Introduction To Corporate Finance (The Capital Asset Pricing Model)

RETURN AND RISK: THE CAPITAL ASSET PRICING MODEL (CAPM) 1. In broad terms, why is some risk diversifiable? Why are some risks non-diversifiable? Does it follow that an investor can control the level of unsystematic risk in a portfolio, but not the level of systematic risk? Answer: Some of the risk in holding any asset is unique to the asset in question. By investing in a variety of assets, this unique portion of the total risk can be eliminated at little cost. On the other hand, there are some risks that affect all investments. This portion of the total risk of an asset cannot be costlessly eliminated. In other words, systematic risk can be controlled, but only by a costly reduction in expected returns. 2. Classify the following events as mostly systematic or mostly unsystematic. Is this distinction clear in every case? a. Short-term interest rates increase unexpectedly. b. The interest rate a company pays on its short-term debt borrowing is increased by its bank. c. Oil prices unexpectedly decline. d. An oil tanker ruptures, creating a large oil spill. e. A manufacturer loses a multimillion-dollar product liability suit. f. A Supreme Court decision substantially broadens producer liability for injuries suffered by product users. Answer: a. b. c. d. e. f. 3. systematic unsystematic both; probably mostly systematic unsystematic unsystematic systematic If a portfolio has a positive investment in every asset, can the expected return on the portfolio be greater than that on every asset in the portfolio? Can it be less than that on every asset in the portfolio? If you answer yes to one or both of these questions, give an example to support your answer. Answer: No to both questions. The portfolio expected return is a weighted average of the asset’s returns, so it must be less than the largest asset return and greater than the smallest asset return. 4. True or false: The most important characteristic in determining the expected return of a welldiversified portfolio is the variances of the individual assets in the portfolio. Explain. Answer: False. The variance of the individual assets is a measure of the total risk. The variance on a welldiversified portfolio is a function of systematic risk only. 5. Is it possible that a risky asset could have a beta of zero? Explain. Based on the CAPM, what is the expected return on such an asset? Is it possible that a risky asset could have a negative beta? What does the CAPM predict about the expected return on such an asset? Can you give an explanation for your answer? Answer: Yes. It is possible, in theory, to construct a zero beta portfolio of risky assets whose return would be equal to the risk-free rate. It is also possible to have a negative beta; the return would be less than the risk-free rate. A negative beta asset would carry a negative risk premium because of its value as a diversification instrument. 6. A broker has advised you not to invest in oil industry stocks because they have high standard deviations. Is the broker’s advice sound for a risk-averse investor like yourself? Why or why not? Answer: The wide fluctuations in the price of oil stocks do not indicate that these stocks are a poor investment. If an oil stock is purchased as part of a well-diversified portfolio, only its contribution to the risk of the entire portfolio matters. This contribution is measured by systematic risk or beta. Since price fluctuations in oil stocks reflect diversifiable plus non-diversifiable risk, observing the standard deviation of price movements is not an adequate measure of the appropriateness of adding oil stocks to a portfolio. 7. What are the portfolio weights for a portfolio that has 135 shares of stock A that sell for $47 per share and 105 shares of stock B that sell ofr $41 per share? Solution: The portfolio weight of an asset is total investment in that asset divided by the total portfolio value. First, we will find the portfolio value, which is: Total value = 135($47) + 105($41) = $10,650 The portfolio weight for each stock is: XA = 135($47)/$10,650 = .5958 XB = 105($41)/$10,650 = .4042 8. You own a portfolio that has $2,100 invested in stock A and $3,200 invested in stock B. If the expected returns on these stocks are 11 percent and 14 percent, respectively, what is the expected return on the portfolio? Solution: The expected return of a portfolio is the sum of the weight of each asset times the expected return of each asset. The total value of the portfolio is: Total value = $2,100 + 3,200 = $5,300 So, the expected return of this portfolio is: E(Rp) = ($2,100/$5,300)(0.11) + ($3,200/$5,300)(.14) = .1281, or 12.81% 9. You have $10,000 to invest in a stock portfolio. Your choices are stock X with an expected return of 14 percent and stock Y with an expected return of 9 percent. If your goal is to create a portfolio with an expected return of 12.9 percent, how much money will you invest in stock X? In stock Y? Solution: Here we are given the expected return of the portfolio and the expected return of each asset in the portfolio and are asked to find the weight of each asset. We can use the equation for the expected return of a portfolio to solve this problem. Since the total weight of a portfolio must equal 1 (100%), the weight of Stock Y must be one minus the weight of Stock X. Mathematically speaking, this means: E(Rp) = .129 = .14XX + .09(1 – XX) We can now solve this equation for the weight of Stock X as: .129 = .14XX + .09 – .10XX .039 = .04XX XX = .7800 So, the dollar amount invested in Stock X is the weight of Stock X times the total portfolio value, or: Investment in X = .7800($10,000) = $7,800 And the dollar amount invested in Stock Y is: Investment in Y = (1 – .7800)($10,000) = $2,200 10. Based on the following information, calculate the expected return and standard deviation: State of Economy Depression Recession Normal Boom Probability of State of Economy .10 .25 .45 .20 Rate of Return if State occurs -.105 .059 .130 .211 Solution: The expected return of an asset is the sum of the probability of each return occurring times the probability of that return occurring. So, the expected return of the stock is: E(RA) = .10(–.105) + .25 (.059) + .45(.130) + .20(.211) = .1050, or 10.50% To calculate the standard deviation, we first need to calculate the variance. To find the variance, we find the squared deviations from the expected return. We then multiply each possible squared deviation by its probability, and then add all of these up. The result is the variance. So, the variance and standard deviation are: 2 =.10(–.105 – .1050)2 + .25(.059 – .1050)2 + .45(.130 – .1050)2 + .20(.211 – .1050)2 = .00747  =(.00747)1/2 = .0864, or 8.64% 11. You own a stock portfolio invested 10 percent in stock Q, 35 percent in stock R, 20 percent in stock S, and 35 percent in stock T. The betas for these four stocks are .75, 1.90, 1.38, and 1.16 respectively. What is the portfolio beta? Solution: The beta of a portfolio is the sum of the weight of each asset times the beta of each asset. So, the beta of the portfolio is: p = .10(.75) + .35(1.90) + .20(1.38) + .35(1.16) = 1.42 12. A stock has a beta of 1.15, the expected return on the market is 11 percent, and the risk-free rate is 5 percent. What must the expected return on this stock be? Solution: CAPM states the relationship between the risk of an asset and its expected return. CAPM is: E(Ri) = Rf + [E(RM) – Rf] × i Substituting the values we are given, we find: E(Ri) = .05 + (.11 – .05)(1.15) = .1190, or 11.90% 13. A stock has an expected return of 10.2 percent, the risk-free rate is 4 percent, and the market risk premium is 7 percent. What must the beta of this stock be? Solution: We are given the values for the CAPM except for the  of the stock. We need to substitute these values into the CAPM, and solve for the  of the stock. One important thing we need to realize is that we are given the market risk premium. The market risk premium is the expected return of the market minus the risk-free rate. We must be careful not to use this value as the expected return of the market. Using the CAPM, we find: E(Ri) = .102 = .04 + .07i i = .89 CAPITAL STRUCTURE: BASIC CONCEPTS 1. In a world with no taxes, no transaction costs, and no costs of financial distress, is the following statement true, false, or uncertain? If a firm issues equity to repurchase some of its debt, the price per share of the firm’s stock will rise because the shares are less risky. Explain. Answer: False. A reduction in leverage will decrease both the risk of the stock and its expected return. Modigliani and Miller state that, in the absence of taxes, these two effects exactly cancel each other out and leave the price of the stock and the overall value of the firm unchanged. 2. Explain what is meant by business and financial risk. Suppose firm A has greater business risk than firm B. Is it true that firm A also has a higher cost of equity capital? Explain. Answer: Business risk is the equity risk arising from the nature of the firm’s operating activity, and is directly related to the systematic risk of the firm’s assets. Financial risk is the equity risk that is due entirely to the firm’s chosen capital structure. As financial leverage, or the use of debt financing, increases, so does financial risk and, hence, the overall risk of the equity. Thus, Firm B could have a higher cost of equity if it uses greater leverage. 3. Is there an easily identifiable debt-equity ratio that will maximize the value of a firm? Why or why not? Answer: Because many relevant factors such as bankruptcy costs, tax asymmetries, and agency costs cannot easily be identified or quantified, it is practically impossible to determine the precise debt/equity ratio that maximizes the value of the firm. However, if the firm’s cost of new debt suddenly becomes much more expensive, it’s probably true that the firm is too highly leveraged. 4. Why is the use of debt financing referred to as financial “leverage”? Answer: It’s called leverage (or “gearing” in the UK) because it magnifies gains or losses. 5. Money, Inc., has no debt outstanding and a total market value of $275,000. Earnings before interest and taxes, EBIT, are projected to be $21,000 if economic conditions are normal. If there is strong expansion in the economy, then EBIT will be 25 percent higher. If there is recession, then EBIT will be 40 percent lower. Money is considering a $99,000 debt issue with an interest rate of 8 percent. The proceeds will be used to repurchase shares of stock. There are currently 5,000 shares outstanding. Ignore taxes for this problem. a. Calculate earnings per share, EPS, under each of the three economic scenarios before any debt is issued. Also calculate the percentage changes in EPS when the economy expands or enters into a recession. b. Repeat part (a) assuming Money goes through with the recapitalization. What do you observe? Solution: a. A table outlining the income statement for the three possible states of the economy is shown below. The EPS is the net income divided by the 5,000 shares outstanding. The last row shows the percentage change in EPS the company will experience in a recession or an expansion economy. EBIT Interest NI EPS %EPS b. Recession $12,600 0 $12,600 $ 2.52 –40 Normal $21,000 0 $21,000 $ 4.20 ––– Expansion $26,250 0 $26,250 $ 5.25 +25 If the company undergoes the proposed recapitalization, it will repurchase: Share price = Equity / Shares outstanding Share price = $275,000/5,000 Share price = $55 Shares repurchased = Debt issued / Share price Shares repurchased =$99,000/$45 Shares repurchased = 1,800 The interest payment each year under all three scenarios will be: Interest payment = $99,000(.08) = $7,920 The last row shows the percentage change in EPS the company will experience in a recession or an expansion economy under the proposed recapitalization. EBIT Interest NI EPS %EPS 6. Recession $12,600 7,920 $ 4,680 $1.46 –64.22 Normal $21,000 7,920 $13,080 $ 4.09 ––– Expansion $26,250 7,920 $18,330 $ 5.73 +40.14 Weston Industries has a debt-equity ratio of 1.5. Its WACC is 11 percent, and cots of debt is 7 percent. The corporate tax rate is 35 percent. a. what is Weston’s cost of equity capital? b. what is weston’s unlevered cost of equity capital? c. what would the cost of equity be if the debt-equity ratio were 2? What if it were 1? What if it were zero? Solution: a. With the information provided, we can use the equation for calculating WACC to find the cost of equity. The equation for WACC is: WACC = (S/V)RS + (B/V)RB(1 – tC) The company has a debt-equity ratio of 1.5, which implies the weight of debt is 1.5/2.5, and the weight of equity is 1/2.5, so WACC = .11 = (1/2.5)RS + (1.5/2.5)(.07)(1 – .35) RS = .2068, or 20.68% b. To find the unlevered cost of equity, we need to use M&M Proposition II with taxes, so: RS = R0 + (R0 – RB)(B/S)(1 – tC) .2068 = R0 + (R0 – .07)(1.5)(1 – .35) R0 = .1392, or 13.92% c. To find the cost of equity under different capital structures, we can again use M&M Proposition II with taxes. With a debt-equity ratio of 2, the cost of equity is: RS = R0 + (R0 – RB)(B/S)(1 – tC) RS= .1392 + (.1392 – .07)(2)(1 – .35) RS = .2293, or 22.93% With a debt-equity ratio of 1.0, the cost of equity is: RS= .1392 + (.1392 – .07)(1)(1 – .35) RS = .1842, or 18.42% And with a debt-equity ratio of 0, the cost of equity is: RS= .1392 + (.1392 – .07)(0)(1 – .35) RS = R0 = .1392, or 13.92% 7. Bruce and Co. expects its EBIT to be $185,000 every year forever. The firm can borrow at 9 percent. Bruce currently has no debt, and its cost of equity is 16 percent. If tax rate is 35 percent, what is the value of the firm? What will the value be if Bruce borrows $135,000 and uses the proceeds to repurchase the shares? Solution: a. The value of the unlevered firm is: V = EBIT(1 – tC)/R0 V = $185,000(1 – .35)/.16 V = $751,562.50 b. The value of the levered firm is: V = VU + tCB V = $751,562.50 + .35($135,000) V = $798,812.50 8. Acetate, Inc., has equity with a market value of $23 million and debt with a market value if $7 million. Treasury bills that mature in one year yield 5 percent per year, and the expected return on the market portfolio is 12 percent. The beta of Acetate’s equity is 1.15. the firm pays no taxes. a. what is Acetate’s debt-equity ratio? b. what is the firm’s weighted average cost of capital? c. what is the cost of capital for an otherwise identical all-equity firm? Solution: a. A firm’s debt-equity ratio is the market value of the firm’s debt divided by the market value of a firm’s equity. So, the debt-equity ratio of the company is: Debt-equity ratio = MV of debt / MV of equity Debt-equity ratio = $7,000,000 / $23,000,000 Debt-equity ratio = .30 b. We first need to calculate the cost of equity. To do this, we can use the CAPM, which gives us: RS = RF + [E(RM) – RF] RS = .05 + 1.15(.12 – .05) RS = .1305, or 13.05% We need to remember that an assumption of the Modigliani-Miller theorem is that the company debt is risk-free, so we can use the Treasury bill rate as the cost of debt for the company. In the absence of taxes, a firm’s weighted average cost of capital is equal to: RWACC = [B / (B + S)]RB + [S / (B + S)]RS RWACC = ($7,000,000/$30,000,000)(.05) + ($23,000,000/$30,000,000)(.1305) RWACC = .1117, or 11.17% c. According to Modigliani-Miller Proposition II with no taxes: RS = R0 + (B/S)(R0 – RB) .1305 = R0 + (.30)(R0 – .05) R0 = .1117, or 11.17% This is consistent with Modigliani-Miller’s proposition that, in the absence of taxes, the cost of capital for an all-equity firm is equal to the weighted average cost of capital of an otherwise identical levered firm. DIVIDENDS AND OTHER PAYOUTS 1. What is the impact of stock repurchase on a company’s debt ratio? Does this suggest another use for excess cash? Answer: A stock repurchase reduces equity while leaving debt unchanged. The debt ratio rises. A firm could, if desired, use excess cash to reduce debt instead. This is a capital structure decision. 2. On Tuesday, December 8, Hometown Power Co.’s board of director declares a dividend of 75 cents per share payable on Wednesday, January 17, to shareholders of record as of Wednesday, January 3. When is the ex-dividend date? If a shareholder buys stock before that date, who gets the dividend on those shares – the buyer or the seller? Answer: Friday, December 29 is the ex-dividend day. Remember not to count January 1 because it is a holiday, and the exchanges are closed. Anyone who buys the stock before December 29 is entitled to the dividend, assuming they do not sell it again before December 29. 3. If increases in dividends tend to be followed by (immediate) increases in share prices, how can it be said that dividend policy is irrelevant? Answer: The change in price is due to the change in dividends, not due to the change in dividend policy. Dividend policy can still be irrelevant without a contradiction. 4. Lee Ann, Inc., has declared a $9.50 per share dividend. Suppose capital gains are not taxed, but dividends are taxed at 15 percent. New IRS regulation requires that taxes be withheld when dividend is paid. Lee Ann sells for $115 per share, and the stock is about to go ex-dividend. What do you think the ex-dividend price will be? Solution: The aftertax dividend is the pretax dividend times one minus the tax rate, so: Aftertax dividend = $9.50(1 – .15) = $8.08 The stock price should drop by the aftertax dividend amount, or: Ex-dividend price = $115 – 8.08 = $106.93 5. The owner’s equity accounts for Hexagon International are shown here: Common stock ($1 par value) $30,000 Capital surplus 185,000 Retained earnings 627,500 Total owner’s equity $842,500 a. If Hexagon stock currently sells for $37 per share and a 10 percent stock dividend is declared, how many new shares will be distributed? Show how the equity accounts would change. b. If Hexagon declared a 25 percent stock dividend, how would the accounts change? Solution: a. The shares outstanding increases by 10 percent, so: New shares outstanding = 30,000(1.10) = 33,000 New shares issued = 3,000 Since the par value of the new shares is $1, the capital surplus per share is $36. The total capital surplus is therefore: Capital surplus on new shares = 3,000($36) = $108,000 Common stock ($1 par value) Capital surplus Retained earnings b. $ 33,000 293,000 516,500 $842,500 The shares outstanding increases by 25 percent, so: New shares outstanding = 30,000(1.25) = 37,500 New shares issued = 7,500 Since the par value of the new shares is $1, the capital surplus per share is $36. The total capital surplus is therefore: Capital surplus on new shares = 7,500($36) = $270,000 Common stock ($1 par value) Capital surplus Retained earnings 6. $ 37,500 455,000 350,000 $842,500 For the company in the previous problem, show how the equity accounts will change if: a. Hexagon declares a four-for-one stock split. How many shares are outstanding now? What is the new par value per share? b. Hexagon declares a one-for-five reverse stock split. How many shares are outstanding now? What is the new par value per share? Solution: a. To find the new shares outstanding, we multiply the current shares outstanding times the ratio of new shares to old shares, so: New shares outstanding = 30,000(4/1) = 120,000 The equity accounts are unchanged except that the par value of the stock is changed by the ratio of new shares to old shares, so the new par value is: New par value = $1(1/4) = $0.25 per share. b. To find the new shares outstanding, we multiply the current shares outstanding times the ratio of new shares to old shares, so: New shares outstanding = 30,000(1/5) = 6,000. The equity accounts are unchanged except that the par value of the stock is changed by the ratio of new shares to old shares, so the new par value is: New par value = $1(5/1) = $5.00 per share. 7. The balance sheet for Levy Corp. is shown here in market value terms. There are 12,000 shares of stock outstanding. Cash Fixed assets Total $55,000 410,000 $465,000 Market value Balance Sheet Equity Total $465,000 $465,000 The company has declared a dividend of $1.90 per share. The stock goes ex-dividend tomorrow. Ignoring any tax effects, what is the stock selling for today? What will it sell for tomorrow? What will the balance sheet look like after the dividends are paid? Solution: The stock price is the total market value of equity divided by the shares outstanding, so: P 0 = $465,000 equity/12,000 shares = $38.75 per share Ignoring tax effects, the stock price will drop by the amount of the dividend, so: P X = $38.75 – 1.90 = $36.85 The total dividends paid will be: $1.90 per share(12,000 shares) = $22,800 The equity and cash accounts will both decline by $22,800. 8. In the previous problem, suppose Levy has announced it is going to repurchase $22,800 worth of stock. What effect will this transaction have on the equity of the firm? How many shares will be outstanding? What will the share price be after the repurchase? Ignoring tax effects, show how the share repurchase is effectively the same as a cash dividend. Solution: Repurchasing the shares will reduce shareholders’ equity by $22,800. The shares repurchased will be the total purchase amount divided by the stock price, so: Shares bought = $22,800/$38.75 = 588 And the new shares outstanding will be: New shares outstanding = 12,000 – 588 = 11,412 After repurchase, the new stock price is: Share price = $442,200/11,412 shares = $38.75 The repurchase is effectively the same as the cash dividend because you either hold a share worth $38.75 or a share worth $36.85 and $1.90 in cash. Therefore, you participate in the repurchase according to the dividend payout percentage; you are unaffected. 9. The company with the common equity accounts shown here has declared a stock dividend of 15 percent when the market value of its stock is $45 per share. What effects on the equity accounts will the distribution of the stock dividend have? Common stock ($1 par value) $ 410,000 Capital surplus 2,150,000 Retained earnings 5,320,000 Total owner’s equity $7,880,000 Solution: With a stock dividend, the shares outstanding will increase by one plus the dividend amount, so: New shares outstanding = 410,000(1.15) = 471,500 The capital surplus is the capital paid in excess of par value, which is $1, so: Capital surplus for new shares = 61,500($44) = $2,706,000 The new capital surplus will be the old capital surplus plus the additional capital surplus for the new shares, so: Capital surplus = $2,150,000 + 2,706,000 = $4,856,000 The new equity portion of the balance sheet will look like this: Common stock ($1 par value) Capital surplus Retained earnings $ 471,500 4,856,000 2,552,500 $7,880,000 RAISING CAPITAL 1. In the aggregate, debt offerings are much more common than equity offerings and typically much larger as well. Why? Answer: A company’s internally generated cash flow provides a source of equity financing. For a profitable company, outside equity may never be needed. Debt issues are larger because large companies have the greatest access to public debt markets (small companies tend to borrow more from private lenders). Equity issuers are frequently small companies going public; such issues are often quite small. Additionally, to maintain a debt-equity ratio, a company must issue new bonds when the current bonds mature. 2. Why are the costs of selling equity so much larger than the costs of selling debt? Answer: From the previous question, economies of scale are part of the answer. Beyond this, debt issues are simply easier and less risky to sell from an investment bank’s perspective. The two main reasons are that very large amounts of debt securities can be sold to a relatively small number of buyers, particularly large institutional buyers such as pension funds and insurance companies, and debt securities are much easier to price. 3. Why do non-investment grade bonds have much higher direct costs than investment grade issues? Answer: They are riskier and harder to market from an investment bank’s perspective. 4. Again, Inc., is proposing a rights offering. Presently there are 550,000 shares outstanding at $85 each. There will be 85,000 new shares offered at $81 each. a. What is the new market value of the company? b.How many rights are associated with one of the new shares? c. What is the ex-rights price? d.What is the value of a right? e. Why might a company have a rights offering rather than a general cash offer? Solution: a. The new market value will be the current shares outstanding times the stock price plus the rights offered times the rights price, so: New market value = 550,000($87) + 85,000($81) = $54,735,000 b. The number of rights associated with the old shares is the number of shares outstanding divided by the rights offered, so: Number of rights needed = 550,000 old shares/85,000 new shares = 6.47 rights per new share c. The new price of the stock will be the new market value of the company divided by the total number of shares outstanding after the rights offer, which will be: PX = $54,735,000/(550,000 + 85,000) = $86.20 d. The value of the right Value of a right = $87.00 – 86.20 = $.80 e. 5. A rights offering usually costs less, it protects the proportionate interests of existing shareholders and also protects against underpricing. The St. Anger Corp. needs to raise $45 million to finance its expansion into new markets. The company will sell new shares via a general cash offering to raise the needed funds. If the offer price is $31 per share and the company’s underwriters charge a spread of 7 percent, how many shares need to be sold? Solution: Using X to stand for the required sale proceeds, the equation to calculate the total sale proceeds, including flotation costs is: X(1 – .07) = $45,000,000 X = $48,387,097 required total proceeds from sale. So the number of shares offered is the total amount raised divided by the offer price, which is: Number of shares offered = $48,387,097/$31 = 1,560,874 6. In the previous problem, if the SEC filing fee and associated administrative expenses of the offering are $1,900,000, how many shares need to be sold? Solution: This is basically the same as the previous problem, except we need to include the $1,900,000 of expenses in the amount the company needs to raise, so: X(1 – .07) = $46,900,000 X = $50,430,108 required total proceeds from sale. Number of shares offered = $50,430,108/$31 = 1,626,778 7. The Newton Company has 50,000 shares of stock that each sell for $40. Suppose the company issues 9,000 shares of new stock at the following prices: $40, $20, and $10. What is the effect of each of the alternative offering prices on the existing price per share? Solution: In general, the new price per share after the offering will be: P= Current market val ue Proceeds from offer Old shares New shares The current market value of the company is the number of shares outstanding times the share price, or: Market value of company = 50,000($40) Market value of company = $2,000,000 If the new shares are issued at $40, the share price after the issue will be: $2,000,000 9,000($40) 50,000 9,000 P = $40.00 P= If the new shares are issued at $20, the share price after the issue will be: $2,000,000 9,000($20) 50,000 9,000 P = $36.95 P= If the new shares are issued at $10, the share price after the issue will be: $2,000,000 9,000($10) 50,000 9,000 P = $35.42 P=