Mutual fund managers who work at approximately 8,000 mutual funds in the United States, plus other mutual funds throughout the rest of the world. Security analysts working at mutual funds in the United States, plus others throughout the rest of the world.
THE PORTFOLIO MANAGEMENT PROCESS
The purpose of this book is to simplify investor choices by treating the countably infinite number of stocks, bonds, and other individual assets as components of portfolios. The individual assets that go into a portfolio are inputs, but they are not the objects of choice for an investor to focus on.
THE SECURITY ANALYST’S JOB
The security analyst's forecast should be in terms of the holding period rate of return, denoted r1. The security analyst must construct a probability distribution of returns for each individual security that is an investment candidate.
PORTFOLIO ANALYSIS
Conversely, investors prefer lower risk to higher risk for any given rate of return. First, many investors simply do not predict asset prices or investment rates of return.
PORTFOLIO SELECTION
All the assumptions underlying the portfolio analysis turned out to be simplistic and in some cases oversimplified. People need to behave only if they have been described with the assumptions that the theory is valid.4.
THE MATHEMATICS IS SEGREGATED
Portfolio selection has become more difficult because security prices change as the latest information becomes constantly available.
TOPICS TO BE DISCUSSED
When this happens, the portfolio must be revised to maintain its superiority over alternative investments. Portfolio theory recognizes this and suggests that the portfolio manager identifies the best portfolio by evaluating all portfolios in terms of their risk and expected returns and then choosing the one that best suits his or her preferences.
APPENDIX: VARIOUS RATES OF RETURN
The rate of return defined by equation (A1.1) is often called the investor's holding period return (HPR). The continuously compounded return is always less than the return during the holding period; that is, r˙t The expected value of a constant multiplied by a random variable of 1 is equal to the constant multiplied by the expected value of the random variable. The expected value of the sum of two independent random variables, X and Y, is simply the sum of their expected values. The expected value of the sum of n constants times n independent random variables is simply the sum of n constants times their expected values. In words, the variance (σX2) is the sum of the products of the squared deviations of the expected value times their probabilities. The math terms variance and standard deviation measure the distribution of results around the expected value. The variance of the $5 gamble is $25 "squared." To translate this measure of risk into more intuitively appealing terms, the standard deviation (σX) will be used. RISK OF A SECURITY Since the covariance is negative, it is expected that securities A and B tend to move in opposite directions. The correlation coefficient for the return on securities A and B in example 2.1 is calculated as follows: The following are monthly return observations on the Excelon (X), Jorgenson (J) and Standard & Poor's 500 Index (M) for the period from January to December. DATA INPUT REQUIREMENTS Equation (2.11) is simply the well-known balance sheet identity where equity is defined as 100 percent=1.0 and total assets have a total weight equal to the sum of the various variables. Assuming that the portfolio has no liabilities, this means that the total assets equal the equity – as shown in equation (2.11). A PORTFOLIO’S EXPECTED RETURN In fact, the portfolio variance is equal to the sum of all four elements of the (2×2) matrix on the right-hand side in equation (2.15). Because the portfolio variance is equal to the sum of all nine elements of the (3×3) matrix on the right-hand side in equation (2.18),. A random variable is a rule or function that assigns a value to each outcome of an experiment. For example, in the coin toss, the random variable is X and can take on two values, −1 for x1 and +1 for x2. The semivariance (SV) of returns, defined in equation (3.2), is a quantitative proxy for risk that measures the area under E(r) in the probability distribution of returns:. 3.2) where ri represents below average returns. These rates of return are less than E(r) (ie, ri Analyzing a company's returns alone may seem too simple compared to fundamental safety analysis techniques that many financial ratios use to analyze financial statements, management interviews, industry forecasts, and the economic outlook.3 However, these two approaches are not contradictory. After the fundamental security analyst has completed her job, all she has to do is convert the estimates into different possible rates of return and add probability estimates to each. Plotting such a figure would generate a set of investment opportunities that could take the escalated quarter-moon shape of Figure 3.9A. Each point on the edge and within the escalated quarter moon shape in Figure 3.9A is a possible investment opportunity. This is true regardless of how risky the portfolio's assets are when analyzed individually. Although the expected return of this portfolio is fixed at 8.3 percent with these proportions of A and B regardless of the correlation coefficient, ρAB, the risk of the portfolio varies with ρAB. The investor's utility function is used to determine the numerical values for each alternative investment. Utility theory argues that the investor should act to maximize expected utility, where expected utility is a numerical value assigned to a portfolio's probabilistic rates of return. BASIC UTILITY AXIOMS THE UTILITY OF WEALTH FUNCTION This means that the investor's wealth utility curve shown in Figure 4.1 is simply a positive linear transformation of that investor's rates of return. And the wealth investor's utility and return utility functions will yield identical preference orders. Equation (4.2) shows that the investor's return is simply a linear transformation of the investor's wealth and vice versa. Investor C prefers the investment with the highest risk because investor C's utility function, equation (4.6), represents risky behavior. Suppose, the investor's utility after playing the risky game is equal to the utility of holding a certainty equivalent (CE) amount of cash, symbolically E [U(W)]=U(CE). Assume again the facts used in Example 4.2, except that the investor's utility function is quadratic; U(W)=W2. Mathematically, decreasing absolute risk aversion is represented by A(W) <0, where A(W) is the first derivative of A(W) with respect to wealth. If investment in risky assets is reduced as the investor's wealth increases, this investor is said to exhibit increasing absolute risk aversion, represented by A(W) >0. If an investor increases (decreases) the percentage of assets invested in risky investments as his assets increase, the investor exhibits a decreasing (increasing) relative risk appetite. The log utility function has absolute risk aversion A(W)=1 and relative risk aversion R(W)=1. Thus, utility analysis can focus on the term that includes returns. without being affected by the level of the investor's initial wealth. A quadratic wealth function implies a quadratic return function, but the coefficients of the return function depend on the level of the investor's initial wealth. When the utility function is quadratic, expected utility is exactly a function of the mean and variance of wealth. As first noted by Levy and Markowitz (1979), if investors with unidentified utility functions choose a mean-variance efficient frontier, they can approx. they maximize their expected utility without even knowing what their utility function is; in fact, without anyone even mentioning the "maxim of expected utility" to them. In contrast, Figure 4.15 shows that as the level of wealth increases, the slope of the indifference curves steepens, indicating more risk aversion and increasing absolute risk aversion. The indifference curves in Figure 4.16 remain parallel with respect to wealth change, representing constant absolute risk aversion.21. Each investor can be represented by a set (family) of indifference curves, and each indifference curve represents a certain level of expected utility. It was shown that indifference curves representing rational decisions are positively sloped and convex for risk-averse investors, negatively sloped and concave for risk-seeking investors, and horizontal for risk-neutral investors. A disadvantage of graphical analysis is that it cannot handle portfolios containing more than a few securities. Thus, the portfolio can be optimized with a large number of different types of constraints that the portfolio manager may wish to impose. In other words, w1+w This equality means that the total net worth of the portfolio has been taken into account. The variables w1 and w2 are weights or percentages of the portfolio owner's equity invested in each security. Because the isomic line E1 intersects the budget constraint line at B, the magnitude of the expected return implied by the isomic line E1 can be obtained by investing w1 in security 1 and w2 in security 2. The expected return implied by this latter isomic line, cannot be achieved without violating the budget constraint. The achievable portfolios on the budget constraint line in Figure 5.4 can be mapped in the mean standard deviation plane. The lower part of the budget constraint line in Figure 5.4, BCb, is mapped into the upper part of the investment opportunity given in Figure 5.5. Since the intersection points A and C in Figure 5.4 are on the same ellipse, they have the same variance, σS2. It is impossible to graph isovariance ellipses for variances less than the MVP variance. LEGITIMATE PORTFOLIOS REPRESENTING CONSTRAINTS GRAPHICALLY Markowitz's portfolio analysis technique, which is explained in this chapter, contrasts with the concept of portfolio management used by interior decorators. For example, portfolio analysis may indicate that a person seeking to minimize risk should hold a portfolio of only two stocks that are individually very risky. As new information is constantly arriving, the security analysts must re-evaluate the statistical input provided to the portfolio analyst. The input statistics change every time the outlook for an actual or a potential investment changes. The first part of the second equation indicates the sum of deviations, and the second part represents the sum of. Note that the terms in the matrix containing identical subscripts (for example wiwiσii) form a diagonal pattern from the upper left corner to the lower right corner. To use Markowitz's portfolio theory, a security analyst must exogenously estimate all of these parameters (plus 100 expected returns). Since wiwjσij=wjwiσji, the matrix is symmetric and each covariance is repeated twice in the matrix. All possible portfolios that can be formed from those securities lie either on or within the boundary of the opportunity set shown in Figure 6.3A. When short selling and/or borrowing is allowed, all possible portfolios lie on or behind the opportunity frontier set in Figure 6.3B. EFFICIENT FRONTIER WITHOUT THE RISK-FREE ASSET Consider a situation where the efficient boundary is not concave.3 The efficient boundary in Figure 6-6 is convex between points D and T. The efficient boundary in Figure 6-5 is bounded at one end by point E and at the other end by point F . State the proportions held in the risky asset and the risk-free asset asw1 and 1−w1, respectively. This optimal portfolio O* in Figure 6-11 includes positive investments in both the risk-free assets and the tangent portfolio. SUMMARY AND CONCLUSIONS These optimal weights of the three assets in equation (7.20) are the same as in example 7.1. Several important implications arise from the property of the weights for an efficient portfolio shown in equation (7.22). What are the optimal weights of the three risky assets and the risk-free asset for the efficient portfolio. Note again that adding the risk-free asset to the portfolio consisting of risky assets does not change portfolio variance. Based on these actual weights, we calculate the expected return and variance of the tangent portfolio. Therefore, the optimal weights of the three risky assets for the efficient portfolio can be obtained by assigning wmin to the three assets according to the weights for the tangent portfolio obtained in Example 7.5. Note that the slight difference between these answers and those in Example 7.4 is due to rounding. For a different target expected return, the optimal weights for the efficient portfolio can similarly be obtained using equation (7.43). The original version of the single index model of equation (8.1) and its excess return version, equation (8.5), are very similar. A variance decomposition similar to earlier equation (8.3) also applies to the excess return version of the single index model. The term on the left-hand side (LHS), σp2, is known as the total risk of the portfolio. The portfolio analysis problem itself can be reformulated in terms of the market index variables E.NOTES
MATHEMATICAL EXPECTATION
WHAT IS RISK?
EXPECTED RETURN
COVARIANCE OF RETURNS
CORRELATION OF RETURNS
USING HISTORICAL RETURNS
PORTFOLIO WEIGHTS
PORTFOLIO RISK
SUMMARY OF NOTATIONS AND FORMULAS
RECONSIDERING RISK
UTILITY THEORY
PRINCIPLES BOX: DEFINITION OF DOMINANCE
RISK-RETURN SPACE
PRINCIPLES BOX: DEFINITION OF AN EFFICIENT INVESTMENT
DIVERSIFICATION
CONCLUSIONS
UTILITY OF WEALTH AND RETURNS
EXPECTED UTILITY OF RETURNS
RISK ATTITUDES
ABSOLUTE RISK AVERSION
RELATIVE RISK AVERSION
MEASURING RISK AVERSION
PORTFOLIO ANALYSIS
INDIFFERENCE CURVES
SUMMARY AND CONCLUSIONS
APPENDIX: RISK AVERSION AND INDIFFERENCE CURVES
DELINEATING EFFICIENT PORTFOLIOS
PORTFOLIO ANALYSIS INPUTS
TWO-ASSET ISOMEAN LINES
TWO-ASSET ISOVARIANCE ELLIPSES
THREE-ASSET PORTFOLIO ANALYSIS
5.7 “UNUSUAL” GRAPHICAL SOLUTIONS DON’T EXIST
THE INTERIOR DECORATOR FALLACY
SUMMARY
APPENDIX: QUADRATIC EQUATIONS
RISK AND RETURN FOR TWO-ASSET PORTFOLIOS
THE OPPORTUNITY SET
Security 2
PRINCIPLE BOX: DEFINITION OF MARKOWITZ DIVERSIFICATION
PRINCIPLES BOX: DEFINITION OF EFFICIENT PORTFOLIOS
PRINCIPLES BOX: DEFINITION OF THE EFFICIENT FRONTIER THEOREM
INTRODUCING A RISK-FREE ASSET
EFFICIENT PORTFOLIOS WITHOUT A RISK-FREE ASSET
EFFICIENT PORTFOLIOS WITH A RISK-FREE ASSET
IDENTIFYING THE TANGENCY PORTFOLIO
SUMMARY AND CONCLUSIONS
APPENDIX: MATHEMATICAL DERIVATION OF THE EFFICIENT FRONTIER
SINGLE-INDEX MODELS
S. Treasury Bill Rate and CPI Inflation Rate
