15 Solution 1.4: The dividend growth model says that! DIV1 = \$6.00! k = 12.0%! g = 4.0% The expected stock price = P0 = \$6 / (12% 4%) = \$75.

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1 1 The present value of the exercise price does not change in one hour if the risk-free rate does not change, so the change in call put is the change in the stock price. The change in call put is \$4, so the new stock price is \$68 \$4 = \$64. 2 Solution 1.2: We use the put call parity relation: c + PV(X) = p + S: put + \$80 = \$12 + \$ ½ or put = \$ ½ \$68 = \$15.93 Jacob: Can you explain the put call parity relation intuitively? Rachel: Suppose an investor buys and a put option with a strike price of \$90, and sells a call option with a strike price of \$90. At the expiration date, the stock price is either above or below \$90. ~ If the stock price is above \$90, the person who bought the call option exercises it, and the investor gets \$90 but must give up the stock. ~ If the stock price is below \$90, the investor exercises the put option and gives up the stock for \$90. Either way, the investor has \$90 and no stock. This means that the investor s portfolio is now worth the present value of \$90, or stock + put call = present value (\$90). The \$90 is the strike price. Substituting X for the \$90 gives the put call parity relation. 3 Let the stock price be S. By the put call parity relation: call + present value of exercise price = put + stock price, so 1 put minus 1 call = the present value of the exercise price the stock price. For an exercise price (strike price) of \$110, put call = \$0 = \$110 / 1.11 S A S = \$110 / 1.11 = \$ For an exercise price (strike price) of \$115, put call = put \$7.50 = \$115 / 1.11 \$99.10 A put = \$115 / 1.11 \$ \$7.50 = \$ We can simplify the solution as follows:! The change in the exercise price is +\$5.00.! The change in the present value of the exercise price is \$5.00 / 1.11 = \$4.50.! The change in the value of 1 put minus 1 call is \$4.50.! The call value decreases by \$2.50, so the put value increases by \$2 to \$ The following explanation is another perspective: The difference between the call and put prices at the two exercise prices is the difference in the present value of the exercise prices. This is the present value of (\$115 \$110) = \$5, for one year at an 11% discount rate: \$5 / 1.11 = \$4.50. The change in(put call) between the two exercise prices is \$4.50 and the change in the call option price is \$2.50, so the change in the put option price is \$4.50 \$2.50 = \$2. The put option price with a \$115 exercise price is \$ \$2 = \$ By the put call parity relation, call put = stock price present value of exercise price. The stock price has not changed, so the only effect on the option portfolio is the change in the present value of the exercise price. \$65 10,000 (1 / /12 1 / /12) = \$5, \$6,000. This is an increase in the present value of the exercise price, so it is a decrease in the value of the option portfolio: \$60,000 \$6,000 = \$54, Higher volatility of the common stock price raises the value of both call and put options. We show this by intuition and by the Black-Scholes formula. Intuition: The option value rises if the stock price moves favorably: up for a call option and down for a put option. The decrease in value is bounded at zero if the stock price moves adversely. Suppose the stock price is \$100, the strike price (exercise price) is \$100, the risk-free interest rate is zero, and the option has one year to maturity. The stock price moves up by Z% or down by Z% over the year with equal probability.

2 The interest rate and probability are chosen for convenience. To be exact, we should speak of the risk-neutral probability of moving up. If the risk-free rate is not zero, the riskneutral probability of moving up is not 50% and the arithmetic differs slightly, but the intuition remains the same. If Z% = 1%, the value of the call option at expiration is \$1 if the stock price moves up and \$0 if the stock price moves down. The expected value of the call option at expiration is \$0.50. The risk-free interest rate is zero, so this is also the present value of the call option. If Z% = 10%, the value of the call option at expiration is \$10 if the stock price moves up and \$0 if the stock price moves down. The expected value of the call option at expiration is \$5.00. The risk-free interest rate is zero, so this is also the present value of the call option. For any Z, the value of the call option at expiration is \$Z if the stock price moves up and \$0 if the stock price moves down. The expected value of the call option at expiration is ½ \$Z. The risk-free interest rate is zero, so this is also the present value of the call option. For the put option, we inter-chance up and down, with the same result. The result is true for any stock price and exercise price. In practice, we use a lognormal distribution of stock price movements, or a normal distribution of rates of return. The Black-Scholes formula has four elements: S, PV(X), d1, and d2. As the volatility F increases, d1 increases and d2 decreases. For a call option, N(d1) increases and N(d2) decreases. S N(d1) increases and PV(X) N(d2) decreases, so the value of the call option increases. For a put option, N( d1) decreases and N( d2) increases. S N( d1) decreases and PV(X) N( d2) increases, so the value of the put option increases. 6 By the put call parity relation: call + present value of exercise price = put + stock price. The portfolio of 1 call minus 1 put = stock price the present value of the exercise price. The stock price, the exercise price, and the risk-free rate have not changed, so the value of the portfolio does not change. 7 Compare two American call or put options, Y and Z, with the same attributes except that option Y has 3 months to maturity and option Z has 1 month to maturity.! The investor can exercise option Z any day in the next month.! The investor can exercise option Y any day in the next three months. Option Y has all the possible exercise dates as option Z plus two more months of possible exercise dates. Option Y is worth more than option Z, whether it is a call or put option. This comparison is valid only for American options, not for European options, which can be exercised only on the expiration date. 8 The Modigliani and Miller arbitrage argument shows that the value of the firm in perfect capital markets does not depend on the capital structure.! The dollar return on assets is the return before payments to capital providers and does not depend on the capital structure.! The percentage return is the dollar return divided by assets, so its also doesn t change. The risk of the equity depends on the systematic risk of the firm. Debt payments are fixed; they do not depend on the income of the firm. We assume the firm has! high income when markets do well in prosperous years and! low income when markets do poorly in recession years. Suppose the firm has assets of \$20 million and earns \$1 million in bad years and \$3 million in good years. If bad and good years are equally probable, the firm makes an average 10% return on assets. The return is 5% in bad years and 15% in good years.! If the firm is all equity financed, the return on equity capital is 10% on average: 5% in bad years and 15% in good years.! If the firm is 50% financed with 8% debt, it pays \$800,000 each year to creditors on \$10 million of debt. The shareholders receive the remaining income: \$200,000, or 2%, in bad years.

3 \$2,200,000, or 22%, in good years. The variability in shareholder returns with a 50%-50% capital structure is! 3 to 1 in the all equity financed firm and! 11 to 1 in the 50% debt financed firm. The required return to creditors also increases as the percentage of debt increases. Suppose the firm is a high-tech start-up or a pharmaceutical firm, whose assets have no value in bankruptcy. The firm has a high cost of bankruptcy, and its cost of debt capital increases as the percentage of debt increases. Suppose the firm has \$20 million of assets, which are used to fund research that pays for new products. The assets are funded by a combination of equity and debt. The risk-free interest rate is 10% per annum. The firm s income is uniformly distributed over (\$0.1 million, \$5.1 million). The firm s expected income is \$2.6 million, and its return on assets is 13%.! If the firm borrows \$1 million, it can surely repay the interest at year end. Creditors require 10% on their capital.! If the firm borrows \$5 million at a 10% coupon rate, it must earn at least \$0.5 million for the interest payment. The probability that it earns less than \$0.5 million is (\$0.5 million \$0.1 million) / (\$5.1 million \$0.1 million) = 8.00%. Creditors demand a return greater than 10% to offset the chance of not receiving their money. 9 With the old capital structure, the return on assets is 20% 7% + 80% 15% = 13.40%. With the new capital structure, we have 13.4% = 40% 7% + 60% R A R = (13.4% 40% 7%) / 60% = 17.67%. 10 The abnormal return is the actual return minus the expected return. The expected return is " + \$ the market return: % = 6.60%. Since the stock actually rose by 6.0%, the abnormal rate of return is 6.0% 6.6% = 0.60%. Jacob: How are the " and \$ in this problem related to the CAPM \$? Rachel: The abnormal return section of the textbook is similar to the CAPM.! The abnormal return equation says a stock s expected return is " + \$N rm.! The CAPM says that a stock s expected return is rf + \$ (E[rm] rf).! Equating the two formulas gives \$N = \$ and " = rf (1 \$). 11 The abnormal return is the actual return on the stock minus the expected return. The expected return is " + \$ market return = % = 7.75%. The stock rises by 7.0%, so the abnormal rate of return is 7.0% 7.75% = 0.75%. 12 The dividend growth rate is the return on book value times the plow-back ratio, which is one minus the payout ratio. The annual earnings are the return on book value times book value per share, which is \$100. The dividend is the annual earnings times the payout ratio.! Investor A: dividend growth rate = 60% 20% = 12%; stock price = (20% \$100 40%) / (20% 12%) = \$100.! Investor B: dividend growth rate = 40% 20% = 8%; stock price = (20% \$100 60%) / (20% 8%) = \$100. We could answer this exercise intuitively. The return on book equity equals the assumed market capitalization rate, so the dividend yield has no effect on the firm s stock price. Jacob: Isn t it always true that the dividend yield does not affect the stock price? Isn t this one of the Miller and Modigliani propositions? Rachel: This should be true in perfect capital markets, where the tax rate is zero and the dividend yield does not convey information to investors. Jacob: Why do you note that the return on book equity equals the capitalization rate? Rachel: If the return on book equity is high, the firm is profitable. If markets are competitive, other firms enter the market and compete away the excess profits. In the long-run, each firm s profits exactly match its risk.

4 We compare two firms with a book value per share of \$100.! Firm Y pays 20% of earnings as dividends.! Firm Z pays 80% of earnings as dividends. Each firm earns a 50% return on book equity and the market capitalization rate is 41%. The dividend growth rates are! Firm Y: 80% plow-back ratio 50% ROE = 40%! Firm Z: 20% plow-back ratio 50% ROE = 10% Next year s dividend per share is! Firm Y: 20% plow-back ratio \$50 = \$10! Firm Z: 80% plow-back ratio \$50 = 40% The expected stock prices are! Firm Y: \$10 / (41% 40%) = \$1,000! Firm Z: \$40 / (41% 10%) = \$129 It seems that Firm Y has the higher stock price. But competitive markets don t allow this. It must be that the return on book equity reflects the systematic risk of the firm. Other firms do not enter the market because the 50% return on book equity is highly volatile and correlated with overall market returns. By paying its earnings as dividends, Firm Z reduces its risk, and its market capitalization rate decreases. The expected stock price is given by the dividend growth model only if the firm operates optimally. Suppose a firm has a monopoly with a high demand for its products. If the firm earns a return on book equity greater than its market capitalization rate, it should pay low dividends (or no dividends). If the firm pays high dividends and does not allow its owners to contribute more capital, it is not giving its owners the profits they could get from growth of the firm. In a perfect capital market, the shareholders would replace the present management with other managers. If they can not do this, the stock price declines. 13 The dividend growth rate is the return on book equity (ROE) times the plow back ratio: 80% 15% = 12.00% Following the figures helps you understand this. The book value per share is \$7.50 / 15% = \$50. The dividend is \$7.50 (1 80%) = \$1.50. The new book value per share is the old book value the retained earnings, of \$50 + (\$7.50 \$1.50) = \$56. A year later, the earnings are 15% \$56 = \$8.40 and the dividend is \$8.40 (1 80%) = \$ The dividend growth rate is \$1.680 / \$ = 12.00%. 14 The dividend growth model says that A! The dividend yield = \$4.20 / \$74 = 5.68%.! The dividend growth rate is 17% (1 45%) = 9.35%.! The market capitalization rate is 5.68% % = 15.03%. 15%. We verify with the dividend growth model: \$4.20 / (15% 17% 55%) = \$ Solution 1.4: The dividend growth model says that! DIV1 = \$6.00! k = 12.0%! g = 4.0% The expected stock price = P0 = \$6 / (12% 4%) = \$ We determine the following values: earnings next year, dividend next year, plow-back ratio, dividend growth rate, and stock price. The dividend growth model says that P0 = DIV1 / (k g), where P0 is the current stock price, DIV1 is next year s dividend, k is the capitalization rate, and g is the dividend growth rate. The dividend growth rate is the return on book equity times the plow-back ratio; the plow-back ratio is the complement of the payout ratio. The dividend growth model is! The earnings next year are \$60 14% = \$8.40.! The dividend is \$ % = \$4.20.! The plow-back ratio is 1 50% = 50%.

5 ! The dividend growth rate is 14% 50% = 7%.! The stock price is \$4.20 / (19% 7%) = \$ dividend yield = 3/60 = 5%; dividend growth rate = 15% (1 40%) = 9.00% 18! The dividend is \$6.! The dividend growth rate is 15% 60% = 9%.! The old stock price is \$6 / (20% 9%) = \$54.55! The new stock price is \$6 / (26% 9%) = \$35.29.! The change in the stock price is \$54.55 \$35.29 = \$19.26.! The percentages change is \$19.26 / \$54.55 = 35.31% Statement A is true for only the strong form of the efficient market hypothesis. The other four statements are false for all forms of the efficient market hypothesis. Statement A: The strong form of the efficient market hypothesis says that all information is immediately incorporated into the stock price. The semi-strong form says only public information is immediately incorporated into the stock price. Statement B: All three forms of the efficient market hypothesis say that prices adjust quickly, not slowly, to incorporate past stock price data. Statements C and D: Actual returns are affected by numerous stochastic factors. Statement E: Only the standard deviation of systematic return affects the expected return. The standard deviation stemming from unique risk is diversified by investors and does not affect the expected return. 21 Insider information is not public. David and Jonathan argue whether non-public information is immediately incorporated into the stock price. The strong form of the efficient market hypothesis says that all information is immediately incorporated into the stock price. The semi-strong form says only public information is immediately incorporated into the stock price. 22 Statement A: The return on the portfolio is value weighted average of the returns on the stocks: ½ (12% + 15%) = 13.5%. Statements C and E: The variance of the portfolio depends on the correlation, which may range from 1 to +1. The minimum variance occurs when the correlation is 1: ½ ½ 60%2 + ½ ½ 50%2 ½ ½ 2 60% 50% 1 = 0.250% The maximum variance occurs when the correlation is +1: ½ ½ 60%2 + ½ ½ 50%2 + ½ ½ 2 60% 50% 1 = % Statements B and D: The standard deviation is the square root of the variance. It ranges from %½ = 5.00% to %½ = 55.00%. 23 The expected return of the portfolio is the weighted average of the expected returns on the two stocks. We work out the weights from the expected return of the portfolio: We derive the weights from the expected return of the portfolio: 12% + (1 ) 15% = 14% A " (12% 15%) = 14% 15% A " = a We work out the variance of the portfolio from the standard deviations and correlation. The variance of each stock is the square of its standard deviation and the covariance of the two stocks is the correlation times their standard deviations. variance = a b a b = The standard deviation of the portfolio is the square root of the variance: 0.204½ = %. 24 ( % %2 + 2 D % 50%)½ = 46.1% A D = [ ( % %2 ) ] / ( % 50%) =

6 40.01% We verify as ( % % % % 50%)½ = 46.1% 25 The price elasticity of demand is the percentage change in the quantity demanded divided by the percentage change in the price. As the price increases, the quantity demanded decreases, so the price elasticity of demand is negative. Capital markets are competitive. In a perfectly competitive market, as the price increases a small amount, the quantity demanded decreases a large amount. At the limit, the price elasticity of demand is 100% / +, ÿ For a price weighted average, the weights are the stock prices, not the market value of the firms. For a value weighted average, the weights are the market values of the firms.! For the price weighted average, the weights are \$80 vs \$40, or b vs a. The weighted return is b +10% + a 10% = 3.33%.! For the value weighted average, the weights are \$160 million vs \$230 million, or a vs b. The weighted return is a +10% + b 10% = 3.33%. 27 The standard deviation of the return on the portfolio is the square root of the variance of the return on the portfolio. The variance of an equally weighted portfolio of two stocks is ½ ½ F + 2 ½ ½ D 2 F F + ½ ½ F2 = ½ F2 + ½ D F2 The variance of an equally weighted portfolio of N stocks is N (1/N 1/N F 2) + 2 N(N-1)/2 1/N 1/N D F F = 1/N F2 + (N-1)/N D F2 = [Variance + (N-1) Covariance] / N = Variance (1 + (N-1) D) / N Each of the N stocks has a weight of 1/N and a variance of F2 / N2. ~ If the N stocks are independent, D = 0 and the covariance is zero. We add the variances of the stocks to get N F2 / N2 = F2 / N. ~ If the N stocks are perfectly correlated, D = 1 and the covariance equals the variance. The N stocks are like N shares of the same stock, so the variance of the portfolio is F2. ~ If the N stocks are partly correlated, we use the full formula. The variance of the portfolio is [70% (1 + 2,999 50%) ] / 3,000 = The standard deviation is 0.350½ = The variance of an equally weighted portfolio of N stocks is N (1/N 1/N F2) + 2 N(N-1)/2 1/N 1/N D F F = 1/N F2 + (N-1)/N D F2 = [Variance + (N-1) Covariance] / N = Variance (1 + (N-1) D) / N The variance of the portfolio before the change is [60% (1 + 2,999 45%) ] / 3,000 = The standard deviation is 0.27½ = = 52%. The variance of the portfolio after the change is [68% (1 + 2,999 40%) ] / 3,000 = The standard deviation is 0.272½ = %. 29 The value of a share of stock is the value of the firm divided by the number of shares. A stock dividend increases the number of shares but does not change the value of the firm. A cash dividend does not change the number of shares, but the value of the firm (and the value of each share) declines by the cash distributed. A stock split does not change the aggregate value of the shares. With a 2 for 1 stock split, the number of shares doubles from 100,000 to 200,000 and the value per share is divided in half, from \$50 to \$25. After a cash dividend, the share price declines by the amount of the dividend: \$25 \$3 = \$22. ~ January 16: 100,000 shares \$50 a share = \$5 million. ~ January 17: 200,000 shares \$25 a share = \$5 million. ~ January 18: 200,000 shares \$22 a share = \$4.4 million. The ending stock value + the cash received = the beginning stock value:! The cash received = the cash dividend the number of shares: \$3 200,000 =

7 \$600,000.! \$600,000 + \$4.4 million = \$5.0 million. 30 The value of a share of stock is the value of the firm divided by the number of shares. A stock dividend increases the number of shares but does not change the value of the firm. A cash dividend does not change the number of shares, but the value of the firm (and the value of each share) declines by the cash distributed.! January 15: 1 share \$80 a share = \$80! January 16: 1 share dividend of \$2 per share = \$2; share price = \$80 \$2 = \$78! January 17: \$78 / 3 = \$26; 1 share 3 = 3 shares The ending stock value + the cash received = the beginning stock value:! The cash received = the cash dividend the number of shares: \$2 1 = \$2.! \$2 + 3 \$26 = \$ The investor has not removed any of his or her equity from the firm. The beginning value is \$240,000, so the ending value is \$240, With perpetual debt that does not vary, the present value of the tax shields is the corporate tax rate times the market value of the debt. 34 The tax shield of perpetual debt is the market value of the debt times the tax rate, so the market value is the tax shield divided by the tax rate: \$77,000 / 35% = \$220, ! The old present value of the tax shields is 35% \$80,000 = \$28,000.! The new present value of the tax shields is 40% \$80,000 = \$32,000. The increase is \$32,000 \$28,000 = \$4, The weighted average cost of capital is the after-tax return the firm must achieve to pay its creditors and investors for the use of their capital. The WACC is after the firm s corporate income taxes and before the personal income taxes of the equity providers. The WACC says: What must be the firm s after-tax return to provide the returns expected by bondholders and shareholders? We refer to this as an after-tax cost of capital.! If the investors demand a 15% return on equity capital, the firm must earn 15% aftertax, or 15% / (1 35%) = 23.08% pre-tax.! If the creditors demand an 8% return on debt capital, the first earn 8% pre-tax, since debt payments are a tax deduction. Its after-tax return must be 8% (1 35%) = 5.20%. The weighted average cost of capital is Ra = " Rd (1 J) + (1 ") Re, where " is the percentage of debt, J is the corporate tax rate, Ra is the return on debt, Rd is the return on debt, and Re is the return on equity. We adjust the pre-tax return on debt by the complement of the tax rate, since the debt payments are tax deductible Before the change in the risk-free rate, the present value of the exercise price is \$80 / 1.02 = \$ The put call parity relation is \$10 + \$78.43 = stock price + \$4 A stock price = \$ After the change in the risk-free rate, the present value of the exercise price is \$80 / 1.03 = \$ The put call parity relation is \$ \$77.67 = \$ put A put = \$ \$77.67 \$84.43 = \$3.49. \$3.50. We can solve this more simply as call put = stock price present value of exercise price A the change in (call put) = the change in (stock price present value of exercise price) The stock price remains the same, and the change in the present value of the exercise price is (\$80 / 1.03 \$80 / 1.02) = \$0.76. The present value of the exercise price

8 decreases, so the value of call put increases by \$0.76. The value of the call option increases by \$0.25, so the value of the put option decreases by \$0.76 \$0.25 = \$0.51. The value of the put option is \$4.00 before the change in the risk-free rate, so it is \$4.00 \$0.51 = \$3.49 after the change. 39 Let the stock price be S. By the put call parity relation: call + present value of exercise price = put + stock price, so a portfolio of 1 call minus 1 put = stock price the present value of the exercise price.! At 10:00 am, 1 call 1 put = \$10 \$2 = \$8! At 11:00 am, 1 call 1 put = \$8 \$4 = \$4 Between 10:00 am and 11:00 am, the value of 1 call 1 put declines by \$4, so the stock price must have declined by \$4. The new stock price is \$68 \$4 = \$ Let the stock price be S. By the put call parity relation: call + present value of exercise price = put + stock price. The interest rate for three months is 3% at 10:00 am and 4% at 11:00 am.! At 10:00 am, \$10 (call) + \$65 / 1.03 = \$4 (put) + S A S = \$6 + \$65/1.03 = \$69.11! At 11:00 am, \$11.60 (call) + \$65 / 1.04 = put + \$69.11 A put = \$65/ \$11.60 \$69.11 = \$4.99. \$5. 41 A higher risk-free rate increases the value of the call option and decreases the value of the put option. Intuition: Suppose the stock price is \$80, the exercise price of a call option is \$80, and the option has three months to maturity. The option has two benefits for the investor: ~ Volatility value: An investor who buys the stock in the free market pays \$80. With the option, the investor can wait three months: if the stock price increases, he buys the stock for \$80, and if the stock price decreases, he can buy the stock in the free market for less. ~ Interest value: The investor gains from the option even if the stock price remains \$80. The investor can invest the cash at the risk-free rate for three months before paying for the stock. An increase in the risk-free rate that is not accompanied by an increase in the stock s return or its volatility does not affect the volatility value of the option. But it increases the interest value by the additional interest income over the term of the call option. A put option has the same volatility value as the call option. But the interest value for a put option is negative. If the stock price remains \$80 over the term of the put option, an investor who sells the stock immediately in the free market can invest the exercise price for the term of the option. An investor who sells the stock by exercising the put option doesn t receive the cash until the expiration date. Binomial lattice pricing model: The risk-neutral probability of an upward movement in the stock price is p = (rf d) / (u d). Suppose the investor exercises the call option only if the stock price moves up. The expected payoff at the expiration date is p (S u X), and the present value of the call option is p (S u X) / (1 + rf). We examine the partial derivative of the call option value with respect to the risk-free rate. (S u X) does not depend on the risk-free rate, nor does the denominator of the riskneutral probability, (u d). We examine the partial derivative of (rf d) / (1 + rf). 42 We can view this scenario as a change in the exercise price or a change in the stock price. The following portfolios are equivalent: ~ Four options with an exercise price of \$50 and a stock price of \$45. ~ Five options with an exercise price of \$40 and a stock price of \$36. The 25% increase in the exercise price, from \$40 to \$50, is like a 20% decrease in the stock price, from \$45 to \$36, and a 25% increase in the number of options. 43 Statement A: The stock s capitalization rate is determined from the risk-free rate and the

9 stock s systematic risk, not its total volatility. Options pricing uses total volatility, not only the systematic risk of the stock. Statement B: Many types of investors buy options. Some institutional investors buy options to hedge risks. Some private investors are attracted to options as a form of gambling, but not all private investors (or even most investors) in options are risk-loving. Financial economists assume investors are risk averse, whether they trade stocks, bonds, or derivatives. Statement C: The difference between the average market capitalization rate and the riskfree rate is the market risk premium, which is about 7% to 8%, or about 2% for a three month term. The market value of the underlying securities on which options are traded is many trillions of dollars, so the difference in the interest rates is tens of billions of dollars. Statement D: If the option delta is ), the option price changes by \$) when the stock price changes by \$1. The combination ) stock price option price does not change when the stock price changes, so it is risk-free. The Black-Scholes formula prices the combination ) stock price option price, so it uses the risk-free rate. Statement E: Options are very risk; the return can be very high or very low. Banks or other institutions which sell options to private investors often hedge their risks, so they have relatively risk-free option portfolios on which they collect sales margins and brokerage fees. This has nothing to do with the use of the risk-free rate to price the option. 44 When a cash dividend is paid to shareholders, the stock price declines by the amount of the dividend, adjusted for tax effects (if any).! When the stock price declines, the value of the call option declines. It may be worthwhile to exercise the call option right before the dividend is paid.! When the stock price declines, the value of the put option increases. The investor would always wait until after the dividend is paid to exercise an American put option. Illustration: Suppose the call option expires on September 30 and the exercise price is \$50. On September 15, the stock pays a dividend of \$3 per share, and the stock price is \$100 before the dividend and \$97 after the dividend. It is unlikely that the share price will decline below the exercise price, so the investor will almost surely exercise the option. Exercising before September 15 gives the investor the \$3 dividend. Illustration: Suppose the call option expires on September 30, the exercise price is \$100, and the risk-free interest rate is 12% per annum. On July 15 (not September 15), the stock pays a dividend of \$1 per share, and the stock price is \$ before the dividend and \$99.50 after the dividend. Exercising the call option before the dividend is paid gives \$0.50. Even if the stock price increases above \$100 by September 30 and the investor exercises the call option then, the value of waiting 2½ months is the investment income on the \$100 exercise price: \$100 ( /12 1) = \$2.39. The present value of this investment income at the dividend date is \$2.39 / /12 = \$2.33. The extra investment income alone makes it worthwhile to wait until September 30 to exercise the call option. In addition, if the stock price declines between July 15 and September 30, the investors avoids the loss by waiting and not exercising the call option. It is unlikely that the share price will decline below the exercise price, so the investor will almost surely exercise the option. Exercising before September 15 gives the investor the \$3 dividend. 45 We use the risk-free rate in the Black-Scholes formula and the binomial tree pricing method, not the actual return on the underlying security. The rational is as follows:! The options pricing formulas give the value of the option as a percentage of the value of the underlying security (the stock). The capitalization rate of the stock is embedded in the value of the stock.! The options pricing formulas price a risk-free combination of ) shares 1 option, where ) is the option delta. To price a risk-free portfolio, we use the risk-free rate. In practice, if the stock price s expected return changes from 11.5% to 12.5%, and the riskfree

10 rate has not changed, the stock has become more risky. If the expected cash flows for the stock have not changed, the stock price declines. 46 Statement A: We use discounted cash flow techniques for stock prices, business projects, insurance policies, and other securities with stochastic cash flows. We use the expected cash flows; we need not know the future cash flows with certainty. Statement B: The traders (investors) in stocks, business projects, insurance policies, and most securities that we price with discounted cash flow techniques are risk averse. We use a risk adjustment capitalization rate based on the systematic risk of the security. Statement C: The systematic risk of the option is a multiple of the systematic risk of the underlying security. When the option is in-the-money or out-of-the-money, the multiple is low. When the option is at-the-money, the multiple is high. As the stock price moves, the multiple changes, so the capitalization rate for the option changes. Statements D and E: Options pricing gives the market value; it is neither conservative nor aggressive. 47 Rachel s stock has a higher price volatility, so the put and call options are worth more. We examine this relation several ways: intuition, binomial tree pricing method (binomial lattice), Black-Scholes formula. Intuition: Rachel s stock price may increase a lot or decrease a lot. If it increases a lot, Rachel has a large gain from her call option; if it decreases a lot, she does not lose anything on her call option. From her put option, Rachel gains a lot if the stock price decreases a lot but doesn t lose if the stock price increases a lot. Jacob s stock price may increase less or decrease less. If it increases, Jacob has a smaller gain from the call option; if it decreases, he does not lose anything on the call option. From the put option, Jacob gains less if the stock price decreases but doesn t lose if the stock price increases. Binomial Lattice: Working out a six period lattice with pencil and paper is cumbersome. To simplify, we examine a one month binomial lattice, and we use a risk-free interest rate of 1% per month. The differences are details; with a spread-sheet, we can work out a six month binomial lattice with a 7% risk-free rate per annum. The present value of the European call option is the value at the expiration date if the stock price moves up the risk-neutral probability of moving up / the risk-free rate ~ For Jacob, if the stock price moves up 10%, the call option is worth \$88 \$80 = \$8. The risk-neutral probability of moving up is ( ) / ( ) = 55.00%. The present value of the call option is \$8 55% / 1.01 = \$4.36. ~ For Rachel, if the stock price moves up 20%, the call option is worth \$96 \$80 = \$16. The risk-neutral probability of moving up is ( ) / ( ) = 52.50%. The present value of the call option is \$ % / 1.01 = \$8.32. Rachel s call option is worth almost twice as much, since if the stock price moves up, her call option pays off twice as much. But the risk-neutral probability that her stock moves up is slightly lower. Black-Scholes formula: A higher stock price volatility F increases d1 and decreases d2. ~ For a call option, this increases S N(d1) and decreases PV(X) N(d2), so it increases the present value of the call option. ~ For a put option, this decreases S N( d1) and increases PV(X) N( d2), so it increases the present value of the put option. 48 The expected return on the stock by the CAPM formula is the risk-free rate + \$ the market risk premium = R + 1 7%. The expected return on the stock by the probabilities of the increase vs decrease is f 20% + c 20% = 15.00% Equating the two expected returns gives R + 7% = 15% A R = 8%.

11 49 The risk-neutral probability of an upward movement in the stock price is. The expected return on the stock by the CAPM formula is R + 1 7%. The expected return on the stock by the probabilities of the increase vs decrease is f 20% + c 20% =15.00% Equating the two expected returns gives R + 7% = 15% A R = 8%. The risk-neutral probability is ( ) / ( ) = 27/40 = 67.50% 50 We solve either by the formula or by the relation that the risk-neutral probability upward movement + (1 the risk-neutral probability) downward movement = the risk-free return, or P U + (1 P) D = Rf. The risk-neutral probability of an upward movement in the stock price is. We solve for Z when the risk-free interest rate is 8% per annum: p = (8% Z) / (+Z Z) = 90% A 8% Z = 90% 2Z A 8% = 80% Z A Z = 10%. If the risk-free rate changes to 6% per annum, the risk-neutral probability changes to p = [6% ( 10%)] / [10% ( 10%)] = 80.00% 51 We evaluate the value of the portfolio at the five stock prices.! If the stock price is \$70 or less, the short put options are worth \$70 S and \$90 S. Together, they are worth \$160 2S. The investor who sells the put options must pay this amount. The two long put options are worth 2 (\$80 S) = \$160 2S. This offsets the loss to the investor from the sale of the two put options at \$70 and \$90.! If the stock price is \$90 or more, all the put options expire worthless, and the net payoff to the investor is zero.! If the stock price is between \$70 and \$80, the put option at \$70 expires worthless, the two put options at \$80 are worth 2 (\$80 S), and the put option at \$90 is worth \$90 S. The net payoff to the investor is 2 (\$80 S) (\$90 S) = S \$70. Since S is between \$70 and \$80, the net payoff is negative, with a minimum of \$10 at S = \$80.! If the stock price is between \$80 and \$90, the put option at \$70 and \$80 expire worthless, and the put option at \$90 is worth \$90 S. The net payoff to the investor is (\$90 S). Since S is between \$70 and \$80, the net payoff is negative, with a minimum of \$10 at S = \$80. At a final stock price of \$80, the portfolio is worth \$10. The maximum value of a butterfly is at the midpoint; this is a short butterfly. 52 Let S be the stock price at the expiration date. If S is less than \$40, the payoff from the two call option is zero, and the payment to the buyer of the put option is at least \$15. The net profit is less than \$15, so S is not less than \$40. If S is more than \$55, the payoff from the two call option is at least \$15 (from the call option with an exercise price of \$40) plus \$5 (from the call option with an exercise price of \$50). The payment to the buyer of the put option is zero. The net profit is at least \$20, so S is not more than \$55. If S is between \$50 and \$55, the payoff from the two call option is between \$10 and \$20: \$10 to \$15 on the call option with an exercise price of \$40 and \$0 to \$5 on the call option with an exercise price of \$50. The payment to the buyer of the put option is between \$0 and \$5. The net profit is between \$5 and \$20, so S is not between \$50 and \$55. We infer that S is between \$40 and \$50. The payoff from the call option with an exercise price of \$40 is S \$40. The payoff from the call option with an exercise price of \$50 is zero. The payment to the buyer of the put option is \$55 S. The net profit is S \$40 (\$55 S) = 2S \$95 = \$3 A S = (\$95 + \$3) / 2 = \$49 53 We assume the CAPM holds, so the expected return = the risk-free rate + \$ the market risk premium. This equation holds for equity, debt, and assets. The beta moves in the same direction as the expected return. If the expected return increases, the beta

13 ! 14% = risk-free rate market risk premium A ( ) market risk premium = 14% 12% = 2% A market risk premium = 2% / 0.4 = 5% To find the risk-free rate: 12% = risk-free rate % A risk-free rate = 8% The expected return on this stock is 8% % = 15.5% 56 Let the risk-free rate be R, so the market risk premium is also R.! The expected return on Stock W is R R = 2.2 R.! The expected return on Stock Y is R R = 1.8 R. The ratio of the expected returns is 2.2R / 1.8R = The CAPM beta is based on diversifiable risk, which is perfectly correlated among the stocks. The beta of the portfolio is the market value weighted average of the betas of the individual stocks, regardless of their correlations. The market values are \$72,000 for Stock Y and \$72,000 for Stock Z, so the beta of the portfolio is the straight average of the betas of the stocks: ½ ( ) = The beta of a portfolio is the weighted average of the betas of each stock in the portfolio. These stocks are equally weighted, so the beta of the portfolio is a ( ) = The expected return on the portfolio is 6.3% % = 14.00% 59 The returns on each stock in 2005 were 13% and 17%. The beta of the portfolio in 2006 is ( ) / ( ) = The expected return in 2006 is 7% % = % 60 The CAPM beta is the covariance of the stock s return with the market return divided by the variance of the market return. If the covariance decreases by 10% and the variance of the market return does not change, the \$ decreases by 10%. Let R be the risk-free rate.! Before the change in operations, the stock s expected return was R R = 2R.! After the change in operations, the stock s expected return is R R = 1.90R. The change in the expected return is 1.90 / = 5.00%. 61 Statements A and B: Let the risk-free rate be R, so the market risk premium is also R.! The expected return on Stock W is R R = 2.2 R.! The expected return on Stock Y is R R = 1.8 R. The ratio of the expected returns is 2.2R / 1.8R = Statement E: The CAPM \$ is the covariance of returns for the stock with the overall market returns divided by the variance of the overall market returns. If the \$ is 50% higher, this covariance is 50% higher. Statements C and D: We can not determine the ratio of the standard deviations or the variances unless we know the correlations of each stock with the overall market return. 62 The CAPM beta is the covariance of the stock s return with the market return divided by the variance of the market return. The correlation is the covariance divided by the product of the standard deviations of the two variables. The standard deviation is the square root of the variance. s m m s m s m m s m s m \$ = covariance(r, r ) / F (r ) = [ D(r, r ) F(r ) F(r ) ] / 2 F2(r ) = [ D(r, r ) F(r ) ] / F(r )! F(rs) = /49% = 70%! F(rm) = /25% = 50% = D(rs, rm) 0.70 / 0.50 A D(rs, rm) = 0, / 0.70 = 60.00% The Brealey and Myers textbook assumes the CAPM equation can be applied to all risky

14 securities, including corporate bonds. This is a valid perspective, but it is not universal. In this perspective, the yield to maturity is after adjustment for defaults. It is hard to judge this perspective, since it is hard to estimate the covariance of corporate bonds with stock returns. Using the CAPM equation, we get: 7% % = 9.00% 65 The coupon rate is the accounting return; the yield to maturity is the market return. The CAPM formula relates to market returns, not accounting returns. We use the yield to maturity, not the coupon rate. 8.5% = 7% + \$ 6% A \$ = 1.5% / 6% = The stock price declines by the cash dividend. We should use a tax adjusted dividend, since the tax rate on dividends differs from the tax rate on capital gains, but the proper adjustment is unclear. We derive the expected return on the stock after the cash dividend (the capitalization rate) from the beta and the CAPM equation. The expected return is 7% % = 18%. The expected stock price in one year is 1.18 (\$82.50 \$2.00) = \$ \$ The project has an NPV of zero, so its internal rate of return is its capitalization rate. We work out the internal rate of return by a quadratic equation. \$20,000 + \$11,300 / (1+R) + \$12,769 / (1+R)2 = 0 (1+R) = [ 11,300 ± (11, ,000 12,769)0.5 ] / 2 20,000 = The project s capitalization rate is 13%. We derive the \$ as (13% 7%) / 8% = We verify this result by dividing into two investments of \$10,000 each and maturities of one year and two years. 68 The CAPM beta reflects only systematic risk, not all risk of the stock. If the beta is zero, the expected return is the risk-free rate + 0 the market risk premium = the risk-free rate. The stock is not risk-free, since it may have non-systematic risk. 69 The weighted average cost of capital (WACC) is the after-tax income needed to pay for a dollar of capital. ~ Shareholders are paid from after-tax income. If they require an R% return, the firm must earn \$R% (after-tax) for each dollar of capital. ~ Creditors are paid from pre-tax income. If they require an R% return, the firm must earn \$R% (pre-tax) for each dollar of capital, or \$R% (1 tax rate) in after-tax funds. The weighted average cost of capital WACC = Ra = " Rd (1 J) + (1 ") Re, wher " is the percentage of debt, J is the corporate tax rate, Ra is the return on debt, Rd is the return on debt, and Re is the return on equity. 70 The weighted average cost of capital is Ra = " Rd (1 J) + (1 ") Re, where " is the percentage of debt, Ra is the return on debt, Rd is the return on debt, and Re is the return on equity. This implies that the return on equity is Re = [Ra " Rd (1 J) ] / (1 ") For this scenario: (15% 20% 65% 7%) / 80% = 17.61% We verify the solution with the formula for the weighted average cost of capital: 20% 7% (1 35%) + 80% 17.61% = 15.00% 71 Hotels and pharmaceutical firms opposites: hotels have very low cost of bankruptcy, since! Their fixed assets are in desirable locations (large cities and resorts) and can generally be sold for their full values.! Their employees are mostly unskilled (bell-hops, cleaning crews, waiters, clerks, drivers), with little investment by the hotel owners in education or training. Other firms with low costs of bankruptcy are retail stores selling to the general public, supermarkets, and department stores. The more specialized the store s products, the higher its cost of bankruptcy. A higher debt-to-equity ratio increases the value of the debt tax shields, thereby increasing

15 the value of the firm, but it also increases the probability of bankruptcy. If the cost of bankruptcy is high, the higher probability of bankruptcy soon outweighs the value of the tax shields. Suppose the probability of bankruptcy is "2, where " is the percentage of debt.! If the firm has no debt, it has no chance of bankruptcy.! If the firm has a 10% debt ratio, it has a 0.01 (or 1%) chance of bankruptcy.! If the firm is financed 100% by debt, it has a 1.0 (or 100%) chance of bankruptcy. Suppose the cost of bankruptcy is 100% of assets for pharmaceutical firms and 50% for hotels. The present value of the debt tax shield is 35% ". We solve ~ Hotels: 35% " = 50% "2 A " = 70% ~ Pharmaceutical firms: 35% " = 100% "2 A " = 35% *Question 1.4: Debt-to-Equity Ratio Assume debt (interest) payments are from pre-tax funds and stockholder dividends are from after-tax funds (as is now the law), and the marginal tax rate is 35%. Firms in the hotel industry have debt-to-equity ratios of 25% (on average). If the Congress changes the corporate tax rate to 15%, which of the following is true? A. The present value of the debt tax shields rises, and the average debt-to-equity ratio rises. B. The present value of the debt tax shields falls, and the average debt-to-equity ratio falls. C. The present value of the debt tax shields rises, and the average debt-to-equity ratio falls. D. The present value of the debt tax shields falls, and the average debt-to-equity ratio rises. E. The present value of the debt tax shields does not change, and the average debt-toequity ratio falls. Answer 1.4: B The present value of the debt tax shields is the corporate tax rate times the market value of the debt. If the Congress raises the corporate tax rate, the present value of the debt tax shields increases. The equilibrium point where the present value of the debt tax shields equals the cost of bankruptcy is at a higher percentage of debt. 72 The weighted average cost of capital is Ra = " Rd (1 J) + (1 ") Re, where " is the percentage of debt, Ra is the return on debt, Rd is the return on debt, and Re is the return on equity. We solve for " as " = (Ra Re) / (Rd (1 J) Re)! equity has a CAPM beta of 100% A the expected return = 7% + 100% 8% = 15.00%! debt has a CAPM beta of 12.5% A the expected return = 7% % 8% = 8.00%! assets have a CAPM beta of 75% A the expected return = 7% + 75% 8% = 13.00% alpha = (13% 15%) / (8% (1 35%) 15%) = 20.41%. 20%. The percentage of debt is 20%, so the debt-to-equity ratio is 20% / 80% = 25%. 73 The rate to compute the after-tax present value of the depreciation tax shields is the aftertax borrowing rate. Year Percent Depreciation Tax Shield Present Value 1 30% \$210,000 \$73,500 \$68, % \$175,000 \$61,250 \$52, % \$140,000 \$49,000 \$39, % \$105,000 \$36,750 \$27, % \$70,000 \$24,500 \$16,830 Total 100% \$700,000 \$245,000 \$204,046 The depreciation is the percentage given in the problem times the \$700,000 basis. The tax shield is 35% times the depreciation. The after-tax borrowing rate is 12% (1 35%) = 7.80%. The present value of the tax shield is the tax shield divided by raised to the power of the years. The total present value of the five years depreciation is \$204,046. Jacob: What do we mean by the present value of the tax shields? And why do we use the

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