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Risk, Return, and the Opportunity Cost of Capital

Risk, Return, and the Opportunity Cost of Capital

Financial analysts are blessed with an enormous quantity of data on security prices and returns. For example, the University of Chicago’s Center for Research in Security Prices (CRSP) has developed a file of prices and dividends for each month since 1926 for every stock that has been listed on the New York Stock Exchange (NYSE) Other files give data for stocks that are traded on the American Stock Exchange and the over-the-counter market, data for bonds, for options, and so on. But this is supposed to be one easy lesson. We, therefore, concentrate on a study by Ibbotson Associates that measures the historical performance of five portfolios of securities:

1. A portfolio of Treasury bills, i.e., United States government debt securities maturing in less than one year.

2. A portfolio of long-term United States government bonds.

3. A portfolio of long-term corporate bonds.

4. Standard and Poors Composite Index (S&P 500), which represents a portfolio of common stocks of 500 large firms. (Although only a small proportion of the 7,000 or so publicly traded companies are included in the S&P 500, these companies account for over 70 percent of the value of stocks traded.)

5. A portfolio of the common stocks of small firms.

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 government to corporate bonds accepts an additional default risk. An investor who shifts from corporate bonds to common stocks has a direct share in the risks of the enterprise.

Figure 7.1 shows how your money would have grown if you had invested $1 at the start of 1926 and reinvested all dividend or interest income in each of the five portfolios. Figure 7.2 is identical except that it depicts the growth in the real value of the portfolio. We will focus here on nominal values.



Portfolio performance coincides with our intuitive risk ranking. A dollar invested in the safest investment, Treasury bills, would have grown to just over $16 by 2000, barely enough to keep up with inflation. An investment in long-term Treasury bonds would have produced $49, and corporate bonds a pinch more. 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 $2,587. The jackpot, however, went to investors in stocks of small firms, who walked away with $6,402 for each dollar invested.

Ibbotson Associates also calculated the rate of return from these portfolios for each year from 1926 to 2000. This rate of return reflects both cash receipts—dividends or interestand the capital gains or losses realized during the year. Averages of the 75 annual rates of return for each portfolio are shown in Table 7.1.


Since 1926 Treasury bills have provided the lowest average return3.9 percent per year in nominal terms and .8 percent in real terms. In other words, the average rate of inflation over this period was just over 3 percent per year. Common stocks were again the winners. Stocks of major corporations provided on average a risk premium of 9.1 percent a year over the return on Treasury bills. Stocks of small firms offered an even higher premium.

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.

Notice that the average returns shown in Table 7.1 are arithmetic averages. In other words, Ibbotson Associates simply added the 75 annual returns and divided by 75. The arithmetic average is higher than the compound annual return over the period. The 75-year compound annual return for the S&P index was 11.0 percent.

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 Oils 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 percent, +10 percent, or +30 percent (we assume that Big Oil does not pay a dividend). The expected return is 1/3 ( - 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 Oils stock:


The expected return of 10 percent is therefore the correct rate at which to discount the expected cash flow from Big Oils 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 percent in a third of the years, +10 percent in a further third, and +30 percent 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. The average compound annual return on Big Oil stock would be


less than the opportunity cost of capital. Investors would not be willing to invest in a project that offered an 8.8 percent expected return if they could get an expected return of 10 percent 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.

Suppose there is an investment project which you knowdont ask howhas the same risk as Standard and Poors 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 securities. What rate should you use to discount this projects 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 rm. One way to estimate rm.is to assume that the future will be like the past and that todays 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 rm.at 13 percent, the average of past market returns.

Unfortunately, this is not the way to do it; rm. is not likely to be stable over time. Remember that it is the sum of the risk-free interest rate rf and a premium for risk. We know that rfvaries. For example, in 1981 the interest rate on Treasury bills was about 15 percent. It is difficult to believe that investors in that year were content to hold common stocks offering an expected return of only 13 percent.

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 9.1 percent, the average risk premium shown in Table 7.1. For example, as we write this in mid-2001 the interest rate on Treasury bills is about 3.5 percent. Adding on the average risk premium, therefore, 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 75 years of data, we cant estimate the market risk premium exactly; nor can we be sure that investors today are demanding the same reward for risk that they were 60 or 70 years ago. All this leaves plenty of room for argument about what the risk premium really is.

Many financial managers and economists believe that long-run historical returns are the best measure available. Others have a gut instinct that investors dont need such a large risk premium to persuade them to hold common stocks. In a recent survey of financial economists, more than a quarter of those polled believed that the expected risk premium was about 8 percent, but most of the remainder opted for a figure between 4 and 7 percent. The average estimate was just over 6 percent.

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

Reason 1 Over the past 75 years stock prices in the United States have outpaced dividend payments. In other words, there has been a long-term decline in the dividend yield. Between 1926 and 2000 this decline in yield added about 2 percent a year to the return on common stocks. Was this yield change anticipated? If not, it would be more reasonable to take the long-term growth in dividends as a measure of the capital appreciation that investors were expecting. This would point to a risk premium of about 7 percent.

Reason 2 Since 1926 the United States has been among the worlds 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.

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 15 countries and shows the average risk premium in each country between 1900 and 2000. Two points are worth making. Notice first that in the United States the risk premium over 101 years has averaged 7.5 percent, somewhat less than the figure that we cited earlier for the period 1926–2000. The period of the First World War and its aftermath was in many ways not typical, so it is hard to say whether we get a more or less representative picture of investor expectations by adding in the extra years. But the effect of doing so is an important reminder of how difficult it is to obtain an accurate measure of the risk premium.


Now compare the returns in the United States with those in the other countries. There is no evidence here that U.S. investors have been particularly fortunate; the USA was exactly average in terms of the risk premium. Danish common stocks came bottom of the league; the average risk premium in Denmark was only 4.3 percent. Top of the form was Italy with a premium of 11.1 percent. Some of these variations between countries may reflect differences in risk. For example, Italian stocks have been particularly variable and investors may have required a higher return to compensate. 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.

Reason 3 During the second half of the 1990s U.S. equity prices experienced a remarkable boom, with the annual return averaging nearly 25 percent more than the return on Treasury bills. Some argued that this price rise reflected optimism that the new economy would lead to a golden age of prosperity and surging profits, but others attributed the rise to a reduction in the market risk premium.

To see how a rise in stock prices can stem from a fall in the risk premium, suppose that investors in common stocks initially look for a return of 13 percent, made up of a 3 percent dividend yield and 10 percent long-term growth in dividends. If they now decide that they are prepared to hold equities on a prospective return of 12 percent, then other things being equal the dividend yield must fall to 2 percent. Thus a 1 percentage point fall in the risk premium would lead to a 50 percent rise in equity prices. If we include this price adjustment in our measures of past returns, we will be doubly wrong in our estimate of the risk premium. First, we will overestimate the return that investors required in the past. Second, we will not recognize that the return that investors require in the future is lower than in the past.

As stock prices began to slide back from their highs of March 2000, this belief in a falling market risk premium began to wane. It seems that if the risk premium truly did fall in the 1990s, then it also rose again as the new century dawned.

Out of this debate only one firm conclusion emerges: Do not trust anyone who claims to know what returns investors expect. History contains some clues, but ultimately we have to judge whether investors on average have received what they expected. Brealey and Myers have no official position on the market risk premium, but we believe that a range of 6 to 8.5 percent is reasonable for the United States.

Taken from Principles of Corporate Finance by Richard A. Brealey and Stewart C. Myers (7th edition)

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