Portfolio Choice and CAPM PDF

Optimal Portfolio Choice CH APTE R and the Capital Asset Pricing Model 11...

Optimal Portfolio Choice CH APTE R and the Capital Asset Pricing Model 11 NOTATION Ri  return of security IN THIS CHAPTER, WE BUILD ON THE IDEAS WE INTRODUCED IN (or investment) i Chapter 10 to explain how an investor can choose an efficient portfolio. In par- x i  fraction invested in ticular, we will demonstrate how to find the optimal portfolio for an investor who security i wants to earn the highest possible return given the level of volatility he or she is E [ Ri ] expected return willing to accept by developing the statistical techniques of mean-variance port- r f risk-free interest rate folio optimization. Both elegant and practical, these techniques are used ­routinely Ri  average return of security by professional investors, money managers, and financial institutions. We then (or investment) introduce the assumptions of the Capital Asset Pricing Model (CAPM), the most Corr ( Ri , R j ) correlation between important model of the relationship between risk and return. Under these as- returns of i and j sumptions, the efficient portfolio is the market portfolio of all stocks and securi- Cov ( Ri , R j ) covariance between ties. As a result, the expected return of any security depends upon its beta with returns of i and j the market portfolio. SD (R ) standard deviation In Chapter 10, we explained how to calculate the expected return and vola- (volatility) of return R tility of a single stock. To find the efficient portfolio, we must understand how to Var (R ) variance of return R do the same thing for a portfolio of stocks. We begin this chapter by explaining n number of securities in a how to calculate the expected return and volatility of a portfolio. With these sta- portfolio tistical tools in hand, we then describe how an investor can create an efficient R xP  return of portfolio with fraction x invested in portfolio out of individual stocks, and consider the implications, if all investors portfolio P and ( 1 − x ) attempt to do so, for an investment’s expected return and cost of capital. invested in the risk-free In our exploration of these concepts, we take the perspective of a stock mar- security ket investor. These concepts, however, are also important for a corporate financial β iP  beta or sensitivity of manager. After all, financial managers are also investors, investing money on be- the investment i to the half of their shareholders. When a company makes a new investment, financial fluctuations of the managers must ensure that the investment has a positive NPV. Doing so requires portfolio P knowing the cost of capital of the investment opportunity and, as we shall see in β i  beta of security i with the next chapter, the CAPM is the main method used by most major corporations respect to the market portfolio to calculate the cost of capital. ri  required return or cost of capital of security i 395 M11_BERK6318_06_GE_C11.indd 395 27/04/23 11:00 AM  396 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model 11.1 The Expected Return of a Portfolio To find an optimal portfolio, we need a method to define a portfolio and analyze its return. We can describe a portfolio by its portfolio weights, the fraction of the total investment in the portfolio held in each individual investment in the portfolio: Value of investment i xi = (11.1) Total value of portfolio These portfolio weights add up to 1 (that is, ∑ i x i = 1), so that they represent the way we have divided our money between the different individual investments in the portfolio. As an example, consider a portfolio with 200 shares of Dolby Laboratories worth $30 per share and 100 shares of Coca-Cola worth $40 per share. The total value of the portfolio is 200 × $30 + 100 × $40 = $10, 000, and the corresponding portfolio weights x D and x C are 200 × $30 100 × $40 xD = = 60%, xC = = 40% $10,000 $10,000 Given the portfolio weights, we can calculate the return on the portfolio. Suppose x 1 ,... , x n are the portfolio weights of the n investments in a portfolio, and these invest- ments have returns R1 ,... , Rn. Then the return on the portfolio, R P , is the weighted average of the returns on the investments in the portfolio, where the weights correspond to portfolio weights: R P = x 1R1 + x 2 R2 +... + x n Rn = ∑ x i Ri (11.2) i The return of a portfolio is straightforward to compute if we know the returns of the individual stocks and the portfolio weights. EXAMPLE 11.1 Calculating Portfolio Returns Problem Suppose you buy 200 shares of Dolby Laboratories at $30 per share and 100 shares of Coca-Cola stock at $40 per share. If Dolby’s share price goes up to $36 and Coca-Cola’s falls to $38, what is the new value of the portfolio, and what return did it earn? Show that Eq. 11.2 holds. After the price change, what are the new portfolio weights? Solution The new value of the portfolio is 200 × $36 + 100 × $38 = $11,000, for a gain of $1000 or a 10% return on your $10,000 investment. Dolby’s return was 36 30 − 1 = 20%, and Coca- Cola’s was 38 40 − 1 = −5%. Given the initial portfolio weights of 60% Dolby and 40% Coca-Cola, we can also compute the portfolio’s return from Eq. 11.2: R P = x D R D + x C RC = 0.6 × ( 20% ) + 0.4 × ( −5% ) = 10% After the price change, the new portfolio weights are 200 × $36 100 × $38 xD = = 65.45%, xC = = 34.55% $11,000 $11,000 Without trading, the weights increase for those stocks whose returns exceed the portfolio’s return. M11_BERK6318_06_GE_C11.indd 396 27/04/23 11:00 AM  11.2 The Volatility of a Two-Stock Portfolio 397 Equation 11.2 also allows us to compute the expected return of a portfolio. Using the facts that the expectation of a sum is just the sum of the expectations and that the expecta- tion of a known multiple is just the multiple of its expectation, we arrive at the following formula for a portfolio’s expected return: E [ R P ] = E  ∑ x i Ri  = ∑ E [ x i Ri ] = ∑ x i E [ Ri ] (11.3) i i i That is, the expected return of a portfolio is simply the weighted average of the expected returns of the investments within it, using the portfolio weights. EXAMPLE 11.2 Portfolio Expected Return Problem Suppose you invest $10,000 in Meta Platforms (Facebook) stock, and $30,000 in Honeywell International stock. You expect a return of 10% for Facebook and 16% for Honeywell. What is your portfolio’s expected return? Solution You invested $40,000 in total, so your portfolio weights are 10,000 40,000 = 0.25 in Facebook and 30,000 40,000 = 0.75 in Honeywell. Therefore, your portfolio’s expected return is E [ RP ] = x F E [ RF ] + x H E [ RH ] = 0.25 × 10% + 0.75 × 16% = 14.5% CONCEPT CHECK 1. What is a portfolio weight? 2. How do we calculate the return on a portfolio? 11.2 The Volatility of a Two-Stock Portfolio As we explained in Chapter 10, combining stocks in a portfolio eliminates some of their risk through diversification. The amount of risk that will remain depends on the degree to which the stocks are exposed to common risks. In this section, we describe the statistical tools that we can use to quantify the risk stocks have in common and determine the volatility of a portfolio. Combining Risks Let’s begin with a simple example of how risk changes when stocks are combined in a portfolio. Table 11.1 shows returns for three hypothetical stocks, along with their average returns and volatilities. While the three stocks have the same volatility and average return, TABLE 11.1    Returns for Three Stocks, and Portfolios of Pairs of Stocks Stock Returns Portfolio Returns Year North Air West Air Tex Oil 1 2 R N + 1 2 RW 1 2 R W + 1 2 RT 2017 21% 9% −2% 15.0% 3.5% 2018 30% 21% −5% 25.5% 8.0% 2019 7% 7% 9% 7.0% 8.0% 2020 −5% −2% 21% −3.5% 9.5% 2021 −2% −5% 30% −3.5% 12.5% 2022 9% 30% 7% 19.5% 18.5% Average Return 10.0% 10.0% 10.0% 10.0% 10.0% Volatility 13.4% 13.4% 13.4% 12.1% 5.1% M11_BERK6318_06_GE_C11.indd 397 27/04/23 11:00 AM  398 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model the pattern of their returns differs. When the airline stocks performed well, the oil stock tended to do poorly (see 2017–2018), and when the airlines did poorly, the oil stock tended to do well (2021–2022). Table 11.1 also shows the returns for two portfolios of the stocks. The first portfolio consists of equal investments in the two airlines, North Air and West Air. The second portfolio includes equal investments in West Air and Tex Oil. The average return of both portfolios is equal to the average return of the stocks, consistent with Eq. 11.3. However, their volatilities—12.1% and 5.1%—are very different from the individual stocks and from each other. This example demonstrates two important phenomena. First, by combining stocks into a portfolio, we reduce risk through diversification. Because the prices of the stocks do not move identically, some of the risk is averaged out in a portfolio. As a result, both portfolios have lower risk than the individual stocks. Second, the amount of risk that is eliminated in a portfolio depends on the degree to which the stocks face common risks and their prices move together. Because the two airline stocks tend to perform well or poorly at the same time, the portfolio of airline stocks has a volatility that is only slightly lower than that of the individual stocks. The airline and oil stocks, by contrast, do not move together; indeed, they tend to move in opposite directions. As a result, additional risk is canceled out, making that portfolio much less risky. This benefit of diversification is obtained costlessly—without any reduction in the average return. Determining Covariance and Correlation To find the risk of a portfolio, we need to know more than the risk and return of the com- ponent stocks: We need to know the degree to which the stocks face common risks and their returns move together. In this section, we introduce two statistical measures, covariance and correlation, that allow us to measure the co-movement of returns. Covariance. Covariance is the expected product of the deviations of two returns from their means. The covariance between returns Ri and R j is: Covariance Between Returns R i and R j Cov ( Ri , R j ) = E  ( Ri − E [ Ri ] ) ( R j − E [ R j ] )  (11.4) When estimating the covariance from historical data, we use the formula1 Estimate of the Covariance from Historical Data T 1 Cov ( Ri , R j ) = T − 1 t =1 ( ∑ Ri , t − Ri ) ( R j ,t − R j ) (11.5) Intuitively, if two stocks move together, their returns will tend to be above or below average at the same time, and the covariance will be positive. If the stocks move in oppo- site directions, one will tend to be above average when the other is below average, and the covariance will be negative. Correlation. While the sign of the covariance is easy to interpret, its magnitude is not. It will be larger if the stocks are more volatile (and so have larger deviations from their expected returns), and it will be larger the more closely the stocks move in relation to each other. 1 As with Eq. 10.7 for historical volatility, we divide by T − 1 rather than by T to make up for the fact that we have used the data to compute the average returns R, eliminating a degree of freedom. M11_BERK6318_06_GE_C11.indd 398 27/04/23 11:00 AM  11.2 The Volatility of a Two-Stock Portfolio 399 FIGURE 11.1 Correlation Correlation measures how Perfectly Perfectly returns move in relation to Negatively Positively Correlated Uncorrelated Correlated each other. It is between +1 (returns always move together) and −1 (returns 21 0 11 always move oppositely). Independent risks have no Always Tend to No Tend to Always tendency to move together Move Move Tendency Move Move and have zero correlation. Oppositely Oppositely Together Together In order to control for the volatility of each stock and quantify the strength of the relation- ship between them, we can calculate the correlation between two stock returns, defined as the covariance of the returns divided by the standard deviation of each return: Cov ( Ri , R j ) Corr ( Ri , R j ) = (11.6) SD ( Ri ) SD ( R j ) The correlation between two stocks has the same sign as their covariance, so it has a similar interpretation. Dividing by the volatilities ensures that correlation is always between −1 and +1, which allows us to gauge the strength of the relationship between the stocks. As Figure 11.1 shows, correlation is a barometer of the degree to which the returns share common risk and tend to move together. The closer the correlation is to +1, the more the returns tend to move together as a result of common risk. When the correlation (and thus the covariance) equals 0, the returns are uncorrelated; that is, they have no tendency to move either together or in opposition to one another. Independent risks are uncorrelated. Finally, the closer the correlation is to −1, the more the returns tend to move in opposite directions. EXAMPLE 11.3 The Covariance and Correlation of a Stock with Itself Problem What are the covariance and the correlation of a stock’s return with itself ? Solution Let Rs be the stock’s return. From the definition of the covariance, Cov ( Rs , Rs ) = E  ( Rs − E [ Rs ] )( Rs − E [ Rs ] )  = E  ( Rs − E [ Rs ])  2  = Var ( Rs ) where the last equation follows from the definition of the variance. That is, the covariance of a stock with itself is simply its variance. Then, Cov ( Rs , Rs ) Var ( Rs ) Corr ( Rs , Rs ) = = =1 SD ( Rs ) SD ( Rs ) SD ( Rs ) 2 where the last equation follows from the definition of the standard deviation. That is, a stock’s return is perfectly positively correlated with itself, as it always moves together with itself in per- fect synchrony. M11_BERK6318_06_GE_C11.indd 399 27/04/23 11:00 AM  400 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model COMMON MISTAKE Computing Variance, Covariance, and Correlation in Excel The computer spreadsheet program Excel does not compute by multiplying by the number of data points and dividing by the the standard deviation, variance, covariance, and correlation number of data points minus one; i.e., COVAR ∗ T ( T − 1 ). consistently. The Excel functions STDEV and VAR correctly Alternatively, you can use the function COVARANCE.S, use Eq. 10.7 to estimate the standard deviation and variance which correctly computes the covariance from a historical from historical data. But the Excel function COVAR does sample (this function was introduced in Excel 2010). Another not use Eq. 11.5; instead, Excel divides by T instead of T − 1. solution is to use the function CORREL to compute the cor- Therefore, to estimate the covariance from a sample of histori- relation, and then estimate the covariance by multiplying the cal returns using COVAR, you must correct the inconsistency correlation by the standard deviation of each return. EXAMPLE 11.4 Computing the Covariance and Correlation Problem Using the data in Table 11.1, what are the covariance and the correlation between North Air and West Air? Between West Air and Tex Oil? Solution Given the returns in Table 11.1, we deduct the mean return (10%) from each and compute the product of these deviations between the pairs of stocks. We then sum them and divide by T − 1 = 5 to compute the covariance, as in Table 11.2. From the table, we see that North Air and West Air have a positive covariance, indicating a tendency to move together, whereas West Air and Tex Oil have a negative covariance, indicating a tendency to move oppositely. We can assess the strength of these tendencies from the correlation, obtained by dividing the covariance by the standard deviation of each stock (13.4%). The cor- relation for North Air and West Air is 62.4%; the correlation for West Air and Tex Oil is −71.3%. TABLE 11.2   Computing the Covariance and Correlation Between Pairs of Stocks Deviation from Mean    North Air and West Air     West Air and Tex Oil Year ( RN − RN ) ( RW − RW ) ( RT − RT ) ( RN − RN ) ( RW − RW ) ( RW − RW ) ( RT − RT ) 2017 11% −1% −12% −0.0011 0.0012 2018 20% 11% −15% 0.0220 −0.0165 2019 −3% −3% −1% 0.0009 0.0003 2020 −15% −12% 11% 0.0180 −0.0132 2021 −12% −15% 20% 0.0180 −0.0300 2022 −1% 20% −3% −0.0020 −0.0060 Sum = ∑ ( Ri ,t − Ri ) ( R j ,t − Rj )= 0.0558 −0.0642 t 1 Covariance: Cov ( Ri , R j ) = Sum = 0.0112 −0.0128 T −1 Cov ( Ri , R j ) Correlation: Corr ( Ri , R j ) = = 0.624 −0.713 SD ( Ri ) SD ( R j ) M11_BERK6318_06_GE_C11.indd 400 27/04/23 11:00 AM  11.2 The Volatility of a Two-Stock Portfolio 401 TABLE 11.3     Historical Annual Volatilities and Correlations for Selected Stocks (based on monthly returns, 1996–2021) Alaska Southwest Ford General Microsoft HP Air Airlines Motor Kellogg Mills Volatility (Standard Deviation) 31% 35% 37% 32% 46% 19% 17% Correlation with Microsoft 1.00 0.39 0.19 0.21 0.27 0.07 0.10 HP 0.39 1.00 0.29 0.35 0.28 0.10 0.04 Alaska Air 0.19 0.29 1.00 0.47 0.20 0.16 0.17 Southwest Airlines 0.21 0.35 0.47 1.00 0.31 0.17 0.18 Ford Motor 0.27 0.28 0.20 0.31 1.00 0.19 0.09 Kellogg 0.07 0.10 0.16 0.17 0.19 1.00 0.54 General Mills 0.10 0.04 0.17 0.18 0.09 0.54 1.00 When will stock returns be highly correlated with each other? Stock returns will tend to move together if they are affected similarly by economic events. Thus, stocks in the same industry tend to have more highly correlated returns than stocks in different industries. This tendency is illustrated in Table 11.3, which shows the volatility of individual stock returns and the correlation between them for several common stocks. Consider, for example, Microsoft and Hewlett-Packard. The returns of these two technology stocks have a higher correlation with each other (39%) than with any of the non-technology stocks (35% or lower). The same pattern holds for the airline and food-processing stocks—their returns are most highly correlated with the other firm in their industry, and much less correlated with those outside their industry. General Mills and Kellogg have the lowest correlation with stocks outside of the food industry; indeed, Kellogg and Microsoft have a correlation of only 7%, suggesting that these two firms are subject to essentially uncorrelated risks. Note, however, that all of the correlations are positive, illustrating the general tendency of stocks to move together. EXAMPLE 11.5 Computing the Covariance from the Correlation Problem Using the data from Table 11.3, what is the covariance between Microsoft and HP? Solution We can rewrite Eq. 11.6 to solve for the covariance: Cov ( R M , R HP ) = Corr ( R M , R HP )SD ( R M ) SD ( R HP ) = ( 0.39 )( 0.31 )( 0.35 ) = 0.0423 Computing a Portfolio’s Variance and Volatility We now have the tools to compute the variance of a portfolio. For a two-stock portfolio with R P = x 1 R1 + x 2 R 2 : Var ( R P ) = Cov ( R P , R P ) = Cov ( x 1R1 + x 2 R2 , x 1R1 + x 2 R2 ) (11.7) = x 1x 1Cov ( R1 , R1 ) + x 1x 2Cov ( R1 , R2 ) + x 2 x 1Cov ( R2 , R1 ) + x 2 x 2Cov ( R2 , R2 ) M11_BERK6318_06_GE_C11.indd 401 27/04/23 11:00 AM  402 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model In the last line of Eq. 11.7, we use the fact that, as with expectations, we can change the order of the covariance with sums and multiples.2 By combining terms and recognizing, from Example 11.4, that Cov ( Ri , Ri ) = Var ( Ri ), we arrive at our main result of this section: The Variance of a Two-Stock Portfolio Var ( R P ) = x 12Var ( R1 ) + x 22Var ( R2 ) + 2 x 1x 2Cov ( R1 , R2 ) (11.8) As always, the volatility is the square root of the variance, SD ( R P ) = Var ( R P ). Let’s check this formula for the airline and oil stocks in Table 11.1. Consider the port- folio containing shares of West Air and Tex Oil. The variance of each stock is equal to the square of its volatility, 0.134 2 = 0.018. From Example 11.3, the covariance between the stocks is −0.0128. Therefore, the variance of a portfolio with 50% invested in each stock is Var ( 21 RW + 21 RT )= xW2 Var ( RW ) + x T2Var ( RT ) + 2 xW x T Cov ( RW , RT ) ( 21 ) ( 0.018 ) + ( 21 ) ( 0.018 ) + 2 ( 21 )( 21 ) ( −0.0128 ) 2 2 = = 0.0026 The volatility of the portfolio is 0.0026 = 5.1%, which corresponds to the calculation in Table 11.1. For the North Air and West Air portfolio, the calculation is the same except for the stocks’ higher covariance of 0.0112, resulting in a higher volatility of 12.1%. Equation 11.8 shows that the variance of the portfolio depends on the variance of the individual stocks and on the covariance between them. We can also rewrite Eq. 11.8 by cal- culating the covariance from the correlation (as in Example 11.5): Var ( R P ) = x 12 SD ( R1 ) 2 + x 22 SD ( R2 ) 2 + 2 x 1x 2Corr ( R1 , R 2 ) SD ( R1 ) SD ( R 2 ) (11.9) Equations 11.8 and 11.9 demonstrate that with a positive amount invested in each stock, the more the stocks move together and the higher their covariance or correlation, the more variable the portfolio will be. The portfolio will have the greatest variance if the stocks have a perfect positive correlation of +1. EXAMPLE 11.6 Computing the Volatility of a Two-Stock Portfolio Problem Using the data from Table 11.3, what is the volatility of a portfolio with equal amounts invested in Microsoft and Hewlett-Packard stock? What is the volatility of a portfolio with equal amounts invested in Microsoft and Alaska Air stock? Solution With portfolio weights of 50% each in Microsoft and Hewlett-Packard stock, from Eq. 11.9, the portfolio’s variance is Var ( R P ) = x M 2 SD ( R ) 2 + x 2 SD ( R M HP HP ) + 2 x M x HP Corr ( R M , R HP ) SD ( R M ) SD ( R HP ) 2 = ( 0.50 ) 2 ( 0.31 ) 2 + ( 0.50 ) 2 ( 0.35 ) 2 + 2 ( 0.50 )( 0.50 )( 0.39 )( 0.31 )( 0.35 ) = 0.0758 The volatility is therefore SD ( R P ) = Var ( R ) = 0.0758 = 27.5%. 2 That is, Cov ( A + B , C ) = Cov ( A , C ) + Cov ( B , C ) and Cov ( mA , B ) = m Cov ( A , B ). M11_BERK6318_06_GE_C11.indd 402 27/04/23 11:00 AM  11.3 The Volatility of a Large Portfolio 403 For the portfolio of Microsoft and Alaska Air stock, Var ( R P ) = x M 2 SD ( R ) 2 + x 2 SD ( R ) 2 + 2 x x Corr ( R , R ) SD ( R )SD ( R ) M A A M A M A M A = ( 0.50 ) 2 ( 0.31 ) 2 + ( 0.50 ) 2 ( 0.37 ) 2 + 2 ( 0.50 )( 0.50 )( 0.19 )( 0.31 )( 0.37 ) = 0.0691 The volatility in this case is SD ( R P ) = Var ( R ) = 0.0691 = 26.3%. Note that the portfolio of Microsoft and Alaska Air stock is less volatile than either of the individual stocks. It is also less volatile than the portfolio of Microsoft and Hewlett-Packard stock. Even though Alaska Air’s stock returns are more volatile than Hewlett-Packard’s, its lower correlation with Microsoft’s returns leads to greater diversification in the portfolio. CONCEPT CHECK 1. What does the correlation measure? 2. How does the correlation between the stocks in a portfolio affect the portfolio’s volatility? 11.3 The Volatility of a Large Portfolio We can gain additional benefits of diversification by holding more than two stocks in our portfolio. While these calculations are best done on a computer, by understanding them we can obtain important intuition regarding the amount of diversification that is possible if we hold many stocks. Large Portfolio Variance Recall that the return on a portfolio of n stocks is simply the weighted average of the re- turns of the stocks in the portfolio: R P = x 1R1 + x 2 R2 +... + x n Rn = ∑ x i Ri i Using the properties of the covariance, we can write the variance of a portfolio as follows: Var ( R P ) = Cov ( R P , R P ) = Cov ( ∑ x R , R ) = ∑ x Cov ( R , R i i i P i i i P ) (11.10) This equation indicates that the variance of a portfolio is equal to the weighted average covariance of each stock with the portfolio. This expression reveals that the risk of a portfolio depends on how each stock’s return moves in relation to it. We can reduce the formula even further by replacing the second R P with a weighted average and simplifying: Var ( R P ) = ∑ i x iCov ( Ri , RP ) = ∑ i x iCov ( Ri , ∑ j x j R j ) = ∑ i ∑ j x i x j Cov ( Ri , R j ) (11.11) This formula says that the variance of a portfolio is equal to the sum of the covariances of the returns of all pairs of stocks in the portfolio multiplied by each of their portfolio weights.3 That is, the overall variability of the portfolio depends on the total co-movement of the stocks within it. 3 Looking back, we can see that Eq. 11.11 generalizes the case of two stocks in Eq. 11.7. M11_BERK6318_06_GE_C11.indd 403 27/04/23 11:00 AM  404 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Diversification with an Equally Weighted Portfolio We can use Eq. 11.11 to calculate the variance of an equally weighted portfolio, a ­portfolio in which the same amount is invested in each stock. An equally weighted ­portfolio consisting of n stocks has portfolio weights x i = 1 n. In this case, we have the following formula:4 Variance of an Equally Weighted Portfolio of n Stocks 1 Var ( R p ) = ( Average Variance of the Individual Stocks ) n   1 +  1 −  ( Average Covariance Between the Stocks ) (11.12)  n Equation 11.12 demonstrates that as the number of stocks, n, grows large, the variance of the portfolio is determined primarily by the average covariance among the stocks. As an example, consider a portfolio of stocks selected randomly from the stock market. The his- torical volatility of the return of a typical large firm in the stock market is about 40%, and the typical correlation between the returns of large firms is about 25%. Using Eq. 11.12, and calculating the covariance from the correlation as in Example 11.5, the volatility of an equally weighted portfolio varies with the number of stocks, n, as follows: 1 1 SD ( R P ) = ( 0.40 2 ) +  1 −  ( 0.25 × 0.40 × 0.40 ) n  n We graph the volatility for different numbers of stocks in Figure 11.2. Note that the vol- atility declines as the number of stocks in the portfolio grows. In fact, nearly half of the FIGURE 11.2 50% Volatility of an Equally Weighted Portfolio Versus 40% the Number of Stocks Portfolio Volatility The volatility declines as the Elimination of 30% number of stocks in the port- diversifiable risk folio increases. Even in a very large portfolio, however, mar- 20% ket risk remains. Correlated 10% (market) risk 0% 1 10 100 1000 Number of Stocks 4 For an n-stock portfolio, there are n variance terms (any time i = j in Eq. 11.11) with weight x i2 = 1 n 2 on each, which implies a weight of n n 2 = 1 n on the average variance. There are n 2 − n covariance terms (all the n × n pairs minus the n variance terms) with weight x i x j = 1 n 2 on each, which implies a weight of ( n 2 − n ) n 2 = 1 − 1 n on the average covariance. M11_BERK6318_06_GE_C11.indd 404 27/04/23 11:00 AM  11.3 The Volatility of a Large Portfolio 405 volatility of the individual stocks is eliminated in a large portfolio as the result of diversifica- tion. The benefit of diversification is most dramatic initially: The decrease in volatility when going from one to two stocks is much larger than the decrease when going from 100 to 101 stocks—indeed, almost all of the benefit of diversification can be achieved with about 30 stocks. Even for a very large portfolio, however, we cannot eliminate all of the risk. The variance of the portfolio converges to the average covariance, so the volatility declines to 0.25 × 0.4 × 0.4 = 20%.5 EXAMPLE 11.7 Diversification Using Different Types of Stocks Problem Stocks within a single industry tend to have a higher correlation than stocks in different indus- tries. Likewise, stocks in different countries have lower correlation on average than stocks within the United States. What is the volatility of a very large portfolio of stocks within an industry in which the stocks have a volatility of 40% and a correlation of 60%? What is the volatility of a very large portfolio of international stocks with a volatility of 40% and a correlation of 10%? Solution From Eq. 11.12, the volatility of the industry portfolio as n → ∞ is given by Average Covariance = 0.60 × 0.40 × 0.40 = 31.0% This volatility is higher than when using stocks from different industries as in Figure 11.2. Combin- ing stocks from the same industry that are more highly correlated therefore provides less diversifi- cation. We can achieve superior diversification using international stocks. In this case, Average Covariance = 0.10 × 0.40 × 0.40 = 12.6% We can also use Eq. 11.12 to derive one of the key results that we discussed in C­ hapter 10: When risks are independent, we can diversify all of the risk by holding a large portfolio. EXAMPLE 11.8 Volatility When Risks Are Independent Problem What is the volatility of an equally weighted average of n independent, identical risks? Solution If risks are independent, they are uncorrelated and their covariance is zero. Using Eq. 11.12, the volatility of an equally weighted portfolio of the risks is 1 SD (Individual Risk ) SD ( R P ) = Var ( R P ) = Var (Individual Risk ) = n n This result coincides with Eq. 10.8, which we used earlier to evaluate independent risks. Note that as n → ∞, the volatility goes to 0 — that is, a very large portfolio will have no risk. In this case, we can eliminate all risk because there is no common risk. 5 You might wonder what happens if the average covariance is negative. It turns out that while the covari- ance between a pair of stocks can be negative, as the portfolio grows large, the average covariance cannot be negative because the returns of all stocks cannot move in opposite directions simultaneously. M11_BERK6318_06_GE_C11.indd 405 27/04/23 11:00 AM  406 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Anne Martin became Wesleyan INTERVIEW WITH ANSWER: As active investors, many ­University’s Chief Investment Officer in ANNE MARTIN endowments have consistently ­outperformed passive benchmarks. 2010. From 2004 to 2010, she was a Director at the Yale ­Investments Office, Endowments with talented staffs have before which she was a g­ eneral partner been good at ferreting out the s­ martest of private equity firm R­ osewood Capi- managers and investing in more ­ inefficient market sectors—areas that tal in San Francisco, California, and a require true skill and talent to generate managing director at Alex. Brown in its returns. For example, endowments have technology practice. been the investor of choice for private ­equity firms that find high-performing QUESTION: Describe how you manage nonpublic companies. Because these ­Wesleyan’s $1 billion endowment. companies don’t have to manage to quarterly earnings and have better ANSWER: Like many university endow- ­capitalization, they can boost overall port- ments, we manage ours to earn about folio returns. Also, the permanent capital 7.5% to 8% nominal return to meet of universities enables staff to fight the Wesleyan’s risk appetite and budget needs herd mentality and be patient, taking a while continuing to grow the endowment. long time horizon and riding out market This is based on a 4.5% payout to the uni- ­volatility better than most investors. versity plus the higher education inflation rate of about 3%. Currently we contribute QUESTION: With the financial crisis 10 years behind us, what perma- about 18% of Wesleyan’s operating budget. Our core invest- nent lessons did endowment managers learn from that experience? ment principles include an equity orientation, diversifica- tion, and use of non-correlated assets to mitigate portfolio ANSWER: The financial crisis taught us to manage ­liquidity volatility. To earn these returns, we have reduced our fixed tightly. Because tools such as mean variance analysis and income assets and increased our allocation to private equity Monte Carlo simulations don’t deal well with ­liquidity ­issues, and alternative investments that can generate returns in we developed a custom model that shows us ­portfolio re- excess of traditional public markets. turns in various stress scenarios. The model helps us ­calculate At Wesleyan we believe we can achieve our return goals the maximum amount of outstanding uncalled commitments with a standard deviation of returns of 12.5% to 13.5%. the endowment can withstand and still meet its obligations. At this level of volatility we minimize disruption to the Most endowments now stress test portfolios to prepare for University’s operating budget in most market conditions. Once another downturn, incorporating regular liquidity testing and we determine the policy portfolio, we select managers who reporting into their portfolio construction. can drive alpha (a measure of an investment’s performance compared with a suitable market index) within each asset class. QUESTION: What else you would like to share with somebody To develop our strategic asset allocation we use mean studying finance in graduate school? variance analysis, applying long-term data and our judgment to evaluate each asset class for expected return, volatility, ANSWER: Although it’s not a well-known career choice, and covariance with other asset classes. With some asset endowment and foundation investment management is a classes that have short track records compared to public rewarding way to put your financial and management skills equities, such as venture capital, we consider trends in to work. Almost everything I learned in business school has ­determining inputs. We also work to constrain our illiquid been relevant; the broad nature of the work draws on the assets classes (real estate, private equity, natural resources whole MBA curriculum, from negotiations to organization and venture capital) at manageable levels. Once we are behavior to finance. Working in a mission-driven environ- comfortable with our inputs, the mean variance model ment provides major psychic benefits; helping a cause you ­provides an efficient frontier of possible portfolios that believe in is personally very satisfying. Finally, it’s intellec- maximize expected return for risk. tually stimulating; we analyze long-term trends, travel the world, and work closely with incredibly bright investors. I QUESTION: Why do you think university endowments have can’t think of another career in which literally everything you ­performed extremely well ­historically compared to public markets? read in the paper every day has some bearing on your work. M11_BERK6318_06_GE_C11.indd 406 27/04/23 11:00 AM  11.4 Risk Versus Return: Choosing an Efficient Portfolio 407 Diversification with General Portfolios The results in the last section depend on the portfolio being equally weighted. For a port- folio with arbitrary weights, we can rewrite Eq. 11.10 in terms of the correlation as follows: Var ( R P ) = ∑ x i Cov ( Ri , R P ) = ∑ x i SD ( Ri )SD ( R P )Corr ( Ri , R P ) i i Dividing both sides of this equation by the standard deviation of the portfolio yields the following important decomposition of the volatility of a portfolio: Volatility of a Portfolio with Arbitrary Weights Security i ’s contribution to the ˛˚˚˚˝˚˚˚¸  volatility of the portfolio SD ( R P ) = ∑ x i × SD ( Ri ) × Corr ( Ri , R P ) (11.13) i ↑ ↑ ↑ Amount     Total         Fraction of i ’ s of i held risk of i            risk that is                  common to P Equation 11.13 states that each security contributes to the volatility of the portfolio according to its volatility, or total risk, scaled by its correlation with the portfolio, which adjusts for the fraction of the total risk that is common to the portfolio. Therefore, when combining stocks into a portfolio that puts positive weight on each stock, unless all of the stocks have a perfect positive correlation of +1 with the portfolio (and thus with one another), the risk of the portfolio will be lower than the weighted average volatility of the individual stocks: SD ( R P ) = ∑ x i SD ( Ri )Corr ( Ri , R P ) < ∑ x i SD ( Ri ) (11.14) i i Contrast Eq. 11.14 with Eq. 11.3 for the expected return. The expected return of a portfolio is equal to the weighted average expected return, but the volatility of a portfolio is less than the weighted average volatility: We can eliminate some volatility by diversifying. CONCEPT CHECK 1. How does the volatility of an equally weighted portfolio change as more stocks are added to it? 2. How does the volatility of a portfolio compare with the weighted average volatility of the stocks within it? 11.4 R  isk Versus Return: Choosing an Efficient Portfolio Now that we understand how to calculate the expected return and volatility of a portfolio, we can return to the main goal of the chapter: Determine how an investor can create an ef- ficient portfolio.6 Let’s start with the simplest case—an investor who can choose between only two stocks. 6 The techniques of portfolio optimization were developed in a 1952 paper by Harry Markowitz, as well as in related work by Andrew Roy (1952) and Bruno de Finetti (1940) (see Further Reading). M11_BERK6318_06_GE_C11.indd 407 27/04/23 11:00 AM  408 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Efficient Portfolios with Two Stocks Consider a portfolio of Intel and Coca-Cola stock. Suppose an investor believes these stocks are uncorrelated and will perform as follows: Stock Expected Return Volatility Intel 26% 50% Coca-Cola 6% 25% How should the investor choose a portfolio of these two stocks? Are some portfolios preferable to others? Let’s compute the expected return and volatility for different combinations of the stocks. Consider a portfolio with 40% invested in Intel stock and 60% invested in Coca- Cola stock. We can compute the expected return from Eq. 11.3 as E [ R40 − 60 ] = x I E [ R I ] + x C E [ RC ] = 0.40 ( 26% ) + 0.60 ( 6% ) = 14% We can compute the variance using Eq. 11.9, Var ( R40 − 60 ) = x I2 SD ( R I ) 2 + x C2 SD ( RC ) 2 + 2 x I x C Corr ( R I , RC ) SD ( R I ) SD ( RC ) = 0.40 2 ( 0.50 ) 2 + 0.60 2 ( 0.25 ) 2 + 2 ( 0.40 ) ( 0.60 ) ( 0 ) ( 0.50 ) ( 0.25 ) = 0.0625 so that the volatility is SD ( R40 − 60 ) = 0.0625 = 25%. Table 11.4 shows the results for different portfolio weights. Due to diversification, it is possible to find a portfolio with even lower volatility than either stock: Investing 20% in Intel stock and 80% in Coca-Cola stock, for example, has a volatility of only 22.4%. But knowing that investors care about volatility and expected return, we must consider both simultaneously. To do so, we plot the volatility and expected return of each portfolio in Figure 11.3. We labeled the portfolios from Table 11.4 with the portfolio weights. The curve (a hyperbola) represents the set of portfolios that we can cre- ate using arbitrary weights. Faced with the choices in Figure 11.3, which ones make sense for an investor who is concerned with both the expected return and the volatility of her portfolio? Suppose the investor considers investing 100% in Coca-Cola stock. As we can see from Figure 11.3, other portfolios—such as the portfolio with 20% in Intel stock and 80% in Coca-Cola stock—make the investor better off in both ways: (1) They have a higher expected return, and (2) they have lower volatility. As a result, investing solely in Coca-Cola stock is not a good idea. TABLE 11.4    Expected Returns and Volatility for Different Portfolios of Two Stocks Portfolio Weights Expected Return (%) Volatility (%) xI xC E [ RP ] SD [ R P ] 1.00 0.00 26.0 50.0 0.80 0.20 22.0 40.3 0.60 0.40 18.0 31.6 0.40 0.60 14.0 25.0 0.20 0.80 10.0 22.4 0.00 1.00 6.0 25.0 M11_BERK6318_06_GE_C11.indd 408 27/04/23 11:00 AM  11.4 Risk Versus Return: Choosing an Efficient Portfolio 409 FIGURE 11.3 30% Volatility Versus Expected Intel Return for Portfolios of (1, 0) 25% Intel and Coca-Cola Stock Efficient Labels indicate portfolio Portfolios (0.8, 0.2) 20% weights ( x I , xC ) for Intel and Expected Return Coca-Cola stocks. Portfolios (0.6, 0.4) on the red portion of the curve, 15% with at least 20% invested (0.4, 0.6) in Intel stock, are efficient. Those on the blue portion 10% (0.2, 0.8) of the curve, with less than Inefficient 20% invested in Intel stock, Portfolios (0, 1) are inefficient—an investor can 5% Coca-Cola earn a higher expected return with lower risk by choosing an alternative portfolio. 0% 0% 10% 20% 30% 40% 50% 60% An interactive version of this figure is available in the eTextbook and Volatility (standard deviation) MyLab Finance. Identifying Inefficient Portfolios. More generally, we say a portfolio is an inefficient portfolio whenever it is possible to find another portfolio that is better in terms of both expected return and volatility. Looking at Figure 11.3, a portfolio is inefficient if there are other portfolios above and to the left—that is, to the northwest—of it. Investing solely in Coca-Cola stock is inefficient, and the same is true of all portfolios with more than 80% in Coca-Cola stock (the blue part of the curve). Inefficient portfolios are not optimal for an investor seeking high returns and low volatility. Identifying Efficient Portfolios. By contrast, portfolios with at least 20% in Intel stock are efficient (the red part of the curve): There is no other portfolio of the two stocks that offers a higher expected return with lower volatility. But while we can rule out inefficient portfolios as inferior investment choices, we cannot easily rank the efficient ones—investors will choose among them based on their own preferences for return versus risk. For ex- ample, an extremely conservative investor who cares only about minimizing risk would choose the lowest-volatility portfolio (20% Intel, 80% Coca-Cola). A more aggressive investor who is tolerant of risk might choose to invest 100% in Intel stock—even though that approach is riskier, the investor may be willing to take that chance to earn a higher expected return. EXAMPLE 11.9 Improving Returns with an Efficient Portfolio Problem Sally Ferson has invested 100% of her money in Coca-Cola stock and is seeking investment advice. She would like to earn the highest expected return possible without increasing her volatil- ity. Which portfolio would you recommend? M11_BERK6318_06_GE_C11.indd 409 27/04/23 11:00 AM  410 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model Solution In Figure 11.3, we can see that Sally can invest up to 40% in Intel stock without increasing her volatility. Because Intel stock has a higher expected return than Coca-Cola stock, she will earn higher expected returns by putting more money in Intel stock. Therefore, you should recom- mend that Sally put 40% of her money in Intel stock, leaving 60% in Coca-Cola stock. This portfolio has the same volatility of 25%, but an expected return of 14% rather than the 6% she has now. The Effect of Correlation In Figure 11.3, we assumed that the returns of Intel and Coca-Cola stocks are uncorrelated. Let’s consider how the risk and return combinations would change if the correlations were different. Correlation has no effect on the expected return of a portfolio. For example, a 40–60 portfolio will still have an expected return of 14%. However, the volatility of the portfolio will differ depending on the correlation, as we saw in Section 11.2. In particular, the lower the correlation, the lower the volatility we can obtain. In terms of Figure 11.3, as we lower the correlation and therefore the volatility of the portfolios, the curve showing the portfo- lios will bend to the left to a greater degree, as illustrated in Figure 11.4. When the stocks are perfectly positively correlated, we can identify the set of portfolios by the straight line between them. In this extreme case (the red line in Figure 11.4), the volatility of the portfolio is equal to the weighted average volatility of the two stocks— there is no diversification. When the correlation is less than 1, however, the volatility of the portfolios is reduced due to diversification, and the curve bends to the left. The reduction in risk (and the bending of the curve) becomes greater as the correlation decreases. At the other extreme of perfect negative correlation (blue line), the line again becomes straight, FIGURE 11.4 30% Effect on Volatility Intel and Expected Return 25% of Changing the Correlation Between Intel 20% Correlation 5 21 and Coca-Cola Stock Expected Return This figure illustrates correlations of 1, 0.5, 0, 15% Correlation 5 11 −0.5 and −1 The lower the correlation, the lower the risk of the portfolios. 10% 5% Coca-Cola 0% An interactive version of this figure 0% 10% 20% 30% 40% 50% 60% is available in the eTextbook and Volatility (standard deviation) MyLab Finance. M11_BERK6318_06_GE_C11.indd 410 27/04/23 11:00 AM  11.4 Risk Versus Return: Choosing an Efficient Portfolio 411 this time reflecting off the vertical axis. In particular, when the two stocks are perfectly negatively correlated, it becomes possible to hold a portfolio that bears absolutely no risk. Short Sales Thus far, we have considered only portfolios in which we invest a positive amount in each stock. We refer to a positive investment in a security as a long position in the security. But it is also possible to invest a negative amount in a stock, called a short position, by engaging in a short sale, a transaction in which you sell a stock today that you do not own, with the obligation to buy it back in the future. (For the mechanics of a short sale, see the box on page 316 in Chapter 9). As the next example demonstrates, we can include a short position as part of a portfolio by assigning that stock a negative portfolio weight. EXAMPLE 11.10 Expected Return and Volatility with a Short Sale Problem Suppose you have $20,000 in cash to invest. You decide to short sell $10,000 worth of Coca- Cola stock and invest the proceeds from your short sale, plus your $20,000, in Intel. What is the expected return and volatility of your portfolio? Solution We can think of our short sale as a negative investment of −$10,000 in Coca-Cola stock. In addition, we invested +$30,000 in Intel stock, for a total net investment of $30,000 − $10,000 = $20,000 cash. The corresponding portfolio weights are Value of investment in Intel 30,000 xI = = = 150% Total value of portfolio 20,000 Value of investment in Coca-Cola −10,000 xC = = = −50% Total value of portfolio 20,000 Note that the portfolio weights still add up to 100%. Using these portfolio weights, we can cal- culate the expected return and volatility of the portfolio using Eq. 11.3 and Eq. 11.8 as before: E [ R P ] = x I E [ R I ] + x C E [ RC ] = 1.50 × 26% + ( −0.50 ) × 6% = 36% SD ( R P ) = Var ( R P ) = x 12Var ( R I ) + x C2 Var ( RC ) + 2 x I x C Cov ( R I , RC ) = 1.5 2 × 0.50 2 + ( −0.5 ) 2 × 0.25 2 + 2 ( 1.5 )( −0.5 )( 0 ) = 76.0% Note that in this case, short selling increases the expected return of your portfolio, but also its volatility, above those of the individual stocks. Short selling is profitable if you expect a stock’s price to decline in the future. Recall that when you borrow a stock to short sell it, you are obligated to buy and return it in the future. So when the stock price declines, you receive more up front for the shares than the cost to replace them in the future. But as the preceding example shows, short selling can be advantageous even if you expect the stock’s price to rise, as long as you invest the pro- ceeds in another stock with an even higher expected return. That said, and as the example also shows, short selling can greatly increase the risk of the portfolio. M11_BERK6318_06_GE_C11.indd 411 27/04/23 11:00 AM  412 Chapter 11 Optimal Portfolio Choice and the Capital Asset Pricing Model In Figure 11.5, we show the effect on the investor’s choice set when we allow for short sales. Short selling Intel to invest in Coca-Cola is not efficient (blue dashed curve)—other portfolios exist that have a higher expected return and a lower volatility. However, because Intel is expected to outperform Coca-Cola, short selling Coca-Cola to invest in Intel is effi- cient in this case. While such a strategy leads to a higher volatility, it also provides the investor with a higher expected return. This strategy could be attractive to a risk-tolerant investor. Efficient Portfolios with Many Stocks Recall from Section 11.3 that adding more stocks to a portfolio reduces risk through diver- sification. Let’s consider the effect of adding to our portfolio a third stock, Bore Industries, which is uncorrelated with Intel and Coca-Cola but is expected to have a very low return of 2%, and the same volatility as Coca-Cola (25%). Figure 11.6 illustrates the portfolios that we can construct using these three stocks. Because Bore stock is inferior to Coca-Cola stock—it has the same volatility but a lower return—you might guess that no investor would want to hold a long position in Bore. However, that conclusion ignores the diversification opportunities that Bore pro- vides. ­Figure 11.6 shows the results of combining Bore with Coca-Cola or with Intel (light blue curves), or combining Bore with a 50–50 portfolio of Coca-Cola and Intel (dark blue curve).7 Notice that some of the portfolios we obtained by combining only Intel and Coca- Cola (black curve) are inferior to these new possibilities. FIGURE 11.5 40% Portfolios of Intel and (1.5, 20.5) Coca-Cola Allowing 35% for Short Sales (1.2, 20.2) 30% Labels indicate portfolio weights Intel Long Intel, Short Coca-Cola ( x I , xC ) for Intel and Coca-Cola 25% Expected Return stocks. Red indicates efficient portfolios, blue indicates inef- 20% ficient portfolios. The dashed curves indicate positions that re- 15% Long Intel, Long Coca-Cola quire shorting either Coca-Cola (red) or Intel (blue). Shorting 10% Intel to invest in Coca-Cola is 5% Coca-Cola inefficient. Shorting Coca-Cola Short Intel, to invest in Intel is efficient and Long Coca-Cola 0% (20.2, 1.2) might be attractive to a risk-­ tolerant investor who is seeking (20.5, 1.5) 25% high expected returns. 0% 10% 20% 30% 40% 50% 60% 70% 80% An interactive version of this figure Volatility (standard deviation) is available in the eTextbook and MyLab Finance. 7 When a portfolio includes another portfolio, we can compute the weight of each stock by multiplying the portfolio weights. For example, a portfolio with 30% in Bore stock and 70% in the portfolio of (50% Intel, 50% Coca-Cola) has 30% in Bore stock, 70% × 50% = 35% in Intel stock, and 70% × 50% = 35% in Coca-Cola stock. M11_BERK6318_06_GE_C11.indd 412 27/04/23 11:00 AM  11.4 Risk Versus Return: Choosing an Efficient Portfolio 413 FIGURE 11.6 30% Expected Return and Intel Volatility for Selected 25% I1C Portfolios of Intel, Coca-Cola, (50% I, 50% C) and Bore Industries Stocks 20% Expected Return By combining Bore (B) with Intel (I), Coca-Cola (C), and portfolios of B 1 (50% I, 50% C) Intel and Coca-Cola, we introduce 15% B1I new risk and return possibilities. We can also do better than with 10% just Coca-Cola and Intel alone (the black curve). Portfolios of Bore and Coca-Cola Coca-Cola ( B + C ) and Bore and 5% Intel ( B + I ) are shown in light blue in the figure. The dark blue curve B1C Bore is a combination of Bore with a 0% 0% 10% 20% 30% 40% 50% 60% portfolio of Intel and Coca-Cola. Volatility (standard deviation) NOBEL PRIZE Harry Markowitz and James Tobin The techniques of mean-variance portfolio optimization, Italiano degli Attuari, but the work remained in obscurity until which allow an investor to find the portfolio with the high- its recent translation in 2004.† est expected return for any level of variance (or volatility), While Markowitz assumed that investors might choose were developed in an article, “Portfolio Selection,” pub- any portfolio on the efficient frontier of risky investments, lished in the Journal of Finance in 1952 by Harry Marko­witz. James Tobin furthered this theory by considering the impli- Marko­witz’s approach has evolved into one of the main cations of allowing investors to combine risky securities methods of portfolio optimization used on Wall Street. In with a risk-free investment. As we will show in Section 11.5, recognition for his contribution to the field, Markowitz was in that case we can identify a unique optimal portfolio of awarded the Nobel Prize for economics in 1990. risky securities that does not depend on an investor’s toler- Markowitz’s work made clear that it is a security’s ance for risk. In his article “Liquidity Preference as Behavior covariance with an investor’s portfolio that determines its Toward Risk” published in the Review of Economic Studies in incremental risk, and thus an investment’s risk cannot be 1958, Tobin proved a “Separation Theorem,” which applied evaluated in isolation. He also demonstrated that diversifi- Markowitz’s techniques to find this optimal risky portfo- cation provided a “free lunch”—the opportunity to reduce lio. The Separation Theorem showed that investors could risk without sacrificing expected return. In later work Mar- choose their ideal exposure to risk by varying their invest-