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     1.12

 

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    1.152

 

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    1.082        1.083

 

   = $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?

 

      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

 

                $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%

 

 

 

 

 

 

 

 

 

 

 

 

 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.

 

       

                                                         

 

 

 

 

 

 


 

 

 

 

 

 

 

 

 

 

 

 

 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: 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%

    

 


    (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%

 

 

 

 

 

 

 

 

 

(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%

 

 

 

 

 

 

 

 

 

 

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%

 

 

 

 

 

     (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%

 

 

 

 

 

 

 

 

 

 

(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%

 

 

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. 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: 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.