MODERN PORTFOLIO THEORY Copyright 1990-1997 By STAFFORD FINANCIAL SERVICES CORPORATION Distributed By STAFFORD ENTERPRISES Denver, CO Selection of portfolio investments is not just limited to the size of returns. Usually the greater the return, the greater the risk. The objective of portfolio management is to balance expected gain and risk, consistent with gain expectations and risk tolerances of the client. Prior to the work of Harry Markowitz (1990 Nobel Prize Winner), portfolio management theory was limited to picking the highest quality stocks with the best expected returns. Markowitz revolutionized portfolio management by pointing out that the "best stock" theory was contrary to portfolio diversification, since it would logically concentrate funds in a few assets with the greatest expected returns. Portfolio diversification, Markowitz asserted, expressed investor concern with risk. His "mean-variance" theory quantifies risk as the variance from each security's expected return; then combines securities having opposing market movement characteristics to reduce overall portfolio risk. By combining assets in a portfolio with returns that increase or decrease differently (are less than perfectly correlated) portfolio risk can be reduced without sacrificing portfolio return. As the correlation (covariance) between asset returns combined in a portfolio decreases, the volatility (standard deviation) of the return on the entire portfolio decreases. ** Definitions ** To aid understanding a few definitions are in order: o Rate of Return is the average (mean) percent of return generated by a security's appreciation (price increase) plus income. o Standard Deviation is a measure of fluctuation of return from a linear trendline (which is to say that it measures the general ups and downs in return over time). One standard deviation defines the normal boundaries of actual results in either direction from the average return about 68% of the time. Standard deviations can be statistically refined to reflect 80%, 90%, etc. probability. o Covariance and Correlation are measures of the interrelationship between assets and whether they move in sinc or opposed to each other. o Efficient Sets or efficient frontiers maximize risk/return utility by the mathematical technique of quadratic optimization. They provide a higher expected return at the same level of risk or a lower level of risk at the same expected return than any other alternatives. o Portfolio Optimization is achieved by selection of portfolios at given levels of return from within the efficient sets. ** Measuring the Individual Investment ** The total gain of an individual investment during a given period equals the dividends or interest income received plus the increase in the value of the investment. This rate of return can be stated as: R = (D + (Pp - Pb) \ Pb R = Rate of Return during the period D = Dividends Pp = Price at the end of the period Pb = Price at the beginning of the period Historical returns are not expected returns. Expected returns are not always realized. It is this quantification of investment risk or exposure to loss, for which Markowitz is famous. Markowitz' studies in the 1950's quantified investment risk as the variance about an asset's expected return. Given the following probabilities of occurrence and expected returns for each probability, the mean or expected return of this investment's probability distribution (expected return denoted by E) is equal to 8%. Return Probability of Occurrence 15% .40 10% .30 0% .20 -10% .10 ------- 1.00 E (R) = (.40*15%) + (.30x10%) + (.20*0%) + (.10*-10%) = 8% The variance of expected returns measures the dispersion of possible outcomes. In our example, the probability of each occurrence is multiplied by the square of the difference from the mean. The variance would be ,0064. Or stated in its more usual form of conversion to the standard deviation or square root of the variance, it would be 8%. The larger the variance or standard deviation, the greater the potential risk. Var(R) = .40*(15%-8%)2 + .30*(10%-8%)2 + .20*(0%-8%)2 + .10*(-10%-8%)2 = .0064 SD(R) = square root of Var(R) = .08 = 8% This says that 68% of the time, the actual return will be 8% below or 8% above the weighted portfolio return (ie. 0% to 16%). ** Measuring Portfolio Return ** The return of a given portfolio of assets equals the sum of the returns on each asset in the portfolio weighted by its percentage of the portfolio. For example, a portfolio composed of three assets representing 50%, 40% and 10% of the portfolio respectively; with the assets returning 10%, 12% and 14% respectively would result in a portfolio return of 11.2%. R = (.50*10%) + (.40*12%) + (.10*14%) = 11.2% The variance of a portfolio of assets depends not only on the variance of each asset in the portfolio but how the assets track each other asset in the portfolio. This introduces the concept of covariance or correlation; that is to say the degree by which the returns of two assets vary or change together. To determine the variance of a portfolio of assets, the sum of the weighted variances of the individual assets and the sum of the weighted covariances of the assets are added together. Correlation and covariance are analogous. A correlation coefficient of +1.0 notes perfect co-movement in the same direction, while -1.0 notes perfect co-movement in opposite directions. We will see that the selection of investments that move in opposite directions are of primary importance in diversification theory. ** Diversification ** Diversification is intended to assemble a portfolio of assets in order to reduce risk. Systematic or market risk cannot be diversified away. However, unsystematic risk is capable of being reduced by diversification. There are basically two diversification strategies to lessen portfolio risk: naive and Markowitz. The naive strategy ranges from seat-of-the-pants to what many refer to as the interior decorator approach. The seat-of-the-pants approach combines investments because "one is supposed to combine investments". There is a guessed at basis for their combination in the portfolio. The interior decorator approach "designs" portfolios for individuals (such as widows with high current income needs) that concentrate investments in single asset categories that "fit" the individual (such as bonds). These approaches invite the very risk that the asset manager is attempting to avoid. Markowitz diversification combines assets with returns that are less than perfectly correlated in order to lower overall risk without sacrificing overall return. Concentrations in single asset categories are avoided. The magic of Markowitz diversification is the degree of correlation between expected asset returns. Portfolio returns may be maintained while obtaining lower risk through assets with low to negative correlations. ** Efficient Portfolio ** Having established the methodology for combining individual portfolios, the question now shifts to which portfolio is best for the investor. Markowitz defined efficient portfolios as those with the highest expected return at a given level of risk. A mathematical technique called quadratic optimization is coupled with the computer to solve for the portfolio that minimizes risk at each level of return. A set of efficient portfolios is then selected that have higher expected returns and lower risk levels than any other potential portfolio combinations. Portfolios are not included in the efficient set if there exists some other portfolio that would provide a lower level of risk at the same return or a higher anticipated return at the same level of risk. ** Selection of the Efficient Portfolio ** Efficient Portfolio analysis presupposes that investors will only want to hold portfolios in the efficient set. That is to say those with the lowest risk and highest return. Once the efficient sets are determined, the problem is reduced to fitting a specific portfolio from among the efficient sets to the investor. Usually the portfolios with higher returns also carry higher risk. Selecting the portfolio from the efficient sets that best fits the investors risk/reward temperament is professional key to this process of modern portfolio management. ** Betas ** Betas are a adaptation of Markowitz theory advanced by William Sharpe (another 1990 Nobel Prize recipient. I effect they are correlations (volatility) measured RELATIVE TO THE MARKET (usually the S&P 500). Betas vary by the periods (usually 36 months) selected by those setting the beta. Computer statistical programs are used to compute the relative betas. The Market (S&P 500) has a beta of 1.0 or supposedly neutral risk. A bond would have a lower (lower fluctuation/volatility) beta .. for example .65. A merging growth stock might have a beta of 1.50 (higher volatility). When the market's price rises or falls 10%, the bond example would have a tendency to rise or fall only 6.5%, while the example merging growth stock would have tendency to rise or fall 15%. Weighting individual portfolio items by their beta, gives the overall beta for the portfolio (relative to market for the period selected). The investor then determines what risk (portfolio beta) he is willing to take and tries portfolio adjustments accordingly. ** Conclusions ** Risk is often ignored in investing. It's so much more pleasant to talk about those big potential rewards. Unfortunately, the greater the potential gain (reward) the greater the potential pain (risk). Markowitz' mean-variance work and Sharpe's beta market adaptations have brought risk onto equal footing with its more glamorous cousin, return. These techniques are universally recognized. Today's prudent investor should understand their importance and application.