Criticisms Of Indexing Dont Hold Up

Amid the stock market's recent travails, critics are once again taking aim at index funds. But like the firing squad that stands in a circle, they aren't making a whole lot of sense.

Indexing, of course, has never been popular in some quarters. Performance-hungry investors loathe the idea of buying index funds and abandoning all chance of beating the market averages. Meanwhile, most Wall Street firms would love indexing to fall from favor because there isn't much money to be made running index funds.

But the latest barrage of nonsense also reflects today's peculiar stock market. Here is a look at four recent complaints about index funds:

They're undiversified. Critics charge that the most popular index funds, those that track the Standard & Poor's 500-stock index, are too focused on a small number of stocks and a single sector, technology.

S&P 500 funds currently have 25.3% of their money in their 10-largest stockholdings and 31.1% of assets in technology companies. This narrow focus made S&P 500 funds especially vulnerable during this year's market swoon.

But the same complaint could be leveled at actively managed funds. According to Chicago researchers Morn-ingstar Inc., diversified U.S. stock funds have an average 36.2% invested in their 10-largest stocks, with 29.1% in technology. . . .

They're top-heavy. Critics also charge that S&P 500 funds represent a big bet on big-company stocks. True enough. I have often argued that most folks would be better off indexing the Wilshire 5000, which includes most regularly traded U.S. stocks, including both large and small companies.

But let's not get carried away. The S&P 500 isn't that narrowly focused. After all, it represents some 77.2% of U.S. stock-market value.

Whether you index the S&P 500 or the Wilshire 5000, what you are getting is a fund that pretty much mirrors the U.S. market. If you think index funds are undiversi-fied and top-heavy, there can only be one reason: The market is undiversified and top heavy. . . .

They're chasing performance. In recent years, the stock market's return has been driven by a relatively small number of sizzling performers. As these hot stocks climbed in value, index funds became more heavily invested in these companies, while lightening up on lackluster performers.

That, complain critics, is the equivalent of buying high and selling low. A devastating criticism? Hardly. This is what all investors do. When Home Depot's stock climbs 5%, investors collectively end up with 5% more money riding on Home Depot's shares. . . .

You can do better. Sure, there is always a chance you will get lucky and beat the market. But don't count on it.

As a group, investors in U.S. stocks can't outperform the market because, collectively, they are the market. In fact, once you figure in investment costs, active investors are destined to lag behind Wilshire 5000-index funds, because these active investors incur far higher investment costs.

But this isn't just a matter of logic. The proof is also in the numbers. Over the past decade, only 28% of U.S. stock funds managed to beat the Wilshire 5000, according to Vanguard.

The problem is, the long-term argument for indexing gets forgotten in the rush to embrace the latest, hottest funds. An indexing strategy will beat most funds in most years. But in any given year, there will always be some funds that do better than the index. These winners garner heaps of publicity, which whets investors' appetites and encourages them to try their luck at beating the market. . . .

Source: Jonathan Clements, "Criticisms of Indexing Don't Hold Up," The Wall Street Journal, April 25, 2000. Reprinted by permission of The Wall Street Journal, © 2000 Dow Jones & Company, Inc. All Rights Reserved Worldwide.

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 average active investor. (We will elaborate this argument and provide a more comprehensive analysis of the relative success of passive strategies in later chapters.) The above box points out that passive index funds have actually outperformed actively managed funds in the past decade.

To summarize, a passive strategy involves investment in two passive portfolios: virtually risk-free short-term T-bills (or, alternatively, a money market fund) and a fund of common 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 1999 data, the

CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset passive risky portfolio offered an average risk premium of 9.3% and a standard deviation of 20.6%, resulting in a reward-to-variability ratio of .45.

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 1.2 in Chapter 1, we estimate that approximately 74% of net worth is invested in a broad array of risky assets.6 We assume this portfolio has the same reward-risk characteristics as the S&P 500, that is, a risk premium of 9.3% and standard deviation of 20.6% as documented in Table 7.4. Substituting these values in equation 7.5, we obtain


which implies a coefficient of risk aversion of

Of course, this calculation is highly speculative. We have assumed without basis that the average investor holds the naive view that historical average rates of return and standard deviations are the best estimates of expected rates of return and risk, looking to the future. To the extent that the average investor takes advantage of contemporary information in addition to simple historical data, our estimate of A = 3.0 would be an unjustified inference. Nevertheless, a broad range of studies, taking into account the full range of available assets, places the degree of risk aversion for the representative investor in the range of 2.0 to 4.0.7


Suppose that expectations about the S&P 500 index and the T-bill rate are the same as they were in 1999, but you find that today a greater proportion is invested in T-bills than in 1999. What can you conclude about the change in risk tolerance over the years since 1999?


1. Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce risk. Other methods involve diversification of the risky portfolio and hedging. We take up these methods in later chapters.

2. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the standard deviation of real rates on short-term T-bills is small compared to that of other assets such as long-term bonds and common stocks, so for the purpose of our analysis we consider T-bills as the risk-free asset. Money market funds hold, in addition to T-bills, short-term relatively safe obligations such as CP and 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.

3. An investor's risky portfolio (the risky asset) can be characterized by its reward-to-variability ratio, S = [E(rP) - r]/aP. 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

6 We include in the risky portfolio tangible assets, half of pension reserves, corporate and noncorporate equity, mutual fund shares, and personal trusts. This portfolio sums to $36,473 billion, which is 74% of household net worth.

7 See, for example, I. Friend and M. Blume, "The Demand for Risky Assets," American Economic Review 64 (1974), or S. J. Grossman and R. J. Shiller, "The Determinants of the Variability of Stock Market Prices," American Economic Review 71 (1981).

would prefer a steeper-sloping CAL, because that means higher expected return for any level of risk. If the borrowing rate is greater than the lending rate, the CAL will be "kinked" at the point of the risky asset.

4. 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 premium for taking on one additional percentage of standard deviation. More risk-averse investors have steeper indifference curves; that is, they require a greater risk premium for taking on more risk.

5. 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:

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

6. 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 1999 investors 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 = 3.0 for the average investor. This is in line with other studies, which estimate typical risk aversion in the range of 2.0 through 4.0.


capital allocation decision asset allocation decision security selection decision risky asset complete portfolio risk-free asset capital allocation line reward-to-variability ratio certainty equivalent passive strategy capital market line


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%.

1. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. What is the expected value and standard deviation of the rate of return on his portfolio?

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

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

3. What is the reward-to-variability ratio (S) of your risky portfolio? Your client's?

4. 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.

5. 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?

CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset 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?

6. 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?

7. 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 optimized portfolio?

You estimate that a passive portfolio, that is, 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%. Continue to assume that rf = 8%.

8. 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.

9. 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.)

10. Consider the client in problem 7 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)?

11. Look at the data in Table 7.4 on the average risk premium of the S&P 500 over T-bills, and the standard deviation of that risk premium. Suppose that the S&P 500 is your risky portfolio.

a. If your risk-aversion coefficient is 4 and you believe that the entire 1926-1999 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 1980-1999 period is representative?

c. What do you conclude upon comparing your answers to (a) and (b)?

12. 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 7.5.

13. Consider the following information about a risky portfolio that you manage, and a risk-free asset: E(rP) = 11%, aP = 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?

7. Capital Allocation between the Risky Asset and the Risk-Free Asset

PART II Portfolio Theory 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?

Suppose that the borrowing rate that your client faces is 9%. Assume that the S&P 500 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 13.

14. 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 who will invest in both the index fund and a money market fund.

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

16. Solve problems 14 and 15 for a client who uses your fund rather than an index fund.

17. 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)?

Use the following graph to answer problems 18 and 19.

Expected return, E(r)

G. 4









allocation line (CAL)

1 —

Risk, a

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

19. Which point designates the optimal portfolio of risky assets?

20. Given $100,000 to invest, what is the expected risk premium in dollars of investing in equities versus risk-free T-bills based on the following table?

7. Capital Allocation between the Risky Asset and the Risk-Free Asset

CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset 203



Expected Return

Invest in






Invest in

risk-free T-bills


21. The change from a straight to a kinked capital allocation line is a result of the:

a. Reward-to-variability 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.

22. 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?

Expected Return

Standard Deviation of Return













23. What is the reward-to- variability ratio for the equity fund in problem 22?

1. 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 X 54% = 27%, and the dollar value of your position in E is $300,000 X .27 = $81,000.

2. 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-variability 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 oP, 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(rc) = rf + y[E(rp) - rf]


ctc = yap


and therefore the reward-to-variability ratio of this portfolio is

C ac yap ap

which is independent of the proportion y.

3. The lending and borrowing rates are unchanged at rf = 7%, rf = 9%. The standard

deviation of the risky portfolio is still 22%, but its expected rate of return shifts from

15% to 17%.

The slope of the two-part CAL is

E(rp) - rf for the lending range ap

E(rP) - rf for the borrowing range ap

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.

4. a. The parameters are rf = 7, E(rP) = 15, aP = 22. An investor with a degree of risk

aversion A will choose a proportion y in the risky portfolio of

E(rp) - rf

y = .01 X Aa2p

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

15 - 7 55 y .01 X 3 X 484

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%. Ac

cordingly, the expected return and standard deviation of the optimal portfolio in-


E(rc) = 7 + (.55 X 8) = 11.4 (before: 10.28)

ac = .55 X 22 = 12.1 (before: 9.02)

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 least risk-averse of

these investors hold 100% in the risky portfolio (y = 1). We can solve for the de

gree of risk aversion of these "cut off' investors from the parameters of the in-

vestment opportunities:

^ E(rp) - rf 8 y 1 .01 X Aa2p 4.84A

which implies


CHAPTER 7 Capital Allocation between the Risky Asset and the Risk-Free Asset 205

Any investor who is more risk tolerant (that is, A < 1.65) would borrow if the borrowing rate were 7%. For borrowers,

Suppose, for example, an investor has an A of 1.1. When rf= rf =7%, this investor chooses to invest in the risky portfolio:


which means that the investor will borrow an amount equal to 50% of her own investment capital. Raise the borrowing rate, in this case to rf = 9%, and the investor will invest less in the risky asset. In that case:


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 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 goes up, this investor chooses portfolio C2 (in the borrowing range of the kinked CAL), which involves less borrowing than before. This investor is hurt by the increase in the borrowing rate.



5. 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 S&P 500 is not a good proxy for the optimal risky portfolio. Perhaps investors expect a higher real rate on T-bills.

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