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What Level of Risk Is Right for You? W

O R

D S

FR O

M T

H E

S T

R E

E T

No risk, no reward. Most people intuitively understand that they have to bear some risk to achieve an acceptable return on their investment portfolios.

But how much risk is right for you? If your investments turn sour, you may put at jeopardy your ability to retire, to pay for your kid’s college education, or to weather an unexpected need for cash. These worst-case scenarios focus our attention on how to manage our exposure to uncertainty.

Assessing—and quantifying—risk aversion is, to put it mildly, difficult. It requires confronting at least these two big questions.

First, how much investment risk can you afford to take? If you have a steady high-paying job, for example, you have greater ability to withstand investment losses. Conversely, if you are close to retirement, you have less ability to adjust your lifestyle in response to bad investment outcomes.

Second, you need to think about your personality and decide how much risk you can tolerate. At what point will you be unable to sleep at night?

To help clients quantify their risk aversion, many financial firms have designed quizzes to help people determine whether they are conservative, moderate, or aggressive investors. These quizzes try to get at clients’ attitudes toward risk and their capacity to absorb investment losses.

Here is a sample of the sort of questions these quizzes tend to pose to shed light on an investor’s risk tolerance.

MEASURING YOUR RISK TOLERANCE Circle the letter that corresponds to your answer.

!. The stock market fell by more than #$% in %$$&. If you had been holding a substantial stock investment in that year, which of the following would you have done?

a. Sold off the remainder of your investment before it had the chance to fall further.

b. Stayed the course with neither redemptions nor purchases.

c. Bought more stock, reasoning that the market is now cheaper and therefore offers better deals.

%. The value of one of the funds in your '$!(k) plan (your pri- mary source of retirement savings) increased #$% last year. What will you do?

a. Move your funds into a money market account in case the price gains reverse.

b. Sit tight and do nothing.

c. Put more of your assets into that fund, reasoning that its value is clearly trending upward.

#. How would you describe your non-investment sources of income (for example, your salary)?

a. Highly uncertain

b. Moderately stable

c. Highly stable

'. At the end of the month, you find yourself:

a. Short of cash and impatiently waiting for your next paycheck.

b. Not overspending your salary, but not saving very much.

c. With a comfortable surplus of funds to put into your sav- ings account.

(. You are #$ years old and enrolling in your company’s retirement plan, and you need to allocate your contribu- tions across # funds: a money market account, a bond fund, and a stock fund. Which of these allocations sounds best to you?

a. Invest everything in a safe money-market fund.

b. Split your money evenly between the bond fund and stock fund.

c. Put everything into the stock fund, reasoning that by the time you retire the year-to-year fluctuations in stock returns will have evened out.

". You are a contestant on Let’s Make a Deal, and have just won $!,$$$. But you can exchange the winnings for two random payoffs. One is a coin flip with a payoff of $%,($$ if the coin comes up heads. The other is a flip of two coins with a payoff of $",$$$ if both coins come up heads. What will you do?

a. Keep the $!,$$$ in cash.

b. Choose the single coin toss.

c. Choose the double coin toss.

). Suppose you have the opportunity to invest in a start-up firm. If the firm is successful, you will multiply your invest- ment by a factor of ten. But if it fails, you will lose everything. You think the odds of success are around %$%. How much would you be willing to invest in the start-up?

a. Nothing

b. % months’ salary

c. " months’ salary

&. Now imagine that to buy into the start-up you will need to borrow money. Would you be willing to take out a $!$,$$$ loan to make the investment?

a. No

b. Maybe

c. Yes

SCORING YOUR RISK TOLERANCE For each question, give yourself one point if you answered (a), two points if you answered (b), and three points for a (c). The higher your total score, the greater is your risk tolerance, or equivalently, the lower is your risk aversion.

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Estimating Risk Aversion How can we estimate the levels of risk aversion of individual investors? A number of meth- ods may be used. The questionnaire in the nearby box is of the simplest variety and, indeed, can distinguish only between high (conservative), medium (moderate), or low (aggressive) levels of the coefficient of risk aversion. More complex questionnaires, allowing subjects to pinpoint specific levels of risk aversion coefficients, ask would-be investors to choose from various sets of hypothetical lotteries.

Access to the investment accounts of active investors would provide observations of how portfolio composition changes over time. Coupling this information with estimates of the risk and return of these positions would in principle allow us to infer investors’ risk aversion coefficients.

Finally, researchers track the behavior of groups of individuals to obtain average degrees of risk aversion. These studies range from observed purchase of insurance policies to labor supply and aggregate consumption behavior.

6.2 Capital Allocation across Risky and Risk-Free Portfolios History shows us that long-term bonds have been riskier investments than Treasury bills and that stocks have been riskier still. On the other hand, the riskier investments have offered higher average returns. Investors, of course, do not make all-or-nothing choices from these investment classes. They can and do construct their portfolios using securities from all asset classes. Some of the portfolio may be in risk-free Treasury bills, some in high-risk stocks.

The most straightforward way to control the risk of the portfolio is through the fraction of the portfolio invested in Treasury bills or other safe money market securities versus what is invested in risky assets. Most investment professionals consider such broad asset allocation decisions the most important part of portfolio construction. Consider this state- ment by John Bogle, made when he was chairman of the Vanguard Group of Investment Companies:

The most fundamental decision of investing is the allocation of your assets: How much should you own in stock? How much should you own in bonds? How much should you own in cash reserves? .!.!. That decision [has been shown to account] for an astonishing 94% of the differences in total returns achieved by institutionally managed pension funds. .!.!. There is no reason to believe that the same relationship does not also hold true for individual investors.1

Therefore, we start our discussion of the risk–return trade-off available to investors by examining the most basic asset allocation choice: how much of the portfolio should be placed in risk-free money market securities versus other risky asset classes.

We denote the investor’s portfolio of risky assets as P and the risk-free asset as F. We assume for the sake of illustration that the risky component of the investor’s overall

1John C. Bogle, Bogle on Mutual Funds (Burr Ridge, IL: Irwin Professional Publishing, 1994), p. 235.

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portfolio comprises two mutual funds, one invested in stocks and the other invested in long-term bonds. For now, we take the composition of the risky portfolio as given and focus only on the allocation between it and risk-free securities. In the next chapter, we ask how best to determine the composition of the risky portfolio.

When we shift wealth from the risky portfolio to the risk-free asset, we do not change the relative proportions of the various risky assets within the risky portfolio. Rather, we reduce the relative weight of the risky portfolio as a whole in favor of risk-free assets.

For example, assume that the total market value of an initial portfolio is $300,000, of which $90,000 is invested in the Ready Asset money market fund, a risk-free asset for practical purposes. The remaining $210,000 is invested in risky securities—$113,400 in equities (E) and $96,600 in long-term bonds (B). The equities and bond holdings comprise “the” risky portfolio, 54% in E and 46% in B:

E:

w E =

113,400 _______ 210,000

= .54

B:

w B = 96,600 _______ 210,000

= .46

The weight of the risky portfolio, P, in the complete portfolio, including risk-free and risky investments, is denoted by y:

y!=! 210,000 _______ 300,000

!=!.7 (risky assets)

1!"!y!= 90,000 _______ 300,000

!=!.3 (risk-free assets)

The weights of each asset class in the complete portfolio are, therefore, as follows:

E : $113,400 ________ $300,000

= .378

B : $96,600 ________

$300,000 = .322

Risky!portfolio = E + B = .700

The risky portfolio makes up 70% of the complete portfolio.

Suppose the owner of this portfolio wishes to decrease risk by reducing the allocation to the risky portfolio from y = .! to y = ."#. The risky portfolio would then total only ."# # $$%%,%%% = $&#',%%%, requiring the sale of $(),%%% of the original $)&%,%%% of risky holdings, with the proceeds used to purchase more shares in Ready Asset (the money market fund). Total

Example 6.2 The Risky Portfolio

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Nevertheless, it is common practice to view Treasury bills as “the” risk-free asset. Their short-term nature makes their prices insensitive to interest rate fluctuations. Indeed, an investor can lock in a short-term nominal return by buying a bill and holding it to matu- rity. Moreover, inflation uncertainty over the course of a few weeks, or even months, is negligible compared with the uncertainty of stock market returns.

In practice, most investors use a broad range of money market instruments as a risk-free asset. All the money market instruments are virtually free of interest rate risk because of their short maturities and are fairly safe in terms of default or credit risk.

Money market funds hold, for the most part, three types of securities—Treasury bills, other Treasury and U.S. agency securities, and repurchase agreements. The yields to matu- rity on nongovernment money market securities are always somewhat higher than those of T-bills with comparable maturity. Still, we saw in Chapter 2, Figure 2.2, that these yield spreads over T-bills are generally small, despite an occasional spike during periods of financial stress.

While the portfolio composition of money market funds changes over time, in the last few years, T-bills have made up only about 20% of their portfolios.2 Nevertheless, the risk of blue-chip short-term investments is minuscule compared with that of most other assets such as long-term corporate bonds, common stocks, or real estate. Hence we treat money market funds as the most easily accessible risk-free asset for most investors.

2See http://www.icifactbook.org/, Section 4 of Data Tables.

6.4 Portfolios of One Risky Asset and a Risk-Free Asset In this section, we examine the risk–return combinations available to investors once the properties of the risky portfolio have been determined. This is the “technical” part of capi- tal allocation. In the next section we address the “personal” part of the problem—the indi- vidual’s choice of the best risk–return combination from the feasible set.

Suppose the investor has already decided on the composition of the risky portfolio, P. Now the concern is with capital allocation, that is, the proportion of the investment budget, y, to be allocated to P. The remaining proportion, 1 ! y, is to be invested in the risk-free asset, F.

Denote the risky rate of return of P by rP, its expected rate of return by E(rP), and its standard deviation by "P. The rate of return on the risk-free asset is denoted as rf. In our numerical example, we will assume that E(rP) = 15%, "P = 22%, and the risk-free rate is rf = 7%. Thus the risk premium on the risky asset is E(rP) ! rf = 8%.

With a proportion, y, in the risky portfolio, and 1 ! y in the risk-free asset, the rate of return on the complete portfolio, denoted C, is rC where r C #=#y r P #+# ( 1#!#y ) r f (6.2) Taking the expectation of this portfolio’s rate of return, E(rC)#=#yE(rP)#+#(1#!#y)rf =#rf#+#y[E(rP)#!#rf ]#=#7#+#y(15#!#7) (6.3)

This result has a nice interpretation: The base rate of return for any portfolio is the risk- free rate. In addition, the portfolio is expected to earn a proportion, y, of the risk premium of the risky portfolio, E(rP) ! rf .

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Students of calculus will recognize that the maximization problem is solved by setting the derivative of this expression to zero. Doing so and solving for y gives us the optimal position for risk-averse investors in the risky asset, y*, as follows:5

y* = E ( r P ) ! r f _______

A " P 2 (6.7)

This solution shows that the optimal position in the risky asset is inversely proportional to the level of risk aversion and the level of risk (as measured by the variance) and directly proportional to the risk premium offered by the risky asset.

5The derivative with respect to y equals E( r P )#!# r f #!#yA " P 2 . Setting this expression equal to zero and solving for y yields Equation 6.7.

Using our numerical example [rf = !%, E(rP) = "#%, and "P = $$%], and expressing all returns as decimals, the optimal solution for an investor with a coefficient of risk aversion A = % is

y&*'=' ."# ! .(! _______ % $ .$$ $

'='.%"

In other words, this particular investor will invest %"% of the investment budget in the risky asset and #)% in the risk-free asset. As we saw in Figure *.#, this is the value of y at which utility is maximized.

With %"% invested in the risky portfolio, the expected return and standard deviation of the complete portfolio are

E ( r C ) '='!'+' [ .%"'$' ( "#'!'! ) ] '='"(.$+% , C '='.%"'$'$$'=').($%

The risk premium of the complete portfolio is E(rC) ! rf = -.$+%, which is obtained by taking on a portfolio with a standard deviation of ).($%. Notice that -.$+/).($ = .-*, which is the reward- to-volatility (Sharpe) ratio of any complete portfolio given the parameters of this example.

Example 6.4 Capital Allocation

A graphical way of presenting this decision problem is to use indifference curve analy- sis. To illustrate how to build an indifference curve, consider an investor with risk aversion A = 4 who currently holds all her wealth in a risk-free portfolio yielding rf = 5%. Because the variance of such a portfolio is zero, Equation 6.1 tells us that its utility value is U = .05. Now we find the expected return the investor would require to maintain the same level of utility when holding a risky portfolio, say, with " = 1%. We use Equation 6.1 to find how much E(r) must increase to compensate for the higher value of ":

U = E ( r ) ! ! $ A $ " 2

.05 =

E ( r ) ! ! $ 4 $ .01 2

This implies that the necessary expected return increases to

Required#E ( r ) = .05 + ! $ A $ "

2

= .05 + ! $ 4 $ .01 2 = .0502

(6.8)

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by more than a factor of 2, ranging from .29 to .65. The lesson here is that we should be very humble when we use historical data to forecast future returns. Returns and the risk– return trade-off are extremely difficult to predict, and we can have only a loose sense of what that trade-off will be in coming periods.

We call the capital allocation line provided by 1-month T-bills and a broad index of common stocks the capital market line (CML). A passive strategy generates an invest- ment opportunity set that is represented by the CML.

How reasonable is it for an investor to pursue a passive strategy? We cannot answer such a question without comparing the strategy to the costs and benefits accruing to an active portfolio strategy. Some thoughts are relevant even at this point, however.

First, the alternative active strategy is not free. Whether you choose to invest the time and cost to acquire the information needed to generate an optimal active portfolio of risky assets, or whether you delegate the task to a professional who will charge a fee, consti- tution of an active portfolio is more expensive than a passive one. Passive management entails only negligible costs to purchase T-bills and very modest management fees to either an exchange-traded fund or a mutual fund company that operates a market index fund. Vanguard, for example, operates several index portfolios. One, the 500 Index Fund, tracks the S&P 500. It purchases shares of the firms comprising the S&P 500 in proportion to the market values of the outstanding equity of each firm, and therefore essentially replicates the S&P 500 index. It has one of the lowest operating expenses (as a percentage of assets) of all mutual stock funds precisely because it requires minimal managerial effort. Whereas the S&P 500 is primarily an index of large, high-capitalization (large cap) stocks, another Vanguard index fund, the Total Stock Market Index Fund,!is more inclusive and provides investors with exposure to the entire U.S. equity market, including small- and mid-cap stocks as well as growth and value stocks. It is nearly identical to what we have called the U.S. Market Index.

A second reason to pursue a passive strategy is the free-rider benefit. If there are many active, knowledgeable investors who quickly bid up prices of undervalued assets and force down prices of overvalued assets (by selling), we have to conclude that at any time most assets will be fairly priced. Therefore, a well-diversified portfolio of common stock will be a reasonably fair buy, and the passive strategy may not be inferior to that of the aver- age active investor. (We will elaborate on this argument and provide a more comprehen- sive analysis of the relative success of passive strategies in later chapters.) As we saw in Chapter 4, passive index funds have actually outperformed most actively managed funds in the past decades and investors are increasingly responding to the lower costs and better performance of index funds by directing their investments into these products.

To summarize, a passive strategy involves investment in two passive portfolios: virtu- ally risk-free short-term T-bills (or, alternatively, a money market fund) and a fund of com- mon stocks that mimics a broad market index. The capital allocation line representing such a strategy is called the capital market line. Historically, based on 1926 to 2015 data, the passive risky portfolio offered an average risk premium of 8.3% and a standard deviation of 20.59%, resulting in a reward-to-volatility ratio of .40.

Passive investors allocate their investment budgets among instruments according to their degree of risk aversion. We can use our analysis to deduce a typical investor’s risk- aversion parameter. From Table 1.1 in Chapter 1, we estimate that approximately 68.7% of net worth is invested in a broad array of risky assets.7 We assume this portfolio has the 7We include in the risky portfolio the following entries from Table 1.1 of Chapter 1: real assets ($30,979 bil- lion), half of pension reserves ($10,486 billion), corporate and noncorporate equity ($24,050 billion), and half of mutual fund shares ($4,060 billion). This portfolio sums to $69,575 billion, which is 68.7% of household net worth ($101,306 billion).

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obligations such as repurchase agreements or bank CDs. These entail some default risk, but again, the additional risk is small relative to most other risky assets. For convenience, we often refer to money market funds as risk-free assets.

7. An investor’s risky portfolio (the risky asset) can be characterized by its reward-to-volatility or Sharpe ratio, S = [E(rP) ! rf]/"P. This ratio is also the slope of the CAL, the line that, when graphed, goes from the risk-free asset through the risky asset. All combinations of the risky asset and the risk-free asset lie on this line. Other things equal, an investor would prefer a steeper-sloping CAL, because that means higher expected return for any level of risk. If the bor- rowing rate is greater than the lending rate, the CAL will be “kinked” at the point of the risky asset.

8. The investor’s degree of risk aversion is characterized by the slope of his or her indifference curve. Indifference curves show, at any level of expected return and risk, the required risk pre- mium for taking on one additional percentage point of standard deviation. More risk-averse investors have steeper indifference curves; that is, they require a greater risk premium for taking on more risk.

9. The optimal position, y*, in the risky asset, is proportional to the risk premium and inversely proportional to the variance and degree of risk aversion:

y*#=# E ( r P ) ! r f _______

A " P 2

Graphically, this portfolio represents the point at which the indifference curve is tangent to the CAL.

10. A passive investment strategy disregards security analysis, targeting instead the risk-free asset and a broad portfolio of risky assets such as the S&P 500 stock portfolio. If in 2016 inves- tors took the mean historical return and standard deviation of the S&P 500 as proxies for its expected return and standard deviation, then the values of outstanding assets would imply a degree of risk aversion of about A = 2.85 for the average investor. This is in line with other stud- ies, which estimate typical risk aversion in the range of 2.0 through 4.0.

risk premium fair game risk averse utility certainty equivalent rate risk neutral

KEY TERMSrisk lover mean-variance (M-V)

criterion indifference curve complete portfolio risk-free asset

capital allocation line (CAL) reward-to-volatility or Sharpe

ratio passive strategy capital market line (CML)

Utility score: U = E(r)#–#! A " 2

Optimal allocation to risky portfolio: y*#=# E ( r P ) ! r f _________

A " P 2

KEY EQUATIONS

1. Which of the following choices best completes the following statement? Explain. An investor with a higher degree of risk aversion, compared to one with a lower degree, will most prefer investment portfolios a. with higher risk premiums. b. that are riskier (with higher standard deviations). c. with lower Sharpe ratios. d. with higher Sharpe ratios.

PROBLEM SETS

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2. Which of the following statements are true? Explain. a. A lower allocation to the risky portfolio reduces the Sharpe (reward-to-volatility) ratio. b. The higher the borrowing rate, the lower the Sharpe ratios of levered portfolios. c. With a fixed risk-free rate, doubling the expected return and standard deviation of the risky

portfolio will double the Sharpe ratio. d. Holding constant the risk premium of the risky portfolio, a higher risk-free rate will increase

the Sharpe ratio of investments with a positive allocation to the risky asset. 3. What do you think would happen to the expected return on stocks if investors perceived higher

volatility in the equity market? Relate your answer to Equation 6.7. 4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either

$70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio? b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be

the expected rate of return on the portfolio? c. Now suppose that you require a risk premium of 12%. What is the price that you will be will-

ing to pay? d. Comparing your answers to (a) and (c), what do you conclude about the relationship between

the required risk premium on a portfolio and the price at which the portfolio will sell? 5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of

18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills?

6. Draw the indifference curve in the expected return–standard deviation plane corresponding to a utility level of .05 for an investor with a risk aversion coefficient of 3. (Hint: Choose several possible standard deviations, ranging from 0 to .25, and find the expected rates of return provid- ing a utility level of .05. Then plot the expected return–standard deviation points so derived.)

7. Now draw the indifference curve corresponding to a utility level of .05 for an investor with risk aversion coefficient A = 4. Comparing your answer to Problem 6, what do you conclude?

8. Draw an indifference curve for a risk-neutral investor providing utility level .05. 9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the

indifference curve for a utility level of .05 for a risk lover. For Problems 10 through 12: Consider historical data showing that the average annual rate of return on the S&P 500 portfolio over the past 90 years has averaged roughly 8% more than the Treasury bill return and that the S&P 500 standard deviation has been about 20% per year. Assume these values are representative of investors’ expectations for future performance and that the current T-bill rate is 5%. 10. Calculate the expected return and variance of portfolios invested in T-bills and the S&P 500

index with weights as follows:

Wbills Windex

!""" #.! !.$ !.% !.& !.' !.' !.& !.% !.$ #.! !"""

11. Calculate the utility levels of each portfolio of Problem 10 for an investor with A = 2. What do you conclude?

12. Repeat Problem 11 for an investor with A = 3. What do you conclude?

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Use these inputs for Problems 13 through 19: 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%. 13. Your client chooses to invest 70% of a portfolio in your fund and 30% in an essentially risk-free

money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?

14. Suppose that your risky portfolio includes the following investments in the given proportions:

Stock A !"% Stock B #!% Stock C $#%

What are the investment proportions of your client’s overall portfolio, including the position in T-bills?

15. What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client’s? 16. Draw the CAL of your portfolio on an expected return–standard deviation diagram. What is the

slope of the CAL? Show the position of your client on your fund’s CAL. 17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment

budget so that the overall portfolio will have an expected rate of return of 16%. a. What is the proportion y? b. What are your client’s investment proportions in your three stocks and the T-bill fund? c. What is the standard deviation of the rate of return on your client’s portfolio?

18. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio’s standard deviation will not exceed 18%. a. What is the investment proportion, y? b. What is the expected rate of return on the complete portfolio?

19. Your client’s degree of risk aversion is A = 3.5. a. What proportion, y, of the total investment should be invested in your fund? b. What is the expected value and standard deviation of the rate of return on your client’s opti-

mized portfolio? 20. Look at the data in Table 6.7 on the average excess return of the U.S. equity market and the

standard deviation of that excess return. Suppose that the U.S. market is your risky portfolio. a. If your risk-aversion coefficient is A = 4 and you believe that the entire 1926–2015 period

is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity?

b. What if you believe that the 1970–1991 period is representative? c. What do you conclude upon comparing your answers to (a) and (b)?

21. Consider the following information about a risky portfolio that you manage and a risk-free asset: E(rP) = 11%, !P = 15%, rf = 5%. a. Your client wants to invest a proportion of her total investment budget in your risky fund

to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky portfolio, P, and what proportion in the risk-free asset?

b. What will be the standard deviation of the rate of return on her portfolio? c. Another client wants the highest return possible subject to the constraint that you limit his

standard deviation to be no more than 12%. Which client is more risk averse? 22. Investment Management Inc. (IMI) uses the capital market line to make asset allocation recom-

mendations. IMI derives the following forecasts: " Expected return on the market portfolio: 12%

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! Standard deviation on the market portfolio: 20% ! Risk-free rate: 5%

Samuel Johnson seeks IMI’s advice for a portfolio asset allocation. Johnson informs IMI that he wants the standard deviation of the portfolio to equal half of the standard deviation for the market portfolio. Using the capital market line, what expected return can IMI provide subject to Johnson’s risk constraint?

For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 23. Draw a diagram of your client’s CML, accounting for the higher borrowing rate. Superimpose

on it two sets of indifference curves, one for a client who will choose to borrow, and one for a client who will invest in both the index fund and a money market fund.

24. What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which y = 1?

25. Solve Problems 23 and 24 for a client who uses your fund rather than an index fund. 26. What is the largest percentage fee that a client who currently is lending (y < 1) will be willing

to pay to invest in your fund? What about a client who is borrowing (y > 1)? For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, yields an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 27. Draw the CML and your funds’ CAL on an expected return–standard deviation diagram.

a. What is the slope of the CML? b. Characterize in one short paragraph the advantage of your fund over the passive fund.

28. Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio. a. Explain to your client the disadvantage of the switch. b. Show him the maximum fee you could charge (as a percentage of the investment in your

fund, deducted at the end of the year) that would leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of his CAL by reducing the expected return net of the fee.)

29. Consider again the client in Problem 19 with A = 3.5. a. If he chose to invest in the passive portfolio, what proportion, y, would he select? b. Is the fee (percentage of the investment in your fund, deducted at the end of the year) that

you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation decision (i.e., his choice of y)?

Utility Formula Data

Investment Expected Return, E(r) Standard Deviation, "

! ".!# ".$" # ".!% ".%" $ ".#! ".!& ' ".#' ".#!

U = E(r())–)*(A"#, where A = '

Use the following data in answering CFA Problems 1–3:

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1. On the basis of the utility formula above, which investment would you select if you were risk averse with A = 4?

2. On the basis of the utility formula above, which investment would you select if you were risk neutral?

3. The variable (A) in the utility formula represents the: a. Investor’s return requirement. b. Investor’s aversion to risk. c. Certainty equivalent rate of the portfolio. d. Preference for one unit of return per four units of risk.

Use the following graph to answer CFA Problems 4 and 5.

Risk, !

Expected Return, E(r)

H

G

F

E

Capital Allocation Line (CAL)

1

1

2

2

3

3

4

4

0

4. Which indifference curve represents the greatest level of utility that can be achieved by the investor?

5. Which point designates the optimal portfolio of risky assets? 6. Given $100,000 to invest, what is the expected risk premium in dollars of investing in equities

versus risk-free T-bills on the basis of the following table?

7. The change from a straight to a kinked capital allocation line is a result of the: a. Reward-to-volatility (Sharpe) ratio increasing. b. Borrowing rate exceeding the lending rate. c. Investor’s risk tolerance decreasing. d. Increase in the portfolio proportion of the risk-free asset.

8. You manage an equity fund with an expected risk premium of 10% and an expected standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her portfolio in your equity fund and $40,000 in a T-bill money market fund. What is the expected return and standard deviation of return on your client’s portfolio?

9. What is the reward-to-volatility (Sharpe) ratio for the equity fund in CFA Problem 8?

ReturnProbabilityAction

Invest in equities !." $#!,!!! !.$

-$%!,!!! Invest in risk-free T-bills &.! $ #,!!!

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E$INVESTMENTS EXERCISES There is a difference between an investor’s willingness to take risk and his or her ability to take risk. Take the quizzes offered at the Web sites below and compare the results. If they are signifi- cantly different, which one would you use to determine an investment strategy?

https://personal.vanguard.com/us/FundsInvQuestionnaire http://njaes.rutgers.edu:!"!"/money/riskquiz/ www.schwab.com/public/file/P-##!$%#/InvestorProfileQuestionnaire.pdf

SOLUTIONS TO CONCEPT CHECKS 1. The investor is taking on exchange rate risk by investing in a pound-denominated asset. If the

exchange rate moves in the investor’s favor, the investor will benefit and will earn more from the U.K. bill than the U.S. bill. For example, if both the U.S. and U.K. interest rates are 5%, and the current exchange rate is $1.40 per pound, a $1.40 investment today can buy 1 pound, which can be invested in England at a certain rate of 5%, for a year-end value of 1.05 pounds. If the year-end exchange rate is $1.50 per pound, the 1.05 pounds can be exchanged for 1.05 ! $1.50 = $1.575 for a rate of return in dollars of 1 + r = $1.575/$1.40 = 1.125, or r = 12.5%, more than is available from U.S. bills. Therefore, if the investor expects favorable exchange rate movements, the U.K. bill is a speculative investment. Otherwise, it is a gamble.

2. For the A = 4 investor the utility of the risky portfolio is

U"=".20"#"(!"!"4"! .3 2 )"=".02 while the utility of bills is

U"=".07"#"(!"!"4"!"0 )"=".07 The investor will prefer bills to the risky portfolio. (Of course, a mixture of bills and the portfolio

might be even better, but that is not a choice here.) For the less risk-averse investor with"A = 2, the utility of the risky portfolio is

U"=".20"#"(!"!"2"!" .3 2 )"=".11 while the utility of bills is again .07. The less risk-averse investor prefers the risky portfolio. 3. The less risk-averse investor has a shallower indifference curve. An increase in risk requires less

increase in expected return to restore utility to the original level.

P E(rP)

E(r)

!P

!

Less Risk Averse

More Risk Averse

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C H A P T E R ! Capital Allocation to Risky Assets "#$

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4. Holding 50% of your invested capital in Ready Assets means that your investment proportion in the risky portfolio is reduced from 70% to 50%.

Your risky portfolio is constructed to invest 54% in E and 46% in B. Thus the proportion of E in your overall portfolio is .5 ! 54% = 27%, and the dollar value of your position in E is $300,000 ! .27 = $81,000.

5. In the expected return–standard deviation plane all portfolios that are constructed from the same risky and risk-free funds (with various proportions) lie on a line from the risk-free rate through the risky fund. The slope of the CAL (capital allocation line) is the same everywhere; hence the reward-to-volatility (Sharpe) ratio is the same for all of these portfolios. Formally, if you invest a proportion, y, in a risky fund with expected return E(rP) and standard deviation "P, and the remainder, 1 # y, in a risk-free asset with a sure rate rf, then the portfolio’s expected return and standard deviation are

E( r C ) = r f + y [ E( r P ) # r f ] " C

=

y " P

and therefore the Sharpe ratio of this portfolio is

S C $=$ E( r C ) # r f ________

" C $=$

y [ E( r P )$#$ r f ] __________ y " P

$=$ E( r P ) # r f _______

" P

which is independent of the proportion y. 6. The lending and borrowing rates are unchanged at r f $=$7%,$ r f B $=$9% . The standard deviation of

the risky portfolio is still 22%, but its expected rate of return on the risky portfolio increases from 15% to 17%.

The slope of the two-part CAL is

E ( r P ) # r f _______

" P for$the$lending$range

E ( r P ) # r f B ________

" P for$the$borrowing$range

Thus, in both cases, the slope increases: from 8/22 to 10/22 for the lending range and from 6/22 to 8/22 for the borrowing range.

7. a. The parameters are rf = .07, E(rP) = .15, "P = .22. An investor with a degree of risk aversion A will choose a proportion y in the risky portfolio of

y$=$ E( r P ) # r f _______

A " P 2

With the assumed parameters and with A = 3 we find that

y$=$ .15 # .07 ________ 3 ! .0484

= .55

When the degree of risk aversion decreases from the original value of 4 to the new value of 3, investment in the risky portfolio increases from 41% to 55%. Accordingly, both the expected return and standard deviation of the optimal portfolio increase:

E( r C )

= .07 + (.55 ! .08) = .114 (before: .1028)

" C

=

.55 ! .22 = .121 (before:$.0902)

b. All investors whose degree of risk aversion is such that they would hold the risky portfolio in a proportion equal to 100% or less (y % 1.00) are lending rather than borrowing, and so are unaffected by the borrowing rate. The individuals just at the dividing line separating lend- ers from borrowers are the ones who choose to hold exactly 100% of their assets in the risky

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portfolio (y = 1). We can solve for the degree of risk aversion of these “cut off” investors from the parameters of the investment opportunities:

y = 1 =! E( r P ) " r f _______

A # P 2 !=! .08 _______

.0484 A

which implies

A!=! .08 _____ .0484

!=!1.65

Any investor who is more risk tolerant (i.e., A < 1.65) would choose to borrow if the borrowing rate were 7%. These are the investors who are affected by the higher borrowing rate. For borrowers,

y!=! E( r P )!"! r f B ________

A # P 2

Suppose, for example, an investor has A = 1.1. If! r f !=! r f B !=!7% , this investor would have chosen to invest in the risky portfolio:

y!=! .08 __________ 1.1!$!.0484

!=!1.50

which means that the investor would have borrowed an amount equal to 50% of her own investment capital, placing all the proceeds in the risky portfolio. But at the higher borrowing rate,! r f B !=!9% , the investor will choose to borrow less and put less in the risky asset. In this case,

y!=! .06 __________ 1.1!$!.0484

!=!1.13

and “only” 13% of her investment capital will be borrowed. Graphically, the line from rf to the risky portfolio shows the CAL for lenders. The dashed part of the line originating at rf would be relevant if the borrowing rate equaled the lending rate. When the borrowing rate exceeds the lending rate, the CAL is kinked at the point corresponding to the risky portfolio.

The following figure shows indifference curves of two investors. The steeper indifference curve portrays the more risk-averse investor, who chooses portfolio C0, which involves lending. This investor’s choice is unaffected by the borrowing rate. The more risk-tolerant investor is portrayed by the shallower-sloped indifference curves. If the lending rate equaled the borrowing rate, this investor would choose portfolio C1 on the dashed part of the CAL. When the borrowing rate is higher, this investor instead chooses portfolio C2 (in the borrowing range of the kinked

E(r)

E(rP)

rf

C0

C2

C1

rf B

!P

!

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CAL), which involves less borrowing than before. This investor is hurt by the increase in the borrowing rate.

8. If all the investment parameters remain unchanged, the only reason for an investor to decrease the investment proportion in the risky asset is an increase in the degree of risk aversion. If you think that this is unlikely, then you have to reconsider your faith in your assumptions. Perhaps the U.S. equity market is not a good proxy for the optimal risky portfolio. Perhaps investors expect a higher real rate on T-bills.

We digress in this appendix to examine the rationale behind our contention that investors are risk averse. Recognition of risk aversion as central in investment decisions goes back at least to 1738. Daniel Bernoulli, one of a famous Swiss family of distinguished mathemati- cians, spent the years 1725 through 1733 in St. Petersburg, where he analyzed the follow- ing coin-toss game. To enter the game one pays an entry fee. Thereafter, a coin is tossed until the first head appears. The number of tails, denoted by n, that appears until the first head is tossed is used to compute the payoff, $R, to the participant, as

R ( n ) !=! 2 n

The probability of no tails before the first head (n = 0) is 1/2 and the corresponding pay- off is 20 = $1. The probability of one tail and then heads (n = 1) is 1/2 " 1/2 with payoff 21 = $2, the probability of two tails and then heads (n = 2) is 1/2 " 1/2 " 1/2, and so forth.

The following table illustrates the probabilities and payoffs for various outcomes:

APPENDIX A:! Risk Aversion, Expected Utility, and the St. Petersburg Paradox

Tails Probability Payoff = $ R(n) Probability " Payoff

! "/# $" $"/# " "/$ $# $"/# # "/% $$ $"/# & "/"' $% $"/# # # # # n ("/#)n + " $#n $"/#

The expected payoff is therefore

E ( R ) !=! ! n = 0

$

Pr ( n ) R ( n ) !=! 1 ⁄ 2 !+! 1 ⁄ 2 !+!%!=!$

The evaluation of this game is called the “St. Petersburg Paradox.” Although the expected payoff is infinite, participants obviously will be willing to purchase tickets to play the game only at a finite, and possibly quite modest, entry fee.

Bernoulli resolved the paradox by noting that investors do not assign the same value per dollar to all payoffs. Specifically, the greater their wealth, the less their “appreciation” for each extra dollar. We can make this insight mathematically precise by assigning a welfare or utility value to any level of investor wealth. Our utility function should increase as wealth is higher, but each extra dollar of wealth should increase utility by progressively smaller amounts.9 (Modern economists would say that investors exhibit “decreasing

9This utility is similar in spirit to the one that assigns a satisfaction level to portfolios with given risk and return attributes. However, the utility function here refers not to investors’ satisfaction with alternative portfolio choices but only to the subjective welfare they derive from different levels of wealth.

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SOLUTIONS TO CONCEPT CHECK A.1. a.

U ( W ) = !

___ W U ( 50,000 ) = !

______ 50,000 = 223.61

U ( 150,000 ) = 387.30

b. E(U) = (.5 ! 223.61) + (.5 ! 387.30) = 305.45 c. We must find WCE that has utility level 305.45. Therefore

!

____ W CE =

305.45

W CE

=

305.45 2 = $93,301

d. Yes. The certainty equivalent of the risky venture is less than the expected outcome of $100,000.

e. The certainty equivalent of the risky venture to this investor is greater than it was for the log utility investor considered in the text. Hence this utility function displays less risk aversion.

APPENDIX B:!Utility Functions and Risk Premiums

The utility function of an individual investor allows us to measure the subjective value the individual would place on a dollar at various levels of wealth. Essentially, a dollar in bad times (when wealth is low) is more valuable than a dollar in good times (when wealth is high).

Suppose that all investors hold the risky S&P 500 portfolio. Then, if the portfolio value falls in a worse-than-expected economy, all investors will, albeit to different degrees, experience a “low-wealth” scenario. Therefore, the equilibrium value of a dollar in the low-wealth economy would be higher than the value of a dollar when the portfolio per- forms better than expected. This observation helps explain why an investment in a stock portfolio (and hence in individual stocks) has a risk premium that appears to be so high and results in probability of shortfall that is so low. Despite the low probability of under- performing, stocks still do not dominate the lower-return risk-free bond, because if an investment shortfall should transpire, it will coincide with states in which the marginal value of an extra dollar is high.

Does revealed behavior of investors demonstrate risk aversion? Looking at prices and past rates of return in financial markets, we can answer with a resounding yes. With remarkable consistency, riskier bonds are sold at lower prices than are safer ones with otherwise similar characteristics. Riskier stocks also have provided higher average rates of return over long periods of time than less risky assets such as T-bills. For example, over the 1926 to 2015 period, the average rate of return on the S&P 500 portfolio exceeded the T-bill return by around 8% per year.

2. If the cost of insuring your house is $1 per $1,000 of value, what will be the certainty equivalent of your end-of-year wealth if you insure your house at: a. ! its value. b. Its full value. c. 1! times its value.

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It is abundantly clear from financial data that the average, or representative, inves- tor exhibits substantial risk aversion. For readers who recognize that financial assets are priced to compensate for risk by providing a risk premium and at the same time feel the urge for some gambling, we have a constructive recommendation: Direct your gambling impulse to investment in financial markets. As Von Neumann once said, “The stock market is a casino with the odds in your favor.” A small risk-seeking investment may provide all the excitement you want with a positive expected return to boot!

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