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Part Two: Risk

Introduction to Risk and Return

We have managed to go through six chapters without directly addressing the problem of risk, but now the jig is up. We can no longer be satisfied with vague statements like “The opportunity cost of capital depends on the risk of the project.” We need to know how risk is defined, what the links are between risk and the opportunity cost of capital, and how the financial manager can cope with risk in practical situations.

In this chapter, we concentrate on the first of these issues and leave the other two to Chapters 8 and 9. We start by summarizing more than 100 years of evidence on rates of return in capital markets. Then we take a first look at investment risks and show how they can be reduced by portfolio diversification. We introduce you to beta, the standard risk measure for individual securities.

The themes of this chapter, then, are portfolio risk, security risk, and diversification. For the most part, we take the view of the individual investor. But at the end of the chapter, we turn the problem around and ask whether diversification makes sense as a corporate objective.

7-1Over a Century of Capital Market History in One Easy Lesson

Financial analysts are blessed with an enormous quantity of data. There are comprehensive databases of the prices of U.S. stocks, bonds, options, and commodities, as well as huge amounts of data for securities in other countries. We focus on a study by Dimson, Marsh, and Staunton that measures the historical performance of three portfolios of U.S. securities:¹

1. A portfolio of Treasury bills, that is, U.S. government debt securities maturing in less than one year.²

2. A portfolio of U.S. government bonds.

3. A portfolio of U.S. common stocks.

These investments offer different degrees of risk. Treasury bills are about as safe an investment as you can make. There is no risk of default, and their short maturity means that the prices of Treasury bills are relatively stable. In fact, an investor who wishes to lend money for, say, three months can achieve a perfectly certain payoff by purchasing a Treasury bill maturing in three months. However, the investor cannot lock in a real rate of return: There is still some uncertainty about inflation.

By switching to long-term government bonds, the investor acquires an asset whose price fluctuates as interest rates vary. (Bond prices fall when interest rates rise and rise when interest rates fall.) An investor who shifts from bonds to common stocks shares in all the ups and downs of the issuing companies.

Figure 7.1 shows how your money would have grown if you had invested $1 at the end of 1899 and reinvested all dividend or interest income in each of the three portfolios.³ Figure 7.2 is identical except that it depicts the growth in the real value of the portfolio. We focus here on nominal values.

FIGURE 7.1 How an investment of $1 at the end of 1899 would have grown by the end of 2017, assuming reinvestment of all dividend and interest payments

Source: E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton, NJ: Princeton University Press, 2002), with updates provided by the authors.

FIGURE 7.2 How an investment of $1 at the end of 1899 would have grown in real terms by the end of 2017, assuming reinvestment of all dividend and interest payments. Compare this plot with Figure 7.1, and note how inflation has eroded the purchasing power of returns to investors.

Investment performance coincides with our intuitive risk ranking. A dollar invested in the safest investment, Treasury bills, would have grown to $74 by the end of 2017, barely enough to keep up with inflation. An investment in long-term Treasury bonds would have produced $293. Common stocks were in a class by themselves. An investor who placed a dollar in the stocks of large U.S. firms would have received $47,661.

We can also calculate the rate of return from these portfolios for each year from 1900 to 2017. This rate of return reflects both cash receipts—dividends or interest—and the capital gains or losses realized during the year. Averages of the 118 annual rates of return for each portfolio are shown in Table 7.1.

Average Annual Rate of Return
	Nominal	Real	Average Risk Premium (Extra Return versus Treasury Bills)
Treasury bills	3.8	0.9	0
Government bonds	5.3	2.5	1.5
Common stocks	11.5	8.4	7.7

TABLE 7.1 Average rates of return on U.S. Treasury bills, government bonds, and common stocks, 1900–2017 (figures in % per year).

Source: E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns, ( Princeton, NJ: Princeton University Press, 2002), with updates provided by the authors.

Over this period, Treasury bills have provided the lowest average return—3.8% per year in nominal terms and 0.9% in real terms. In other words, the average rate of inflation over this period was about 3% per year. Common stocks were again the winners. Stocks of major corporations provided an average nominal return of 11.5%. By taking on the risk of common stocks, investors earned a risk premium of 11.5 – 3.8 = 7.7% over the return on Treasury bills.

You may ask why we look back over such a long period to measure average rates of return. The reason is that annual rates of return for common stocks fluctuate so much that averages taken over short periods are meaningless. Our only hope of gaining insights from historical rates of return is to look at a very long period.⁴

Arithmetic Averages and Compound Annual Returns

Notice that the average returns shown in Table 7.1 are arithmetic averages. In other words, we simply added the 118 annual returns and divided by 118. The arithmetic average is higher than the compound annual return over the period. The 118-year compound annual return for common stocks was 9.6%.⁵

The proper uses of arithmetic and compound rates of return from past investments are often misunderstood. Therefore, we call a brief time-out for a clarifying example.

Suppose that the price of Big Oil’s common stock is $100. There is an equal chance that at the end of the year the stock will be worth $90, $110, or $130. Therefore, the return could be –10%, +10%, or +30% (we assume that Big Oil does not pay a dividend). The expected return is ⅓ (–10 + 10 + 30) = +10%.

If we run the process in reverse and discount the expected cash flow by the expected rate of return, we obtain the value of Big Oil’s stock:

The expected return of 10% is therefore the correct rate at which to discount the expected cash flow from Big Oil’s stock. It is also the opportunity cost of capital for investments that have the same degree of risk as Big Oil.

Now suppose that we observe the returns on Big Oil stock over a large number of years. If the odds are unchanged, the return will be –10% in a third of the years, +10% in a further third, and +30% in the remaining years. The arithmetic average of these yearly returns is

Thus, the arithmetic average of the returns correctly measures the opportunity cost of capital for investments of similar risk to Big Oil stock.⁶

The average compound annual return⁷ on Big Oil stock would be

(.9 × 1.1 × 1.3)^1/3 − 1 = .088, or 8.8%

which is less than the opportunity cost of capital. Investors would not be willing to invest in a project that offered an 8.8% expected return if they could get an expected return of 10% in the capital markets. The net present value of such a project would be

Moral: If the cost of capital is estimated from historical returns or risk premiums, use arithmetic averages, not compound annual rates of return.⁸

Using Historical Evidence to Evaluate Today’s Cost of Capital

Suppose there is an investment project that you know—don’t ask how—has the same risk as Standard and Poor’s Composite Index. We will say that it has the same degree of risk as the market portfolio, although this is speaking somewhat loosely, because the index does not include all risky investments. What rate should you use to discount this project’s forecasted cash flows?

Clearly you should use the currently expected rate of return on the market portfolio; that is, the return investors would forgo by investing in the proposed project. Let us call this market return r_m. One way to estimate r_m is to assume that the future will be like the past and that today’s investors expect to receive the same “normal” rates of return revealed by the averages shown in Table 7.1. In this case, you would set r_m at 11.5%, the average of past market returns.

Unfortunately, this is not the way to do it; r_m is not likely to be stable over time. Remember that it is the sum of the risk-free interest rate r_f and a premium for risk. We know that r_f varies. For example, in 1981 the interest rate on Treasury bills was about 15%. It is difficult to believe that investors in that year were content to hold common stocks offering an expected return of only 11.5%.

If you need to estimate the return that investors expect to receive, a more sensible procedure is to take the interest rate on Treasury bills and add 7.7%, the average risk premium shown in Table 7.1. For example, suppose that the current interest rate on Treasury bills is 2%. Adding on the average risk premium gives

The crucial assumption here is that there is a normal, stable risk premium on the market portfolio, so that the expected future risk premium can be measured by the average past risk premium.

Even with more than 100 years of data, we can’t estimate the market risk premium exactly; nor can we be sure that investors today are demanding the same reward for risk that they were 50 or 100 years ago. All this leaves plenty of room for argument about what the risk premium really is.⁹

Many financial managers and economists believe that long-run historical returns are the best measure available. Others have a gut instinct that investors don’t need such a large risk premium to persuade them to hold common stocks.¹⁰ For example, surveys of businesspeople and academics commonly suggest that they expect a market risk premium that is somewhat below the historical average.¹¹

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Market risk premium survey

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If you believe that the expected market risk premium is less than the historical average, you probably also believe that history has been unexpectedly kind to investors in the United States and that this good luck is unlikely to be repeated. Here are two reasons that history may overstate the risk premium that investors demand today.

Reason 1 Since 1900, the United States has been among the world’s most prosperous countries. Other economies have languished or been wracked by war or civil unrest. By focusing on equity returns in the United States, we may obtain a biased view of what investors expected. Perhaps the historical averages miss the possibility that the United States could have turned out to be one of those less-fortunate countries.¹²

Figure 7.3 sheds some light on this issue. It is taken from a comprehensive study by Dimson, Marsh, and Staunton of market returns in 20 countries and shows the average risk premium in each country between 1900 and 2017. There is no evidence here that U.S. investors have been particularly fortunate; the United States was just about average in terms of the risk premium.

FIGURE 7.3 Average market risk premiums (nominal return on stocks minus nominal return on bills), 1900–2017

In Figure 7.3, Swiss stocks come bottom of the league; the average risk premium in Switzerland was only 5.5%. The clear winner was Portugal, with a premium of 10.0%. Some of these differences between countries may reflect differences in risk. But remember how difficult it is to make precise estimates of what investors expected. You probably would not be too far out if you concluded that the expected risk premium was the same in each country.¹³

Reason 2 Economists who believe that history may overstate the return that investors expect often point to the fact that stock prices in the United States have for some years outpaced the growth in company dividends or earnings.

Figure 7.4 plots dividend yields in the United States from 1900 to 2017. At the start of the period, the yield was 4.4%. By 1917, it had risen to just over 10.0%, but from then onward, there was a clear, long-term decline. By 2017, yields had fallen to 1.9%. It seems unlikely that investors expected this decline in yields, in which case, some part of the actual return during this period was unexpected.¹⁴

FIGURE 7.4 Dividend yields in the United States 1900–2017

Source: Federal Reserve Bank of St. Louis, Economic Data.

How should we interpret the decline in yields? Suppose that investors expect a steady growth (g) in a stock’s dividend. Then its value is PV = DIV₁/(r − g), and its dividend yield is DIV₁/PV = r – g. In this case, the dividend yield measures the difference between the discount rate and the expected growth rate. So, if we observe that dividend yields decline, it could either be because investors have increased their forecast of future growth or because they are content with a lower expected return.

What’s the answer? Have investors raised their forecast of future dividend growth? One possibility is that they now anticipate a forthcoming golden age of prosperity and surging profits. But a simpler (and more plausible) argument is that companies have increasingly preferred to distribute cash by stock repurchase. As we explain in Chapter 16, the effect of using cash to buy back stock is to reduce the current dividend yield and to increase the future rate of dividend growth. The dividend yield is lower but the expected return is unchanged.

What about the second possibility? Could a decline in risk have caused investors to be satisfied with a lower rate of return? A few years ago, you would likely hear people say that improvements in economic management have made investment in the stock market less risky than it used to be. Since the financial crisis of 2007–2009, investors are less sure that this is the case. But perhaps the growth in mutual funds has made it easier for individuals to diversify away part of their risk, or perhaps pension funds and other financial institutions have found that they also could reduce their risk by investing part of their funds overseas. If these investors can eliminate more of their risk than in the past, they may be content with a lower risk premium.

The effect of any decline in the expected market risk premium is to increase the realized rate of return. Suppose that the stocks in the Standard & Poor’s Index pay an aggregate dividend of $400 billion (DIV₁ = 400) and that this dividend is expected to grow indefinitely at 6% per year (g = .06). If the yield on these stocks is 2%, the expected total rate of return is r = 6 + 2 = 8%. If we plug these numbers into the constant-growth dividend-discount model, then the value of the market portfolio is PV = DIV₁/(r − g) = 400/(.08 − .06) = $20,000 billion, approximately its actual total value in 2017.

The required return of 8%, of course, includes a risk premium. For example, if the risk-free interest rate is 1%, the risk premium is 7%. Suppose that investors now see the stock market as a safer investment than before. Therefore, they revise their required risk premium downward from 7% to 6.5% and the required return from 8% to 7.5%. As a result the value of the market portfolio increases to PV = DIV₁/(r − g) = 400/(.075 − .06) = $26,667 billion, and the dividend yield falls to DIV₁/PV = 400/26,667 = .015 or 1.5%.

Thus a fall of 0.5 percentage point in the risk premium that investors demand would cause a 33% rise in market value, from $20,000 to $26,667 billion. The total return to investors when this happens, including the 2% dividend yield, is 2 + 33 = 35%. With a 1% interest rate, the risk premium earned is 35 – 1 = 34%, much greater than investors expected. If and when this 34% risk premium enters our sample of past risk premiums, we may be led to a double mistake. First, we will overestimate the risk premium that investors required in the past. Second, we will fail to recognize that investors require a lower expected risk premium when they look to the future.

Out of this debate only one firm conclusion emerges: Trying to pin down an exact number for the market risk premium is about as hopeless as eating spaghetti with a one-pronged fork. History contains some clues, but ultimately, we have to judge whether investors on average have received what they expected. Many financial economists rely on the evidence of history and therefore work with a risk premium of about 7%. The remainder generally use a somewhat lower figure. Brealey, Myers, and Allen have no official position on the issue, but we believe that a range of 5% to 8% is reasonable for the risk premium in the United States.

7-2Diversification and Portfolio Risk

You now have a couple of benchmarks. You know the discount rate for safe projects, and you have an estimate of the rate for average-risk projects. But you don’t know yet how to estimate discount rates for assets that do not fit these simple cases. To do that, you have to learn (1) how to measure risk and (2) the relationship between risks borne and risk premiums demanded.

Figure 7.5 shows the 118 annual rates of return for U.S. common stocks. The fluctuations in year-to-year returns are remarkably wide. The highest annual return was 57.6% in 1933—a partial rebound from the stock market crash of 1929–1932. However, there were losses exceeding 25% in six years, the worst being the –43.9% return in 1931.

FIGURE 7.5 The stock market has been a profitable but variable investment

Another way to present these data is by a histogram or frequency distribution. This is done in Figure 7.6, where the variability of year-to-year returns shows up in the wide “spread” of outcomes.

FIGURE 7.6 Histogram of the annual rates of return from the stock market in the United States, 1900–2017, showing the wide spread of returns from investment in common stocks

Source: E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns, (Princeton, NJ: Princeton University Press, 2002), with updates provided by the authors.

Variance and Standard Deviation

The standard statistical measures of spread are variance and standard deviation. The variance of the market return is the expected squared deviation from the expected return. In other words,

Variance (_m) = the expected value of (_m − r_m)²

where _m is the actual return and r_m is the expected return.¹⁵ The standard deviation is simply the square root of the variance:

Standard deviation is often denoted by σ and variance by σ².

Here is a very simple example showing how variance and standard deviation are calculated. Suppose that you are offered the chance to play the following game. You start by investing $100. Then two coins are flipped. For each head that comes up, you get back your starting balance plus 20%, and for each tail that comes up, you get back your starting balance less 10%. Clearly there are four equally likely outcomes:

· Head + head: You gain 40%.

· Head + tail: You gain 10%.

· Tail + head: You gain 10%.

· Tail + tail: You lose 20%.

There is a chance of 1 in 4, or .25, that you will make 40%; a chance of 2 in 4, or .5, that you will make 10%; and a chance of 1 in 4, or .25, that you will lose 20%. The game’s expected return is, therefore, a weighted average of the possible outcomes:

Expected return = (.25 × 40) + (.5 × 10) + (.25 × −20) = +10%

Table 7.2 shows that the variance of the percentage returns is 450. Standard deviation is the square root of 450, or 21. This figure is in the same units as the rate of return, so we can say that the game’s variability is 21%.

TABLE 7.2 The coin-tossing game: calculating variance and standard deviation

If outcomes are uncertain, then more things can happen than will happen. The risk of an asset can be completely expressed, as we did for the coin-tossing game, by writing all possible outcomes and the probability of each. In practice, this is cumbersome and often impossible. Therefore, we use variance or standard deviation to summarize the spread of possible outcomes.¹⁶

These measures are natural indexes of risk.¹⁷ If the outcome of the coin-tossing game had been certain, the standard deviation would have been zero. The actual standard deviation is positive because we don’t know what will happen.

Or think of a second game, the same as the first except that each head means a 35% gain and each tail means a 25% loss. Again, there are four equally likely outcomes:

· Head + head: You gain 70%.

· Head + tail: You gain 10%.

· Tail + head: You gain 10%.

· Tail + tail: You lose 50%.

For this game the expected return is 10%, the same as that of the first game. But its standard deviation is double that of the first game, 42% versus 21%. By this measure the second game is twice as risky as the first.

Measuring Variability

In principle, you could estimate the variability of any portfolio of stocks or bonds by the procedure just described. You would identify the possible outcomes, assign a probability to each outcome, and grind through the calculations. But where do the probabilities come from? You can’t look them up in the newspaper; newspapers seem to go out of their way to avoid definite statements about prospects for securities. We once saw an article headlined “Bond Prices Possibly Set to Move Sharply Either Way.” Stockbrokers are much the same. Yours may respond to your query about possible market outcomes with a statement like this:

The market currently appears to be undergoing a period of consolidation. For the intermediate term, we would take a constructive view, provided economic recovery continues. The market could be up 20% a year from now, perhaps more if inflation continues low. On the other hand, . . .

The Delphic oracle gave advice, but no probabilities.

Most financial analysts start by observing past variability. Of course, there is no risk in hindsight, but it is reasonable to assume that portfolios with histories of high variability also have the least predictable future performance.

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How to calculate variance and standard deviation

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The annual standard deviations and variances observed for our three portfolios over the period 1900–2017 were:¹⁸

Portfolio	Standard Deviation (σ)	Variance (σ²)
Treasury bills	2.9	8.1
Government bonds	9.0	80.6
Common stocks	19.7	388.7

As expected, Treasury bills were the least variable security, and common stocks were the most variable. Government bonds hold the middle ground.

You may find it interesting to compare the coin-tossing game and the stock market as alternative investments. The stock market generated an average annual return of 11.5% with a standard deviation of 19.7%. The game offers 10% and 21%, respectively—slightly lower return and about the same variability. Your gambling friends may have come up with a crude representation of the stock market.

Figure 7.7 compares the standard deviation of stock market returns in 20 countries over the same 118-year period. Portugal occupies high field with a standard deviation of 38.8%, but most of the other countries cluster together with percentage standard deviations in the low 20s.

FIGURE 7.7 The risk (standard deviation of annual returns) of markets around the world, 1900–2017

Of course, there is no reason to suppose that the market’s variability should stay the same over more than a century. For example, Germany, Italy, and Japan now have much more stable economies and markets than they did in the years leading up to and including the Second World War.

Figure 7.8 does not suggest a long-term upward or downward trend in the volatility of the U.S. stock market.¹⁹ Instead there have been periods of both calm and turbulence. In 1995, an unusually tranquil year, the standard deviation of returns was less than 8%. Later, in the financial crisis, the standard deviation spiked at over 40%. By 2017, it had dropped back to its level in 1995.

FIGURE 7.8 Annualized standard deviation of the preceding 52 weekly returns on the Dow Jones Industrial Average, January 1900 to December 2017

Market turbulence over shorter daily, weekly, or monthly periods can be amazingly high. On Black Monday, October 19, 1987, the U.S. market fell by 23% on a single day. The market standard deviation for the week surrounding Black Monday was equivalent to 89% per year. Fortunately, volatility reverted to normal levels within a few weeks after the crash.

How Diversification Reduces Risk

We can calculate our measures of variability equally well for individual securities and portfolios of securities. Of course, the level of variability over 100 years is less interesting for specific companies than for the market portfolio—it is a rare company that faces the same business risks today as it did a century ago.

Table 7.3 presents estimated standard deviations for 10 well-known common stocks for a recent five-year period.²⁰ Do these standard deviations look high to you? They should. The market portfolio’s standard deviation was about 12% during this period. All of our individual stocks had higher volatility. Five of them were more than twice as variable as the market portfolio.

Stock	Standard Deviation σ	Stock	Standard Deviation σ
United States Steel	73.0	Consolidated Edison	16.6
Tesla	57.2	The Travelers Companies	16.4
Newmont	42.2	ExxonMobil	14.0
Southwest Airlines	27.9	Johnson & Johnson	12.8
Amazon	26.6	Coca-Cola	12.6

TABLE 7.3 Standard deviations for selected U.S. common stocks, January 2013–December 2017 (figures in percent per year)

Take a look also at Table 7.4, which shows the standard deviations of some well-known stocks from different countries and of the markets in which they trade. Some of these stocks are more variable than others, but you can see that once again the individual stocks for the most part are more variable than the market indexes.

TABLE 7.4 Standard deviations for selected foreign stocks and market indexes, July 2012–June 2017 (figures in percent per year)

This raises an important question: The market portfolio is made up of individual stocks, so why doesn’t its variability reflect the average variability of its components? The answer is that diversification reduces variability.

Selling umbrellas is a risky business; you may make a killing when it rains, but you are likely to lose your shirt in a heat wave. Selling ice cream is not safe; you do well in the heat wave, but business is poor in the rain. Suppose, however, that you invest in both an umbrella shop and an ice cream shop. By diversifying your business across two businesses, you make an average level of profit come rain or shine.

For investors, even a little diversification can provide a substantial reduction in variability. Suppose you calculate and compare the standard deviations between 2007 and 2017 of one-stock portfolios, two-stock portfolios, five-stock portfolios, and so forth. You can see from Figure 7.9 that diversification can cut the variability of returns by about a third. Notice also that you can get most of this benefit with relatively few stocks: The improvement is much smaller when the number of securities is increased beyond, say, 20 or 30.

FIGURE 7.9 Average risk (standard deviation) of portfolios containing different numbers of stocks. The stocks were selected randomly from stocks traded on the New York Exchange from 2007 through 2017. Notice that diversification reduces risk rapidly at first, then more slowly.

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Try It! Risk and increasing diversification

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Diversification works because prices of different stocks do not move exactly together. Statisticians make the same point when they say that stock price changes are less than perfectly correlated. Look, for example, at Figure 7.10. Panels (a) and (b) show the spread of monthly returns on the stocks of Southwest Airlines and Amazon. Although the two stocks enjoyed a fairly bumpy ride, they did not move in exact lockstep. Often a decline in the value of one stock was offset by a rise in the price of the other.²¹ So, if you had split your portfolio evenly between the two stocks, you could have reduced the monthly fluctuations in the value of your investment. You can see this from panel (c), which shows that if your portfolio had been evenly divided between the two stocks, there would have been many more months when the return was just middling and far fewer cases of extreme returns.

FIGURE 7.10 Diversification reduces risk. Panels (a) and (b) show histograms of the monthly returns on the stocks of Southwest Airlines and Amazon between January 2013 and December 2017. Panel (c) shows a comparable histogram of the returns on a portfolio that was evenly divided between the two stocks. The spread of the portfolio’s returns is markedly less than that of the individual stocks.

The risk that potentially can be eliminated by diversification is called specific risk.²² Specific risk stems from the fact that many of the perils that surround an individual company are peculiar to that company and perhaps its immediate competitors. But there is also some risk that you can’t avoid, regardless of how much you diversify. This risk is generally known as market risk.²³ Market risk stems from the fact that there are other economywide perils that threaten all businesses. That is why stocks have a tendency to move together. And that is why investors are exposed to market uncertainties, no matter how many stocks they hold.

In Figure 7.11, we have divided risk into its two parts—specific risk and market risk. If you have only a single stock, specific risk is very important; but once you have a portfolio of 20 or more stocks, diversification has done the bulk of its work. For a reasonably well- diversified portfolio, only market risk matters. Therefore, the predominant source of uncertainty for a diversified investor is that the market will rise or plummet, carrying the investor’s portfolio with it.

FIGURE 7.11 Diversification eliminates specific risk. But there is some risk that diversification cannot eliminate. This is called market risk.

7-3Calculating Portfolio Risk

We have given you an intuitive idea of how diversification reduces risk, but to understand fully the effect of diversification, you need to know how the risk of a portfolio depends on the risk of the individual shares.

Suppose that 60% of your portfolio is invested in Southwest Airlines and the remainder is invested in Amazon. You expect that over the coming year, Amazon will give a return of 10.0% and Southwest 15.0%. The expected return on your portfolio is simply a weighted average of the expected returns on the individual stocks:²⁴

Expected portfolio return = (.60 × 15) + (.40 × 10) = 13%

Calculating the expected portfolio return is easy. The hard part is to work out the risk of your portfolio. In the past, the standard deviation of returns was 26.6% for Amazon and 27.9% for Southwest Airlines. You believe that these figures are a good representation of the spread of possible future outcomes. At first you may be inclined to assume that the standard deviation of the portfolio is a weighted average of the standard deviations of the two stocks—that is, (.40 × 26.6) + (.60 × 27.9) = 27.4%. That would be correct only if the prices of the two stocks moved in perfect lockstep. In any other case, diversification reduces the risk below this figure.

The exact procedure for calculating the risk of a two-stock portfolio is given in Figure 7.12. You need to fill in four boxes. To complete the top-left box, you weight the variance of the returns on stock by the square of the proportion invested in it Similarly, to complete the bottom-right box, you weight the variance of the returns on stock by the square of the proportion invested in stock

FIGURE 7.12 The variance of a two-stock portfolio is the sum of these four boxes. x₁, x₂ = proportions invested in stocks 1 and 2; variance of stock returns; σ₁₂ = covariance of returns (ρ₁₂ σ₁ σ₂); ρ₁₂ = correlation between returns on stocks 1 and 2.

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How to calculate correlation coefficients

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The entries in these diagonal boxes depend on the variances of stocks 1 and 2; the entries in the other two boxes depend on their covariance. As you might guess, the covariance is a measure of the degree to which the two stocks “covary.” The covariance can be expressed as the product of the correlation coefficient ρ₁₂ and the two standard deviations:²⁵

Covariance between stocks 1 and 2 = σ₁₂ = ρ₁₂ σ₁ σ₂

For the most part stocks tend to move together. In this case the correlation coefficient ρ₁₂ is positive, and therefore the covariance σ₁₂ is also positive. If the prospects of the stocks were wholly unrelated, both the correlation coefficient and the covariance would be zero; and if the stocks tended to move in opposite directions, the correlation coefficient and the covariance would be negative. Just as you weighted the variances by the square of the proportion invested, so you must weight the covariance by the product of the two proportionate holdings x₁ and x₂.

Once you have completed these four boxes, you simply add the entries to obtain the portfolio variance:

The portfolio standard deviation is, of course, the square root of the variance.

Now you can try putting in some figures for Southwest Airlines (LUV) and Amazon (AMZN). We said earlier that if the two stocks were perfectly correlated, the standard deviation of the portfolio would lie 40% of the way between the standard deviations of the two stocks. Let us check this out by filling in the boxes with ρ₁₂ = +1.

The variance of your portfolio is the sum of these entries:

The standard deviation or 60% of the way between 26.6 and 27.9.

Southwest Airlines and Amazon do not move in perfect lockstep. If recent experience is any guide, the correlation between the two stocks is .26. If we go through the same exercise again with ρ₁₂ = .26, we find

The standard deviation is The risk is now less than 60% of the way between 26.6 and 27.9. In fact, it is almost a fifth less than investing in just one of the two stocks.

The greatest payoff to diversification comes when the two stocks are negatively correlated. Unfortunately, this almost never occurs with real stocks, but just for illustration, let us assume it for Amazon and Southwest Airlines. And as long as we are being unrealistic, we might as well go whole hog and assume perfect negative correlation (ρ₁₂ = –1). In this case,

The standard deviation is Risk is almost eliminated. But you can still do better in terms of risk by putting 51.2% of your investment in Amazon and 48.8% in Southwest Airlines.²⁶ In that case, the standard deviation is almost exactly zero. (Check the calculation yourself.)

When there is perfect negative correlation, there is always a portfolio strategy (represented by a particular set of portfolio weights) that will completely eliminate risk. It’s too bad perfect negative correlation doesn’t really occur between common stocks.

General Formula for Computing Portfolio Risk

The method for calculating portfolio risk can easily be extended to portfolios of three or more securities. We just have to fill in a larger number of boxes. Each of those down the diagonal—the red boxes in Figure 7.13—contains the variance weighted by the square of the proportion invested. Each of the other boxes contains the covariance between that pair of securities, weighted by the product of the proportions invested.²⁷

FIGURE 7.13 To find the variance of an N-stock portfolio, we must add the entries in a matrix like this. The diagonal cells contain variance terms (x²σ²) and the off-diagonal cells contain covariance terms (x_ix_jσ_ij).

EXAMPLE 7.1 Limits to Diversification

Did you notice in Figure 7.13 how much more important the covariances become as we add more securities to the portfolio? When there are just two securities, there are equal numbers of variance boxes and of covariance boxes. When there are many securities, the number of covariances is much larger than the number of variances. Thus the variability of a well- diversified portfolio reflects mainly the covariances.

BEYOND THE PAGE

The changing importance of the market factor, 1973–2017

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Suppose we are dealing with portfolios in which equal investments are made in each of N stocks. The proportion invested in each stock is, therefore, 1/N. So in each variance box we have (1/N)² times the variance, and in each covariance box we have (1/N)² times the covariance. There are N variance boxes and N² – N covariance boxes. Therefore,

Notice that as N increases, the portfolio variance steadily approaches the average covariance. If the average covariance were zero, it would be possible to eliminate all risk by holding a sufficient number of securities. Unfortunately common stocks move together, not independently. Thus most of the stocks that the investor can actually buy are tied together in a web of positive covariances that set the limit to the benefits of diversification. Now we can understand the precise meaning of the market risk portrayed in Figure 7.11. It is the average covariance that constitutes the bedrock of risk remaining after diversification has done its work.

BEYOND THE PAGE

Correlations between markets

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Do I Really Have to Add up 36 Million Boxes?

“Adding up the boxes” in Figure 7.13 sounds simple enough, until you remember that there are nearly 6,000 companies listed on the New York and NASDAQ stock exchanges. A portfolio manager who tried to include every one of those companies’ stocks would have to fill up about 6,000 × 6,000 = 36,000,000 boxes! Of course, the boxes above the diagonal line of red boxes in Figure 7.13 match the boxes below. Nevertheless, getting accurate estimates of about 18,000,000 covariances is just impossible. Getting unbiased forecasts of rates of return for about 6,000 stocks is likewise impossible.

Smart investors don’t try. They don’t attempt to forecast portfolio risk or return by “adding up the boxes” for thousands of stocks. But they do understand how portfolio risk is determined by the covariances across securities. (See Example 7.1.) They appreciate the power of diversification, and they want more of it. They want a well-diversified portfolio. Often, they end up holding the entire stock market, as represented by a market index.

You can “buy the market” by purchasing shares in an index fund: a mutual fund or exchange-traded fund (ETF) that invests in the market index that you want to track. Well-run index funds track the market almost exactly and charge very low management fees, often less than 0.1% per year. The most widely used U.S. index is the Standard & Poor’s Composite, which includes 500 of the largest stocks. Index funds have attracted about $5 trillion from investors.

If you have no special information about any of the stocks in the index, it makes sense to be an indexer—that is, to buy the market as a passive rather than active investor. In that case, there is only one box to add up. Just think of the market portfolio as occupying the top-left box in Figure 7.13.

If you want to try out as an active investor, you are well-advised to (1) start with a widely diversified portfolio, for example, a market index fund, and then (2) concentrate on a few stocks as possible additions. You may decide to trade off some investment in the stocks that you are especially fond of against the resulting loss of diversification. In this case, the market index fund occupies the top-left box, and the possible additions occupy a few adjacent boxes.

But our main takeaway so far is this: Smart and serious investors hold widely diversified portfolios; their starting portfolio is often the market itself. How then should such investors assess the risk of individual stocks? Clearly they have to ask how much risk each stock contributes to the risk of a diversified portfolio.

7-4How Individual Securities Affect Portfolio Risk

This brings us to our next major takeaway: The risk of a well-diversified portfolio depends on the market risk of the securities included in the portfolio. Tattoo that statement on your forehead if you can’t remember it any other way. It is one of the most important ideas in this book.

Market Risk Is Measured by Beta

If you want to know the contribution of an individual security to the risk of a well-diversified portfolio, it is no good thinking about how risky that security is if held in isolation—you need to measure its market risk, and that boils down to measuring how sensitive it is to market movements. This sensitivity is called beta (β).

Stocks with betas greater than 1.0 tend to amplify the overall movements of the market. Stocks with betas between 0 and 1.0 tend to move in the same direction as the market, but not as far. Of course, the market is the portfolio of all stocks, so the “average” stock has a beta of 1.0. Table 7.5 reports betas for the 10 well-known common stocks we referred to earlier.

Stock	Beta (β)	Stock	Beta (β)
United States Steel	3.01	ExxonMobil	0.82
Amazon	1.47	Johnson & Johnson	0.81
Southwest Airlines	1.35	Coca-Cola	0.70
The Travelers Companies	1.26	Consolidated Edison	0.11
Tesla	0.94	Newmont	0.10

TABLE 7.5 Estimated betas for selected U.S. common stocks, January 2013– December 2017

Over the five years from January 2013 to December 2017, Amazon had a beta of 1.47. If the future resembles the past, this means that on average, when the market rises an extra 1%, Amazon’s stock price will rise by an extra 1.47%. When the market falls an extra 2%, Amazon’s stock price will fall, on average, an extra 2 × 1.47 = 2.94%. Thus, a line fitted to a plot of Amazon’s returns versus market returns has a slope of 1.47. See Figure 7.14.

FIGURE 7.14 The return on Amazon stock changes on average by 1.47% for each additional 1% change in the market return. Beta is therefore 1.47.

Of course, Amazon’s stock returns are not perfectly correlated with market returns. The company is also subject to specific risk, so the actual returns will be scattered about the line in Figure 7.14. Sometimes, Amazon will head south while the market goes north, and vice versa.

Of the 10 stocks in Table 7.5, U.S. Steel has the highest beta. Newmont Mining is at the other extreme. A line fitted to a plot of Newmont’s returns versus market returns would be less steep: Its slope would be only .10. Notice that many of the stocks that have high standard deviations also have high betas. But that is not always so. For example, Newmont, which has a relatively high standard deviation, is a leading member of the low-beta club in the right-hand column of Table 7.5. It seems that while Newmont is a risky investment if held on its own, it does not contribute to the risk of a diversified portfolio.

Just as we can measure how the returns of a U.S. stock are affected by fluctuations in the U.S. market, so we can measure how stocks in other countries are affected by movements in their markets. Table 7.6 shows the betas for the sample of stocks from other countries.

Stock	Beta (β) Stock		Beta (β)
Tata Motors (India)	1.47	Toronto Dominion Bank (Canada)	1.05
Samsung (Korea)	1.33	Siemens (Germany)	1.01
BP (U.K.)	1.28	Heineken (Netherlands)	0.82
LVMH (France)	1.19	Nestlé (Switzerland)	0.76
Sony (Japan)	1.08	Industrial and Commercial Bank (China)	0.56

TABLE 7.6 Betas for selected foreign stocks, July 2012–June 2017 (beta is measured relative to the stock’s home market)

Why Security Betas Determine Portfolio Risk

Let us review the two crucial points about security risk and portfolio risk:

· Market risk accounts for most of the risk of a well-diversified portfolio.

· The beta of an individual security measures its sensitivity to market movements.

It is easy to see where we are headed: In a portfolio context, a security’s risk is measured by beta. Perhaps we could just jump to that conclusion, but we would rather explain it. Here is an intuitive explanation. We provide a more technical one in footnote 29.

Where’s Bedrock? Look again at Figure 7.11, which shows how the standard deviation of portfolio return depends on the number of securities in the portfolio. With more securities, and therefore better diversification, portfolio risk declines until all specific risk is eliminated and only the bedrock of market risk remains.

Where’s bedrock? It depends on the average beta of the securities selected.

Suppose we constructed a portfolio containing a large number of stocks—500, say—drawn randomly from the whole market. What would we get? The market itself, or a portfolio very close to it. The portfolio beta would be 1.0, and the correlation with the market would be 1.0. If the standard deviation of the market were 20% (roughly its average for 1900–2017), then the portfolio standard deviation would also be 20%. This is shown by the green line in Figure 7.15.

FIGURE 7.15 The green line shows that a well diversified portfolio of randomly selected stocks ends up with β = 1 and a standard deviation equal to the market’s—in this case 20%. The upper red line shows that a well diversified portfolio with β = 1.5 has a standard deviation of about 30%—1.5 times that of the market. The lower blue line shows that a well-diversified portfolio with β = .5 has a standard deviation of about 10%—half that of the market.

Note: In this figure we assume for simplicity that the total risks of individual stocks are proportional to their market risks.

But suppose we constructed the portfolio from a large group of stocks with an average beta of 1.5. Again we would end up with a 500-stock portfolio with virtually no specific risk—a portfolio that moves almost in lockstep with the market. However, this portfolio’s standard deviation would be 30%, 1.5 times that of the market.²⁸ A well-diversified portfolio with a beta of 1.5 will amplify every market move by 50% and end up with 150% of the market’s risk. The upper red line in Figure 7.15 shows this case.

Of course, we could repeat the same experiment with stocks with a beta of .5 and end up with a well-diversified portfolio half as risky as the market. You can see this also in Figure 7.15.

The general point is this: The risk of a well-diversified portfolio is proportional to the portfolio beta, which equals the average beta of the securities included in the portfolio. This shows you how portfolio risk is driven by security betas.

Calculating Beta A statistician would define the beta of stock i as

where σ_im is the covariance between the stock returns and the market returns and is the variance of the returns on the market. It turns out that this ratio of covariance to variance measures a stock’s contribution to portfolio risk.²⁹

BEYOND THE PAGE

Try It! Table 7.7: Calculating Anchovy Queen’s beta

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Here is a simple example of how to do the calculations. Columns 2 and 3 in Table 7.7 show the returns over a particular six-month period on the market and the stock of the Anchovy Queen restaurant chain. You can see that, although both investments provided an average return of 2%, Anchovy Queen’s stock was particularly sensitive to market movements, rising more when the market rises and falling more when the market falls.

TABLE 7.7 Calculating the variance of the market returns and the covariance between the returns on the market and those of Anchovy Queen. Beta is the ratio of the covariance to the variance

Columns 4 and 5 show the deviations of each month’s return from the average. To calculate the market variance, we need to average the squared deviations of the market returns (column 6). And to calculate the covariance between the stock returns and the market, we need to average the product of the two deviations (column 7). Beta is the ratio of the covariance to the market variance, or 76/50.67 = 1.50. A diversified portfolio of stocks with the same beta as Anchovy Queen would be one-and-a-half times as volatile as the market.

7-5Diversification and Value Additivity

We have seen that diversification reduces risk and, therefore, makes sense for investors. But does it also make sense for the firm? Is a diversified firm more attractive to investors than an undiversified one? If it is, we have an extremely disturbing result. If diversification is an appropriate corporate objective, each project has to be analyzed as a potential addition to the firm’s portfolio of assets. The value of the diversified package would be greater than the sum of the parts. So present values would no longer add.

Diversification is undoubtedly a good thing, but that does not mean that firms should practice it. If investors were not able to hold a large number of securities, then they might want firms to diversify for them. But investors can diversify. In many ways they can do so more easily than firms. Individuals can invest in the steel industry this week and pull out next week. A firm cannot do that. To be sure, the individual would have to pay brokerage fees on the purchase and sale of steel company shares, but think of the time and expense for a firm to acquire a steel company or to start up a new steel-making operation.

You can probably see where we are heading. If investors can diversify on their own account, they will not pay any extra for firms that diversify. And if they have a sufficiently wide choice of securities, they will not pay any less because they are unable to invest separately in each factory. Therefore, in countries like the United States, which have large and competitive capital markets, diversification does not add to a firm’s value or subtract from it. The total value is the sum of its parts.

This conclusion is important for corporate finance, because it justifies adding present values. The concept of value additivity is so important that we will give a formal definition of it. If the capital market establishes a value PV(A) for asset A and PV(B) for B, the market value of a firm that holds only these two assets is

PV (AB) = PV (A) + PV (B)

A three-asset firm combining assets A, B, and C would be worth PV(ABC) = PV(A) + PV(B) + PV(C), and so on for any number of assets.

We have relied on intuitive arguments for value additivity. But the concept is a general one that can be proved formally by several different routes.³⁰ The concept seems to be widely accepted, for thousands of managers add thousands of present values daily, usually without thinking about it.

SUMMARY

Our review of capital market history showed that the returns to investors have varied according to the risks they have borne. At one extreme, very safe securities like U.S. Treasury bills have provided an average return over 118 years of only 3.8% a year. The riskiest securities that we looked at were common stocks. The stock market provided an average return of 11.5%, a premium of 7.7% over the safe rate of interest.

This gives us two benchmarks for the opportunity cost of capital. If we are evaluating a safe project, we discount at the current risk-free rate of interest. If we are evaluating a project of average risk, we discount at the expected return on the average common stock. Historical evidence suggests that this return is 7.7% above the risk-free rate, but many financial managers and economists opt for a lower figure. That still leaves us with a lot of assets that don’t fit these simple cases. Before we can deal with them, we need to learn how to measure risk.

Risk is best judged in a portfolio context. Most investors do not put all their eggs into one basket: They diversify. Thus, the effective risk of any security cannot be judged by an examination of that security alone. Part of the uncertainty about the security’s return is diversified away when the security is grouped with others in a portfolio.

Risk in investment means that future returns are unpredictable. This spread of possible outcomes is usually measured by standard deviation. The standard deviation of the market portfolio, as represented by the Standard and Poor’s Composite Index, has averaged around 20% a year.

Most individual stocks have higher standard deviations than this, but much of their variability represents specific risk that can be eliminated by diversification. Diversification cannot eliminate market risk. Diversified portfolios are exposed to variation in the general level of the market.

A security’s contribution to the risk of a well-diversified portfolio depends on how the security is liable to be affected by a general market decline. This sensitivity to market movements is known as beta (β). Beta measures the amount that investors expect the stock price to change for each additional 1% change in the market. The average beta of all stocks is 1.0. A stock with a beta greater than 1 is unusually sensitive to market movements; a stock with a beta below 1 is unusually insensitive to market movements. The standard deviation of a well-diversified portfolio is proportional to its beta. Thus a diversified portfolio invested in stocks with a beta of 2.0 will have twice the risk of a diversified portfolio with a beta of 1.0.

One theme of this chapter is that diversification is a good thing for the investor. This does not imply that firms should diversify. Corporate diversification is redundant if investors can diversify on their own account. Since diversification does not affect the value of the firm, present values add even when risk is explicitly considered. Thanks to value additivity, the net present value rule for capital budgeting works even under uncertainty.

In this chapter, we have introduced you to a number of formulas. They are reproduced in the endpapers to the book. You should take a look and check that you understand them.

Near the end of Chapter 9, we list some Excel functions that are useful for measuring the risk of stocks and portfolios.

FURTHER READING

For international evidence on market returns since 1900, see:

E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton, NJ: Princeton University Press, 2002). More recent data are available in the Credit Suisse Global Investment Returns Yearbook at https://www.credit-suisse.com/media/assets/corporate/docs/about-us/media/media-release/2018/02/giry-summary-2018.pdf.

The Ibbotson Yearbook is a valuable record of the performance of U.S. securities since 1926:

R. Ibbotson, R. J. Grabowski, J. P. Harrington, and C. Nunes, 2017 Stocks, Bonds, Bills, and Inflation SBBI Yearbook (New York: Wiley, 2017).

Useful books and reviews on the equity risk premium include:

P. Fernandez, V. Pershin, and I. Fernandez Acín, “Market Risk Premium and Risk-free Rate Used for 59 Countries in 2018: A Survey,” April 3, 2018. Available at SSRN: https://ssrn.com/abstract=3155709.

W. Goetzmann and R. Ibbotson, The Equity Risk Premium: Essays and Explorations (Oxford, U.K.: Oxford University Press, 2006).

R. Mehra (ed.), Handbook of the Equity Risk Premium (Amsterdam: North-Holland, 2007).

R. Mehra and E. C. Prescott, “The Equity Risk Premium in Retrospect,” in Handbook of the Economics of Finance, eds. G. M. Constantinides, M. Harris, and R. M. Stulz (Amsterdam: North-Holland, 2003) Vol. 1, Part 2, pp. 889–938.

PROBLEM SETS

Select problems are available in McGraw-Hill’s Connect. Please see the preface for more information.

1. Rate of return The level of the Syldavia market index is 21,000 at the start of the year and 25,500 at the end. The dividend yield on the index is 4.2%.

a. What is the return on the index over the year?

b. If the interest rate is 6%, what is the risk premium over the year?

c. If the inflation rate is 8%, what is the real return on the index over the year?

2. Real versus nominal returns The Costaguana stock market provided a rate of return of 95%. The inflation rate in Costaguana during the year was 80%. In Ruritania the stock market return was 12%, but the inflation rate was only 2%. Which country’s stock market provided the higher real rate of return?

3. Arithmetic average and compound returns* Integrated Potato Chips (IPC) does not pay a dividend. Its current stock price is $150 and there is an equal probability that the return over the coming year will be –10%, +20%, or +50%.

a. What is the expected price at year-end?

b. If the probabilities of future returns remain unchanged and you could observe the returns of IPC over a large number of years, what would be the (arithmetic) average return?

c. If you were to discount IPC’s expected price at year-end from part (a) by this number, would you underestimate, overestimate, or correctly estimate the stock’s present value?

d. If you could observe the returns of IPC over a large number of years, what would be the compound (geometric average) rate of return?

e. If you were to discount IPC’s expected price at year-end from part (a) by this number, would you underestimate, overestimate, or correctly estimate the stock’s present value?

Risk premiums* Here are inflation rates and U.S. stock market and Treasury bill returns between 1929 and 1933:

Year	Inflation, %	Stock Market Return, %	T-Bill Return, %
1929	–0.2	–14.5	4.8
1930	–6.0	–28.3	2.4
1931	–9.5	–43.9	1.1
1932	10.3	––9.9	1.0
1933	0.5	57.3	0.3

a. What was the real return on the stock market in each year?

b. What was the average real return?

c. What was the risk premium in each year?

d. What was the average risk premium?

5. Risk Premium Suppose that in year 2030, investors become much more willing than before to bear risk. As a result, they require a return of 8% to invest in common stocks rather than the 10% that they had required in the past. This shift in risk aversion causes a 15% change in the value of the market portfolio.

a. Do stock prices rise by 15% or fall?

b. If you now use past returns to estimate the expected risk premium, will the inclusion of data for 2030 cause you to underestimate or overestimate the return that investors required in the past?

c. Will the inclusion of data for 2030 cause you to underestimate or overestimate the return that investors require in the future?

6. Stocks vs. bonds Each of the following statements is dangerous or misleading. Explain why.

a. A long-term U.S. government bond is always absolutely safe.

b. All investors should prefer stocks to bonds because stocks offer higher long-run rates of return.

c. The best practical forecast of future rates of return on the stock market is a 5- or 10-year average of historical returns.

7. Expected return and standard deviation A game of chance offers the following odds and payoffs. Each play of the game costs $100, so the net profit per play is the payoff less $100.

Probability	Payoff	Net Profit
0.10	$500	$400
0.50	100	0
0.40	0	–100

8. What are the expected cash payoff and expected rate of return? Calculate the variance and standard deviation of this rate of return. (Do not make the adjustment for degrees of freedom described in footnote 15.)

9. Standard deviation of returns The following table shows the nominal returns on Brazilian stocks and the rate of inflation.

a. What was the standard deviation of the market returns? (Do not make the adjustment for degrees of freedom described in footnote 15.)

b. Calculate the average real return.

Year	Nominal Return (%)	Inflation (%)
2012	0.1	5.8
2013	–16.0	5.9
2014	–14.0	6.4
2015	–41.4	10.7
2016	66.2	6.3
2017	26.9	2.9

10. Average returns and standard deviation During the boom years of 2010–2014, ace mutual fund manager Diana Sauros produced the following percentage rates of return. Rates of return on the market are given for comparison.

Calculate the average return and standard deviation of Ms. Sauros’s mutual fund. Did she do better or worse than the market by these measures?

11. Risk and diversification Hippique s.a., which owns a stable of racehorses, has just invested in a mysterious black stallion with great form but disputed bloodlines. Some experts in horseflesh predict the horse will win the coveted Prix de Bidet; others argue that it should be put out to grass. Is this a risky investment for Hippique shareholders? Explain.

12. Risk and diversification Lonesome Gulch Mines has a standard deviation of 42% per year and a beta of +.10. Amalgamated Copper has a standard deviation of 31% a year and a beta of +.66. Explain why Lonesome Gulch is the safer investment for a diversified investor.

13. Diversification* Here are the percentage returns on two stocks.

a. Calculate the monthly variance and standard deviation of each stock. Which stock is the riskier if held on its own?

b. Now calculate the variance and standard deviation of the returns on a portfolio that invests an equal amount each month in the two stocks.

c. Is the variance more or less than half way between the variance of the two individual stocks?

Month	Digital Cheese	Executive Fruit
January	+15%	+7%
February	–3	+1
March	+5	+4
April	+7	+13
May	–4	+2
June	+3	+5
July	–2	–3
August	–8	–2

14.

Risk and diversification In which of the following situations would you get the largest reduction in risk by spreading your investment across two stocks?

a. The two shares are perfectly correlated.

b. There is no correlation.

c. There is modest negative correlation.

d. There is perfect negative correlation

15. Portfolio risk* True or false?

a. Investors prefer diversified companies because they are less risky.

b. If stocks were perfectly positively correlated, diversification would not reduce risk.

c. Diversification over a large number of assets completely eliminates risk.

d. Diversification works only when assets are uncorrelated.

e. Diversification reduces the portfolio beta.

f. A portfolio of stocks, each with a beta of 1.0, will have a beta of less than 1.0 unless the returns are perfectly correlated.

g. A stock with a low standard deviation always contributes less to portfolio risk than a stock with a higher standard deviation.

h. The contribution of a stock to the risk of a well-diversified portfolio depends on its market risk.

i. A well-diversified portfolio with a beta of 2.0 is twice as risky as the market portfolio.

j. An undiversified portfolio with a beta of 2.0 is less than twice as risky as the market portfolio.

16. Portfolio risk To calculate the variance of a three-stock portfolio, you need to add nine boxes:

17. Use the same symbols that we used in this chapter; for example, x₁ = proportion invested in stock 1 and σ₁₂ = covariance between stocks 1 and 2. Now complete the nine boxes.

18. Portfolio risk

a. How many variance terms and how many different covariance terms do you need to calculate the risk of a 100-share portfolio?

b. Suppose all stocks had a standard deviation of 30% and a correlation with each other of .4. What is the standard deviation of the returns on a portfolio that has equal holdings in 50 stocks?

c. What is the standard deviation of a fully diversified portfolio of such stocks?

19. Portfolio risk Suppose that the standard deviation of returns from a typical share is about .40 (or 40%) a year. The correlation between the returns of each pair of shares is about .3.

a. Calculate the variance and standard deviation of the returns on a portfolio that has equal investments in 2 shares, 3 shares, and so on, up to 10 shares.

b. Use your estimates to draw a graph like Figure 7.11. How large is the underlying market variance that cannot be diversified away?

c. Now repeat the problem, assuming that the correlation between each pair of stocks is zero.

18. Portfolio risk Table 7.8 shows standard deviations and correlation coefficients for seven stocks from different countries. Calculate the variance of a portfolio with equal investments in each stock.

TABLE 7.8 Standard deviations of returns and correlation coefficients for a sample of seven stocks

Note: Correlations and standard deviations were calculated using returns in each country’s own currency. In other words, they assume that the investor is protected against exchange risk.

19. Portfolio risk Your eccentric Aunt Claudia has left you $50,000 in BP shares plus $50,000 cash. Unfortunately, her will requires that the BP stock not be sold for one year and the $50,000 cash must be entirely invested in one of the stocks shown in Table 7.8. What is the safest attainable portfolio under these restrictions?

20. Portfolio risk* Hyacinth Macaw invests 60% of her funds in stock I and the balance in stock J. The standard deviation of returns on I is 10%, and on J it is 20%. Calculate the variance and standard deviation of portfolio returns, assuming

a. The correlation between the returns is 1.0.

b. The correlation is .5.

c. The correlation is 0.

21. Stock betas* What is the beta of each of the stocks shown in Table 7.9?

	Stock Return if Market Return Is:
Stock	−10%	+10%
A	0	+20
B	−20	+20
C	−30	0
D	+15	+15
E	+10	−10

22. TABLE 7.9 Stock betas. See Problem 21.

23. Stock betas There are few, if any, real companies with negative betas. But suppose you found one with β = –.25.

a. How would you expect this stock’s rate of return to change if the overall market rose by an extra 5%? What if the market fell by an extra 5%?

You have $1 million invested in a well-diversified portfolio of stocks. Now you receive an additional $20,000 bequest. Which of the following actions will yield the safest overall portfolio return?

i. Invest $20,000 in Treasury bills (which have β = 0).

ii. Invest $20,000 in stocks with β = 1.

iii. Invest $20,000 in the stock with β = –.25.

Explain your answer.

24. Portfolio betas A portfolio contains equal investments in 10 stocks. Five have a beta of 1.2; the remainder have a beta of 1.4. What is the portfolio beta?

a. 1.3.

b. Greater than 1.3 because the portfolio is not completely diversified.

c. Less than 1.3 because diversification reduces beta.

25. Portfolio betas Suppose the standard deviation of the market return is 20%.

a. What is the standard deviation of returns on a well-diversified portfolio with a beta of 1.3?

b. What is the standard deviation of returns on a well-diversified portfolio with a beta of 0?

c. A well-diversified portfolio has a standard deviation of 15%. What is its beta?

d. A poorly diversified portfolio has a standard deviation of 20%. What can you say about its beta?

CHALLENGE

25. Portfolio risk Here are some historical data on the risk characteristics of Ford and Harley Davidson:

	Ford	Harley Davidson
β (beta)	1.26	0.96
Yearly standard deviation of return (%)	18.9	23.1

26. Assume the standard deviation of the return on the market was 9.5%.

a. The correlation coefficient of Ford’s return versus Harley Davidson is 0.30. What is the standard deviation of a portfolio invested half in each share?

b. What is the standard deviation of a portfolio invested one-third in Ford, one-third in Harley Davidson, and one-third in risk-free Treasury bills?

c. What is the standard deviation if the portfolio is split evenly between Ford and Harley Davidson and is financed at 50% margin, that is, the investor puts up only 50% of the total amount and borrows the balance from the broker?

d. What is the approximate standard deviation of a portfolio composed of 100 stocks with betas of 1.26 like Ford? How about 100 stocks like Harley Davidson? [Hint: Part (d) should not require anything but the simplest arithmetic to answer.]

27. Portfolio risk Suppose that Treasury bills offer a return of about 6% and the expected market risk premium is 8.5%. The standard deviation of Treasury-bill returns is zero and the standard deviation of market returns is 20%. Use the formula for portfolio risk to calculate the standard deviation of portfolios with different proportions in Treasury bills and the market. (Note: The covariance of two rates of return must be zero when the standard deviation of one return is zero.) Graph the expected returns and standard deviations.

28. Beta Calculate the beta of each of the stocks in Table 7.8 relative to a portfolio with equal investments in each stock.

FINANCE ON THE WEB

You can download data for questions 1 and 2 from finance.yahoo.com. Refer to the Useful Spreadsheet Functions box near the end of Chapter 9 for information on Excel functions.

1. Download to a spreadsheet the last three years of monthly adjusted stock prices for Coca-Cola (KO), Citigroup (C), and Pfizer (PFE).

a. Calculate the monthly returns.

b. Calculate the monthly standard deviation of those returns (see Section 7-2). Use the Excel function STDEVP to check your answer. Find the annualized standard deviation by multiplying by the square root of 12.

c. Use the Excel function CORREL to calculate the correlation coefficient between the monthly returns for each pair of stocks. Which pair provides the greatest gain from diversification?

d. Calculate the standard deviation of returns for a portfolio with equal investments in the three stocks.

2. Download to a spreadsheet the last five years of monthly adjusted stock prices for each of the companies in Table 7.5 and for the Standard & Poor’s Composite Index (S&P 500).

a. Calculate the monthly returns.

b. Calculate beta for each stock using the Excel function SLOPE, where the “y” range refers to the stock return (the dependent variable) and the “x” range is the market return (the independent variable).

c. How have the betas changed from those reported in Table 7.5?

3. A large mutual fund group such as Fidelity offers a variety of funds. They include sector funds that specialize in particular industries and index funds that simply invest in the market index. Log on to www.fidelity.com and find first the standard deviation of returns on the Fidelity Spartan 500 Index Fund, which replicates the S&P 500. Now find the standard deviations for different sector funds. Are they larger or smaller than the figure for the index fund? How do you interpret your findings?

¹See E. Dimson, P. R. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton, NJ: Princeton University Press, 2002).

²Treasury bills were not issued before 1919. Before that date, the interest rate used is the commercial paper rate.

³Portfolio values are plotted on a log scale. If they were not, the ending values for the common stock portfolio would run off the top of the page.

⁴We cannot be sure that this period is truly representative and that the average is not distorted by a few unusually high or low returns. The reliability of an estimate of the average is usually measured by its standard error. For example, the standard error of our estimate of the average risk premium on common stocks is 1.9%. There is a 95% chance that the true average is within plus or minus 2 standard errors of the 7.7% estimate. In other words, if you said that the true average was between 3.9% and 11.5%, you would have a 95% chance of being right. Technical note: The standard error of the average is equal to the standard deviation divided by the square root of the number of observations. In our case the standard deviation of the risk premium is 20.2%, and therefore the standard error is

⁵This was calculated from (1 + r)¹¹⁸ = 47,661, which implies r = .096. Technical note: For log normally distributed returns the annual compound return is equal to the arithmetic average return minus half the variance. For example, the annual standard deviation of returns on the U.S. market was about .20, or 20%. Variance was therefore .20², or .04. The compound annual return is about .04/2 = .02, or 2 percentage points less than the arithmetic average.

⁶You sometimes hear that the arithmetic average correctly measures the opportunity cost of capital for one-year cash flows, but not for more distant ones. Let us check. Suppose that you expect to receive a cash flow of $121 in year 2. We know that one year hence investors will value that cash flow by discounting at 10% (the arithmetic average of possible returns). In other words, at the end of the year they will be willing to pay PV1 = 121/1.10 = $110 for the expected cash flow. But we already know how to value an asset that pays off $110 in year 1—just discount at the 10% opportunity cost of capital. Thus PV0 = PV1/1.10 = 110/1.1 = $100. Our example demonstrates that the arithmetic average (10% in our example) provides a correct measure of the opportunity cost of capital regardless of the timing of the cash flow.

⁷The compound annual return is often referred to as the geometric average return.

⁸Our discussion assumed that we knew that the returns of –10, +10, and +30% were equally likely. For an analysis of the effect of uncertainty about the expected return see I. A. Cooper, “Arithmetic Versus Geometric Mean Estimators: Setting Discount Rates for Capital Budgeting,” European Financial Management 2 (July 1996), pp. 157–167; and E. Jacquier, A. Kane, and A. J. Marcus, “Optimal Estimation of the Risk Premium for the Long Run and Asset Allocation: A Case of Compounded Estimation Risk,” Journal of Financial Econometrics 3 (2005), pp. 37–55. When future returns are forecasted to distant horizons, the historical arithmetic means are upward-biased. This bias would be small in most corporate-finance applications, however.

⁹Some of the disagreements simply reflect the fact that the risk premium is sometimes defined in different ways. Some measure the average difference between stock returns and the returns (or yields) on long-term bonds. Others measure the difference between the compound rate of return on stocks and the interest rate. As we explained earlier, this is not an appropriate measure of the cost of capital.

¹⁰There is some theory behind this instinct. The high risk premium earned in the market seems to imply that investors are extremely risk-averse. If that is true, investors ought to cut back their consumption when stock prices fall and wealth decreases. But the evidence suggests that when stock prices fall, investors spend at nearly the same rate. This is difficult to reconcile with high risk aversion and a high market risk premium. There is an active research literature on this “equity premium puzzle.” See R. Mehra, “The Equity Premium Puzzle: A Review,” Foundations and Trends in Finance 2 (2006), pp. 11–81; and R. Mehra, ed., Handbook of the Equity Risk Premium (Amsterdam: Elsevier Handbooks in Finance Series, 2008).

¹¹For example, a survey of U.S. CFOs in December 2017 produced an average forecast risk premium of 5.7% over the three-month bill rate. A parallel 2017 survey of academics, analysts, and managers likewise found that the average estimate of the required market risk premium for the United States was 5.4%, though this figure seems to represent the premium over a long-term bond rate. See, respectively, Duke/CFO Magazine, “Global Business Outlook Survey,” Fourth Quarter 2017, http://www.cfosurvey.org/; and P. Fernandez, V. Pershin, and I. Fernández Acín, “Market Risk Premium and Risk-free Rate Used for 59 Countries in 2018: A Survey,” April 4, 2018. Available at SSRN: https://ssrn.com/abstract=3155709.

¹²This possibility was suggested in P. Jorion and W. N. Goetzmann, “Global Stock Markets in the Twentieth Century,” Journal of Finance 54 (June 1999), pp. 953–980.

¹³We are concerned here with the difference between the nominal market return and the nominal interest rate. Sometimes you will see real risk premiums quoted—that is, the difference between the real market return and the real interest rate. If the inflation rate is i, then the real risk premium is (r_m – r_f)/(1 + i). For countries such as Italy that have experienced a high degree of inflation, this real risk premium may be significantly lower than the nominal premium.

¹⁴This argument is made by Fama and French in E. F. Fama and K. R. French, “The Equity Premium,” Journal of Finance 57 (April 2002), pp. 637–659.

¹⁵ One more technical point. When variance is estimated from a sample of observed returns, we add the squared deviations and divide by N – 1, where N is the number of observations. We divide by N – 1 rather than N to correct for what is called the loss of a degree of freedom. The formula is

where is the market return in period t and r_m is the mean of the values of .

¹⁶Which of the two we use is solely a matter of convenience. Since standard deviation is in the same units as the rate of return, it is generally more convenient to use standard deviation. However, when we are talking about the proportion of risk that is due to some factor, it is less confusing to work in terms of the variance.

¹⁷As we explain in Chapter 8, standard deviation and variance are the correct measures of risk if the returns are normally distributed.

¹⁸In discussing the riskiness of bonds, be careful to specify the time period and whether you are speaking in real or nominal terms. The nominal return on a long-term government bond is absolutely certain to an investor who holds on until maturity; in other words, it is risk-free if you forget about inflation. After all, the U.S. government can always print money to pay off its debts. However, the real return on Treasury securities is uncertain because no one knows how much each future dollar will buy.

The bond returns used to construct this table were measured annually. The returns reflect year-to-year changes in bond prices as well as interest received. The one-year returns on long-term bonds are risky in both real and nominal terms.

¹⁹These estimates are derived from weekly rates of return. The weekly variance is converted to an annual variance by multiplying by the number of weeks in the year. That is, the variance of the annual return is 52 times the weekly variance. The longer you hold a security or portfolio, the more risk you have to bear.

This conversion assumes that successive weekly returns are statistically independent. This is, in fact, a good assumption, as we will show in Chapter 13.

Because variance is approximately proportional to the length of time interval over which a security or portfolio return is measured, standard deviation is proportional to the square root of the interval.

²⁰These standard deviations are estimated from monthly data.

²¹ Over this period, the correlation between the returns on the two stocks was .26.

²² Specific risk may be called unsystematic risk, residual risk, unique risk, or diversifiable risk.

²³ Market risk may be called systematic risk or undiversifiable risk.

²⁴Let’s check this. Suppose you invest $40 in Amazon and $60 in Southwest Airlines. The expected dollar return on your Amazon holding is .10 × 40 = $4.00, and on Southwest it is .15 × 60 = $9.00. The expected dollar return on your portfolio is 4.00 + 9.00 = $13.00. The portfolio rate of return is 13.00/100 = .130, or 13.0%.

²⁵Another way to define the covariance is as follows:

Covariance between stocks 1 and 2 = σ₁₂ = expected value of

Note that any security’s covariance with itself is just its variance:

²⁶The standard deviation of Southwest is 27.9/26.6 = 1.049 times the standard deviation of Amazon. Therefore, you have to invest 1.049 times more in Amazon than in Southwest to eliminate all risk in a two-stock portfolio. The portfolio weights that exactly eliminate risk are .512 for Amazon and .488 for Southwest.

²⁷The formal equivalent to “add up all the boxes” is

Notice that when i = j, σ_ij is just the variance of stock i.

²⁸A 500-stock portfolio with β = 1.5 would still have some specific risk. Its actual standard deviation would be a bit higher than 30%. If that worries you, relax; we will show you in Chapter 8 how you can construct a fully diversified portfolio with a beta of 1.5 by borrowing and investing in the market portfolio.

²⁹To understand why, skip back to Figure 7.13. Each row of boxes in Figure 7.13 represents the contribution of that particular security to the portfolio’s risk. For example, the contribution of stock 1 is

x₁x₁ σ₁₁ + x₁ x₂ σ₁₂ + • • • = x₁( x₁ σ₁₁ + x₂ σ₁₂ + • • •)

where x_i is the proportion invested in stock i, and σ_ij is the covariance between stocks i and j (note: σ_ii is equal to the variance of stock i). In other words, the contribution of stock 1 to portfolio risk is equal to the relative size of the holding ( x₁) times the average covariance between stock 1 and all the stocks in the portfolio. We can write this more concisely by saying that the contribution of stock 1 to portfolio risk is equal to the holding size (x₁) times the covariance between stock 1 and the entire portfolio (σ₁ p).

To find stock 1’s relative contribution to risk we simply divide by the portfolio variance to give In other words, it is equal to the holding size (x₁) times the beta of stock 1 relative to the portfolio

We can calculate the beta of a stock relative to any portfolio by simply taking its covariance with the portfolio and dividing by the portfolio’s variance. If we wish to find a stock’s beta relative to the market portfolio we just calculate its covariance with the market portfolio and divide by the variance of the market:

³⁰You may wish to refer to the Appendix to Chapter 31, which discusses diversification and value additivity in the context of mergers.

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