Introduction to Risk, Return and the Opportunity Cost of Capital

Transcription

1 Introduction to Risk, Return and the Opportunity Cost of Capital Alexander Krüger, Definitions and Forulas Investent risk There are three basic questions arising when we start thinking about investent risk. The first question arising is how the risk is defined; the second, how it should be calculated for an investent; and the third exaines how to prevent or lower the risk for an investent. Intuitionally, we would answer, first, risk is the possible difference between our expected outcoe and the real outcoe in the future. Second, we should copare our investent with other investents done in the past to calculate how risky it is. And third, we should diversify our investent instead of putting all our eggs in one basket. Surprisingly, that is exactly how it is done. Before we start easuring risk, we will introduce the atheatical basics needed to do so. Arithetic Averages and Copound Annual Returns Arithetical average of n annual rates of return r to r n : r+ r rn n Copound (geoetric) average of n annual rates of return: n ( + ) ( + )... ( + ) r r r n When we speak of averages, we ean, if not otherwise stated, arithetical averages. Without proofing, we assue, that if the cost of capital is estiated fro historical returns or risk preius, arithetical averages correctly easure the cost of capital as opposed to copound averages. Variance, Standard Deviation, Covariance Our first basic assuption for the stock arket is that the rates of return are norally distributed. The variance of the arket return is defined as: σ = the expected value of ( ~ r ), ~ r r

2 r~ denotes the actual return and r the expected return. Variance estiated fro a saple of n returns: n σ ~ r = ( ~ rt r ) n t= where r~ t is the arket return in period t and r is the ean of the values of r~ t. The Standard Deviation is the square of the variance: σ ~ r = σ ~ r The covariance between stocks and is defined as: σ = ρσσ, where ρ denotes the correlation factor between stocks and. Portfolio Variance and Beta Holding a portfolio of stocks, the portfolio variance is: i= j= x x i jσ ij where xn denotes the proportion of stock n copared to the portfolio. The beta relative to arket portfolio (or, ore siply, beta) is defined as: σ i β i =, σ where σ i is the covariance between stock and the arket returns and σ is the variance of the returns on the arket. Risk preiu and Historical Review of Capital Markets A Century of Capital Market History We have taken a look at the perforance of two portfolios of U.S. securities since 900, inforation based on a study by Dison, Marsh, and Staunton.. A portfolio of Treasury bills, ie, U.S. governent debt securities aturing in less than one year (before 99 the coercial paper rate is used in this study).. A portfolio of U.S. coon stocks. Treasury bills offer an investent as safe as you can ake, without a risk of default, and relatively stable prices caused by the short aturity. The only uncertainty is the inflation and the resulting real rate of return. We only focus on the noinal rate of return, so we need not bother about inflation; we will siply keep it in ind. The consequence of investing in coon stocks

3 is that we share all of the up- and downswings inherent to the. The average annual rate of return fro 900 to 006 was about 4% for U.S. treasury bills and.7% for U.S. coon stocks. We define the extra return of coon stocks versus treasury bills (7.6% in this case due to rounding) as the average (historical) risk preiu. You ay ask why we look back such a long period and why we only focus on the U.S. arket to define our historical risk preiu. The only reasonable approach to correctly easuring annual rates of return on coon stocks is to look at a very extensive period due to the high fluctuation. Furtherore, the aforeentioned study shows that our risk preiu is about average copared with evaluated risk preius in other industrial nations for the sae period of tie. Using Historical Evidence to Evaluate Today s Cost of Capital How to estiate the cost of capital today? Let's suppose there is an investent project as risky as the Standard and Poor s Coposite Index. We will suppose, to take things a bit loosely, it has the sae degree of risk as the arket portfolio. 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 arket portfolio. This will be tered arket return r. Assuing that the future will be coparable to the past and that today s investors expect the sae noral rates of return as in the past, you would set r at.7%, the average of past arket returns. Do you believe that investors would be content to hold coon stocks offering an expected return of.7% when Treasury bills were offering 5% as they did in 98, for exaple? That s why r is defined as the su of the risk-free interest rate and the risk preiu. An exaple: In id 006, Treasury bills were about 5%. Adding on the risk preiu 7.6% yields r ( 006) = 5% + 7.6% =.6% The crucial assuption here is that there is a noral, stable risk preiu on the arket portfolio, so that the expected future risk can be easured by the average past risk preiu. Even with over 00 years of data, we can t estiate the arket risk preiu; nor can we be sure that investors today are deanding the sae reward for risk that they were 50 or 00 years ago. All this leaves plenty of roo for arguents about what the risk preiu really is. Soe reasons why today s expected risk preiu ight be less than the historical average are the fact that today the econoies and arkets see to be ore stable than they were in the past and today's investors can ore easily diversify their investents, thus diinishing their 3

4 risk. There are few who would subscribe to the notion that today s risk preiu is higher than the historical one. Dividend Yields and the Risk Preiu Another way to deterine the risk preiu is to have a closer look at past dividend yields. If stock prices are expected to keep pace with the growth in dividends, then the expected arket return is equal to the dividend yield plus the expected dividend growth. Dividend yields in the United States have averaged 4.4% since 900, and the annual growth in dividends has averaged about 5.6%. The expected arket value return over this period was DIV r = + g = = 0% Ρ0 This is 6% above the risk-free interest rate and.6% lower than our realized risk preiu before. Furtherore, a reduction in the dividend yields would appear to herald a reduction in the risk preiu that investors can expect over the subsequent few years. Therefore when yields are relatively low, copanies ay be justified in shaving their estiate of required returns over the next few years. However, changes in the dividend yield tell copanies next to nothing about the expected risk preiu over the next 0 or 0 years. There is only one fir conclusion eerging fro this debate: Do not trust anyone who clais to know what returns investors expect. Many financial econoists rely on the evidence of history and work with a risk preiu of 7.5%. Others use a soewhat lower figure. Portfolio Risk You have already learned the discount rate for safe projects and for average-risk projects. But you still don t know how to estiate discount rates for assets that do not fit into these straightforward categories. Measuring Variability In principle, you could estiate the variability of any portfolio of stocks or bonds using the procedure described in the following exaple. You would identify the possible outcoe, and do the nuber-crunching calculations. For exaple: Suppose you are offered to play the following gae. You start by investing $00. Then two coins are flipped. If both coins turn up heads, you get $40 back. If both coins turn up tails, you get $80 back. Or if they each show soething different, you get $0 back. Exp. return = (. 5 40) + (.5 ( 0)) + (0.5 0) = 0% 4

5 σ = (40 0).5 + (0 0).5 + ( 0 0).5 = 450 σ = 450 This gae s expected return is 0%, the variance of the percentage return is 450 and the standard deviation is. The standard deviation is in the sae units as the rate of return, eaning that the gae s variability is %. The risk of an asset can be copletely expressed by all possible outcoes and the probability of each. In practice, this is often ipossible, therefore variance and standard deviation are used for suarizing the spread of possible outcoes. Unfortunately, the probability of each outcoe is not fixed and cannot be found in a newspaper or elsewhere. So we ust resort to observing the past once again. Of course, hindsight entails no risk, but it is reasonable to assue that portfolios with histories of high variability also have the least predictable future perforance. The annual standard deviation for Treasury bills fro was.8 and 9.8 for coon stocks. Just as today s preiu risk ay be different fro historical risk, today s arket variability ay be different as well. Diversification The standard deviation of the U.S. arket portfolio during the five-year period fro July, 00 June, 006 was about 6%. Wouldn t you expect that the standard deviation of one single stock should be close to 6% on average for the sae period? ie, the standard deviation of one single stock is higher than the standard deviation of the arket portfolio because diversification reduces variability. Diversification works because stocks are not perfectly correlated, eaning prices of different stocks do not ove in exactly the sae ways. Even a little diversification can provide a substantial reduction in variability. For exaple: The standard deviation for IBM during the five-year period of July, 00 June, 006 was about 9.7% and for Boeing about 9.8%. The standard deviation of a portfolio evenly distributed between IBM and Boeing was about.5%. This is 8% less than holding just one of those stocks. The iproveent is uch saller when the nuber of securities is increased beyond, say 0 or 30. The risk that can be potentially eliinated by diversification is called unique risk (or unsysteatic, residual, specific, diversifiable risk). Unique risk stes fro the fact that any of the perils that surround an individual copany are peculiar to that copany and perhaps its iediate copetitors. But there is also soe risk that you cannot avoid regardless of how uch you diversify. 5

6 This risk is called arket risk (or systeatic, undiversifiable risk). Market risk stes fro the fact that there are perils that threaten all businesses. That is why stocks have a tendency to ove together. Calculating Portfolio Risk We now have an intuitive idea of how diversification reduces risk. The next step is to ake a forulary definition of portfolio variance and to practice soe calculations. The forula for portfolio variance is: i= j= x x i jσ ij Inserting the covariance for stock i and stock j ( σ = ρ σ σ ): Portfolio variance = x x i jρ ijσ iσ j i= j= To calculate portfolio risk we need to know the correlation coefficients of the stocks and their single standard deviation. For exaple: Suppose that 60% of your portfolio is invested in Wal-Mart (stock ) and the reainder is invested in IBM (stock ). You expect that over the coing year Wal-Mart will give a return of 0% and IBM, 5%. The expected return, r p, on your portfolio is: r p = (. 6 0) + (.4 5) = % In the past, the standard deviation of IBM was about 9.7% and for Wal-Mart about 9.8%. The past correlation was about.35. You believe that these figures are a good representation of the spread of possible future outcoes. σ p = xixjρijσ iσ j = x σ + xσ + i= j= = = 38. ij ij ( x x ρ σ σ ) (. 6) (9.8) + (.4) (9.7) + ( i j 9.7) σ = % The expected standard variation is 9.5%. Copared to investing in Wal-Mart only, we reduced risk and iproved our expected return by adding a stock with a higher risk and a higher expected return due to diversification. How Individual Securities Affect Portfolio Risk The risk of a well-diversified portfolio depends on the arket risk of the securities included in the portfolio. As we entioned before, the risk of a security is the su of unique risk and arket risk. While unique risk can be eliinated by diversification, arket risk cannot. Market risk of a security is the sensitivity to arket oveents, called beta (β ). The beta 6

7 of the arket portfolio is.0. A stock with a beta higher than.0 tends to aplify the overall oveents of the arket. A stock with a beta between 0 and.0 tends to ove in the sae direction as the arket but not as far. Often stocks with high standard deviation have high betas. The risk of a welldiversified portfolio is proportional to the portfolio beta, which equals the average beta of the securities included in the portfolio. Hence, the risk of a well-diversified portfolio with a beta of 0.5 is half of the risk of the arket portfolio and a third of the risk of a well-diversified portfolio with a beta of.5. For exaple: The following table shows the returns over a particular sixth-onth period on the arket and the stock of Anchovy Queen restaurant chain. Although both investents provided an average return of %, Anchovy Queen s stock was particularly sensitive to arket oveents, β =. 5. A diversified portfolio of stocks with the sae beta as Anchovy Queen would be one-anda-half ties as volatile as the arket. Deviation Deviation Squared Product of fro fro average deviation deviations Market Anchovy Q average Anchovy Q fro average fro average Month return return arket return return arket return returns -8% -% % 8% % 9% % -3% % 3% % 6% Average % % Total Variance = σ ² = 304/6 = Covariance = σ i = 456/6 = 76 Beta (β) = σ i /σ ² = 76/50.67 =.5 Calculating the variance of the arket returns and the covariance between the returns on the arket and those of Anchovy Queen. Beta is the ratio of the variance to the covariance. Diversification and Value Additivity Diversification reduces risk and, therefore, akes sense for investors. Undoubtedly, diversification is a good thing but that does not ean that firs should practice it. As long as investors have the easy option, as they do today, to diversify on their own account, they do not pay any ore for a fir that offers the diversification. An individual can invest in the steel industry this week and pull out next week. A fir cannot do that. To be sure, the individual would have to pay brokerage fees on the purchase and sale of steel copany shares, but think of the tie and expense for a fir to acquire a steel copany or to start up a new steel-aking operation. Another aspect is that if investors have a sufficiently broad choice of securities they will not pay any less because they are unable to invest separately in each factory. 7

8 In countries which have large and copetitive capital arkets diversification does not add or subtract fro the value of a fir. The total value is the su of its parts. This concept is called value additivity. If the capital arket establishes a value PV(A) for asset A and PV(B) for asset B, the arket value of a fir holding only these two assets is PV(AB) = PV(A)+PV(B) 8