/(1 + i) + C2/(1 + i)2
+ (C3 + FV)/(1 + i)3
C1 = C2 = C3 = $1,000 = (.1)($10,000)
PV(3TBND) = $909.09 + $826.45 + $8,264.46 =
$10,000
Note that when the yield to maturity is equal
to the annual coupon rate, the price of
the bond is equal to FV.
c. Suppose the yield to maturity on Treasury
bills and bonds rises to 20%. Redo the computations in parts a and b. Explain
what you find.
a.(1) PV(1TB) = $1,000/(1.2) = $833.33
a.(2) PV(2TB) = $1,000/(1.2)2 =
$694.44
a.(3) PV(3TB) = $1,000/(1.2)3 =
$578.70
a.(4) PV(3TB) = $10,000/(1.2)3 =
$5,787.04
b.
PV(3TBND) = $833.33 + $694.44 + $578.70 + $5,787.04
= $7,893.51
NOTE: The rise in the market interest rate
(10%) causes the prices of shorter term bills to vary less than the prices of
longer term bills. For example, one year bills lose about 17% of their value
when market yields rise by 10% but three year bills lose about 43% of their
value when the same change occurs.
d. Suppose that the price of the three year
Treasury bond in part c. was $8,200 after the increase in the interest rate.
How could a person earn an equivalent amount of income at a lower price?
Explain.
If PV(3TBND) = $8,200 > $7,893.51, then we
could purchase the right to an equivalent stream of income by buying a one year
Treasury bill with a face value of $1,000, a two year Treasury bill with a face
value of $1,000, and a three year Treasury bill with a face value of $1,000,
and a three year Treasury bill for $10,000. These Treasury bills have a current
price (PV) which sums to a value of $7,893.51.
6. Suppose A 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 nominal
and real value of this bond.
Year
0 1 2
3 4 5
Price
Index
100 102.0 105.06
109.26 114.72 121.60
The nominal value of the 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
The real value of the bond is
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
0 1 2
3 4 5
Price
Index
100 102.0 105.06
109.26 114.72 121.60
Indexed
Coupon $510
$525.30 $546.30 $573.60 $608.00
Payment
Indexed
Face
$12,160.00
Value
The nominal value of the bond is
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
The real value of the bond is
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.
7. Use
the concepts from the chapter (Chapter 5) on the behavior of interest rates to
answer the following:
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.
Note:
The arrows next to the axes indicates 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.
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.
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 stock prices would reduce
their rate of return relative to bonds (or increase the relative rate of return
on bonds), resulting in an increase in demand for bonds which lowers interest
rates.
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.
