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Investments 10th Canadian Edition By Zvi Bodie, Alex Kane, Alan Marcus, Lorne Switzer, Maureen Stapleton, Dana Boyko, Christine Panasian (Solutions Manual All Chapters, 100% Original Verified, A+ Grade) All Chapters Solutions Manual Supplement files download link at the end of this file. CHAPTER 1: THE INVESTMENT ENVIRONMENT PROBLEM SETS: 1.

While it is ultimately true that real assets determine the material well-being of an economy, financial innovation in the form of bundling and unbundling securities creates opportunities for investors to form more efficient portfolios. Both institutional and individual investors can benefit when financial engineering creates new products that allow them to manage their portfolios of financial assets more efficiently. Bundling and unbundling create financial products with new properties and sensitivities to various sources of risk that allows investors to reduce volatility by hedging particular sources of risk more efficiently.

Securitization requires access to a large number of potential investors. To attract these investors, the capital market needs: 1. A safe system of business laws and low probability of confiscatory taxation/regulation; 2. A well-developed investment banking industry; 3. A well-developed system of brokerage and financial transactions; and 4. A well-developed media, particularly financial reporting. These characteristics are found in (indeed make for) a well-developed financial market.

Securitization leads to disintermediation; that is, securitization provides a means for market participants to bypass intermediaries. For example, mortgage-backed securities channel funds to the housing market without requiring that banks or thrift institutions make loans from their own portfolios. Securitization works well and can benefit many, but only if the market for these securities is highly liquid. As securitization progresses, however, and financial intermediaries lose opportunities, they must increase other revenue-generating activities such as providing short-term liquidity to consumers and small business and financial services.

The existence of well-developed capital markets and the liquid trading of financial assets make it easy for large firms to raise the capital needed to finance their investments in real assets. If Suncor Energy, for example, could not issue stocks or bonds to the general public, it would have a far more difficult time raising capital. Contraction of the supply of financial assets would make financing more difficult, thereby increasing the cost of capital. A higher cost of capital makes investments in real assets less profitable/viable leading to lower real growth.

Even if the firm does not need to issue stock in any particular year, the stock market is still important to the financial manager. The stock price provides important information about how the market values the firm's investment projects. For example, if the stock price rises considerably, managers might conclude that the market believes the firm's future prospects Bodie et al. Investments 10th Canadian Edition Solutions Manual © 2022 McGraw-Hill Education Ltd. 1-1

are bright. This might be a useful signal to the firm to proceed with an investment such as an expansion of the firm's business. In addition, shares that can be traded in the secondary market are more attractive to initial investors since they know that they will be able to sell their shares. This in turn makes investors more willing to buy shares in a primary offering and thus improves the terms on which firms can raise money in the equity market. Remember that stock exchanges like those in New York, Toronto, and London are the heart of capitalism, in which firms can raise capital quickly in primary markets because investors know there are liquid secondary markets.

a. No. The increase in price does not add to the productive capacity of the economy. b. Yes, the value of the equity held in these assets has increased. c. Future homeowners as a whole are worse off, since mortgage liabilities have also increased. In addition, this housing price bubble will eventually burst and society as a whole (and most likely taxpayers) will suffer the damage.

a. The bank loan is a financial liability for Lanni, and a financial asset for the bank. The cash Lanni receives is a financial asset. The new financial asset created is Lanni's promissory note to repay the loan. b. Lanni transfers financial assets (cash) to the software developers. In return, Lanni receives the completed software package, which is a real asset. No financial assets are created or destroyed; cash is simply transferred from one party to another. c. Lanni exchanges the real asset (the software) for a financial asset, which is 1,250 shares of Microsoft stock. If Microsoft issues new shares in order to pay Lanni, then this would represent the creation of new financial assets. d. By selling its shares in Microsoft, Lanni exchanges one financial asset (1,250 shares of stock) for another ($125,000 in cash). Lanni uses the financial asset of $50,000 in cash to repay the bank loan and retire its promissory note. The bank must return the promissory note (financial asset) to Lanni. The loan is now "destroyed" in the transaction, since it is retired when paid off and no longer exists.

a. Liabilities & Shareholders’ Equity Cash $ 70,000 Bank loan $ 50,000 Computers 30,000 Shareholders’ equity 50,000 Total $100,000 Total $100,000 Ratio of real assets to total assets = $30,000/$100,000 = 0.30 Assets

b. Assets Software product* Computers Total

$ 70,000 30,000 $100,000

Liabilities & Shareholders’ Equity Bank loan $ 50,000 Shareholders’ equity 50,000 Total $100,000

*Valued at cost Ratio of real assets to total assets = $100,000/$100,000 = 1.0 c. Assets Microsoft shares Computers Total

$125,000 30,000 $155,000

Liabilities & Shareholders’ Equity Bank loan $ 50,000 Shareholders’ equity 105,000 Total $155,000

Ratio of real assets to total assets = $30,000/$155,000 = 0.19 Conclusion: when the firm starts up and raises working capital, it is characterized by a low ratio of real assets to total assets. When it is in full production/operation, it has a high ratio of real assets to total assets. When the project "shuts down" and the firm sells it off for cash, financial assets once again replace real assets. 9.

a. This is a primary market transaction in which gold certificates are being offered to public investors for the first time by an underwriting syndicate led by JW Korth Capital. b. The certificates are derivative assets because they represent an investment in physical gold, but each investor receives a certificate and no gold. Note that investors can convert the certificate into gold during the four-year period.

10.

a. A fixed salary means that compensation is (at least in the short run) independent of the firm's success. This salary structure does not tie the manager’s immediate compensation to the success of the firm, so a manager might not feel too compelled to work hard to maximize firm value. However, the manager might view this as the safest compensation structure and therefore value it more highly. b. A salary that is paid in the form of stock in the firm means that the manager earns the most when the shareholders’ wealth is maximized. Five years of vesting helps align the interests of the employee with the long-term performance of the firm. This structure is therefore most likely to align the interests of managers and shareholders. If stock compensation is overdone, however, the manager might view it as overly risky since the manager’s career is already linked to the firm, and this undiversified exposure would be exacerbated with a large stock position in the firm. Bodie et al. Investments 10th Canadian Edition Solutions Manual © 2022 McGraw-Hill Education Ltd. 1-3

c. A profit-linked salary creates great incentives for managers to contribute to the firm’s success. However, a manager whose salary is tied to short-term profits will be risk seeking, especially if these short-term profits determine salary or if the compensation structure does not bear the full cost of the project’s risks. Shareholders, in contrast, bear the losses as well as the gains on the project and might be less willing to assume that risk. 11.

Even if an individual shareholder could monitor and improve managers’ performance and thereby increase the value of the firm, the payoff would be small, since the ownership share in a large corporation would be very small. For example, if you own $10,000 of Loblaw stock and you can increase the value of the firm by 5%, a very ambitious goal, you benefit by only: 0.05  $10,000 = $500. The cost, both personal and financial to an individual investor, is likely to be prohibitive and would typically easily exceed any accrued benefits, in this case $500. In contrast, a creditor, such as a bank that has a multimillion-dollar loan outstanding to the firm, has a big stake in making sure that the firm can repay the loan. It is clearly worthwhile for the bank to spend considerable resources to monitor the firm.

12.

Mutual funds accept funds from small investors and invest, on behalf of these investors, in the domestic and international securities markets. Pension funds accept funds and then invest in a wide range of financial securities, on behalf of current and future retirees, thereby channeling funds from one sector of the economy to another. Venture capital firms pool the funds of private investors and invest in start-up firms. Banks accept deposits from customers and loan those funds to businesses or use the funds to buy securities of large corporations.

13.

Treasury bills serve a purpose for investors who prefer a low-risk investment. The lower average rate of return compared to stocks is the price investors pay for higher liquidity and the predictability of investment performance and portfolio value.

14.

With a top-down investing style, you focus on asset allocation or the broad composition of the entire portfolio, which is the major determinant of overall performance. Moreover, topdown management is the natural way to establish a portfolio with a level of risk consistent with your risk tolerance. The disadvantage of an exclusive emphasis on top-down issues is that you may forfeit the potential high returns that could result from identifying and concentrating in undervalued securities or sectors of the market. With a bottom-up investing style, you try to benefit from identifying undervalued securities. The disadvantage is that investors might tend to overlook the overall composition of your portfolio, which may result in a non-diversified portfolio or a portfolio with a risk level inconsistent with the appropriate level of risk tolerance. In addition, this technique tends to require more active management, thus generating more transaction costs. Finally, the bottomup analysis may be incorrect, in which case there will be a fruitlessly expended effort and money attempting to beat a simple buy-and-hold strategy. Bodie et al. Investments 10th Canadian Edition Solutions Manual © 2022 McGraw-Hill Education Ltd. 1-4

15.

You should be skeptical. If the author actually knows how to achieve such returns, one must question why the author would then be so ready to sell the secret to others. Financial markets are very competitive; one of the implications of this fact is that riches do not come easily. High expected returns require bearing some risk, and obvious bargains are few and far between. Odds are that the only one getting rich from the book is its author.

16.

Financial assets provide for a means to acquire real assets as well as an expansion of these real assets. Financial assets provide a measure of liquidity to real assets and allow for investors to more effectively reduce risk through diversification.

17.

Allowing traders to share in the profits increases the traders’ willingness to assume risk. Traders will share in the upside potential directly in the form of higher compensation but only in the downside indirectly in the form of potential job loss if performance is bad enough. This scenario creates a form of agency conflict known as moral hazard, in which the owners of the financial institution share in both the total profits and losses, while the traders will tend to share more of the gains than the losses.

18.

Answers may vary; however, students should touch on the following: increased transparency, regulations to promote capital adequacy by increasing the frequency of gain or loss settlement, incentives to discourage excessive risk taking, and the promotion of more accurate and unbiased risk assessment.

CHAPTER 2: FINANCIAL MARKETS, ASSET CLASSES, AND FINANCIAL INSTRUMENTS PROBLEM SETS: 1.

Money market securities are called “cash equivalents” because of their great liquidity. The prices of money market securities are very stable, and they can be converted to cash (i.e., sold) on very short notice and with very low transaction costs. a. rBEY

1,000 − P 365  n P

rBEY

1,000 − 960 365  = .083562, or 8.36% 960 182

b. One reason is that the discount yield is computed by dividing the dollar discount from par by the par value, $10,000, rather than by the bill’s price, $9,600. A second reason is that the discount yield is annualized by a 360-day rather than a 365-day year. 3.

P = $1,000 [1 – rBD (n/360)] where rBD is the discount yield. Pask = $1,000[1 – .0681(60/360)] = $988.65 Pbid = $1,000 [1 – .0690(60/360)] = $988.50

rBEY

1,000 − P 365  n P

1,000 − 988.65 365  = 6.98%, 60 988.65

which exceeds the discount yield, rBD = 6.81%. To obtain the effective annual yield, rEAY, note that the 60-day growth factor for invested funds 1,000 is = 1.01148. Annualizing this growth rate results in 988.65 1 + rEAY = ( 5.

1,000 365/60 ) = 1.0719 which implies that rEAY = 7.19%. 988.65

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P = $10,000/[1 + 0.05 × (91/365)] = $ 9,876.88. 4

a. i.

1 + r = ($10,000/$9,764) = 1.1002 r = 10.02% 2

ii.

1 + r = ($10,000/$9,539) = 1.0990 r = 9.90% The three-month bill offers a higher effective annual yield.

b. i.

rBD =

1,000 − 976.4 360  91 = 0.0934 = 9.34% 1,000

ii.

rBD =

1,000 − 953.9 360  182 = 0.0912 = 9.12% 1,000

90 a. Price = $1,000 × [1 – 0.03 × 360 ] = $992.5 b. 90-day return =

1,000 − 992.5 = 0.007557 = 0.7557% 992.5

365 c. rBEY = 0.7557% × 90 = 3.06% d. Effective annual yield = (1.007557)365/90 – 1 = 0.0310 = 3.10% 8.

The bill has a maturity of one half-year (180 days), and an annualized discount of 9.18%. Therefore, its actual percentage discount from par value is half of 9.18% = 9.18% × 1/2 = 4.59%. The bill will sell for $100,000 × (1– 0.0459) = $95,410.

The total before-tax income is $4. Since the dividend income is fully excluded from taxable income for corporations, the after-tax income is also $4, for a rate of return of $4/$40 = 10%.

10. a. The index at t = 0 is ($60 + $80 + $20)/3 = $53.33, or 53.33. At t = 1, it is ($70 + $70 + $25)/3 = $55, or 55, for a rate of return of 3.13%. Please note that index values are unit free, therefore we have used 53.33 instead of $53.33. b. Stock Q P0 Market Value P1 Market Value at Time t = 0 at Time t = 1 (Q * P0) (Q * P1) A 200 $60 $12,000 $70 $14,000 B 500 $80 $40,000 $70 $35,000 C 600 $20 $12,000 $25 $15,000 Total Market Capitalization $64,000 $64,000 Bodie et al. Investments 10th Canadian Edition Solutions Manual © 2022 McGraw-Hill Ryerson Ltd.

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Let’s arbitrarily choose a starting (t = 0) value of the value-weighted index to 100. The change in total market capitalization in t = 1 is 0 (=64000/64000 - 1). That means rate of return is zero. Therefore, the index value for t =1 would be 100 x (1+0.00) = 100.

c. Stock A B C

Before Splits P0 Q $60 200 $80 500 $20 600

After Splits P0 Q $30 400 $20 2,000 $20 600

P1 $35 $17.5 $25

After the splits the index has to remain unchanged so the divisor (which initially was 3) has to be reset. The sum of the three prices after the split is 70, while the index value before splits was 53.33. Therefore, the divisor has to reset in such a way that index value remains unchanged, i.e. $70/d = 53.33 and the new divisor must be 1.3125. The index at t = 1 is ($35 + $17.5 + $25)/1.3125 = 59.05 for a return of 10.71%. d. The total market value of A and B as well as the total market capitalization has remained unchanged after the two splits so that the return on the value-weighted index is not affected by the splits (and it is zero). 11. a. The index at t = 0 is ($90 + $50 + $100)/3 = 80. At t = 1, it is $250/3 = 83.333, for a rate of return of 4.17%. b. In the absence of a split, stock C would sell for 110, and the index would be 250/3 = 83.333. After the split, stock C sells at 55. Therefore, we need to set the divisor d such that 83.333 = (95 + 45 + 55)/d, meaning that d = 2.34. c. The index remains unchanged, as it should, since the return on each stock separately equals zero. Note: Total market capitalization Time (t) Market Cap. t=0 $39,000 t=1 $40,500 t=2 $40,500 If we set index value for t =0 to 100*39,000/39,000 = 100, then Index value for t = 1 would be 100*40,500/39,000 = 103.846 Index value for t = 2 would be 100*40,500/39,000 = 103.846 Therefore, percentage change in index from t=1 to t=2 is zero Bodie et al. Investments 10th Canadian Edition Solutions Manual © 2022 McGraw-Hill Ryerson Ltd.

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12. a. Total market value at t = 0 is ($9,000 + $10,000 + $20,000) = $39,000. Market value at t = 1 is ($9,500 + $9,000 + $22,000) = 40,500. The corresponding indices are 100 and 103.846 for t = 0 and t = 1 respectively. Therefore, the rate of return = $40,500/$39,000 – 1 = 3.85%, or 103.846/100 -1 = 3.85%. b. The return on each stock is as follows: rA = 95/90 – 1 = 0.0556 rB = 45/50 – 1 = –0.10 rC = 110/100 – 1 = 0.10 The equally-weighted index return = (0.0556-0.10+0.10)/3 =0.0185 = 1.85% 13. a. Since these two bonds are identical except in their coupon rate, the bond with higher coupon rate should be selling at higher price. b. The call with the lower exercise price because there is an inverse relationship between value of call option and exercise price. c. The put on the lower priced stock because put option becomes worthier when stock price departures below from its exercise price. d. As there is inverse relationship between T-bill yield and T-bill price, the bill with the lower yield should be selling at higher price. 14.

Preferred stock is like a long-term debt in which the firm (or issuer) typically promises a fixed dividend payment each year. Preferred stock, also, does not give the holder voting rights in the firm.

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Preferred stock is like equity (common stock) in which the firm is under no contractual obligation to make the dividend payments. Failure to make payments does not set off corporate bankruptcy. With respect to the priority of claims to the assets of the firm in the event of corporate bankruptcy, preferred stock has a higher priority than common equity but a lower priority than bonds. Generally preferred stocks like common stocks are perpetual. 15.

Value of call at expiration a. $0 b. $0 c. $0 d. $5 e. $10

–

Initial cost $4 $4 $4 $4 $4

Profit $–4 $–4 $–4 $1 $6

Value of put at expiration

–

Initial cost

Profit

a. b. c. d. e.

$10 $5 $0 $0 $0

$6 $6 $6 $6 $6

$4 $–1 $–6 $–6 $–6

16.

Generally, there are some chances (probabilities) that the option will be in-the-money at some point prior to expiration. Investors will pay something for these chances of positive payoffs.

17.

A call option conveys the right but not the obligation to buy the underlying asset at the exercise price. A long position in a futures contract carries an obligation to buy the underlying asset at the predetermined price.

18.

A put option conveys the right but not the obligation to sell the underlying asset at the exercise price. A short position in a futures contract carries an obligation to sell the underlying asset at the predetermined price.

19.

Individual response. However, on the day that we tried this experiment, 18 of the 25 stocks met this criterion, leading us to conclude that returns on stock investments can be quite volatile.

20.

The spread will widen. Deterioration of the economy increases credit risk, that is, the likelihood of default. Investors will demand a greater premium on debt securities subject to default risk.

21. a. Because the stock price exceeds the exercise price, you will choose to exercise. The payoff on the option will be $25 − $20 = $5. The option originally cost $1.92, so the gain is $5.00 − $1.92 = $3.08. Since the contracts are for 100 shares, your gain is $308.00. b. If the exercise price is $20, and the stock price $19, you would not exercise. The loss on the call would be the initial cost, which was $1.92. Your total loss is therefore $192.00. Bodie et al. Investments 10th Canadian Edition Solutions Manual © 2022 McGraw-Hill Ryerson Ltd.

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c. If the put has an exercise price of $22, you would exercise if the stock price is $19 at expiration. The put would be exercised for a gain of $22 – $19 = $3 which would give a total profit of ($3 gain – $2.29 cost)*100 = $71.00. 22. a. Enbridge closed at $39.21. b. Assuming that you buy at the closing price, you could buy $5,000/$39.21 = 127.52 shares, which we will round to 128. c. The annual dividend is 4*.$81= $3.24 per share. Your total dividend income is therefore $3.24*128=$414.72 annually. d. While the earnings per share (EPS) has been provided in the figure 2.8 as $.95, we can find it using the price-to-earnings (P/E) and the stock price. EPS = Price/ P/E ratio = $39.21/41.3 = $.95. 23. a. You bought the contract when the futures price (index points) was 962.9 (the settlement price). The contract closes at an index points of 990, which is 27.1 higher than the original futures index points. The contract multiplier is $200. Therefore, you will incur a gain of 27.1  $200 = $5420. Note: you can find more information about SFX contract it from the Montreal Exchange website: https://www.m-x.ca/produits_indices_sxf_en.php. b. Open interest (the number of outstanding contracts) on the index is 291,708 contracts. CFA PROBLEMS 1.

(d) There are tax advantages for corporations that own preferred shares and a large majority of institutional investors such as pension funds invest in preferred stocks.

(a) Writing a call entails unlimited potential losses as the stock price rises.

The equivalent taxable yield is: .0675/(1 − 0.34) = 10.23%

The taxable bond. With a zero tax bracket, the after-tax yield for the taxable bond is the same as the before-tax yield (5%), which is greater than the yield on the municipal bond.

The taxable bond. The after-tax yield for the taxable bond is: 0.05  (1 – 0.10) = 4.5%

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The after-tax yield is the same as that of the municipal bond. d.

The municipal bond offers the higher after-tax yield for investors in tax brackets above 20%.

If the after-tax yields are equal, then: 0.056 = 0.08 × (1 – t) This implies that t = 0.30 =30%.

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CHAPTER 3 TRADING ON SECURITIES MARKETS PROBLEMS: 1.

Individual solution—answers to this problem will vary. In general, full-service brokers charge more fees/commission on trading than the discount brokers because full-service entails cost for the resources such as personnel (financial advisors), facilities and technologies.

2. a.

In principle, potential losses are unbounded, growing directly with increases in the share price of Restaurant Brands.

If the stop-buy order can be filled at $78, the maximum possible loss per share is $8. If Restaurant Brand’s shares go above $78, the stop-buy order is executed, limiting the losses from the short sale.

3. a.

The stock is purchased for 300  $40 = $12,000. Borrowed funds are $4,000. Therefore, the investor put up equity or margin of $8,000.

If the share price falls to $30, the total value of the stocks falls to $9,000. The amount of the loan owed to the broker grows to $4,000  1.08 = $4,320. Therefore, remaining margin is $9,000 − $4,320 = $4,680. The percentage margin is now $4,680/$9,000 = 0.52 = 52%, so there will not be a margin call. c. The rate of return on investment over the years is (Ending value of account − Initial equity)/Initial equity = ($4,680 − $8,000)/$8,000 = −0.415 = −41.5%.

a. The initial margin was 0.50  1,000  $40 = $20,000. Old Economy Traders loses $10  1,000 = $10,000 due to the increase in the stock price so margin falls by $10,000. Moreover, the firm must pay the dividend of $2 per share, which means the margin account falls by an additional $2,000. So, the remaining margin is $8,000. b. The percentage margin is $8,000/$50,000 = 0.16 = 16%, so there will be a margin call. c. The margin in the account fell from $20,000 to $8,000 in one year, for a rate of return of −$12,000/$20,000 = −0.60 = −60%. Bodie et al. Investments 10th Canadian Edition Solutions Manual ..

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The stop-loss order will be executed as soon as the stock price hits the limit price. If the stock price later rebounds, the investor does not participate in the gains because the stock has been sold. In contrast, the put option need not be exercised when the stock price falls below the exercise price. An investor who owns a share of stock and a put option can hold on to both securities. If the stock price never rebounds, the put can be exercised eventually, and the stock sold for the exercise price. This provides the same downside protection as the stop-loss order. If the price does rebound, however, the investor benefits because the stock is still held. This advantage of the put over the stop-loss order justifies the cost of the put.

Calls are options to purchase a stock at any time prior to expiration. Stop-buys require purchase as soon as the stock price hits the limit. The advantage of the call over the stop-buy is that the investor need not commit to buying until expiration. If the stock price later falls, the holder of the call can choose not to purchase.

Placing a stop-loss order to sell at $38, you are telling your broker to sell Barrick stock as soon as a sale takes place at a price of $38 or less. Here, the broker will attempt to execute your order considering the bid price. Since the bid price now is $37.80 which is below $38, the broker executes your order (at current market price) and sell the stock at $37.80.

The broker is instructed to attempt to sell your Kinross stock as soon as the Kinross stock trades at a bid price of $11.50 or less. Here, the broker will attempt to execute, but may not be able to sell at $11.50, since the bid price is now $11.47. The price at which you sell may be more or less than $11.50 because the stop-loss becomes a market order to sell at current market prices. If the bid has sufficient quantity you are likely to get $11.47, however.

9. a.

The buy order will be filled at the best limit-sell order, $50.25.

At the next-best price, $51.50.

You should increase your position. There is considerable buy pressure at prices just below $50, meaning that downside risk is limited. In contrast, sell pressure is sparse, meaning that a moderate buy order could result in a substantial price increase.

10. The system expedites the flow of market orders or limit orders from exchange members to the specialists. It allows members to send computerized orders directly to the floor of the exchange, which allows the nearly simultaneous sale of each stock in a large portfolio. This capability is necessary for program trading. 11. The dealer (or market maker). Spreads should be higher on inactive stocks and lower on Bodie et al. Investments 10th Canadian Edition Solutions Manual ..

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CHAPTER 6: CAPITAL ALLOCATION TO RISKY ASSETS PROBLEM SETS 1.

(d) While a higher or lower Sharpe ratios are not an indication of an investor's tolerance for risk, any investor will always prefer investment portfolios with higher Sharpe ratios. The Sharpe ratio is simply a tool to absolutely measure the return premium earned per unit of risk.

(b) A higher borrowing rate is a consequence of the risk of the borrowers’ default. In perfect markets with no additional cost of default, this increment would equal the value of the borrower’s option to default, and the Sharpe measure, with appropriate treatment of the default option, would be the same. However, in reality there are costs to default so that this part of the increment lowers the Sharpe ratio. Also, notice that answer (c) is not correct because doubling the expected return and standard deviation given the fixed risk-free rate will lead to more than double the risk premium which results into a higher Sharpe ratio but won’t double the Sharpe ratio because the divisor in the Sharpe ratio (portfolio standard deviation) would also be doubled.

Assuming no change in risk tolerance, that is, an unchanged risk-aversion coefficient (A), higher perceived volatility increases the denominator of the equation for the optimal investment in the risky portfolio (Equation 6.7). The proportion invested in the risky portfolio will therefore decrease.

The expected cash flow is: (0.5 × $70,000) + (0.5 × 200,000) = $135,000. With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%. Therefore, the present value of the portfolio is: $135,000/1.14 = $118,421

If the portfolio is purchased for $118,421 and provides an expected cash inflow of $135,000, then the expected rate of return [E(r)] is as follows: $118,421 × [1 + E(r)] = $135,000 Therefore, E(r) = 14%. The portfolio price is set to equate the expected rate of return with the required rate of return.

If the risk premium over T-bills is now 12%, then the required return is: 6% + 12% = 18% The present value of the portfolio is now: Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-1

$135,000/1.18 = $114,407 d.

For a given expected cash flow, portfolios that command greater risk premiums must sell at lower prices. The extra discount from expected value is a penalty for risk.

When we specify utility by U = E(r) – 0.5Aσ2, the utility level for T-bills is: 0.07 The utility level for the risky portfolio is: U = 0.12 – 0.5 × A × (0.18)2 = 0.12 – 0.0162 × A In order for the risky portfolio to be preferred to bills, the following must hold: 0.12 – 0.0162A > 0.07  A < 0.05/0.0162 = 3.09 Therefore, the risk aversion coefficient, A must be less than 3.09 for the risky portfolio to be preferred to bills.

Points on the curve are derived by solving for E(r) in the following equation: U = 0.05 = E(r) – 0.5Aσ2 = E(r) – 1.5σ2 The values of E(r), given the values of σ2, are therefore:  0.00 0.05 0.10 0.15 0.20 0.25

2 0.0000 0.0025 0.0100 0.0225 0.0400 0.0625

E(r) 0.05000 0.05375 0.06500 0.08375 0.11000 0.14375

The bold line in the graph on the next page (labeled Q6, for Question 6) depicts the indifference curve. 7.

Repeating the analysis in Problem 6 but with risk aversion coefficient, A=4, the utility equation is now: U = E(r) – 0.5Aσ2 = E(r) – 2.0σ2 = 0.05 The equal-utility combinations of expected return and standard deviation are presented in the table below. The indifference curve is the upward sloping line in the graph on the next page, labeled Q7 (for Question 7).  0.00 0.05 0.10

2 0.0000 0.0025 0.0100

E(r) 0.0500 0.0550 0.0700

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-2

0.15 0.20 0.25

0.0225 0.0400 0.0625

0.0950 0.1300 0.1750

The indifference curve in Problem 7 differs from that in Problem 6 in slope. When A increases from 3 to 4, the increased risk aversion results in a greater slope for the indifference curve since more expected return is needed in order to compensate for additional σ.

E(r) U(Q7,A=4)

U(Q6,A=3)

U(Q8,A=0) U(Q9,A<0)

The coefficient of risk aversion for a risk neutral investor is zero. Therefore, the corresponding utility is equal to the portfolio’s expected return. The corresponding indifference curve in the expected return-standard deviation plane is a horizontal line, labeled Q8 in the graph above (see Problem 6).

A risk lover, rather than penalizing portfolio utility to account for risk, derives greater utility as variance increases. This amounts to a negative coefficient of risk aversion. The corresponding indifference curve is downward sloping in the graph above (see Problem 6) and is labeled Q9. Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-3

10.

The portfolio expected return and variance are computed as follows: rPortfolio = WBill* rBills + WIndex* rIndex Portfolio = WIndex* Index | Bills=0 (1) WBills 0.0 0.2 0.4 0.6 0.8 1.0

11.

(2) rBills 5% 5 5 5 5 5

(3) WIndex 1.0 0.8 0.6 0.4 0.2 0.0

(4) rIndex 13.0% 13.0 13.0 13.0 13.0 13.0

rPortfolio (1)×(2)+(3)×(4) 13.0% = 0.130 11.4% = 0.114 9.8% = 0.098 8.2% = 0.082 6.6% = 0.066 5.0% = 0.050

Portfolio (3) × 20% 20% = 0.20 16% = 0.16 12% = 0.12 8% = 0.08 4% = 0.04 0% = 0.00

 2 Portfolio 0.0400 0.0256 0.0144 0.0064 0.0016 0.0000

Computing utility for an investor with A =2. From U = E(r) – 0.5 × Aσ2 = E(r) – σ2, we arrive at the values in the column Utility in the following table: WBills 0.0 0.2 0.4 0.6 0.8 1.0

WIndex 1.0 0.8 0.6 0.4 0.2 0.0

rPortfolio 0.130 0.114 0.098 0.082 0.066 0.050

Portfolio 0.20 0.16 0.12 0.08 0.04 0.00

2Portfolio 0.0400 0.0256 0.0144 0.0064 0.0016 0.0000

Utility 0.0900 0.0884 0.0836 0.0756 0.0644 0.0500

The column Utility implies that investors with A = 2 prefer a portfolio that is invested 100% in the market index to any of the other portfolios in the table.

12.

Computing utility for an investor with A=3. From U = E(r) – 0.5Aσ2 = E(r) – 1.5σ2, we derive utility values as follows: WBills 0.0 0.2 0.4 0.6 0.8 1.0

WIndex 1.0 0.8 0.6 0.4 0.2 0.0

rPortfolio 0.130 0.114 0.098 0.082 0.066 0.050

Portfolio 0.20 0.16 0.12 0.08 0.04 0.00

2Portfolio 0.0400 0.0256 0.0144 0.0064 0.0016 0.0000

Utility .0700 .0756 .0764 .0724 .0636 .0500

The more risk averse investors prefer the portfolio that is invested 40% in the market, Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-4

rather than the 100% market weight preferred by investors with A = 2.

13.

Expected return = (0.7 × 18%) + (0.3 × 8%) = 15% Standard deviation = 0.7 × 28% = 19.6%

14.

Investment proportions in overall portfolio:

30.0% in T-bills 17.5% in Stock A 22.4% in Stock B 30.1% in Stock C 100% (total weights in overall portfolio)

0.7 × 25% = 0.7 × 32% = 0.7 × 43% =

15.

Reward-to-volatility (Sharpe) ratio of your (risky) portfolio: S =

.18 − .08 = 0.3571 .28

Reward-to-volatility (Sharpe) ratio of your client’s (overall) portfolio : .15 − .08 S= = 0.3571 .196

16. 30

CAL (Slope = 0.3571)

25 20 E(r)% 15

Client

10 5 0 0

 () 17.

E(rC) = rf + y × [E(rP) – rf] = 8 + y × (18 − 8) Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-5

If the expected return for the overall or complete portfolio is 16%, then: 16% = 8% + 10% × y  y =

.16 − .08 = 0.8 .10

Therefore, in order to have a portfolio with expected rate of return equal to 16%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills. b. Client’s investment proportions in complete portfolio: 0.8 × 25% = 0.8 × 32% = 0.8 × 43% =

18.

σC = 0.8 × σP = 0.8 × 28% = 22.4%

σC = y × 28%

20.0% in T-bills 20.0% in Stock A 25.6% in Stock B 34.4% in Stock C

If your client prefers a standard deviation of at most 18%, then: y = 18/28 = 0.6429 = 64.29% invested in the risky portfolio.

19.

E(rc) = .08+.1*y = .08+(.6429*.1)=0.1443 = 14.43%

y* =

E(rP ) − r f Aσ

2 P

0.18 − 0.08 0.10 = = 0.3644 2 0.2744 3.5  0.28

Therefore, the client’s optimal proportions are: 36.44% invested in the risky portfolio and 63.56% invested in T-bills. b.

E(rC) = 0.08 + 0.10 × y* = 0.08 + (0.3644 × 0.1) = 0.1164 or 11.64% C = 0.3644 × 28 = 10.20%

20.

If the period 1957–2019 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity: A = 4, E(rM) =10.25%, rf = 5.59%, σM = 16.56% (we use the standard deviation of the risk premium from Table 6.7). Then y* is given by: y* = (E(rM) − rf) / Aσ2M = 0.0466/ (4 x 0.16562) = 0.4248 That is, 42.48% of the portfolio should be allocated to equity and 57.52% should be allocated to T-bills. Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-6

If the period 1985–1998 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity: A = 4, E(rM) − rf = 3.27%, σM = 14.18% and y* is given by:

y* = (E(rM) − rf) / Aσ2M = 0.0327/ (4 x 0.14182) = 0.4066 Therefore, 40.66% of the complete portfolio should be allocated to equity and 59.34% should be allocated to T-bills.

21.

22.

In part (b), the market risk premium as well as the standard deviation of market returns are lower than in part (a). The reward-to-volatility ratio in part (b) is 0.231 compared to 0.281 in the part (a). These results indicate that the greater proportion invested in T-bills in part (b).

E(rC) = 8% = 5% + y × (11% – 5%)  y =

σC = y × σP = 0.50 × 15% = 7.5%

The first client is more risk averse, preferring investments that have less risk as evidenced by the lower standard deviation.

.08 − .05 = 0.5 .11 − .05

Johnson requests the portfolio standard deviation to equal one half the market portfolio standard deviation. The market portfolio  M = 20% , which implies  P = 10% . The intercept of the CML equals rf = 0.05 and the slope of the CML equals the Sharpe ratio for the market portfolio (7%/20% = 0.35). Therefore using the CML: E (rP ) = rf +

E (rM ) − rf

M

 P = 0.05 + 0.35  0.10 = 0.085 = 8.5%

23. Data: rf = 5%, E(rM) = 13%, σM = 25%, and rfB = 9% The CML and indifference curves are as follows:

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-7

24.

For y to be less than 1.0 (that the investor is a lender), risk aversion (A) must be large enough such that: y=

E(rM ) − r f Aσ

2 M

1  A 

0.13 − 0.05 = 1.28 0.25 2

For y to be greater than 1 (the investor is a borrower), A must be small enough: y=

E (rM ) − r f Aσ

2 M

1  A 

0.13 − 0.09 = 0.64 0.25 2

For values of risk aversion within this range, the client will neither borrow nor lend but will hold a portfolio composed only of the optimal risky portfolio: y = 1 for 0.64 ≤ A ≤ 1.28

25.

The graph for Problem 23 has to be redrawn here, with: E(rP) = 11% and σP = 15%

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-8

0.11 − 0.05 = 2.67 0.15 2 0.11 − 0.09 = 0.89 For a borrowing position: A  0.15 2 For a lending position: A 

Therefore, y = 1 for 0.89 ≤ A ≤ 2.67

26.

The maximum feasible fee, denoted f, depends on the reward-to-variability ratio. For y < 1, the lending rate, 5%, is viewed as the relevant risk-free rate, and we solve for f as follows:

.11 − .05 − f .13 − .05 .15  .08 = .012, or 1.2% =  f = .06 − .15 .25 .25 For y > 1, the borrowing rate, 9%, is the relevant risk-free rate. Then we notice that, even without a fee, the active fund is inferior to the passive fund because: .11 – .09 – f .13 – .09 = 0.13 < = 0.16 → f = –.004 .15 .25 More risk tolerant investors (who are more inclined to borrow) will not be clients of the fund. We find that f is negative: that is, you would need to pay investors to choose your active fund. These investors desire higher risk–higher return complete portfolios and thus are in the borrowing range of the relevant CAL. In this range, the reward-to-variability ratio of the index (the passive fund) is better than that of the managed fund.

27.

Slope of the CML =

.13 − .08 = 0.20 .25

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-9

The diagram follows.

CML and CAL 18 16

CAL: Slope = 0.3571

Expected Retrun

14 12 10

CML: Slope = 0.20

8 6 4 2 0 0

Standard Deviation

28.

My fund allows an investor to achieve a higher mean for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.

With 70% of his money invested in my fund’s portfolio, the client’s expected return is 15% per year with a standard deviation of 19.6% per year. If he shifts that money to the passive portfolio (which has an expected return of 13% and standard deviation of 25%), his overall expected return becomes: E(rC) = rf + 0.7 × [E(rM) − rf] = .08 + [0.7 × (.13 – .08)] = .115, or 11.5% The standard deviation of the complete portfolio using the passive portfolio would be: σC = 0.7 × σM = 0.7 × 25% = 17.5% Therefore, the shift entails a decrease in mean from 15% to 11.5% and a decrease in standard deviation from 19.6% to 17.5%. Since both mean return and standard deviation decrease, it is not yet clear whether the move is beneficial. The disadvantage of the shift is that, if the client is willing to accept a mean return on his total portfolio of 11.5%, he can achieve it with a lower standard deviation using my fund rather than the passive portfolio. To achieve a target mean of 11.5%, we first write the mean of the complete portfolio as a function of the proportion invested in my fund (y): E(rC) = .08 + y × (.18 − .08) = .08 + .10 × y Our target is: E(rC) = 11.5%. Therefore, the proportion that must be invested in my fund is determined as follows: Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-10

.115 = .08 + .10 × y  y =

.115 − .08 = 0.35 .10

The standard deviation of this portfolio would be: σC = y × 28% = 0.35 × 28% = 9.8% Thus, by using my portfolio, the same 11.5% expected return can be achieved with a standard deviation of only 9.8% as opposed to the standard deviation of 17.5% using the passive portfolio. b.

The fee would reduce the reward-to-volatility ratio, i.e., the slope of the CAL. The client will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let f denote the fee: Slope of CAL with fee =

.18 − .08 − f .10 − f = .28 .28

Slope of CML (which requires no fee) =

.13 − .08 = 0.20 .25

Setting these slopes equal we have:

.10 − f = 0.20  f = 0.044 = 4.4% per year .28

29.

The formula for the optimal proportion to invest in the passive portfolio is: y* =

E (rM ) − r f Aσ 2M

Substitute the following: E(rM) = 13%; rf = 8%; σM = 25%; A = 3.5:

y* = b.

0.13 − 0.08 = 0.2286, or 22.86% in the passive portfolio 3.5  0.252

You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. You have a client with a degree of risk aversion A = 3.5. Then leave the questions (a and b) as they are.

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-11

CFA PROBLEMS 1.

Utility for each investment = E(r) – 0.5 × 4 × σ2 We choose the investment with the highest utility value, Investment 3. Expected return Investment E(r) 1 0.12 2 0.15 3 0.21 4 0.24

Standard deviation  0.30 0.50 0.16 0.21

Utility U -0.0600 -0.3500 0.1588 0.1518

When investors are risk neutral, then A = 0; the investment with the highest utility is Investment 4 because it has the highest expected return.

(b)

Indifference curve 2 because it is tangent to the CAL.

Point E. Note that the tangent point F is the optimal complete portfolio.

(0.6 × $50,000) + [0.4 × (−$30,000)] − $5,000 = $13,000

(b) Higher borrowing rates will reduce the total return to the portfolio and this results in a part of the line that has a lower slope.

Expected return for equity fund = T-bill rate + Risk premium = 6% + 10% = 16% Expected rate of return of the client’s portfolio = (0.6 × 16%) + (0.4 × 6%) = 12% Expected return of the client’s portfolio = 0.12 × $100,000 = $12,000 (which implies expected total wealth at the end of the period = $112,000) Standard deviation of client’s overall portfolio = 0.6 × 14% = 8.4%

Reward-to-volatility ratio =

.10 = 0.71 .14

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-12

CHAPTER 6: APPENDIX 1.

By year-end, the $50,000 investment in risk-free asset will grow to: $50,000 × 1.06 = $53,000 Without insurance, the probability distribution of end-of-year wealth is: No fire Fire

Probability 0.999 0.001

Wealth $253,000 53,000

For this distribution, expected utility is computed as follows: E[U(W)] = [0.999 × ln(253,000)] + [0.001 × ln(53,000)] = 12.439582 The certainty equivalent is: WCE = e 12.439582 = $252,604.85 With fire insurance, at a cost of $P, the investment in the risk-free asset is: $(50,000 – P) Year-end wealth will be certain (since you are fully insured) and equal to: [$(50,000 – P) × 1.06] + $200,000 Solve for P in the following equation: [$(50,000 – P) × 1.06] + $200,000 = $252,604.85  P = $372.78 This is the most you are willing to pay for insurance. Note that the expected loss is “only” $200, so you are willing to pay a substantial risk premium over the expected value of losses. The primary reason is that the value of the house is a large proportion of your wealth. 2.

With insurance coverage for one-half the value of the house, the premium is $100, and the investment in the safe asset is $49,900. By year-end, the investment of $49,900 will grow to: $49,900 × 1.06 = $52,894 If there is a fire, your insurance proceeds will be $100,000, and the probability distribution of end-of-year wealth is: No fire Fire

Probability 0.999 0.001

Wealth $252,894 152,894

For this distribution, expected utility is computed as follows: E[U(W)] = [0.999 × ln(252,894)] + [0.001 × ln(152,894)] = 12.4402225 The certainty equivalent is: WCE = e 12.4402225 = $252,766.77

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-13

With insurance coverage for the full value of the house, costing $200, end-of-year wealth is certain, and equal to: [($50,000 – $200) × 1.06] + $200,000 = $252,788 Since wealth is certain, this is also the certainty equivalent wealth of the fully insured position.

With insurance coverage for 1½ times the value of the house, the premium is $300, and the insurance pays off $300,000 in the event of a fire. The investment in the safe asset is $49,700. By year-end, the investment of $49,700 will grow to: $49,700 × 1.06 = $52,682 The probability distribution of end-of-year wealth is: No fire Fire

Probability 0.999 0.001

Wealth $252,682 352,682

For this distribution, expected utility is computed as follows: E[U(W)] = [0.999 × ln(252,682)] + [0.001 × ln(352,682)] = 12.4402205 The certainty equivalent is: WCE = e 12.440222 = $252,766.27 Therefore, full insurance dominates both over- and underinsurance. Overinsuring creates a gamble (you actually gain when the house burns down). Risk is minimized when you insure exactly the value of the house.

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 6-14

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS PROBLEM SETS 1.

(a) and (e). Short-term rates and labor issues are factors that are common to all firms and therefore must be considered as market risk factors. The remaining three factors are unique to this corporation and are not a part of market risk.

(a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash, and real estate. Portfolio variance now includes a variance term for real estate returns and a covariance term for real estate returns with returns for each of the other three asset classes. Therefore, portfolio risk is affected by the variance (or standard deviation) of real estate returns and the correlation between real estate returns and returns for each of the other asset classes. (Note that the correlation between real estate returns and returns for cash is most likely zero.)

(a) Answer (a) is valid because it provides the definition of the minimum variance portfolio.

The parameters of the opportunity set are: E(rS) = 20%, E(rB) = 12%, σS = 30%, σB = 15%, ρ = 0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS , rB ) =    S   B ]: Bonds Stocks

Bonds 225 45

Stocks 45 900

The minimum-variance portfolio for two risky asset can be computed as follows: wMin(S) =

 B2 − Cov(rS ,rB ) 225 − 45 = = 0.1739 2 2  S +  B − 2Cov(rS ,rB ) 900 + 225 − (2  45)

wMin(B) = 1 − 0.1739 = 0.8261 The expected return or mean and standard deviation of the minimum variance portfolio are: E(rMin) = (0.1739 × .20) + (0.8261 × .12) = .1339 = 13.39% σMin = [ wS2 S2 + wB2  B2 + 2wS wB Cov (rS , rB )]1 / 2 Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-1

= [(0.17392  900) + (0.82612  225) + (2  0.1739  0.8261  45)]1/2 = 13.92% 5. Proportion in Stock Fund

Proportion in Bond Fund

Expected Return

Standard Deviation

0.00%

100.00%

12.00%

15.00%

17.39 20.00 40.00

82.61 80.00 60.00

13.39 13.60 15.20

13.92 13.94 15.70

45.16 60.00 80.00 100.00

54.84 40.00 20.00 0.00

15.61 16.80 18.40 20.00

16.54 19.53 24.48 30.00

Minimum variance portfolio

Tangency portfolio

Graph shown below.

25.00

INVESTMENT OPPORTUNITY SET CML

20.00

Tangency Portfolio

Efficient frontier of risky assets

15.00

Minimum Variance Portfolio

10.00

rf = 8.00 5.00

0.00 0.00

5.00

10.00

15.00

20.00

25.00

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-2

30.00

The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. See numerical solutions to next problem to derive these values.

The proportion the stock fund in the optimal risky portfolio is given by:

wS = =

[ E (rS ) − rf ]   B2 − [ E (rB ) − rf ]  Cov(rS , rB ) [ E (rS ) − rf ]   B2 + [ E (rB ) − rf ]   S2 − [ E (rS ) − rf + E (rB ) − rf ]  Cov(rS , rB ) [(.20 − .08)  225] − [(.12 − .08)  45] = 0.4516 [(.20 − .08)  225] + [(.12 − .08)  900] − [(.20 − .08 + .12 − .08)  45]

wB = 1 − 0.4516 = 0.5484

The mean and standard deviation of the optimal risky portfolio are: E(rP) = (0.4516 × .20) + (0.5484 × .12) = .1561 = 15.61% σp = [(0.45162  900) + (0.54842  225) + (2  0.4516  0.5484 × 45)]1/2 = 16.54% 8.

The reward-to-volatility ratio of the optimal CAL is: E (rp ) − rf

p 9.

.1561 − .08 = 0.4601 .1654

If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is:

E (rC ) = rf +

E (rp ) − rf

P

 C = .08 + 0.4601 C

If E(rC) is equal to 14%, then the standard deviation of the portfolio is 13.04%. b.

To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the risky portfolio P. The mean of any portfolio along the optimal CAL is: E (rC ) = (1 − y )  rf + y  E (rP ) = rf + y  [ E (rP ) − rf ] = .08 + y  (.1561 − .08)

Setting E(rC) = 14% we find: y = 0.7884 and (1 − y) = 0.2119 (the proportion Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-3

invested in the T-bill fund). To find the proportions of investments in stocks and bonds in the complete portfolio, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7884  0.4516 = 0.3560 Proportion of bonds in complete portfolio = 0.7884  0.5484 = 0.4323 Note that the sum of the proportions in bonds and stock should be 0.7884. Here the sum is 0.7883 which is due to rounding to four decimal places. 10. Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund (wS) and the appropriate proportion in the bond fund (wB = 1 − wS) as follows: 0.14 = 0.20 × wS + 0.12 × (1 − wS) = 0.12 + 0.08 × wS  wS = 0.25 So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: σP = [(0.252  900) + (0.752  225) + (2  0.25  0.75  45)]1/2 = 14.13% This is considerably greater than the standard deviation of 13.04% achieved using T-bills and the optimal risky portfolio. 11.

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-4

Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low or negative, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio.

12.

If the correlation between gold and stocks equals +1, then no one would hold gold. The optimal CAL would be composed of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), these combinations would be dominated by the stock portfolio.

Of course, this situation in part (b) above could not persist. If no one desires gold, its price would fall and its expected rate of return would increase until it become sufficiently attractive to include in a portfolio.

Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio [with the proportion wA invested in Stock A and wB = (1 – wA ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation is: σP = Absolute value [wAσA − wBσB] Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-5

0 = 5 × wA − [10  (1 – wA)]  wA = 2/3 = 0.6667 The expected rate of return for this risk-free portfolio is: E(r) = (0.6667 × 10) + (0.3333 × 15) = 11.667% Therefore, the risk-free rate is: 11.667% 13.

False. If the borrowing and lending rates are not identical, then, depending on the tastes of the individuals (that is, the shape of their indifference curves), borrowers and lenders could have different optimal risky portfolios.

14.

False. The portfolio standard deviation equals the weighted average of the component-asset standard deviations if and only if all assets in the portfolio are perfectly positively correlated. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights.

15.

The probability distribution is: Probability 0.6 0.4

Rate of Return 100% −50

Mean = [0.6 × 100%] + [0.4 × (-50%)] = 40% Variance = [0.6 × (100 − 40)2] + [0.4 × (-50 − 40)2] = 5400 Standard deviation = (5425)1/2 = 73.48%

16.

σP = 30 = y × σ = 40 × y  y = 0.75 E(rP) = 12 + 0.75(30 − 12) = 25.5%

17.

The correct choice is (c). Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the stock that has the lowest correlation with Stock A. More formally, we note that when all stocks have the same expected rate of return, the optimal portfolio for any risk-averse investor is the global minimum variance portfolio (G). When the portfolio is restricted to Stock A and one additional stock, the objective is to find G for any pair that includes Stock A, and then select the combination with the lowest variance. With two stocks, I and J, the formula for the Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-6

weights in G is:

 J2 − Cov(rI , rJ ) wMin ( I ) = 2  I +  J2 − 2Cov(rI , rJ ) wMin ( J ) = 1 − wMin ( I )

Since all standard deviations are equal to 20%: Cov(rI , rJ ) =  I  J = 400  and wMin ( I ) = wMin ( J ) = 0.5

This intuitive result is an implication of a property of any efficient frontier, namely, that the covariances of the global minimum variance portfolio with all other assets on the frontier are identical and equal to its own variance. (Otherwise, additional diversification would further reduce the variance.) In this case, the standard deviation of G(I, J) reduces to:

 Min (G ) = [200  (1 +  IJ )]1/2 This leads to the intuitive result that the desired addition would be the stock with the lowest correlation with Stock A, which is Stock D. The optimal portfolio is equally invested in Stock A and Stock D, and the standard deviation is 17.03%. 18. No, the answer to Problem 17 would not change, at least as long as investors are not risk lovers. Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8%. 19. Yes, the answers to Problems 17 and 18 would change. The efficient frontier of risky assets is horizontal at 8%, so the optimal CAL runs from the risk-free rate through G. This implies risk-averse investors will just hold Treasury bills. 20.

Rearrange the table (converting rows to columns) and compute serial correlation results in the following table: Nominal Rates Small Company Stocks

Large Company Stocks

Long-Term Government Bonds

Treasury Bills

Inflation

-3.72

18.36

3.98

3.56

-1.00

1930s

7.28

-1.25

4.60

0.30

-2.04

1940s

20.63

9.11

3.59

0.37

5.36

1950s

19.01

19.41

0.25

1.87

2.22

1960s

13.72

7.84

1.14

3.89

2.52

1970s

8.75

5.90

6.63

6.29

7.36

1920s

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-7

1980s

12.46

17.60

11.50

9.00

1990s

13.84

2000s

6.70

Serial Correlation

0.34

-0.35

5.10

18.20

8.60

5.02

2.93

-1.00

5.00

2.70

2.50

0.55

0.59

0.23

For example: to compute serial correlation in decade nominal returns for largecompany stocks, we set up the following two columns in an Excel spreadsheet. Then, use the Excel function “CORREL” to calculate the correlation for the data. 1930s 1940s 1950s 1960s 1970s 1980s 1990s 2000s

Decade Previous -1.25% 18.36% 9.11% -1.25% 19.41% 9.11% 7.84% 19.41% 5.90% 7.84% 17.60% 5.90% 18.20% 17.60% -1.00% 18.20%

Note that each correlation is based on only seven observations, so we cannot arrive at any statistically significant conclusions. Looking at the results, however, it appears that, with the exception of large-company stocks, there is persistent positive serial correlation or autocorrelation. 21.

The table for real rates (using the approximation of subtracting a decade’s average inflation from the decade’s average nominal return) is: Real Rates Small Company Stocks

Large Company Stocks

Long-Term Government Bonds

Treasury Bills

1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s 2000s

-2.72 9.32 15.27 16.79 11.20 1.39 7.36 10.91 4.20

19.36 0.79 3.75 17.19 5.32 -1.46 12.50 15.27 -3.5

4.98 6.64 -1.77 -1.97 -1.38 -0.73 6.40 5.67 2.5

4.56 2.34 -4.99 -0.35 1.37 -1.07 3.90 2.09 0.2

Serial Correlation

0.20

-0.38

0.37

0.00

While the serial correlation in decade nominal returns seems to be positive, it appears that real rates are serially less correlated for Small Cap stocks, Large Cap stocks and Long-term T-bonds. The coefficients are relatively small. However, in the case of T-bills, there is no serial correlation. The decade time series (although again too short for any definitive conclusions) suggest that real rates of return are Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-8

independent from decade to decade. 22. The risk premium for the S&P portfolio is: (1 + .05)1 − 1 = 0.05 The risk premium for the hedge fund portfolio is (1 + .1)1 − 1 = 0.1 The S&P standard deviation is: 0.2  1 = 0.20 . The hedge fund standard deviation is: 0.35  1 = 0.35 S&P Sharpe ratio is 5/20 = 0.25 The hedge fund Sharpe ratio is 10/35 = 0.2857. With a (S&P,Hedge) = 0, the optimal asset allocation is 5  352 − 10  (0  20  35) WS & P = = 0.6049 5  352 + 10  202 − (5 + 10)  (0  20  35) WHedge = 1 − 0.6049 = 0.3951 . With these weights, E (rP ) = 0.6049  5 + 0.395110 = 0.0698 = 6.9753% 23.

 P = .60492  202 + .39512  352 + 2  .6049  .3951 (0  20  35) = .1837 = 18.3731% The resulting Sharpe ratio is 6.9753/18.3731= 0.3796 24. Greta has a risk aversion of A=3, Therefore, she will invest .06975 y= = 0.6888 = 68.88% 3  .1837 2 of her wealth in this risky portfolio and remaining 31.11% in Risk-free asset. The resulting investment composition will be S&P: 0.6888 .6049 = 41.67% Hedge: 0.6888 .3951 = 27.21%. Risk-free asset: 31.11% Total 100% 25. With ρ = 0.3, the annual covariance is .3  .2  .35 = 0.021 . 26.

With a ρ = .3, the optimal asset allocation is

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-9

5  352 − 10  (0.3  20  35) = 0.5771 5  352 + 10  202 − (5 + 10)  (0.3  20  35) WHedge = 1 − 0.5771 = 0.4229 . With these weights, E (rP ) = 0.5771 5 + 0.4229 10 = 0.0711 = 7.1147% WS & P =

 P = .57712  202 + .42292  352 + 2  .5771 .4229  (.3  20  35) = .2133 The resulting Sharpe ratio is 7.11/21.33 = 0.3336. 27. Greta has a risk aversion of A=3, Therefore, she will invest 0.07115 y= = 0.5214 = 52.14% 3  .21332 of her wealth in this risky portfolio. The resulting investment composition will be S&P: 0.5214 0.5771 = 30.09% and Hedge: .5214 .4229 = 22.05%. The remaining 47.86% will be invested in the risk-free asset.

CFA PROBLEMS 1.

Restricting the portfolio to 20 stocks, rather than 40, will increase the risk of the portfolio, but it is possible that the increase in risk will be minimal. Suppose that, for instance, the 40 stocks in a universe have the same standard deviation () and the correlations between each pair are identical, with correlation coefficient ρ. Then, the covariance between each pair of stocks would be ρσ2, and the variance of an equally weighted portfolio would be: σ 2P =

1 2 n −1 2 σ + ρσ n n

The effect of the reduction in n on the second term on the right-hand side would be relatively small (since 39/40 is close to 19/20 and ρσ2 is smaller than σ2), but the denominator of the first term would be 20 instead of 40. For example, if σ = 45% and ρ = 0.2, then the standard deviation with 40 stocks would be 21.11%, and would rise to 22.05% when only 20 stocks are held. Such an increase might be acceptable if the expected return is increased sufficiently. b.

Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio. This entails maintaining a low correlation among the remaining stocks. For example, in part (a), with ρ = 0.2, the increase in portfolio risk was minimal. Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-10

As a practical matter, this means that Hennessy would have to spread his portfolio among many industries; concentrating on just a few industries would result in higher correlations among the included stocks. 2.

Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. Rather, the incremental benefits from additional diversification are most important when you are least diversified. Restricting Hennessy to 10 instead of 20 issues would increase the risk of his portfolio by a greater amount than would a reduction in the size of the portfolio from 30 to 20 stocks. In our example, restricting the number of stocks to 10 will increase the standard deviation to 23.81%. The 1.76% increase in standard deviation resulting from giving up 10 of 20 stocks is greater than the .94% increase that results from giving up 20 of 40 stocks.

The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessy’s portfolio is only one of six welldiversified portfolios and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous.

Portfolio Y cannot be efficient because it is dominated by another portfolio. For example, Portfolio X has both higher expected return and lower standard deviation.

10. Since we do not have any information about expected returns, we focus exclusively on reducing variability. Stocks A and C have equal standard deviations, but the correlation of Stock B with Stock C (0.10) is less than that of Stock A with Stock B (0.90). Therefore, a portfolio composed of Stocks B and C will have lower total risk than a portfolio composed of Stocks A and B. Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-11

11.

Fund D represents the single best addition to complement Stephenson's current portfolio, given his selection criteria. Fund D’s expected return (14.0 percent) has the potential to increase the portfolio’s return somewhat. Fund D’s relatively low correlation with his current portfolio (+0.65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio. The other three funds have shortcomings in terms of expected return enhancement or volatility reduction through diversification. Fund A offers the potential for increasing the portfolio’s return but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits but is expected to generate a return well below the current portfolio’s return. Fund C has the greatest potential to increase the portfolio’s return but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification.

12. a.

Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9  0.67) + (0.1  1.25) = 0.728% ii. Cov = ρ  OP  ABC = 0.40  2.37  2.95 = 2.7966  2.80 iii. NP = [wOP2 OP2 + wABC2 ABC2 + 2 wOP wABC (CovOP , ABC)]1/2 = [(0.9 2  2.372) + (0.12  2.952) + (2  0.9  0.1  2.80)]1/2 = 2.2673%  2.27%

Subscript OP refers to the original portfolio, GS to government securities, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9  0.67) + (0.1  0.42) = 0.645% ii. Cov = ρ  OP  GS = 0  2.37  0 = 0 iii. NP = [wOP2 OP2 + wGS2 GS2 + 2 wOP wGS (CovOP , GS)]1/2 = [(0.92  2.372) + (0.12  0) + (2  0.9  0.1  0)]1/2 = 2.133%  2.13%

Adding the risk-free government securities would result in a lower beta for the new portfolio. The new portfolio beta will be a weighted average of the individual security betas in the portfolio; the presence of the risk-free securities Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-12

would lower that weighted average. d.

The comment is not correct. Although the respective standard deviations and expected returns for the two securities under consideration are equal, the covariances between each security and the original portfolio are unknown, making it impossible to draw the conclusion stated. For instance, if the covariances are different, selecting one security over the other may result in a lower standard deviation for the portfolio as a whole. In such a case, that security would be the preferred investment, assuming all other factors are equal.

i. Grace clearly expressed the sentiment that the risk of loss was more important to her than the opportunity for return. Using variance (or standard deviation) as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements. ii. Two alternative risk measures that could be used instead of variance are: Range of returns, which considers the highest and lowest expected returns in the future period, with a larger range being a sign of greater variability and therefore of greater risk. Semivariance can be used to measure expected deviations of returns below the mean, or some other benchmark, such as zero. Either of these measures would potentially be superior to variance for Grace. Range of returns would help to highlight the full spectrum of risk she is assuming, especially the downside portion of the range about which she is so concerned. Semivariance would also be effective, because it implicitly assumes that the investor wants to minimize the likelihood of returns falling below some target rate; in Grace’s case, the target rate would be set at zero (to protect against negative returns).

13. a.

Systematic risk refers to fluctuations in asset prices caused by macroeconomic factors that are common to all risky assets; hence systematic risk is often referred to as market risk. Examples of systematic risk factors include the business cycle, inflation, monetary policy, fiscal policy, and technological changes. Firm-specific risk refers to fluctuations in asset prices caused by factors that are independent of the market, such as industry characteristics or firm characteristics. Examples of firm-specific risk factors include litigation, patents, management, operating cash flow changes, and financial leverage.

Trudy should explain to the client that picking only the top five best ideas would most likely result in the client holding a much more risky portfolio. The total risk of a portfolio, or portfolio variance, is the combination of systematic risk and firm-specific risk. The systematic component depends on the sensitivity of the individual assets Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-13

to market movements as measured by beta. Assuming the portfolio is well diversified, the number of assets will not affect the systematic risk component of portfolio variance. The portfolio beta depends on the individual security betas and the portfolio weights of those securities. On the other hand, the components of firm-specific risk (sometimes called nonsystematic risk) are not perfectly positively correlated with each other and, as more assets are added to the portfolio, those additional assets tend to reduce portfolio risk. Hence, increasing the number of securities in a portfolio reduces firm-specific risk. For example, a patent expiration for one company would not affect the other securities in the portfolio. An increase in oil prices is likely to cause a drop in the price of an airline stock but will likely result in an increase in the price of an energy stock. As the number of randomly selected securities increases, the total risk (variance) of the portfolio approaches its systematic variance.

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 7-14

CHAPTER 8 INDEX MODELS l. a. To optimize this portfolio one would need: n

estimates of means

estimates of variances

n2 – n 2

1770

estimates of covariances

________ _____________________________ n2 + 3n 2

= 1,890

total estimates

b. In a single-index model: ri – rf = i + i(rM – rf) + ei the variance of the rate of return on each stock can be decomposed into the components: 2

(l) i M The variance due to the common market factor (2) 2(ei) The variance due to firm specific unanticipated events 2

In this model Cov(ri,rj) = ijM . The number of parameter estimates required would be: n = 60 estimates of the mean E(ri) n = 60 estimates of the sensitivity coefficient i, n = 60 estimates of the firm-specific variance 2(ei), and 1 estimate of the market mean E(rM) 2

1 estimate for the market variance M –––––––––––––––––––––––––––––––––––––––––––– 182 estimates Thus, the single-index model reduces the total number of required parameter estimates from 1,890 to 182 , and in general from (n2 + 3n)/2 to 3n + 2. 2. a. The standard deviation of each individual stock is given by 2

i = [i M + 2(ei) ]1/2 Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-1

Since A = .8, B = 1.2, (eA) = 30%, (eB) = 40%, and M = 22% we get A = (.82  .222 + .302)1/2 = 34.78% B = (1.22  .222 + .402)1/2 = 47.93% b. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(rp) = wAE(rA) + wBE(rB) + wfrf where wA, wB, and wf are the portfolio weights of stock A, stock B, and Tbills, respectively. Substituting in the formula we get E(rp) = .30  .13 + .45  .18 + .25  .08 = 14% The beta of a portfolio is similarly a weighted average of the betas of the individual securities: P = wAA + wBB + wff The beta of T-bills (f ) is zero. The beta of the portfolio is therefore P = .30  .8 + .45  1.2 + 0 = .78 The variance of this portfolio is 2

P = P M + 2(eP) where P M is the systematic component and 2(eP) is the nonsystematic component. Since the residuals, ei are uncorrelated, the nonsystematic variance is 2

2(eP) = wA 2 (eA) +wB 2(eB) + wf 2(ef) = .302  .302 + .452  .402 + .252  0 = .0405 where 2(eA) and 2(eB) are the firm-specific (nonsystematic) variances of

stocks A and B, and 2(ef), the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus (eP) = (.0405)1/2 = 20.12% The total variance of the portfolio is then Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-2

P = .782  .222 + .0405 = .069947 and the standard deviation is 26.45%. 3. a. The two figures depict the stocks’ security characteristic lines (SCL). Stock A has a higher firm-specific risk because the deviations of the observations from the SCL are larger for A than for B. Deviations are measured by the vertical distance of each observation from the SCL. b. Beta is the slope of the SCL, which is the measure of systematic risk. Stock B’s SCL is steeper, hence stock B’s systematic risk is greater. c. The R2 (or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock’s residual variance). 2 2

R2 =

i M 2 2

i M + 2(ei) 2

Since stock B’s explained variance is higher (its explained variance is B M , which is greater since its beta is higher), and its residual variance 2(eB) is smaller, its R2 is higher than stock A’s. d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas stock B has a negative alpha; hence stock A’s alpha is larger. e. The correlation coefficient is simply the square root of R2, so stock B’s correlation with the market is higher. 4. a. Firm-specific risk is measured by the standard deviation of residuals. Thus, stock A has more firm-specific risk: 10.3% > 9.1%. b. Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 > .8. c. R2 measures the fraction of total variance of return explained by the market return. A’s R2 is larger than B’s: .576 > .436. d. The average rate of return in excess of that predicted by the CAPM is measured by alpha, the intercept of the SCL. A = 1% is larger than B = –2%. e. Rewriting the SCL equation in terms of total return (r) rather than excess return (R): rA – rf =  + (rM – rf) Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-3

rA =  + rf(1 – ) + rM The intercept is now equal to  + rf(1 – ) = 1 + rf (l – 1.2) Since rf = 6%, the intercept would be: 1 – 1.2 = –.2%. 5. The standard deviation of each stock can be derived from the following equation for R2: 2 2

Ri =

i M 2

i

Explained variance Total variance

Therefore, 2 2

A =

A  M 2

= (.72*.202)/.20 = .0980

A = 31.30% For stock B, 2

B = (1.22*.202)/.12 = .04800 B = 69.28% 6. The systematic risk for A is 2

A M = .702  .202 = .0196 and the firm-specific risk of A (the residual variance) is the difference between A’s total risk and its systematic risk, .0980 – .0196 = .0784 B’s systematic risk is 2

B M = 1.22  .202 = .0576 and B’s firm-specific risk (residual variance) is .4800 – .0576 = .4224 7. The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated) 2

Cov(rA,rB) = A B M = .70  1.2  .0400 = .0336 Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-4

The correlation coefficient between the returns of A and B is AB =

Cov(rA,rB) AB

=0.0336/(0.3130*0.6928)= .1549

8. Note that the correlation coefficient is the square root of R2:  = R2 . Cov(rA,rM) = AM = .201/2  .3130  .20 = .0280 Cov(rB,rM) = BM = .121/2  .6928  .20 = .0480 9. The nonzero alphas from the regressions are inconsistent with the CAPM. The question is whether the alpha estimates reflect sampling errors or real mispricing. To test the hypothesis of whether the intercepts (3 percent for A, and –2 percent for B) are significantly different from zero, we would need to compute t-values for each intercept. 10. For portfolio P we can compute: P = [.62  .0980 + .42  .48 + 2  .4  .6  .0336]1/2 = [.128208]1/2 = 35.81% P = .6  .70 + .4  1.2 = .90 2

2(eP) = P – P M = .128208 – .902  .0400 = .095808 2

Cov(rP, rM) = PM = .90  .0400 = .0360 This same result can also be attained using the covariances of the individual stocks with the market:

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-5

Cov(rP, rM) = Cov(.6rA+.4rB, rM) = .6 Cov(rA, rM) + .4 Cov(rB, rM) = .6  .0280 + .4  .0480 = .0360 11. Note that the variance of T-bills and their covariance with any asset are zero. Therefore, for portfolio Q 2

Q = wP P + wM M + 2  wP  wM  Cov(rP, rM) Q = [.52  .128208 + .32  .0400 + 2  .5  .3  .0360]1/2 = [.046452] 1/2 = 21.55% Q = .5  .90 + .3  1 + 0 = .75 2

2(eQ) = Q – Q M = .046452 – .752  .0400 = .023952 2

Cov(rQ,rM) = QM = .75  .0400 = .0300 12. a. (2/3) × 1.24 + (1/3) × 1 = 1.16 b) βt+1 = .3 + .7 × 1.24 = 1.168 13. For stock A:

 A = rA − [rf +  A  (rM − rf )] = .11 − [.06 + 0.8  (.12 − .06)] = 0.2% For stock B:

 B = rB − [rf +  B  (rM − rf )] = .14 − [.06 + 1.5  (.12 − .06)] = −1% Stock A would be a good addition to a well-diversified portfolio. A short position in stock B may be desirable. CFA® Problems

1. ABC is , a negative-alpha security is overpriced and, other things equal, its portfolio weight should be reduced , or perhaps a short position should be taken in it. XYZ has higher total risk -both systematic and unsystematic than ABC based on monthly data; based on weekly data XYZ shows higher systematic risk; XYZ (ABC) is a more (less) volatile stock. Since the beta of XYZ is greater than 1, it will increase the systematic risk of the portfolio. 2. The R2 of the regression is .702 = .49. Therefore, 51% of total variance is unexplained by the market; this is unsystematic risk. Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-6

3. Applying the equation rC = rf + βC(rM – rf) to Charlottesville International Fund, we get 9% = 3% + βC(11% – 3%), from which we get βC = .75. 4. d 5. b.

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 8-7

CHAPTER 9: THE CAPITAL ASSET PRICING MODEL PROBLEM SETS 1.

E (rP ) = rf + β P  [ E (rM ) − rf ] .18 = .06 + β P  [.14 − .06] → β P =

.12 = 1.5 .08

If the security’s correlation coefficient with the market portfolio doubles (with all other variables such as variances unchanged and the correlation coefficient is still less than 1), then beta, and therefore the risk premium, will also double. The current risk premium is: 16% – 8% = 8% The new risk premium would be 16%, and the new discount rate for the security would be: 16% + 8% = 24% If the stock pays a constant perpetual dividend, then we know from the original data that the dividend (D) must satisfy the equation for the present value of a perpetuity: Price = Dividend/Discount rate 50 = D/0.16  D = 50  0.16 = $8.00 At the new discount rate of 24%, the stock would be worth: $8/0.24 = $33.33 The increase in stock risk has lowered its value by $16.67 or 33.33%.

False. β = 0 implies E(r) = rf , not zero.

False. Investors require a risk premium only for bearing systematic (undiversifiable or market) risk. Total volatility, as measured by the standard deviation, includes diversifiable risk.

False. Your portfolio should be invested 75% in the market portfolio and 25% in T-bills. Then: β P = (0.75 1) + (0.25  0) = 0.75

The expected return is the return predicted by the CAPM for a given level of systematic risk. E (ri ) = rf + βi  [ E (rM ) − rf ] E (r$1 Discount ) = .04 + 1.5  (.10 − .04) = .13, or 13% E (rEverything $5 ) = .04 + 1.0  (.10 − .04) = .10, or 10% Bodie et al. Investments10th Canadian Edition Solutions Manual .. 9-1

According to the CAPM, $1 Discount Stores requires a return of 13% based on its systematic risk level of β = 1.5. However, the forecasted return is only 12%. Therefore, the security is currently overvalued. Everything $5 requires a return of 10% based on its systematic risk level of β = 1.0. However, the forecasted return is 11%. Therefore, the security is currently undervalued.

Correct answer is choice a. The expected return of a stock with a β = 1.0 must, on average, be the same as the expected return of the market which also has a β = 1.0.

Correct answer is choice a. Beta is a measure of systematic risk. Since only systematic risk is rewarded, it is safe to conclude that the expected return will be higher for Kaskin’s stock than for Quinn’s stock.

The appropriate discount rate for the project is: rf + β × [E(rM ) – rf ] = .08 + [1.8  (.16 – .08)] = .224, or 22.4% Using this discount rate: 10

$15 = −$40 + [$15  Annuity factor (22.4%, 10 years)] = $18.09 t t =1 1.224

NPV = −$40 + 

The internal rate of return (IRR) for the project is 35.73%. Recall from your introductory finance class that NPV is positive if IRR > discount rate (or, equivalently, hurdle rate). The highest value that beta can take before the hurdle rate exceeds the IRR is determined by: .3573 = .08 + β × (.16 – .08)  β = .2773/.08 = 3.47

Call the aggressive stock A and the defensive stock D. Beta is the sensitivity of the stock’s return to the market return, i.e., the change in the stock return per unit change in the market return. Therefore, we compute each stock’s beta by calculating the difference in its return across the two scenarios divided by the difference in the market return:

Bodie et al. Investments10th Canadian Edition Solutions Manual .. 9-2

βA =

−.02 − .38 = 2.00 .05 − .25

βD =

.06 − .12 = 0.30 .05 − .25

With the two scenarios equally likely, the expected return is an average of the two possible outcomes: E(rA ) = 0.5  (–.02 + .38) = .18 = 18% E(rD ) = 0.5  (.06 + .12) = .09 = 9%

The SML is determined by the market expected return of [0.5 × (.25 + .05)] = 15%, with βM = 1, and rf = 6% (which has βf = 0). See the following graph: Expected Return - Beta Relationship 40

SML

Expected Return

30 25

A

5 0 0

0.5

1.5

2.5

Beta

The equation for the security market line is: E(r) = .06 + β × (.15 – .06) d.

Based on its risk, the aggressive stock has a required expected return of: E(rA ) = .06 + 2.0 × (.15 – .06) = .24 = 24% The analyst’s forecast of expected return is only 18%. Thus the stock’s alpha is: αA = actually expected return – required return (given risk) = 18% – 24% = –6% Similarly, the required return for the defensive stock is: E(rD) = .06 + 0.3 × (.15 – .06) = 8.7% Bodie et al. Investments10th Canadian Edition Solutions Manual .. 9-3