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Essentials of Investments, 12e Zvi Bodie, Alex Kane, Alan Marcus (Solutions Manual All Chapters, 100% Original Verified, A+ Grade) All Chapters Solutions Manual Supplement files download link at the end of this file. CHAPTER 01 INVESTMENTS: BACKGROUND AND ISSUES

1. Equity is a lower-priority claim on earnings (expressed as dividends) that represents an ownership share in a corporation. Fixed-income (debt) security is a higher-priority claim that legally obligates the issuer to pay the holder of the debt, but does not have an ownership interest. Fixed-income securities typically pay a specified cash flow at precontracted time intervals until the last payment on the maturity date. Shares of equity have an indefinite life.

2. A primary (financial) asset has a claim on the real assets of a firm, whereas a derivative asset provides a payoff that depends on the prices of a primary asset but does not include the claim on the real assets.

3. Asset allocation is the allocation of an investment portfolio across broad asset classes. Security selection is the choice of specific securities within each asset class.

4. Agency problems are conflicts of interest between managers and stockholders. They can be addressed through corporate governance mechanisms, such as the design of executive compensation, oversight by the Board, and monitoring from the institutional investors.

5. Real assets have productive capacity; they are assets used to produce goods and services. Real assets can be tangible (e.g., machinery) or intangible (e.g., a patent). Financial assets are claims on real assets or the income generated by them.

6. Investment bankers are firms specializing in the sale of new securities to the public, typically by underwriting the issue. Commercial banks accept deposits and lend the money to other borrowers. After the Glass-Steagall Act was repealed in 1999, some commercial banks started transforming to “universal banks” which provide the services of both commercial banks and investment banks. With the passage of the Dodd–Frank Wall Street Reform and Consumer Protection Act in 2010, Glass-Steagall was partially restored via the Volcker Rule (which generally prohibits commercial banks from conducting certain investment activities with their own accounts and investing in hedge funds and private equity funds). In 2018, Congress passed The Economic Growth, Regulatory Relief, and Consumer Protection Act established a threshold ($10 billion) for banks to be exempt from the Volcker Rule.

Chapter 01 - Investments: Background and Issues

7. Financial and Real Assets a. Toyota creates a real asset—the factory. The loan is a financial asset that is created in the transaction. b. When the loan is repaid, the financial asset is destroyed but the real asset continues to exist. c. The cash is a financial asset that is traded in exchange for a real asset, inventory.

8. Real Estate as a Real Asset a. No. The real estate in existence has not changed, only the perception of its value has. b. Yes. The financial asset value of the claims on the real estate has changed, and thus the balance sheet of individual investors has been reduced. c. The difference between these two answers reflects the difference between real and financial asset values. Real assets still exist, yet the value of the claims on those assets or the cash flows they generate do change. Thus, there is the difference.

9. Real and Financial Assets a. The bank loan is a financial liability for Lanni. Lanni's $50,000 IOU is the bank's financial asset. The cash Lanni receives is a financial asset. The new financial asset created is Lanni's promissory note held by the bank. b. The cash paid by Lanni (both the loan and its own cash) is the transfer of a financial asset to the software developer. In return, Lanni gets a real asset, the completed software. No financial assets are created or destroyed. Cash is simply transferred from one firm to another. c. Lanni sells the software, which is a real asset, to Microsoft. In exchange Lanni receives a financial asset, 1,000 shares of Microsoft stock. If Microsoft issues new shares in order to pay Lanni, that would be the creation of a new financial asset. d. In selling 1,000 shares of stock for $140,000, Lanni is exchanging one financial asset for another. In paying off the IOU with $50,000, Lanni is exchanging financial assets. The loan is "destroyed" in the transaction since it is retired when paid.

Chapter 01 - Investments: Background and Issues

10. a.

Cash Computers

$70,000 30,000

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

Total

$100,000

Total

Assets

Ratio of real to total assets =

$100,000

$30,000 = 0.3 $100,000

Assets Software product* Computers

$70,000 30,000

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

Total

$100,000

Total

$100,000

*Value at cost

Ratio of real to total assets =

$100,000 = 1.0 $100,000

Assets Microsoft shares ($70/share) Computers

$140,000 30,000

Bank loan Shareholders’ equity

$50,000 120,000

Total

$170,000

Total

$170,000

Liabilities &

Ratio of real to total assets =

$30,000 = 0.2 $155,000

Conclusion: When the firm starts up and raises working capital, it will be characterized by a low ratio of real to total assets. When it is in full production, it will have a high ratio of real assets. When the project "shuts down" and the firm sells it, the percentage of real assets to total assets goes down again because the product is again exchanged into financial assets. Copyright © 2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 01 - Investments: Background and Issues

11. Passed in 2010, the Dodd-Frank Wall Street Reform and Consumer Protection Act proposed several mechanisms to mitigate systemic risk. The act attempts to limit the risky activities in which the banks can engage and calls for stricter rules for bank capital, liquidity, and risk management practices, especially as banks become larger and their potential failure becomes more threatening to other institutions. The act seeks to unify and clarify the lines of regulatory authority and responsibility in government agencies and to address the incentive issue by forcing employee compensation to reflect longer-term performance. It also mandates increased transparency, especially in derivatives markets. 12. a. For commercial banks, the ratio is:

$184.2 = 0.0102 $18, 090.1

b. For non-financial firms, the ratio is:

$23,907 = 0.5238 $45, 643

c. The difference should be expected since the business of financial institutions is to make loans that are financial assets.

13. National wealth is a measurement of the real assets used to produce GDP in the economy. Financial assets are claims on those assets held by individuals. Financial assets owned by households represent their claims on the real assets of the issuers, and thus show up as wealth to households. Their interests in the issuers, on the other hand, are obligations to the issuers. At the national level, the financial interests and the obligations cancel each other out, so only the real assets are measured as the wealth of the economy. The financial assets are important since they drive the efficient use of real assets and help us allocate resources, specifically in terms of risk return trade-offs.

14. Compensation and Agency Problems a. A fixed salary means 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, and thus allows the manager to envision and seek the sustainable operation of the company. However, since the compensation is secured and not tied to the performance of the firm, the manager might not be motivated to take any risk to maximize the value of the company.

Chapter 01 - Investments: Background and Issues

b. A salary paid in the form of stock in the firm means the manager earns the most when shareholder wealth is maximized. When the stock must be held for five years, the manager has less of an incentive to manipulate the stock price. This structure is most likely to align the interests of managers with the interests of the shareholders. If stock compensation is used too much, the manager might view it as overly risky since the manager’s career is already linked to the firm. This undiversified exposure would be exacerbated with a large stock position in the firm. c. When executive salaries are linked to firm profits, the firm creates incentives for managers to contribute to the firm’s success. However, this may also lead to earnings manipulation or accounting fraud, such as divestment of its subsidiaries or unreasonable revenue recognition. That is what audits and external analysts will look out for.

15. Even if an individual investor has the expertise and capability to monitor and improve the managers’ performance, the payoffs would not be worth the effort, since his ownership in a large corporation is so small compared to that of institutional investors. For example, if the individual investor owns $10,000 of IBM stock and can increase the value of the firm by 5%, a very ambitious goal, the benefit would only be: $10,000 x 5% = $500. In contrast, a bank that has a multimillion-dollar loan outstanding to the firm has a big stake in making sure the firm can repay the loan. It is clearly worthwhile for the bank to spend considerable resources to monitor the firm.

16. Since the traders benefited from profits but did not get penalized by losses, they were encouraged to take extraordinary risks. Since traders sell to other traders, there also existed a moral hazard since other traders might facilitate the misdeed. In the end, this represents an agency problem.

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

Chapter 01 - Investments: Background and Issues

18. Progress in securitization facilitates the shifting of default risk from the intermediates to the investors of such a security. Since the intermediates no longer bear the default risk, their role and motivation in assessing and monitoring the quality of the borrowers is mitigated. For example, when the national market in mortgage-backed securities becomes highly developed, local banks can easily sell their claims on mortgages to the issuers of mortgage-backed securities and then use the money they receive to create more mortgages because the local banks make profits both from making loans and selling loans to the issuers of mortgage-backed securities. This way the local banks are incentivized by the volume of the loan that they lend out and not by the quality of the loan, and thus they become less cautious in originating subprime mortgages.

19. (answers will vary) Mutual funds accept funds from small investors and invest, on behalf of these investors, in the national and international securities markets. Pension funds accept funds and then invest, 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.

20. 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. If the stock price rises considerably, managers might conclude that the market believes the firm's future prospects are bright (and generally supports the actions of management). This might be a useful signal to the firm to proceed with an investment such as an expansion of the firm's business. Since shares can be easily traded in the secondary market it makes them more attractive to investors since investors know that they will be able to sell their shares quickly. This 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.

21. 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 predictability of investment performance and portfolio value.

Chapter 01 - Investments: Background and Issues

22. 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 this book is its author.

Chapter 02 - Asset Classes and Financial Instruments

CHAPTER 2 ASSET CLASSES AND FINANCIAL INSTRUMENTS

1. Common stock is an ownership share in a publicly held corporation. Common shareholders have voting rights and may receive dividends (but are not contractually obligated to do so). Preferred stock represents nonvoting shares in a corporation, usually paying a fixed stream of dividends (but are not contractually obligated to do so). While corporate bonds are long-term debt issued by corporations, the bonds typically pay semi-annual coupons (and are contractually obligated to pay them) and return the face value of the bond at maturity.

2. While the DJIA has 30 large corporations in the index, it does not represent the overall market nearly as well as the approximate 3,500 stocks contained in The Wilshire 5000 index. The DJIA is simply too small.

3. Money market securities are short-term, relatively low risk, and highly liquid. Also, their unit value almost never changes.

4. The major components of the money market are Treasury bills, certificates of deposit, commercial paper, bankers’ acceptances, Eurodollars, repos and reverses, federal funds, and brokers’ calls.

5. American Depositary Receipts, or ADRs, are certificates traded in U.S. markets that represent ownership in shares of a foreign company. Investors may also purchase shares of foreign companies on foreign exchanges. Lastly, investors may use international mutual funds to own shares indirectly.

6. The coupons paid by municipal bonds are exempt from federal income tax and from state tax in many states. Therefore, the higher the tax bracket that the investor is in, the more valuable the tax-exempt feature to the investor.

7. The London Interbank Offer Rate (LIBOR)—a key reference rate in the money market—is the rate at which large banks in London are willing to lend money among them. The Federal funds rate is the rate of interest on very short-term loans among financial institutions in the U.S.A.

Chapter 02 - Asset Classes and Financial Instruments

8. General obligation bonds are backed by the taxing power of the local governments, while revenue bonds have proceeds attached to specific projects. A revenue bond has fewer guarantees, it is riskier in terms of default, and, therefore, you expect it to have a higher yield.

9. Corporations may exclude 50% of dividends received from domestic corporations in the computation of their taxable income.

10. Limited liability means that the most shareholders can lose in event of the failure of the corporation is their original investment (which differs from owners of unincorporated businesses).

11. (a) A repurchase agreement is the sale of a security with a commitment to repurchase the same security at a specified future date and a designated price. 12. Money market securities are referred to as “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.

13. Equivalent taxable yield =

Rate on municipal bond 1－ Tax rate

rm 1－ t

.0225 1 － 0.35

= .0346 or

3.46% 14. After-tax yield = Rate on the taxable bond x (1 － Tax rate) a. 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 4% yield on the municipal bond. b. The taxable bond. The after-tax yield for the taxable bond is: 0.05 x (1 – 0.10) = 0.045 or 4.50%. c. Neither. The after-tax yield for the taxable bond is: 0.05 x (1 – 0.20) = 0.04 or 4%. The after-tax yield of taxable bond is the same as that of the municipal bond. d. The municipal bond. The after-tax yield for the taxable bond is: 0.05 x (1 – 0.30) = 0.035 or 3.5%. The municipal bond offers the higher after-tax yield for investors in tax brackets above 20%.

Chapter 02 - Asset Classes and Financial Instruments

15. The after-tax yield on the corporate bonds is: 0.09 x (1 – 0.30) = 0.063 or 6.3%. Therefore, the municipals must offer at least 6.3% yields. 16. Using the formula of Equivalent taxable yield (r) = a. r =

b. r =

c. r =

d. r =

0.04 1－0

rm 1－ t

, we get:

= 0.04 or 4.00%

0.04 1 － 0.10 0.04 1 － 0.20 0.04 1 － 0.30

= 0.0444 or 4.44%

= 0.05 or 5.00%

= 0.0571 or 5.71%

17. a. You would have to pay the asked price of: 128.212 = 128.212% of par = $1,282.12 b. The coupon rate is 3.500%, implying coupon payments of $35.00 annually or, more precisely, $17.50 (= 35.00/2) semiannually. c. Given the asked price and coupon rate, we can calculate current yield with the formula: Annual coupon income 3.5 Current yield = = = 0.0273 = 2.73% Price 128.212 18. a. The closing price today is $224.15, which is $1.44 above yesterday’s price. Therefore, yesterday’s closing price was: $224.15 － $1.44 = $222.71. b. You would buy 22 shares: $5,000/$224.15 = 22.31 (round down for 22 shares) c. Your annual dividend income on 22 shares would be 22 x $5.44 = $119.68. d. Earnings per share can be derived from the price-earnings (PE) ratio: Given price/Earnings = 22.36 and Price = $224.15, we know that Earnings per $224.13 = $10.02 Share = 22.36 19. a. At t = 0, the value of the index is: ($90 + $50 + $100)/3 = 80 Copyright © 2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 02 - Asset Classes and Financial Instruments

At t = 1, the value of the index is: ($95 + $45 + $110)/3 = 83.33 V1 The rate of return is: － 1 = (83.33/80) – 1 = 0.0417 or 4.17% V0 b. In the absence of a split, stock C would sell for $110, and the value of the index would be the average price of the individual stocks included in the index: ($95 + $45 + $110)/3 = $83.33. After the split, stock C sells at $55; however, the value of the index should not be affected by the split. We need to set the divisor (d) such that: $95 + $45 + $55 $83.33 = → d d = 2.34 c. The rate of return is zero. The value of the index remains unchanged since the return on each stock separately equals zero. 20. a. Total market value at t = 0 is: ($90 x 100) + ($50 x 200) + ($100 x 200) = $39,000 Total market value at t = 1 is: ($95 x 100) + ($45 x 200) + ($110 x 200) = $40,500 V1 Rate of return = － 1 = ($40,500/$39,000) – 1 = 0.0385 or 3.85% V0 b. The return on each stock is as follows: V1 RA = － 1 = ($95/$90) – 1 = 0.0556 or 5.56% V0 RB = RC =

V1 V0 V1 V0

－ 1 = ($45/$50) – 1 = –0.10 or –10.00% － 1 = ($110/$100) – 1 = 0.10 or 10.00%

The equally-weighted average is

5.56% + ( –10.00% ) + 10.00% = 1.85% 3

21. The fund would require constant readjustment since every change in the price of a stock would bring the fund asset allocation out of balance.

Chapter 02 - Asset Classes and Financial Instruments

22. In this case, the value of the divisor will increase by an amount necessary to maintain the index value on the day of the change. For example, if the index was comprised of $115 − $50 only one stock, it would increase by = 1.30. $50

23. Bank discount of 87 days: 0.034 x

87 days 360 days

= 0.008217

a. Price: $10,000 x (1 – 0.008217) = $9,917.83 b. Bond equivalent yield =

Face value － Purchase price Purchase price x T

$10,000 － $9,917.83 87 days

$9,917.83 x 365 days

= 0.0348 or 3.48%

24. a. The higher coupon bond: The 10-year T-bond with a 6% coupon b. The call with the lower exercise price: The call with the exercise price of $35 c. The put option on the lower priced stock: The put on the stock selling at $50

25. The December maturity futures price is $3.9725 per bushel. If the contract closes at $4.00 per bushel in December, your profit / loss on each contract (for delivery of 5,000 bushels of corn) will8 be: ($4.00 – $3.9725) x 5000 = $ 137.50 gain.

26. a. Yes. As long as the stock price at expiration exceeds the exercise price, it makes sense to exercise the call. Gross profit is: ($144 – $140) x 100 shares = $400 Net profit = ($4.00 – $5.32) x 100 shares = $1.32 loss Rate of return = –$1.32/$5.32 = –0.2481 or 24.81% loss b. Yes, exercise. Gross profit is: ($144 – $135) x 100 shares = $900 Net profit = ($9.00 – $8.12) x 100 shares = $88 gain Rate of return = $.88/$8.12 = 0.1084 or 10.84 % gain c. A put with an exercise price of $140 would expire worthless for any stock price equal to or greater than $140 (in this case $144). An investment in such a put would have a rate of return over the holding period of –100%. Copyright © 2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 02 - Asset Classes and Financial Instruments

27. a. b. c. d.

Long call Long put Short put Short call

28. There is always a chance that the option will expire in the money. Investors will pay something for this chance of a positive payoff.

29. Long call for $4:

a. b. c. d. e.

Value of call at expiration 0 0 0 5 10

Initial Cost

Profit

4 4 4 4 4

-4 -4 -4 1 6

Initial Cost

Profit

6 6 6 6 6

4 -1 -6 -6 -6

Long put for $6: Value of put at expiration a. 10 b. 5 c. 0 d. 0 e. 0

30. 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.

31. Six of seven stocks have a 52-week high at least 40% above the 52-week low (and three out of seven are at 50% or more). It can be concluded that individual stocks are much more volatile than a group of stocks. 52-wk high

52-wk low

61.77

33.62

Price ratio (High-Low)/Low

84%

Chapter 02 - Asset Classes and Financial Instruments

161.58 74.81 17.27 229.27 31.04 178.47

99.15 35.59 12.09 158.09 22.87 123.48

63% 110% 43% 45% 36% 45%

32. The total before-tax income is $4.00. The corporations may exclude 50% of dividends received from domestic corporations in the computation of their taxable income; the taxable income is therefore: $4.00 x 50% = $2.00. Income tax in the 30% tax bracket: $2.00 x 21% = $0.42 After-tax income = $4.00 – $0.42 = $3.58 After-tax rate of return = $3.58/$40.00 = 0.0895 or 8.95%

33. A put option conveys the right 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 futures price.

34. A call option conveys the right 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 futures price. CFA 1 Answer: c. Taxation

Problem 5-4 Your firm invested $2,500,000 in 270-day commercial paper today. At the end of the investment period (in 270 days) the firm will receive $2,585,000. a. What is the 270-day holding period rate of return on the investment? (Round your answer to 2 decimal places.) HPR

<Ans: 3.40 +/-.5%> %

b. How many 270-day periods are there in one year? (Use a 365-day year. Round your answer to 4 decimal places.) Number of periods <Ans: 1.3519 +/-.5%> c. What is the annual percentage rate APR earned on the investment? (Round your answer to 2 decimal places.) APR

<Ans: 4.60 +/-.5%> %

d. What is the effective annual rate (EAR)? (Round your answer to 2 decimal places.) EAR

<Ans: 4.62 +/-.5%> %

Explanation: a. The HPR equals (2,585,0000 – 2,500,000) / 2,500,000 = 3.40%. b. There are 365/270 = 1.3519 (rounded) 270-day periods in one year. c. The annual percentage rate APR is the 270-day rate times the number of 270day periods in one year. APR = 3.40% × 1.35185 = 4.60% d. The EAR may be calculated either as 1 + EAR = (1 + rate per period) n = (1 + .034 )

1.35185

= 1.0462 ,

EAR = 4.62%

or as n

.046   APR   1 + EAR = 1 +  = 1 +  n    1.35185 

1.35185

= 1.0462 ,

EAR = 4.62%

Problem 5-5 The following data represent the probability distribution of the holding period returns for an investment in Lazy Rapids Kayaks (LARK) stock. State of the Economy Boom Normal growth Recession

Scenario #(s) 1 2 3

Probability, HPR p(s) 0.36 29.2% 0.45 8.40% 0.19 -18.30%

a. What is the expected return on LARK? (Round your answer to 2 decimal places.) Expected return

<Ans: 10.82 +/-.5%> %

b. What is the standard deviation of the returns on LARK? (Round your answer to 2 decimal places.) Standard deviation <Ans: 16.89 +/-.5%> %

Explanation: a.The expected return is 3

E (r ) =  p( s )r( s ) = .36  29.2% + .45  8.40% + .19  ( −18.30%) = 10.82% s =1

b.The standard deviation is the square root of the variance. The variance is given by the formula 3

VAR( r ) =  p( s )[r( s ) − E( r )] 2 s =1

To find the variance of the returns it is helpful to set up a table for the calculations. Each column in the table below has a column number at the top (indicated in parentheses), which is used to indicate how the column is used in calculations.

(1)

(2)

(3)

(4)

(5)

(6)

State of the Economy

Probability, p(s)

HPR

HPR - E(r)

(4) squared

(2) × (5)

Boom

0.36

29.2%

18.39%

0.0338008

0.01217

Normal growth

0.45

8.40%

-2.42%

0.000583

0.00026

Recession

0.19

-18.30%

-29.12%

0.084768

0.01611

VAR = sum of (6) =

0.02854

VAR =

16.89%

SD =

Problem 5-6 The common stock of Perforated Pool Liners, Inc. now sells for $50.00 per share. The table below shows the anticipated stock price and the dividend to be paid one year from now. Both the price and the dividend will depend on the level of growth experienced by the firm. State

Probability, p(s)

End-of-Year Price

Annual Dividend

Super high growth High growth Normal growth Low growth No growth

0.1 0.2 0.4 0.2 0.1

$62.00 $58.00 $56.00 $50.00 $46.00

$3.00 $3.00 $2.00 $2.00 $0.00

a. Calculate the holding period return (HPR) for each of the possible states, assuming a one-year holding period. (Use a negative sign to indicate a negative answer. Round your answer to 2 decimal places.) Super high growth <Ans: 30.00 +/-.5%> % High growth

<Ans: 22.00 +/-.5%> %

Normal growth

<Ans: 16.00 +/-.5%> %

Low growth

<Ans: 4.00 +/-.5%> %

No growth

<Ans: -8.00 +/-.5%> %

b. What is the expected return for a holder of Perforated Pool Liners stock? (Round your answer to 2 decimal places.) Expected HPR

<Ans: 13.80 +/-.5%> %

c. What is the standard deviation of the returns? (Round your answer to 2 decimal places.) Standard deviation <Ans: 10.41 +/-.5%> %

Explanation: a. The holding period return (HPR) for each state is calculated as

HPR =

End of year price + Dividend - Beginning of year price . Beginning of year price

The table below shows the calculations based on the beginning stock price of $50.00. State

End-of-Year Price

Annual Dividend

Super high growth High growth Normal growth Low growth No growth

$62.00 $58.00 $56.00 $50.00 $46.00

$3.00 $3.00 $2.00 $2.00 $0.00

HPR (62+3-50)/50 = 30.00% (58+3-50)/50 = 22.00% (56+2-50)/50 = 16.00% (50+2-50)/50 = 4.00% (46+0-50)/50 = -8.00%

b. The expected return is 5

E( r ) =  p( s )r( s ) s =1

= . (1 30%) + (.2  22%) + (.4  16%) + (.2  4%) + (.1  (-8%)) = 13.80% c. The standard deviation is the square root of the variance. The variance is given by the formula 5

VAR( r ) =  p( s )[r( s ) − E( r )] 2 s =1

since there are 5 possible outcomes. The calculations are shown in the table below. Each column in the table has a column number at the top (indicated in parentheses), which is used to indicate how the column is used in calculations. (1)

(2) Probability p(s)

(3) HPR (from part a)

Super high growth

0.1

High growth

State

(4)

(5)

(6)

HPR - E(r) (4) squared

(2) × (5)

30.00%

16.20%

0.02624

0.00262

0.2

22.00%

8.20%

0.00672

0.00134

Normal growth

0.4

16.00%

2.20%

0.00048

0.00019

Low growth

0.2

4.00%

-9.80%

0.00960

0.00192

No growth

0.1

-8.00%

-21.80%

0.04752

0.00475

Variance

0.01084

Std. deviation 10.41%

Problem 5-7 You earned a nominal rate of return equal to 11.7% on your investments last year. The annual inflation rate was 2.3%. a. What was your approximate real rate of return? (Round your answer to 2 decimal places.) Approximate real rate of return

<Ans: 9.40 +/-.5%> %

b. What was your exact real rate of return? (Round your answer to 2 decimal places.) Real rate of return

<Ans: 9.19 +/-.5%> %

Explanation: a. The approximate rate of return equals the nominal rate minus the inflation rate: r  R – i = 11.7% − 2.3% = 9.4%. b. The exact real rate of return is calculated from the equation 1+ r =

1 + R 1.117 = = 1.0919 1 + i 1.023

r = 9.19%.

Problem 5-8 If the real interest rate is 6.5% per year and the expected inflation rate is 2.2%, what is the nominal interest rate according to the Fisher equation? (Round your answer to 2 decimal places.) Nominal interest rate

<Ans: 8.70 +/-.5%> %

Explanation: The Fisher equation shows the relationship R = r + E(i). In this case, R = 6.5% + 2.2% = 8.7%.

Problem 5-9 A portfolio earned a rate of return equal to 18% last year with a standard deviation of 27%. Treasury Bills returned 3%. a. What is the portfolio’s excess return (Round your answer to 2 decimal places.) Approximate real rate of return

<Ans: 15.00 +/-.5%> %

b. What is the portfolio’s Sharpe Ratio? (Round your answer to 2 decimal places.) Real rate of return

<Ans: 0.56 +/-.5%>

Explanation: a. The excess return equals rate minus the risk-free rate (Treasury Bills): Excess Return = 18% - 3% = 15% b. The Sharpe Ratio is calculated from the equation Sharpe Ratio : SP =

rP − rf

P

SP = (.18 - .03)/.27 = 0.56.

Chapter 05 – Risk, Return, and the Historical Record

CHAPTER 05 RISK, RETURN, AND THE HISTORICAL RECORD 1. The 1% VaR will be less (more negative) than –30%. As the percentile declines the magnitude of that negative return increases. Thus, a 1% probability implies a great value at risk than a 5% probability. Said another way, ending up in the bottom 1% requires a larger potential loss than ending up in the bottom 5%.

2. If inflation increases from 3% to 5%, according to the Fisher equation there will be a concurrent increase in the nominal rate to offsets the increase in expected inflation. This gives investors an unchanged growth of purchasing power.

3. The Sharpe ratio is a statistic designed to rank portfolios, so its use will be the same whether nominal or real are used as long as you are consistent in the calculations: you can use either real rates for the returns on both the portfolio and the risk-free asset, or nominal rate for each. Just don’t mix and match! Apply the same approach to each portfolio in the comparison. Similarly, the standard deviation of the excess return also will be unaffected, again as long as you are consistent across portfolios (and benchmarks).

4. Decrease. Typically, standard deviation exceeds return. Thus, an underestimation of 1.05 will imply a necessary increase in the amount of risk in your desired portfolio. To return to the proper risk return relationship the portfolio will need to decrease the amount of risk free investments.

5. Using Equation 5.10, calculate the mean of the HPR (holding period return) as: 𝑆

E(r) = ∑𝑠=1 p(s) r(s) = (0.3  0.44) + (0.4  0.14) + [0.3  (–0.16)] = 0.14 or 14% Using Equation 5.11, calculate the variance: 𝑆

Var(r) = 2 = ∑𝑠=1 p(s) [ r(s) – E(r)]2 = [0.3  (0.44 – 0.14)2] + [0.4  (0.14 – 0.14)2] + [0.3  (–0.16 – 0.14)2] = 0.054

Using Equation 5.12, derive standard deviation from variance: SD(r) =  = √Var(r) = √0.054 = 0.2324 or 23.24% [Standard Deviation] Copyright © 2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 05 – Risk, Return, and the Historical Record

6. We use the below equation to calculate the holding period return of each scenario: HPR =

Ending Price － Beginning Price + Cash Dividend Beginning Price

a. The holding period returns for the three scenarios are: Boom:

(50 – 40 + 2)/40 = 0.30 = 30%

Normal:

(43 – 40 + 1)/40 = 0.10 = 10%

Recession:

(34 – 40 + 0.50)/40 = –0.1375 = –13.75% 𝑆

E(HPR) = ∑𝑠=1 p(s) r(s) = [(1/3)  0.30] + [(1/3)  0.10] + [(1/3)  (–0.1375)] = 0.0875 or 8.75% 𝑆

Var(HPR) = ∑𝑠=1 p(s) [ r(s) – E(r)]2 = [(1/3)  (0.30 – 0.0875)2] + [(1/3)  (0.10 – 0.0875)2] + [(1/3) (–0.1375 – 0.0875)2] = 0.031979 SD(r) =  = √Var(r) =

319 .79 = 0.1788 or 17.88%

b. E(r) = (0.5  8.75%) + (0.5  4%) = 6.375%  = 0.5  17.88% = 8.94%

7. a. Time-weighted average returns are based on year-by-year rates of return. Year

Return = [(Capital gains + Dividend)/Price]

2018-2019 2019-2020 2020-2021

(110 – 100 + 4)/100 = 0.14 or 14.00% (90 – 110 + 4)/110 = –0.1455 or –14.55% (95 – 90 + 4)/90 = 0.10 or 10.00%

Arithmetic mean: [0.14 + (–0.1455) + 0.10]/3 = 0.0315 or 3.15% 3 Geometric mean: √(1 + 0.14)  [1 + (–0.1455)]  (1 + 0.10) – 1 = 0.0233 or 2.33%

b. Net Cash Flow

2018 –300

2019 –208

Date 2020 110

2021 396

Chapter 05 – Risk, Return, and the Historical Record

Time

Net Cash flow

0 1

–300 –208

110

396

Explanation Purchase of three shares at $100 per share Purchase of two shares at $110, plus dividend income on three shares held Dividends on five shares, plus sale of one share at $90 Dividends on four shares, plus sale of four shares at $95 per share

The dollar-weighted return is the internal rate of return that sets the sum of the present value of each net cash flow to zero: –$208 $110 $396 0 = –$300 + + 2 + 1+ IRR (1+ IRR) (1+ IRR)3 Dollar-weighted return = Internal rate of return = –0.1661%

8. a. Given that A = 4 and the projected standard deviation of the market return = 20%, we can use the below equation to solve for the expected market risk premium: A=4=

Average(rM)－ rf Sample M2

Average(rM)－ rf (20%)2

E(rM) – rf = AM2 = 4  (0.20) = 0.16 or 16% b. Solve E(rM) – rf = 0.09 = AM2 = A  (0.20) , we can get A = 0.09/0.04 = 2.25 c. Increased risk tolerance means decreased risk aversion (A), which results in a decline in risk premia. 9. From Table 5.3, we find that for the period 1927 – 2018, the mean excess return for S&P 500 over 1-month T-bills is 8.34%. E(r) = Risk-free rate + Risk premium = 3% + 8.34% = 11.34%

10. To answer this question with the data provided in the textbook, we look up the historical average for Treasury Bills, Treasury Bonds and stocks for 1927-2018 from Table 5.3 Arithmetic Average, Nominal Returns T-bills: 3.38% T-bonds: 5.83% Stocks: 11.72%

Chapter 05 – Risk, Return, and the Historical Record

To estimate the real rate of return, use historical inflation rate 3.01% from Table 5.2: The relationship between real rates and nominal rates/inflation is: 1 + r =

1+ R 1 + .0338 −1 = − 1 = .0036 = 0.36% 1+ i 1 + .0301 1+ R 1 + .0583 −1 = − 1 = .0274 = 2.74% T-bonds: r = 1+ i 1 + .0301 1+ R 1 + .1172 −1 = − 1 = .0846 = 8.46% Stocks: r = 1+ i 1 + .0301

1+ R 1+ i

T-bills: r =

11. a. The expected cash flow is: (0.5  $50,000) + (0.5  $150,000) = $100,000 With a risk premium of 10%, the required rate of return is 15%. Therefore, if the value of the portfolio is X, then, in order to earn a 15% expected return: Solving X  (1 + 0.15) = $100,000, we get X = $86,957 b. If the portfolio is purchased at $86,957, and the expected payoff is $100,000, then the expected rate of return, E(r), is:

$100,000 − $86,957 = 0.15 = 15% $86,957 The portfolio price is set to equate the expected return with the required rate of return. c. If the risk premium over T-bills is now 15%, then the required return is: 5% + 15% = 20% The value of the portfolio (X) must satisfy: X  (1 + 0.20) = $100, 000  X = $83,333 d. For a given expected cash flow, portfolios that command greater risk premiums must sell at lower prices. The extra discount in the purchase price from the expected value is to compensate the investor for bearing additional risk.

12. a. Allocating 70% of the capital in the risky portfolio P, and 30% in risk-free asset, the client’s calculated rate of return equals the sum of the expected return of the risky proportion (y) plus the expected return of the risk-free proportion (1 - y): E(rC) = y  E(rP) + (1 – y)  rf = (0.7  0.17) + (0.3  0.07) = 0.14 or 14% per year

Chapter 05 – Risk, Return, and the Historical Record

The standard deviation of the portfolio equals the standard deviation of the risky fund times the fraction of the complete portfolio invested in the risky fund:

C = y P = 0.7  0.27 = 0.189 or 18.9% per year b. The investment proportions of the client’s overall portfolio is the proportion of risky portfolio in the complete portfolio times the proportion allocated in each stock. Security T-Bills Stock A Stock B Stock C

0.7  27% = 0.7  33% = 0.7  40% =

Investment Proportions 30.0% 18.9% 23.1% 28.0%

c. We calculate the reward-to-variability ratio (Sharpe ratio) using Equation 5.14. For the risky portfolio: S =

E(rP) － rf 0.17 － 0.07

P

For the client’s overall portfolio: S =

0.27

E(rC) － rf

C

= 0.3704

0.14 － 0.07 0.189

= 0.3704

d. E(r)

% P

CAL(slope=.3704)

14 client

 18.9

Chapter 05 – Risk, Return, and the Historical Record

13. a. E(rC) = y  E(rP) + (1 – y)  rf = y  0.17 + (1 – y)  0.07 = 0.15 or 15% per year Solving for y, we get y =

0.15 － 0.07 0.10

= 0.8

Therefore, in order to achieve an expected rate of return of 15%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills. b. The investment proportions of the client’s overall portfolio can be calculated by the proportion of risky asset in the whole portfolio times the proportion allocated in each stock. Security T-Bills Stock A Stock B Stock C

0.8  27% = 0.8  33% = 0.8  40% =

Investment Proportions 20.0% 21.6% 26.4% 32.0%

c. The standard deviation of the complete portfolio is the standard deviation of the risky portfolio times the fraction of the portfolio invested in the risky asset:

C = y P = 0.8  0.27 = 0.216 or 21.6% per year 14. a. Standard deviation of the complete portfolio = C = y  0.27 If the client wants the standard deviation to be equal or less than 20%, then: y = (0.20/0.27) = 0.7407 = 74.07% He should invest, at most, 74.07% in the risky fund. b. E(rC) = rf + y  [E(rP) – rf] = 0.07 + 0.7407  0.10 = 0.1441 or 14.41%

Chapter 05 – Risk, Return, and the Historical Record

15. a. See the diagram below:

b. Slope of the CML =

E(rM) － rf

M

0.13 － 0.07 0.25

= 0.24

c. Your fund allows an investor to achieve a higher expected rate of return for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk. 16. a. With 70% of his money in your fund's portfolio, the client has an expected rate of return of 14% per year and a standard deviation of 18.9% per year. If he shifts that money to the passive portfolio (which has an expected rate of return of 13% and standard deviation of 25%), his overall expected return and standard deviation would become: E(rC) = rf + 0.7  E(rM) – rf] In this case, rf = 7% and E(rM) = 13%. Therefore: E(rC) = 0.07 + (0.7  0.06) = 0.112 or 11.2% The standard deviation of the complete portfolio using the passive portfolio would be:

C = 0.7 M = 0.7  0.25 = 0.175 or 17.5% Therefore, the shift entails a decline in the mean from 14% to 11.2% and a decline in the standard deviation from 18.9% to 17.5%. Since both mean return and standard deviation fall, it is not yet clear whether the move is beneficial. If your client is willing to accept an expected return on his total portfolio of 11.2%, he can achieve that return with a lower standard deviation using your fund portfolio rather than the passive portfolio. To achieve Copyright © 2020 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 05 – Risk, Return, and the Historical Record

a target mean of 11.2%, we first write the mean of the complete portfolio as a function of the proportions invested in your fund portfolio, y: E(rC) .11= 7% + y  (17% – 7%) = 7% + 10%  y Because our target is E(rC) = 11.2%, the proportion that must be invested in your fund is determined as follows: 11.2% = 7% + 10%  y  y =

11.2% － 7% 10%

= 0.42

The standard deviation of the portfolio would be:

C = y  27% = 0.42  27% = 11.34% Thus, by using your portfolio, the same 11.2% expected rate of return can be achieved with a standard deviation of only 11.34% as opposed to the standard deviation of 17.5% using the passive portfolio. b. The fee would reduce the reward-to-variability ratio, i.e., the slope of the CAL. Clients will be indifferent between your fund and the passive portfolio if the slope of the afterfee CAL and the CML are equal. Let f denote the fee: Slope of CAL with fee =

17% － 7% － f 27%

Slope of CML (which requires no fee) =

10% － f

27% 13% － 7% 25%

= 0.24

Setting these slopes equal and solving for f: 10% － f 27%

= 0.24

10% – f = 27%  0.24 = 6.48% f = 10% − 6.48% = 3.52% per year 17. Assuming no change in tastes, that is, an unchanged risk aversion, investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before. If we assume that the risk-free rate is unaffected, the increase in the risk premium would require a higher expected rate of return in the equity market.

18. Expected return for your fund = T-bill rate + risk premium = 6% + 10% = 16% Expected return of client’s overall portfolio = (0.6  16%) + (0.4  6%) = 12% Standard deviation of client’s overall portfolio = 0.6  14% = 8.4%

Chapter 05 – Risk, Return, and the Historical Record

19. Reward to volatility ratio =

Portfolio Risk Premium Standard Deviation of Portfolio Excess Return

10% 14%

= 0.7143

20.

1927-2018 1927-1949 1950-1972 1973-1995 1996-2018

Excess returns statistics for Growth indices Average Excess Returns Standard Deviation Sharpe Ratio Big Small Big Small Big Small 8.07 8.99 18.35 26.06 0.44 0.34 8.72 13.04 25.47 35.74 0.34 0.36 10.41 10.72 12.95 17.90 0.80 0.60 4.60 5.20 17.49 23.45 0.26 0.22 8.50 6.95 15.07 23.93 0.56 0.29

1927-2018 1927-1949 1950-1972 1973-1995 1996-2018

Excess returns statistics for Value indices Average Excess Returns Standard Deviation Sharpe Ratio Big Small Big Small Big Small 11.69 15.38 24.70 28.21 0.47 0.55 14.64 20.55 40.37 46.53 0.36 0.44 14.29 15.28 15.42 16.70 0.93 0.92 10.18 13.97 15.57 19.10 0.65 0.73 7.70 11.76 18.28 19.48 0.42 0.60

a. In three out of four subperiods, the Big/Growth index has a higher Sharpe Ratio than the Small/Growth index. Note the opposite is true for value—in three out of four subperiods, the Small/Value index provides the better Sharpe Ratios. b. The Small/Value index has a higher Sharpe Ratio than the Small/Growth index for all subperiods.

1927-2018 1927-1949 1950-1972 1973-1995 1996-2018

Small Cap Sharpe Ratio Growth Value 0.34 0.55 0.36 0.44 0.60 0.92 0.22 0.73 0.29 0.60

21. The real returns are the ratio of returns to monthly inflation, pulled from the St. Louis Federal Reserve Branch (https://fred.stlouisfed.org/series/CPIAUCNS). Real returns are

Chapter 05 – Risk, Return, and the Historical Record

1 + rnom − 1 and are annualized before the standard deviation is 1+ i calculated (this is the method used to calculate the standard deviation in Table 5.3). calculated as rreal =

The real risk premium is difference between the annualized arithmetic average of the real returns and the annualized arithmetic average of Treasury bills. The standard deviation of the real returns is slightly less than the standard deviation of the excess returns. Risk and Real Return of Stocks, 1927-2018 Average (Geometric) 6.65 Average (Arithmetic) 8.60 Real Risk Premium 5.60 Standard Deviation 19.96 Max 56.11 Min -38.29

22. The standard deviation is calculated from the indices’ monthly returns and monthly data from the St. Louis Federal Reserve (https://fred.stlouisfed.org/series/CPIAUCNS) Unlike problem 21, the standard deviation is calculated first and then the data are annualized (the methodology used in Tables 5.4 and 5.4). The volatility of real returns and the volatility of excess returns are very similar. Standard Deviations for Growth indices Big/Growth Small/Growth Real Excess Real Excess 18.37 18.35 25.96 1927-2018 26.06 25.48 25.47 35.60 1927-1949 35.74 13.10 12.95 18.03 1950-1972 17.90 17.57 17.49 23.37 1973-1995 23.45 14.94 15.07 23.72 1996-2018 23.93 Standard Deviations for Value indices Big/Value Small/Value Real Excess Real Excess 24.65 24.70 28.11 1927-2018 28.21 40.27 40.37 46.38 1927-1949 46.53 15.50 15.42 16.77 1950-1972 16.70 15.58 15.57 19.04 1973-1995 19.10 18.17 18.28 19.31 1996-2018 19.48

Chapter 05 – Risk, Return, and the Historical Record

23. a.-e. Results Arithmetic Average Geometric Average Standard Deviation Skew of return Kurtosis of Return VAR,5%

T-Bill 3.43% 3.38% 3.14% 1.02 0.99 0.02%

S&P 500* 8.26% 6.27% 20.03% -0.30 -0.05 -25.76%

Market* 8.48% 6.43% 20.29% -0.33 0.01 -28.03%

* Excess Returns

Comparison The combined market index represents the Fama-French market factor (Mkt). It is better diversified than the S&P 500 index since it contains approximately ten times as many stocks. The total market capitalization of the additional stocks, however, is relatively small compared to the S&P 500. As a result, the performance of the value-weighted portfolios is expected to be quite similar, and the correlation of the excess returns very high. Even though the sample contains 90 observations, the standard deviation of the annual returns is relatively high, but the difference between the two indices is very small. When comparing the continuously compounded excess returns, we see that the difference between the two portfolios is indeed quite small, and the correlation coefficient between their returns is 0.9912. Both deviate from the normal distribution as seen from the negative skew and mixed kurtosis. As a result of all this, we expect the risk premium of the two portfolios to be similar, as we find from the sample. It is worth noting that the excess return of both portfolios has a small negative correlation with the risk-free rate. Since we expect the risk-free rate to be highly correlated with the rate of inflation, this suggests that equities are not a perfect hedge against inflation. More rigorous analysis of this point is important, but beyond the scope of this question.

CFA 1 Answer: V(12/31/2020) = V(1/1/2014)  (1 + g)7 = $100,000  (1.05)7 = $140,710.04

CF 2 Answer: a. and b. are true. The standard deviation is non-negative.

CFA 3 Answer: c. Determines most of the portfolio’s return and volatility over time.

Chapter 05 – Risk, Return, and the Historical Record

CFA 4 Answer: Investment 3. For each portfolio: Utility = E(r) – (0.5  4  2) Investment 1 2 3 4

E(r) 0.12 0.15 0.21 0.24

 0.30 0.50 0.16 0.21

Utility -0.0600 -0.3500 0.1588 0.1518

Choose the portfolio with the highest utility value. CFA 5 Answer: Investment 4. When an investor is risk neutral, A = 0 so that the portfolio with the highest utility is the portfolio with the highest expected return.

CFA 6 Answer: b. Investor’s aversion to risk.

CFA 7 Answer: E(rX) = [0.2  (–0.20)] + (0.5  0.18) + (0.3  0.50) = 0.20 or 20% E(rY) = [0.2  (–0.15)] + (0.5  0.20) + (0.3  0.10) = 0.10 or 10%

CFA 8 Answer:

X2 = [0.2  (–0.20 – 0.20)2] + [0.5  (0.18 – 0.20)2] + [0.3  (0.50 – 0.20)2] = 0.0592

X = 0.2433 = 24.33% Y2 = [0.2  (–0.15 – 0.10)2] + [0.5  (0.20 – 0.10)2] + [0.3  (0.10 – 0.10)2] = 0.0175

Y = 0.1323 = 13.23% CFA 9 Answer: E(r) = (0.9  0.20) + (0.1  0.10) = 0.19 or 19%

Chapter 05 – Risk, Return, and the Historical Record

CFA 10 Answer: The probability is 0.5 that the state of the economy is neutral. Given a neutral economy, the probability that the performance of the stock will be poor is 0.3, and the probability of both a neutral economy and poor stock performance is: 0.3  0.5 = 0.15 CFA 11 Answer: E(r) = (0.1  0.15) + (0.6  0.13) + (0.3  0.07) = 0.114 or 11.4%

Chapter 06 - Efficient Diversification

CHAPTER 06 EFFICIENT DIVERSIFICATION 1. So long as the correlation coefficient is below 1.0, the portfolio will benefit from diversification because returns on component securities will not move in perfect lockstep. The portfolio standard deviation will be less than a weighted average of the standard deviations of the component securities. The smaller the correlation, the bigger the gains from diversification. The upper limit of these gains occurs when the correlation coefficient hits its lower bound of -1.0.

2. The covariance with the other assets is more important. Diversification is accomplished via correlation with other assets and covariance is a function of correlation. Note that covariance is based on the assets’ standard deviations as well as correlation (e.g., Cov(rs , rB ) =  SB   S   B ), but the contribution to decreased risk comes from the correlation component, not the standard deviations.

3. a and b will have the same impact of increasing the Sharpe ratio from .40 to .45. SP =

rP − rf

P

.12 − .04 = .40 .20

.13 − .04 = .45 .20 .12 − .03 b. S P = = .45 .20 .12 − .04 c. S P = = .42 .19 a. S P =

4. The expected return of the portfolio will be impacted if the asset allocation is changed. Since the expected return of the portfolio is the first item in the numerator of the Sharpe ratio, the ratio will be changed.

5. Total variance = Systematic variance + Residual variance = β2×Var(rM) + Var(e) When β = 1.5 and σ(e) = .3, variance = 1.52 × .22 + .32 = .18. In the other scenarios:

s(e)

0.2 0.2

0.3 0.33

1.65 1.5

Total Correlation Variance Coefficient 0.1989 0.1989

0.7399 0.6727

Chapter 06 - Efficient Diversification

a. Both will have the same impact. Total variance will increase from .18 to .1989. b. Even though the increase in the total variability of the stock is the same in either scenario, the increase in residual risk will have less impact on portfolio volatility. This is because residual risk is diversifiable. In contrast, the increase in beta increases systematic risk, which is perfectly correlated with the marketindex portfolio, is non-diversifiable, and therefore has a greater impact on portfolio risk. 6. a. Without doing any math, the severe recession is worse and the boom is better. Thus, there appears to be a higher variance, yet the mean is probably the same since the spread is equally large on both the high and low side. The mean return, however, should be higher since there is higher probability given to the higher returns. b. Calculation of mean return and variance for the stock fund: (A)

(B)

(C)

Rate of Scenario Probability Return Severe recession 0.05 -40 Mild recession 0.25 -14 Normal growth 0.40 17 Boom 0.30 33 Expected Return =

(D)

(E) (F) Deviation Col. B from × Expected Squared Col. C Return Deviation -2.0 -51.2 2621.44 -3.5 -25.2 635.04 6.8 5.8 33.64 9.9 21.8 475.24 Variance = 11.2 Standard Deviation =

(G) Col. B  Col. F 131.07 158.76 13.46 142.57 445.86 21.12

c. Calculation of covariance: (A)

Scenario Severe recession Mild recession Normal growth Boom

(B)

Probability 0.05 0.25 0.40 0.30

Bond Fund -14 10 3 -10

(E)

(F)

Col. C  Col. D 716.8 -252 17.4 -218 Covariance =

Col. B  Col. E 35.84 -63.00 6.96 -65.40 -85.6

Chapter 06 - Efficient Diversification

Covariance has increased because the stock returns are more extreme in the recession and boom periods. This makes the tendency for stock returns to be poor when bond returns are good (and vice versa) even more dramatic.

7. a. One would expect variance to increase because the probabilities of the extreme outcomes are now higher. b. Calculation of mean return and variance for the stock fund:

Scenario Severe recession Mild recession Normal growth Boom

Stock Rate of Probability Return 0.10 -0.37 0.20 -0.11 0.35 0.14 0.35 0.30 Expected Return =

Deviation Col. B from Col. B  Expected Squared  Col. C Return Deviation Col. F -0.037 -0.465 0.2162 0.0216 -0.022 -0.205 0.0420 0.0084 0.049 0.045 0.0020 0.0007 0.105 0.205 0.0420 0.0147 0.095 Variance = 0.0454 Standard Deviation = 0.2132

c. Calculation of covariance Deviation from Mean Return

Scenario Severe recession Mild recession Normal growth Boom

Stock Probability Fund 0.1 -0.465 0.2 -0.205 0.35 0.045 0.35 0.205 Expected return =

Bond Fund -0.122 0.119 0.049 -0.082 -0.036

Col. C  Col. D 0.05673 -0.024395 0.002205 -0.01681 Covariance =

Col. B  Col. E 0.00567 -0.0049 0.00077 -0.0059 -0.0043

Covariance has decreased because the probabilities of the more extreme returns in the recession and boom periods are now higher. This gives more weight to the extremes in the mean calculation, thus making their deviation from the mean less pronounced.

Chapter 06 - Efficient Diversification

8. The parameters of the opportunity set are: E(rS) = 15%, E(rB) = 9%, S = 32%, B = 23%,  = 0.15, rf = 5.5% From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS, rB) = SB]: Bonds Stocks Bonds 529.0 110.4 Stocks 110.4 1024.0 The minimum-variance portfolio proportions are:

B2 － Cov(rS, rB) 529 － 110.4 wMin(S) = 2 = = .3142 2 S + B － 2Cov(rS, rB) 1,024 + 529 － (2 ×110.4) wMin(B) = 1 – .3142 = .6858 The mean and standard deviation of the minimum variance portfolio are: E(rMin) = ( .3142  15%) + ( .6858  9%) = 10.89%

Min = [wS S + wB B + 2 wS wB Cov(rS, rB)]1/2 2

= [( .31422  1024) + ( .68582  529) + (2  .3142  .6858  110.4)]1/2 = 19.94% % in bonds Exp. Return 1.00 0.09 0.80 0.10 0.6858 0.1089 0.60 0.11 0.40 0.13 0.3534 0.1288 0.20 0.14 0.00 0.15

Std dev. Sharpe Ratio 0.23 0.15 0.20 0.23 0.1994 0.2701 Min. Var. Portfolio 0.20 0.29 0.23 0.3155 0.233382 0.3162 Tangency Portfolio 0.27 0.31 0.32 0.30

Investment Opportunity Set

Expected Return (%)

20 15 10 5 0 0

Standard Deviation (%)

Chapter 06 - Efficient Diversification

9. Investment Opportunity Set

Expected Return (%)

20 18 16 14 12 10 8 6 4 2 0 0

Standard Deviation (%)

The graph approximates the points: Minimum variance portfolio Tangency portfolio

E(r) 10.89% 12.88%

 19.94% 23.3382%

10. The Sharpe ratio of the optimal CAL is:

E(rP) － rf 12.88% － 5.50%

P

23.34%

= .3162

11. a. The equation for the CAL is: E(rC) = rf +

E(rP) － rf

P

C = 5.50% + .3162C

Setting E(rC) equal to 12% yields a standard deviation of 20.56%. b. The mean of the complete portfolio as a function of the proportion invested in the risky portfolio (y) is: E(rC) = rf + y × [E(rP) − rf] = 5.50% + y × (12.88% − 5.50%) Setting E(rC) = 12%  y = .8808 (88.08% in the risky portfolio) 1 − y = .1192 (11.92% in T-bills)