Econ 422 - Summer 2012 PROBLEM SET 3

PCBR (Interest Rates) pages 102 106 Problems 25, 27, 29 - 32 5-25. In 1975, interest rates were 7.85% and the rate of inflation was 12.3% in the United States.

What was the real interest rate in 1975? How would the purchasing power of your savings have changed over the year?

rr i 7.85% 12.3%r 3.96%1 i 1.123

= = = +

The purchasing power of your savings declined by 3.96% over the year. 5-27. Can the nominal interest rate available to an investor be significantly negative? (Hint:

Consider the interest rate earned from saving cash under the mattress.) Can the real interest rate be negative? Explain.

By holding cash, an investor earns a nominal interest rate of 0%. Since an investor can always earn at least 0%, the nominal interest rate cannot be negative. The real interest rate can be negative, however. It is negative whenever the rate of inflation exceeds the nominal interest rate.

5-29. Suppose the term structure of risk-free interest rates is as shown below:

a. Calculate the present value of an investment that pays $1000 in two years and $2000 in

five years for certain.

b. Calculate the present value of receiving $500 per year, with certainty, at the end of the next five years. To find the rates for the missing years in the table, linearly interpolate between the years for which you do know the rates. (For example, the rate in year 4 would be the average of the rate in year 3 and year 5.)

c. Calculate the present value of receiving $2300 per year, with certainty, for the next 20 years. Infer rates for the missing years using linear interpolation. (Hint : Use a spreadsheet.)

a. Timeline: 0 1 2 3 4 5

1,000 2,000

Since the opportunity cost of capital is different for investments of different maturities, we must use the cost of capital associated with each cash flow as the discount rate for that cash flow:

( ) ( )2 51, 000 2, 000

PV $2,652.15.1.0241 1.0332

= + =

b. Timeline:

0 1 2 3 4 5 500 500 500 500 500

Since the opportunity cost of capital is different for investments of different maturities, we must use the cost of capital associated with each cash flow as the discount rate for that cash flow. Unfortunately, we do not have a rate for a 4-year cash flow, so we linearly interpolate.

( ) ( )41 1r 2.74 3.32 3.032 2

= + =

( ) ( ) ( ) ( )2 3 4 5

500 500 500 500 500PV $2,296.431.0199 1.0241 1.0274 1.0303 1.0332

= + + + + =

c. Timeline: 0 1 2 3 20

2,300 2,300 2,300 2,300

Since the opportunity cot of capital is different for investments of different maturities, we must use the cost of capital associated with each cash flow as the discount rate for that cash flow. Unfortunately, we do not have a rate for a number of years, so we linearly interpolate.

( ) ( )

( ) ( )

( ) ( )

4

6

8

1 1r 2.74 3.322 23.031 1r 3.32 3.762 23.542 1r 3.76 4.133 33.883

= +

=

= +

=

= +

=

( ) ( )

( ) ( )

( ) ( )

9

11

12

1 2r 3.76 4.133 34.00679 1r 4.13 4.9310 104.218 2r 4.13 4.9310 104.29

= +

=

= +

=

= +

=

13

14

15

16

17

18

19

r 4.37r 4.45r 4.53r 4.61r 4.64r 4.77r 4.85

=

=

=

=

=

=

=

( ) ( ) ( )

( )

2 3 201 2 3 20

20

2,300 2,300 2,300 2,300...1 1 1 12,300 2,300 2,300 2,300...1.0199 1.0241 1.0274 1.0493$30,636.56

PV r r r r= + + + +

+ + + +

= + + + +

=

5-30. Using the term structure in Problem 29, what is the present value of an investment that pays $100 at the end of each of years 1, 2, and 3? If you wanted to value this investment correctly using the annuity formula, which discount rate should you use?

PV = 100 / 1.0199 + 100 / 1.02412 + 100 / 1.02743 =$285.61.

To determine the single discount rate that would compute the value correctly, we solve the following for r:

PV = 285.61 = 100/(1 + r) + 100 / (1 + r)2 + 100/(1 + r)3 = $285.61.

This is just an IRR calculation. Using trial and error or the annuity calculator, r = 2.50%. Note that this rate is between the 1, 2, and 3-yr rates given.

5-31. What is the shape of the yield curve given the term structure in Problem 29? What expectations are investors likely to have about future interest rates?

The yield curve is increasing. This is often a sign that investors expect interest rates to rise in the future.

5-32. Suppose the current one-year interest rate is 6%. One year from now, you believe the economy will start to slow and the one-year interest rate will fall to 5%. In two years, you expect the economy to be in the midst of a recession, causing the Federal Reserve to cut interest rates drastically and the one-year interest rate to fall to 2%. The one-year interest rate will then rise to 3% the following year, and continue to rise by 1% per year until it returns to 6%, where it will remain from then on.

a. If you were certain regarding these future interest rate changes, what two-year interest rate would be consistent with these expectations?

b. What current term structure of interest rates, for terms of 1 to 10 years, would be consistent with these expectations?

c. Plot the yield curve in this case. How does the one-year interest rate compare to the 10-year interest rate?

a. The one-year interest rate is 6%. If rates fall next year to 5%, then if you reinvest at this rate over two years you would earn (1.06)(1.05) = 1.113 per dollar invested. This amount corresponds to an EAR of (1.113)1/2 1 = 5.50% per year for two years. Thus, the two-year rate that is consistent with these expectations is 5.50%.

b. We can apply the same logic for future years:

c. We can plot the yield curve using the EARs in (b); note that the 10-year rate is below the 1-year rate (yield curve is inverted).

PCBR (Valuing Bonds) pages 196-202 Problems 1-4, 10, 13, 17-19, 28

8-1. A 30-year bond with a face value of $1000 has a coupon rate of 5.5%, with semiannual payments.

a. What is the coupon payment for this bond?

b. Draw the cash flows for the bond on a timeline.

a. The coupon payment is:

Coupon Rate Face Value 0.055 $1000 $27.50.Number of Coupons per Year 2

CPN = = =

b. The timeline for the cash flows for this bond is (the unit of time on this timeline is six-month periods):

2P 100/(1.055) $89.85= =

8-2. Assume that a bond will make payments every six months as shown on the following timeline (using six-month periods):

a. What is the maturity of the bond (in years)?

b. What is the coupon rate (in percent)?

c. What is the face value?

1

$27.50

0

2

$27.50

3

$27.50

60

$27.50 + $1000

Year Future Interest Rates FV from reinvesting EAR1 6% 1.0600 6.00%2 5% 1.1130 5.50%3 2% 1.1353 4.32%4 3% 1.1693 3.99%5 4% 1.2161 3.99%6 5% 1.2769 4.16%7 6% 1.3535 4.42%8 6% 1.4347 4.62%9 6% 1.5208 4.77%10 6% 1.6121 4.89%

a. The maturity is 10 years.

b. (20/1000) x 2 = 4%, so the coupon rate is 4%.

c. The face value is $1000.

8-3. The following table summarizes prices of various default-free, zero-coupon bonds (expressed as a percentage of face value):

a. Compute the yield to maturity for each bond.

b. Plot the zero-coupon yield curve (for the first five years).

c. Is the yield curve upward sloping, downward sloping, or flat?

a. Use the following equation. 1/ n

nn

FV1 YTM

P

+ =

1/1

1 11001 YTM YTM 4.70%95.51

+ = =

1/ 2

1 11001 YTM YTM 4.80%91.05

+ = =

1/ 3

3 31001 YTM YTM 5.00%86.38

+ = =

1/ 4

4 41001 YTM YTM 5.20%81.65

+ = =

1/ 5

5 51001 YTM YTM 5.50%76.51

+ = =

b. The yield curve is as shown below.

c. The yield curve is upward sloping.

4.6 4.7 4.8 4.9

5 5.1 5.2 5.3 5.4 5.5 5.6

0 2 4 6

Yiel

d to

Mat

urity

Maturity (Years)

Zero Coupon Yield Curve

8-4. Suppose the current zero-coupon yield curve for risk-free bonds is as follows:

a. What is the price per $100 face value of a two-year, zero-coupon, risk-free bond?

b. What is the price per $100 face value of a four-year, zero-coupon, risk-free bond?

c. What is the risk-free interest rate for a five-year maturity?

a. 2P 100(1.055) $89.85= =

b. 4P 100/(1.0595) $79.36= =

c. 6.05% 8-10. Suppose a seven-year, $1000 bond with an 8% coupon rate and semiannual coupons is

trading with a yield to maturity of 6.75%.

a. Is this bond currently trading at a discount, at par, or at a premium? Explain.

b. If the yield to maturity of the bond rises to 7% (APR with semiannual compounding), what price will the bond trade for?

a. Because the yield to maturity is less than the coupon rate, the bond is trading at a premium.

b. 2 14

40 40 40 1000 $1,054.60(1 .035) (1 .035) (1 .035)

++ + + =

+ + +L

NPER Rate PV PMT FV Excel Formula Given: 14 3.50% 40 1,000 Solve For PV: (1,054.60) =PV(0.035,14,40,1000)

8-13. Consider the following bonds:

a. What is the percentage change in the price of each bond if its yield to maturity falls from

6% to 5%?

b. Which of the bonds AD is most sensitive to a 1% drop in interest rates from 6% to 5% and why? Which bond is least sensitive? Provide an intuitive explanation for your answer.

a. We can compute the price of each bond at each YTM using Eq. 8.5. For example, with a 6% YTM, the price of bond A per $100 face value is

15

100P(bond A, 6% YTM) $41.73.1.06

= =

The price of bond D is

10 10

1 1 100P(bond D, 6% YTM) 8 1 $114.72..06 1.06 1.06

= + =

One can also use the Excel formula to compute the price: PV(YTM, NPER, PMT, FV).

Once we compute the price of each bond for each YTM, we can compute the % price change as

Percent change = ( ) ( )( )

Price at 5% YTM Price at 6% YTM.

Price at 6% YTM

The results are shown in the table below.

Coupon Rate Maturity Price at Price at Percentage Change(annual payments) (years) 6% YTM 5% YTM

A 0% 15 $41.73 $48.10 15.3%B 0% 10 $55.84 $61.39 9.9%C 4% 15 $80.58 $89.62 11.2%D 8% 10 $114.72 $123.17 7.4%

Bond

b. Bond A is most sensitive, because it has the longest maturity and no coupons. Bond D is the least sensitive. Intuitively, higher coupon rates and a shorter maturity typically lower a bonds interest rate sensitivity.

8-17. What is the price today of a two-year, default-free security with a face value of $1000 and an annual coupon rate of 6%? Does this bond trade at a discount, at par, or at a premium?

2 21 2

60 60 1000... $1032.091 (1 .04)(1 ) (1 ) (1 .043)NN

CPN CPN CPN FVPYTM YTM YTM

+ += + + + = + =

+ ++ + +

This bond trades at a premium. The coupon of the bond is greater than each of the zero coupon yields, so the coupon will also be greater than the yield to maturity on this bond. Therefore it trades at a premium

8-18. What is the price of a five-year, zero-coupon, default-free security with a face value of $1000?

The price of the zero-coupon bond is

5

1000 $791.03(1 ) (1 0.048)NN

FVPYTM

= = =+ +

8-19. What is the price of a three-year, default-free security with a face value of $1000 and an annual coupon rate of 4%? What is the yield to maturity for this bond?

The price of the bond is

2 2 31 2

40 40 40 1000... $986.58.1 (1 .04)(1 ) (1 ) (1 .043) (1 .045)NN

CPN CPN CPN FVPYTM YTM YTM

+ += + + + = + + =

+ ++ + + +

The yield to maturity is

2 ...1 (1 ) (1 )N

CPN CPN CPN FVPYTM YTM YTM

+= + + +

+ + +

2 3

40 40 40 1000$986.58 4.488%(1 ) (1 ) (1 )

YTMYTM YTM YTM

+= + + =

+ + +

The following table summarizes the yields to maturity on several one-year, zero-coupon securities:

a. What is the price (expressed as a percentage of the face value) of a one-year, zero-

coupon corporate bond with a AAA rating?

b. What is the credit spread on AAA-rated corporate bonds?

c. What is the credit spread on B-rated corporate bonds?

d. How does the credit spread change with the bond rating? Why?

a. The price of this bond will be

100 96.899.1 .032

P = =+

b. The credit spread on AAA-rated corporate bonds is 0.032 0.031 = 0.1%.

c. The credit spread on B-rated corporate bonds is 0.049 0.031 = 1.8%.

d. The credit spread increases as the bond rating falls, because lower rated bonds are riskier.

PCBR (Valuing Stocks) pages 239-243 Problems 1-4, 6, 8, 21 9-1. Assume Evco, Inc., has a current price of $50 and will pay a $2 dividend in one year, and its

equity cost of capital is 15%. What price must you expect it to sell for right after paying the dividend in one year in order to justify its current price?

We can use Eq. (9.1) to solve for the price of the stock in one year given the current price of $50.00, the $2 dividend, and the 15% cost of capital.

2501.1555.50

X

X

+=

=

At a current price of $50, we can expect Evco stock to sell for $55.50 immediately after the firm pays the dividend in one year.

9-2. Anle Corporation has a current price of $20, is expected to pay a dividend of $1 in one year, and its expected price right after paying that dividend is $22.

a. What is Anles expected dividend yield?

b. What is Anles expected capital gain rate?

c. What is Anles equity cost of capital?

a. Div yld = 1/20 = 5%

b. Cap gain rate = (22-20)/20 = 10%

c. Equity cost of capital = 5% + 10% = 15%

9-3. Suppose Acap Corporation will pay a dividend of $2.80 per share at the end of this year and $3 per share next year. You expect Acaps stock price to be $52 in two years. If Acaps equity cost of capital is 10%:

a. What price would you be willing to pay for a share of Acap stock today, if you planned to hold the stock for two years?

b. Suppose instead you plan to hold the stock for one year. What price would you expect to be able to sell a share of Acap stock for in one year?

c. Given your answer in part (b), what price would you be willing to pay for a share of Acap stock today, if you planned to hold the stock for one year? How does this compare to you answer in part (a)?

a. P(0) = 2.80 / 1.10 + (3.00 + 52.00) / 1.102 = $48.00

b. P(1) = (3.00 + 52.00) / 1.10 = $50.00

c. P(0) = (2.80 + 50.00) / 1.10 = $48.00

9-4. Krell Industries has a share price of $22 today. If Krell is expected to pay a dividend of $0.88 this year, and its stock price is expected to grow to $23.54 at the end of the year, what is Krells dividend yield and equity cost of capital?

Dividend Yield = 0.88 / 22.00 = 4%

Capital gain rate = (23.54 22.00) / 22.00 = 7%

Total expected return = rE = 4% + 7% = 11%

9-6. Summit Systems will pay a dividend of $1.50 this year. If you expect Summits dividend to grow by 6% per year, what is its price per share if its equity cost of capital is 11%?

P = 1.50 / (11% 6%) = $30 9-8. Kenneth Cole Productions (KCP), suspended its dividend at the start of 2009. Suppose you

do not expect KCP to resume paying dividends until 2011.You expect KCPs dividend in 2011 to be $0.40 per year (paid at the end of the year), and you expect it to grow by 5% per year thereafter. If KCPs equity cost of capital is 11%, what is the value of a share of KCP at the start of 2009?

P(2010) = Div(2011)/(r g) = 0.40/(.11 .05) = 6.67

P(2009) = 6.67/1.112 = $5.41 9-21. Consider the valuation of Kenneth Cole Productions in Example 9.7.

a. Suppose you believe KCPs initial revenue growth rate will be between 4% and 11% (with growth slowing in equal steps to 4% by year 2011). What range of share prices for KCP is consistent with these forecasts?

b. Suppose you believe KCPs EBIT margin will be between 7% and 10% of sales. What range of share prices for KCP is consistent with these forecasts (keeping KCPs initial revenue growth at 9%)?

c. Suppose you believe KCPs weighted average cost of capital is between 10% and 12%. What range of share prices for KCP is consistent with these forecasts (keeping KCPs initial revenue growth and EBIT margin at 9%)?

d. What range of share prices is consistent if you vary the estimates as in parts (a), (b), and (c) simultaneously?

a. $22.85 - $25.68

b. $19.60 - $27.50

c. $22.24 --- $28.34

d. $16.55 --- $32.64

1. The following is a list of spot rates from zero coupon bonds of various maturities (STRIP data taken from finance.yahoo.com) Maturity (years) US zero coupon spot rate (annualized rate) 1 0.0210 2 0.0249 3 0.0235 4 0.0275 a. What are the prices of US zero coupon bonds with maturities 1, 2, 3, and 4 years. Assume these bonds have a face value of $1,000.

b. Calculate the implied 1-year forward rates, ft-1,t , for t=2, 3, 4.

c. Calculate the implied 2-year forward rates f1,3 and f2,4.

d. If the expectations hypothesis of the term structure holds, what does the information in the current term structure say about the course of future 1 year interest rates? The expectations hypothesis says that forward rates are unbiased predictors of future spot rates. Based on the 1 yr forward rates, the expectations hypothesis says that future rates will rise slightly in one year, then fall slightly after two years and then rise sharply in year 3. e. Suppose the US government decides to issue a 4 year coupon bond today with face value $1,000 and an annual coupon rate of 3%. Using the term structure of interest rates, give the no arbitrage market price of the bond.

2. The following table gives the yields on four zero coupon Treasury bonds: Maturity Date Years from now Yield Jan 1997 0.5 5.41 July 1997 1.0 5.77 Jan 1998 1.5 6.07 July 1998 2.0 6.20 Note that the yields are annual yields, but they assume semi annual compounding. a. What is the price of a two year, 6.0 percent coupon Treasury bond, paying semi-annual coupons? [Hint: Think term structure!] b. Define the yield to maturity of a bond. Is the yield to maturity of the bond in part a higher of lower than the coupon rate? Explain. c. Suppose that in 1994 you invest $10,000 in an asset and that in 1995, one year later, you invest another $5,000. At the present time, two years after the initial investment, the asset is worth $18,144. What has been your rate of return on the investment? d. What is the July 1997 to July 1998 forward interest rate?