# Zero Coupon Bonds: Value of Option to Default on the Bonds

2. Assume your firm has zero coupon bonds maturing in 10 years with a face value of $183.75 million.

a) If the assets are worth $150 million, what is the value of the stocks and the bonds?

b) If the assets are worth $400 million, what is the value of the stocks and the bonds?

c) If the assets are worth $400 million, the standard deviation is 0.60 and the risk-free rate is the T-bond rate, what is the value of the option to default on the bonds? What is the value of the stockholders' call option? Use Black and Scholes to get the answer to part c.

Note: For parts a and b, assume the options would be exercised immediately. Be sure to label your answer.

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#### Solution Preview

See the attached file.

2. Assume your firm has zero coupon bonds maturing in 10 years with a face value of $183.75 million.

a) If the assets are worth $150 million, what is the value of the stocks and the bonds?

Equity is a call option. If this option is exercised today, shareholders will receive their share only when after satisfying the financial claims of the debt holders.

The payoff to equity investors on liquidation are

= V - D if V > D

= 0 if V <D

where,

V = Value of the firm

D = Face Value of the outstanding debt ...

#### Solution Summary

Shows how to calculate the value of the option to default on the bonds, value of the stockholders' call option using Black and Scholes.

20 multiple choice questions about bonds, stocks and options

Question 1

Moerdyk Corporation's bonds have a 10-year maturity, a 6.25% semiannual coupon, and a par value of $1,000. The going interest rate (rd) is 4.75%, based on semiannual compounding. What is the bond's price?

$1063.09

$1090.35

$1118.31

$1146.27

$1174.93

Question 2

Call options on XYZ Corporation's common stock trade in the market. Which of the following statements is most correct, holding other things constant?

The price of these call options is likely to rise if XYZ's stock price rises.

The higher the strike price on XYZ's options, the higher the option's price will be.

Assuming the same strike price, an XYZ call option that expires in one month will sell at a higher price than one that expires in three months.

If XYZ's stock price stabilizes (becomes less volatile), then the price of its options will increase.

If XYZ pays a dividend, then its option holders will not receive a cash payment, but the strike price of the option will be reduced by the amount of the dividend.

Question 3

Assume that investors have recently become more risk averse, so the market risk premium has increased. Also, assume that the risk-free rate and expected inflation have not changed. Which of the following is most likely to occur?

The required rate of return for an average stock (beta = 1) will increase by an amount equal to the increase in the market risk premium.

The required rate of return will decline for stocks whose betas are less than 1.0.

The required rate of return on the market, rM, will not change as a result of these changes.

The required rate of return for each individual stock in the market will increase by an amount equal to the increase in the market risk premium.

The required rate of return on a riskless bond will decline.

Question 4

Which of the following statements is CORRECT?

If the maturity risk premium (MRP) is greater than zero, then the yield curve must have an upward slope.

If inflation is expected to increase in the future, and if the maturity risk premium (MRP) is greater than zero, then the yield curve will have an upward slope.

Because long-term bonds are riskier than short-term bonds, yields on long-term Treasury bonds will always be higher than yields on short-term T-bonds.

If the maturity risk premium (MRP) equals zero, the yield curve must be flat.

The yield curve can never be downward sloping.

Question 5

Companies can issue different classes of common stock. Which of the following statements concerning stock classes is CORRECT?

All common stocks fall into one of three classes: A, B, and C.

All common stocks, regardless of class, must have the same voting rights.

All firms have several classes of common stock.

All common stock, regardless of class, must pay the same dividend.

Some class or classes of common stock may be entitled to more votes per share than other classes.

Question 6

Rick Kish has a $100,000 stock portfolio. $32,000 is invested in a stock with a beta of 0.75 and the remainder is invested in a stock with a beta of 1.38. These are the only two investments in his portfolio. What is his portfolio's beta?

1.18

1.24

1.30

1.36

1.43

Question 7

Stock X has a beta of 0.6, while Stock Y has a beta of 1.4. Which of the following statements is CORRECT?

A portfolio consisting of $50,000 invested in Stock X and $50,000 invested in Stock Y will have a required return that exceeds that of the overall market.

Stock Y must have a higher expected return and a higher standard deviation than Stock X.

If expected inflation increases (but the market risk premium is unchanged), the required return on both stocks will decrease by the same amount.

If the market risk premium decreases but expected inflation is unchanged, the required return on both stocks will decrease, but the decrease will be greater for Stock Y.

If expected inflation decreases (but the market risk premium is unchanged), the required return on both stocks will decrease but the decrease will be greater for Stock Y.

Question 8

Which of the following statements is CORRECT?

A zero coupon bond of any maturity will have more interest rate price risk than any coupon bond, even a perpetuity.

a 10-year coupon bond would have more interest rate price risk than a 5-year bond, assuming the same coupon rate for both.

A 10-year coupon bond would have more reinvestment rate risk than a 5-year coupon bond (assuming the same coupon rate for both), but all 10-year coupon bonds have the same amount of reinvestment rate risk.

A 5-year coupon bond would have more interest rate price risk than a 10-year coupon bond, assuming the same coupon rate for both.

If their maturities and other characteristics were the same, a 5% coupon bond would have less interest rate price risk than a 10% coupon bond.

Question 9

For a stock to be in equilibrium, that is, for there to be no consistent pressure for its price to depart from its current level, then

The expected future return must be less than the most recent past realized return.

The past realized return must be equal to the expected return during the same period.

The required return must equal the realized return in all periods.

The expected return must be equal to both the required future return and the past realized return.

The expected future returns must be equal to the required return.

Question 10

Keys Corporation's 5-year bonds yield 7.00%, and 5-year T-bonds yield 5.15%. The real risk-free rate is r* = 3.0%, the inflation premium for 5-year bonds is IP = 1.75%, the liquidity premium for Keys' bonds is LP = 0.75% versus zero for T-bonds, and the maturity risk premium for all bonds is found with the formula MRP = (t ï? 1) ï?´ 0.1%, where t = number of years to maturity. What is the default risk premium (DRP) on Keys' bonds?

0.99%

1.10%

1.21%

1.33%

1.46%

Question 11

Which of the following statements is CORRECT?

The constant growth model takes into consideration the capital gains investors expect to earn on a stock.

Two firms with the same expected dividend and growth rate must also have the same stock price.

It is appropriate to use the constant growth model to estimate stock value even if the growth rate is never expected to become constant.

If a stock has a required rate of return rs = 12%, and if its dividend is expected to grow at a constant rate of 5%, this implies that the stock's dividend yield is also 5%.

The price of a stock is the present value of all expected future dividends, discounted at the dividend growth rate.

Question 12

E. M. Roussakis Inc.'s stock currently sells for $45 per share. The stock's dividend is projected to increase at a constant rate of 3.75% per year. The required rate of return on the stock, rs, is 15.50%. What is Roussakis' expected price 5 years from now?

$45.00

$46.69

$51.42

$54.09

$56.11

Question 13

The current price of a stock is $50, the annual risk-free rate is 6%, and a 1-year call option with a strike price of $55 sells for $7.20. What is the value of a put option, assuming the same strike price and expiration date as for the call option?

$7.33

$7.71

$8.12

$8.55

$9.00

Question 14

If D1 = $1.25, g (which is constant) = 5.5%, and P0 = $44, what is the stock's expected total return for the coming year?

7.54%

7.73%

7.93%

8.34%

8.53%

Question 15

You observe the following information regarding Companies X and Y:

Company X has a higher expected return than Company Y.

Company X has a lower standard deviation of returns than Company Y.

Company X has a higher beta than Company Y.

Given this information, which of the following statements is CORRECT?

Company X has more company-specific risk than Company Y.

Company X has a lower coefficient of variation than Company Y.

Company X has less market risk than Company Y.

Company X's returns will be negative when Y's returns are positive.

Company X's stock is a better buy than Company Y's stock.

Question 16

Stock A has a beta of 0.8, Stock B has a beta of 1.0, and Stock C has a beta of 1.2. Portfolio P has equal amounts invested in each of the three stocks. Each of the stocks has a standard deviation of 25%. The returns on the three stocks are independent of one another (i.e., the correlation coefficients all equal zero). Assume that there is an increase in the market risk premium, but the risk-free rate remains unchanged. Which of the following statements is CORRECT?

The required return of all stocks will remain unchanged since there was no change in their betas.

The required return on Stock A will increase by less than the increase in the market risk premium, while the required return on Stock C will increase by more than the increase in the market risk premium.

The required return on the average stock will remain unchanged, but the returns of riskier stocks (such as Stock C) will increase while the returns of safer stocks (such as Stock A) will decrease.

The required returns on all three stocks will increase by the amount of the increase in the market risk premium.

The required return on the average stock will remain unchanged, but the returns of riskier stocks (such as Stock C) will decrease while the returns on safer stocks (such as Stock A) will increase.

Question 17

Which of the following statements is CORRECT?

If two bonds have the same maturity, the same yield to maturity, and the same level of risk, the bonds should sell for the same price regardless of the bond's coupon rates.

All else equal, an increase in interest rates will have a greater effect on the prices of short-term than long-term bonds.

All else equal, an increase in interest rates will have a greater effect on higher-coupon bonds than it will have on lower-coupon bonds.

If a bond's yield to maturity exceeds its coupon rate, the bond's price must be less than its maturity value.

If a bond's yield to maturity exceeds its coupon rate, the bond's current yield must be less than its coupon rate.

Question 18

Which of the following statements is CORRECT?

Assume that two bonds have equal maturities and are of equal risk, but one bond sells at par while the other sells at a premium above par. The premium bond must have a lower current yield and a higher capital gains yield than the par bond.

A bond's current yield must always be either equal to its yield to maturity or between its yield to maturity and its coupon rate.

If a bond sells at par, then its current yield will be less than its yield to maturity.

If a bond sells for less than par, then its yield to maturity is less than its coupon rate.

A discount bond's price declines each year until it matures, when its value equals its par value.

Question 19

A stock just paid a dividend of D0 = $1.75. The required rate of return is rs = 12.0%, and the constant growth rate is g = 4.0%. What is the current stock price?

$20.56

$21.09

$21.63

$22.18

$22.75

Question 20

Suppose you believe that Delva Corporation's stock price is going to decline from its current level of $82.50 sometime during the next 5 months. For $510.25 you could buy a 5-month put option giving you the right to sell 100 shares at a price of $85 per share. If you bought this option for $510.25 and Delva's stock price actually dropped to $60, what would your pre-tax net profit be?

-$510.25 (you had a loss)

$1989.75

$2089.24

$2193.70

$2303.38

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