Economics 0281 Berger

Homework
#1: Answers

1. Suppose
you can get a $1,000 fixed payment car loan that requires you to make payments
of $600 for the next two years.

a. Calculate the present value of the loan payments
when the interest rate is 10%.

$600 $600

PV = ─────
+ ─────
= $1,041.32

1.1 1.1^{2}

b. Will the yield to maturity be above or
below 10%? Explain.

The yield to maturity will be above 10%
because a higher interest rate will lower the PV toward $1,000. (Remember the
definition of yield to maturity.)

c. Calculate the present value of the loan
payments when the interest rate is 15%.

$600 $600

PV = ─────
+ ──────
= $975.42

1.15 1.15^{2}

d. Will the yield to maturity be above or
below 15%? Explain.

The yield to maturity will be below 15%
because a lower interest rate will raise the PV toward a $1000.

2. 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 of this bond
when the interest rate is 8%.

$100 $100
$100 + $1,000

PV = ─────
+ ──────
+ ──────────────
= $1,051.54

1.08 1.08^{2} 1.08^{3}

=
$92.59 + $85.73 + $873.22 = $1,051.54

c. Will the yield to maturity be above or
below 8%? Explain.

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.

3. 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?

r_{t,t+1}
= (C + P_{t+1} - P_{t})/P_{t}

PV = $100/0.10 = $1,000 = P_{t}

PV = $100/0.05 = $2,000 = P_{t+1}

r_{t,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 = P_{t}

1.10 1.10^{2}

$100 + $1,000

PV = ─────────────
= $1,047.62 = P_{t+1}

1.05

r_{t,t+1}
= (C + P_{t+1} - P_{t})/P_{t}

=
($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?

P_{t }=
$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?

P_{t+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?

r_{t,t+1} = (C + P_{t+1} - P_{t})/P_{t}

_{ }

r_{t,t+1} = ($8,000 + $105,572 -
$105,344)/$105,344

r_{t,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 + 1/1.07^{2} +
1/1.07^{3} + 1/1.07^{4} + 1/1.07^{5}]

+ $10,000/1.07^{5}

= ($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.

8. Use
the concepts from the chapter 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: i_{t} = 3%

2 YR: i_{2t} = (i_{t} + i^{e}_{t+1})/2
= (3% + 5%)/2 = 4%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3

= (3% + 5% + 7%)/3 = 5%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4

= (3% + 5% + 7% + 9%)/4 = 6%

5 YR: i_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5

= (3% + 5% + 7% + 9% + 11%)/5 = 7%

(2)
5%, 5%, 5%, 5%, 5%

1 YR: 5%

2 YR: i_{2t} = (i_{t} + i^{e}_{t+1})/2

= (5% + 5%)/2 = 5%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3

= (5% + 5% + 5%)/3 = 5%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4

= (5% + 5% + 5% + 5%)/4 = 5%

5 YR: i_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5

= (5% + 5% + 5% + 5% + 5%)/5
= 5%

(3) 7%, 6%, 5%, 4%, 3%

1 YR: 7%

2 YR: i_{2t} = (i_{t} + i^{e}_{t+1})/2

= (7% + 6%)/2 = 6.5%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3

= (7% + 6% + 5%)/3 = 6%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4

= (7% + 6% + 5% + 4%)/4 = 5.5%

5 YR: i_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5

= (7% + 6% + 5% + 4% + 3%)/5
= 5%

b. Now assume that investors prefer short
term securities to long-term securities with the term premium k_{nt}
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: i_{2t} = (i_{t} + i^{e}_{t+1})/2
+ k_{2t}

= (3% + 5%)/2 + 2% = 6%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3 + k_{3t}

= (3% + 5% + 7%)/3 + 3% = 8%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4 + k_{4t}

= (3% + 5% + 7% + 9%)/4 + 4%
= 10%

5 YR: i_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5
+ k_{5t}

= (3% + 5% + 7% + 9% + 11%)/5
+ 5% = 12%

(2) 5%, 5%, 5%, 5%, 5%

1 YR: 6%

2 YR: i_{2t} = (i_{t} + i^{e}_{t+1})/2
+ k_{2t}

= (5% + 5%)/2 + 2% = 7%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3 + k_{3t}

= (5% + 5% + 5%)/3 + 3% = 8%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4 + k_{4t}

= (5% + 5% + 5% + 5%)/4 + 4%
= 9%

5 YR: i_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5
+ k_{5t}

= (5% + 5% + 5% + 5% + 5%)/5
+ 5% = 10%

(3) 7%, 6%, 5%, 4%, 3%

1 YR: 8%

2 YR: i_{2t} = (i_{t} + i^{e}_{t+1})/2
+ k_{2t}

= (7% + 6%)/2 + 2% = 8.5%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3 + k_{3t}

= (7% + 6% + 5%)/3 + 3% = 9%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4 + k_{4t}

= (7% + 6% + 5% + 4%)/4 + 4%
= 9.5%

5 YR: i_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5
+ k_{5t}

= (7% + 6% + 5% + 4% + 3%)/5
+ 5% = 10%

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.

k_{nt} 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: i_{2t} = (i_{t} + i^{e}_{t+1})/2
+ k_{2t}

= (12% + 9%)/2 + 2% = 12.5%

3 YR: i_{3t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2})/3 + k_{3t}

= (12% + 9% + 6%)/3 + 3% =
12.0%

4 YR: i_{4t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3})/4 + k_{4t}

= (12% + 9% + 6% + 3%)/4 +
4% = 11.5%

_{5t} = (i_{t} + i^{e}_{t+1}
+ i^{e}_{t+2} + i^{e}_{t+3} + i^{e}_{t+4})/5
+ k_{5t}

= (12% + 9% + 6% + 3% +
0%)/5 + 5% = 11.0%

d. For answer to Q.6, p. 139 see back of
text.

e. 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: r_{t} +P^{e}_{t}
= 3% + 1% = 4%

2 YR: r_{2t} = (r_{t} +P^{e}_{t}
+ r^{e}_{t+1 }+ P^{e}_{t+1})/2

= (3% + 1% + 5% + 1.5%)/2 =
5.25%

3 YR: r_{3t} = (r_{t} +P^{e}_{t}
+ r^{e}_{t+1 }+ P^{e}_{t+1} + r^{e}_{t+2}
+ P^{e}_{t+2})/3

= (3% + 1% + 5% + 1.5% + 7%
+2%)/3 = 6.5%

4 YR: r_{4t} = (r_{t} + P^{e}_{t}
+ r^{e}_{t+1} + P^{e}_{t+1} + r^{e}_{t+2}
+ P^{e}_{t+2} +

r^{e}_{t+3 }+
P^{e}_{t+3})/4

= (3% + 1% + 5% + 1.5% + 7% + 2% + 9% +
2.5%)/4

= 7.75%

5 YR: r_{5t} = (r_{t} + P^{e}_{t}
+ r^{e}_{t+1} + P^{e}_{t+1} + r^{e}_{t+2}
+ P^{e}_{t+2} +

r^{e}_{t+3 }+
P^{e}_{t+3} + r^{e}_{t+4 }+ P^{e}_{t+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: r_{t} +P^{e}_{t}
= 5% + 1% = 6%

2 YR: r_{2t} = (r_{t} +P^{e}_{t}
+ r^{e}_{t+1 }+ P^{e}_{t+1})/2

= (5% + 1% + 5% + 1.5%) =
6.25%

3 YR: r_{3t} = r_{t} +P^{e}_{t}
+ r^{e}_{t+1 }+ P^{e}_{t+1} + r^{e}_{t+2}
+ P^{e}_{t+2})/3

= (5% + 1% + 5% + 1.5% + 5% +
2%)/3 = 6.5%

4 YR: i_{4t} = (r_{t} + P^{e}_{t}
+ r^{e}_{t+1} + P^{e}_{t+1} + r^{e}_{t+2}
+ P^{e}_{t+2} +

r^{e}_{t+3 }+
P^{e}_{t+3})/4

= (5% + 1% + 5% + 1.5% + 5% +
2% + 5% + 2.5%)/4

= 6.75%

5 YR: r_{5t} = (r_{t} + P^{e}_{t}
+ r^{e}_{t+1} + P^{e}_{t+1} + r^{e}_{t+2}
+ P^{e}_{t+2} +

r^{e}_{t+3 }+ P^{e}_{t+3}
+ r^{e}_{t+4 }+ P^{e}_{t+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.