Ch06 Portfolio Theory.docx

PORTFOLIO THEORY Measuring Risk: Risk is often associated with the dispersion in the likely outcomes. Dispersion refers to variability. Risk is assumed to arise out of variability', which is consistent with our definition of risk as the chance that the actual outcome of an investment will differ from the expected outcome. If an asset's return has no variability, in effect it has no risk. Thus, a one-year treasury bill purchased to yield 10 percent and held to maturity will, in fact, yield (a nominal) 10 percent No other outcome is possible, barring default by the U.S. government, which is not considered a reasonable possibility. Consider an investor analyzing a series of returns (TRs) for the major types of financial asset over some period of years. Knowing the mean of .this series is not enough; the investor also needs to know something about the variability in the returns. Relative to the other assets, common stocks show, the largest variability (dispersion) in returns, with small common stocks showing f ten greater variability. Corporate bonds have a much smaller variability and therefore a more compact distribution of returns. Of course, Treasury bills are the least risky. The "dispersion of annual returns for bills is compact. Standard Deviation: The risk of distributions' can be measured with an absolute measure of dispersion, or variability. The most commonly used measure of dispersion over some period of years is the standard deviation, which measures the deviation of each observation from the arithmetic mean of the observations and is a reliable measure of variability, because all the information in a sample is used. The standard deviation is a measure of the total risk of an asset or a portfolio. It captures the total variability in the assets or portfolios return whatever the source of that variability. The standard deviation can be calculated from the variance, which is calculated as: n ?2 = ?(X - X) 2901950-20193000 i = 1 2471420-1079500n - 1 Where; ?2 = the variance of a set of values X = each value in the set X = the mean of the observations n = the number of returns in the sample ?2 = (?2) 1 / 2 = standard deviation Knowing the returns from the sample, we can calculate the standard deviation quite easily. Dealing with Uncertainty: Realized returns are important for several reasons. For example, investors need to know how their portfolios have performed. Realized returns, also can be particularly important in helping investors to form expectations about future returns, because investors must concern themselves with their best estimate of return over the next year, or six months, or whatever. -6358509000 203 Investment Analysis & Portfolio Management 0-444500 How do we go about estimating returns, which is what investors must actually do in managing their portfolios? The total return measure, TR, is applicable whether one is measuring realized returns; or estimating, future (expected) returns. Because it includes everything the investor can expect to receive over any specified future period, the TR is useful in conceptualizing the estimated returns from securities. Similarly, the variance, or its square root, the standard deviation, is an accepted measure of variability for both realized returns and expected returns. We will calculate both the variance and the standard deviation below and use them interchangeably as the situation dictates. Sometimes it is preferable to use one and sometimes the other. Using Probability Distributions: The return an investor will earn from investing is not known; it must be estimated. Future return is an expected return and may or may not actually be-realized. An investor may expect the TR on a particular security to be 0.10 for the coming year, but in truth this is only a "point estimate." Risk, or the chance that some unfavorable event will occur, is involved when investment decisions are made. Investors are often overly optimistic about expected returns. Probability Distributions: To deal with the uncertainty of returns, investors need to think explicitly about a: security's distribution of probable TRs. ln other words, investors need to keep in mind that, although they may expect a security to return 10 percent, for example, this is only a one-point estimate of the entire range of possibilities. Given that investors must deal with the uncertain future, a number of possible returns can, and will, occur. In the case of a Treasury bond paying fixed rate of interest, the interest payment will be made with l00 -percent certainty barring a financial collapse of the economy. The probability of occurrence is 1.0; because no other outcome is possible. With the possibility of two or more outcomes, which is the norm for common stocks, each possible likely outcome must be considered and a probability of its occurrence assessed. The probability for a particular outcome is simply the chance that the specified outcome will occur. The result of considering these outcomes and their probabilities together is a probability distribution consisting of the specification of the likely outcomes that may occur and the probabilities associated with these likely outcomes. Probabilities represent the likelihood of various outcomes and are typically expressed as a decimal. The sum of the probabilities of all possible outcomes must be 1.0, because they must completely describe all the (perceived) likely occurrences. How are these probabilities and associated outcomes obtained? In the final analysis, investing for some future period involves uncertainty, and therefore subjective estimates. Although past occurrences (frequencies) may be relied on heavily to estimate the probabilities the past must be modified for any changes expected in the future. 075501500 204 Investment Analysis & Portfolio Management -635-444500 Calculating Expected Return for a Security: To describe the single most likely outcome from a particular probability distribution, it is necessary to calculate its expected value. The expected value is the weighted average of'all possible return outcomes, where each outcome is weighted by its respective probability of occurrence. Since investors are interested in returns, we will-call this expected value the expected rate of return, or simply expected-return, and for any security, it is calculated as; m E (R) = ?Ri pri i = 1 Where; E (R) = the expected return on a security' Ri = the ith possible return pri = the probability of the ith return Ri m = the number of possible returns Calculating Risk for a Security: Investors must be able to quantify and measure risk. To calculate the total risk associated with the expected return, the variance or standard deviation is used, the variance and, its square root, standard deviation, are measures of the spread or dispersion in the probability distribution; that is, they measure the dispersion of a random variable around its mean. The larger this dispersion, the larger the variance or standard deviation. To calculate the variance or standard deviation from the probability distribution, first calculate the expected return of the distribution. Essentially, the same procedure used to measure risk, but now the probabilities associated with the outcomes must be included, m The variance of returns = ?2 = ?- [Ri – E (R)]2pri i = 1 And The standard deviation of returns = ?= (?2)1/2 Portfolio Expected Return: The expected return on any portfolio is easily calculated as a weighted average of the individual securities expected returns. The percentages of a portfolio’s total value that are invested in each portfolio asset are referred to as portfolio weights, which will denote by w. The combined portfolio weights are assumed to sum to 100 percent of, total investable funds, or 1.0, indicating that all portfolio funds are invested. That is, n w1 + w2 + … + wn = ?wi = 1.0 i = 1 -63576898500 205 Investment Analysis & Portfolio Management 0-444500 Portfolio Risk: The remaining computation in investment analysis is that of the risk of the portfolio. Risk is measured by the variance (or standard deviation) of the portfolio's return, exactly as in the case of each individual security. Typically, portfolio risk is stated in terms of standard deviation which is simply the square root of the variance. It is at this point that the basis of modern portfolio theory emerges, which can be stated as follows: Although the expected return of a portfolio is a weighted average of its expected returns, portfolio risk (as measured by the variance or standard deviation) is not a weighted average of the risk of the individual securities in the portfolio. Symbolically, n (Rp) = ?wi E (Ri) = 1 But n ?2p ??wi ?2i i = 1 Precisely, investors can reduce the risk of a portfolio beyond what it would be if risk were, in fact, simply a weighted average of the individual securities' risk. In order to see how this risk reduction can be accomplished, we must analyze portfolio risk in detail. 0487235500 PORTFOLIO THEORY Contd… ANALYZING PORTFOLIO RISK: Risk Reduction: The Insurance Principle: To begin our analysis of how a portfolio of assets can reduce risk, assume that all risk sources in a portfolio of securities are independent. As we add securities to this portfolio, the exposure to any particular source of risk becomes small. According to the Law of Large Numbers, the larger the sample size, the more likely it is that the sample mean will be close to the population expected value. Risk reduction in the case of independent risk sources can be thought of as the insurance principle, named for the idea that an insurance company reduces its risk by writing many policies against many independent sources of risk. We are assuming here that rates of return on individual securities are statistically independent such that any one security's rate of return is unaffected by another's rate of: return. In this situation, the standard deviation of the portfolio is given by, ?p = ?i/ n 1/2 Diversification: The insurance principle illustrates the concept of attempting to diversify the risk involved in a portfolio of assets (or liabilities). In fact, diversification is the key to the management of portfolio risk, because it allows investors; significantly to lower portfolio risk without adversely affecting return. Random Diversification: Random or naive diversification refers to the act of randomly diversifying without regard to relevant investment characteristics such as expected return and industry classification. An investor simply selects a relatively large number of securities randomly—the proverbial "throwing a dart at the Wall Street Journal page showing stock quotes. For simplicity, we assume equal dollar amounts are invested in each stock. Markowitz Portfolio Theory: Before Markowitz, investors dealt loosely with the concepts of return and risk. Investors have known intuitively for many years that it is smart to diversify; that is, not to "put all of your eggs in one basket? Markowitz however, was the first .to develop the concept of portfolio diversification in a formal way— he quantified the concept of diversification. He showed quantitatively why and how portfolio diversification works to reduce the risk of a portfolio to an investor. Markowitz sought to organize the existing thoughts and practices into, a more formal framework and to answer a basic question. Does the risk of a portfolio equal to the sum of the risks of the individual securities comprising it? Markowitz was the first to develop a specific measure of portfolio risk and to derive the Expected return and risk for a portfolio based on covariance relationships. We consider covariances in detail in the discussion below. -6359651900 207 Investment Analysis & Portfolio Management 0-444600 Portfolio risk is not simply a weighted average of the individual security risks. Rather, as Markowitz first showed, we must account for the interrelationships among, security returns in order to calculate portfolio risk, and in order to reduce portfolio risk to its minimum level for any given level of return. The reason we need to consider these, interrelationships, or comovements, among security return. In order to remove the inequality sign from Equation and develop that will calculate the risk of a portfolio as measured by the variance or standard deviation, we must account for two factors; Weighted individual security risks (i.e. the variance of each individual security, weighted by the percentage of investable funds placed in each individual security.) Weighted comovements between securities returns (i.e., the coyariance between, the securities returns, again weighted by the percentage of investable funds placed in each security). Measuring Comovements in Security Returns: Covariance is an absolute measure of the comovements between security returns used in the calculation of portfolio risk. We need the actual covariance between securities in a portfolio in order to calculate portfolio variance or standard deviation. Before considering covariance, however, we can easily illustrate how security returns move together by considering the correlation coefficient, a relative measure' of association learned in statistics. Correlation Coefficient: As used in portfolio theory, the correlation coefficient ?ij (pronounced "rho") is a statistical measure of the relative comovernents between security returns. It measures the extent to which the returns on any two securities are related, however, it denotes only association, not causation. It is a relative measure of association that is bounded by +1.0 'and—1.0, with; ?ij = +1.0 = perfect positive correlation ?ij = -1.0 = perfect negative (inverse) correlation ?ij = 0.0 = zero correlation Covariance: Given the significant amount of correlation among security returns, we must measure the actual amount of comovement and incorporate it into any measure of portfolio risk, because such comovements affect the portfolio's variance (or standard deviation). The Covariance measure does this. The covariance is an absolute measure of the degree of association between the returns for a pair of securities. Covariance is defined as the extent to which two random variables covary (move together) over time. As is true throughout our discussion, the variables in question are the returns (TRs) on two securities. As in the case of the correlation coefficient, the covariance can be: 026288900 208 Investment Analysis & Portfolio Management -635-444600 Positive, indicating that the returns on the two securities tend to move in the same direction at the same time; when one increases (decreases), the other tends to do the same. When the covariance is positive, the correlation coefficient will also be positive. Negative, indicating that the returns on the two securities tend to move inversely; when one increases (decreases), the other tends to decrease (increase), When the covariance is negative, the correlation coefficient will also be negative. Zero, indicating that the returns on two securities are independent and have no tendency to move in the same or opposite directions together. The formula for calculating covariance on an expected basis is; m ?AB = ?[RA,i – E ( RA)] [RB,i – E(RB)] pri i = 1 Where; ?AB = the covariance between securities A and B RA = one possible return on 'security A E ( RA) = the expected value of the return on security A , m = the number of likely outcomes for a security for the period Covariance is the expected value of the product of deviations from the mean. The size of the covariance measure depends upon the units of the variables involved and usually changes when these units are changed. Therefore, the, covariance primarily provides information about whether the association between variables is positive, negative, or zero because simply observing the number itself is not very useful. Relating the Correlation Coefficient and the Covariance: The covariance and the correlation coefficient can be related in the following manner: ?AB = ?AB / ?A?B This equation shows that the correlation coefficient is simply the covariance standardized by dividing by the product of the two standard deviations of returns. Given this definition of the correlation coefficient, the covariance can be written as; ?AB = ?AB?A?B Therefore, knowing the correlation coefficient, we can calculate the covariance because the standard deviations of the assets rates of return will already be available. Knowing the covariance, we can easily calculate the correlation coefficient. Calculating Portfolio Risk: Co variances account for the comovements in security returns; we are ready to calculate portfolio risk. First, we will consider the simplest possible case, two securities, in order to see what is happening in the portfolio risk equation. We will then consider the case of many securities, where the calculations soon become too large and complex lo analyze with any -63525018900 209 Investment Analysis & Portfolio Management means other than a computer. THE n-SECURITY CASE: The two-security case can be generalized to the n-security case. Portfolio risk can be reduced by combining assets with less than perfect positive correlation. Furthermore, the smaller the positive correlation, the better. Portfolio risk is a function of each individual security's risk and the covariances between the returns on the individual securities. Stated in terms of variances portfolio risk is; n nn ?2p = ?wi2 ?i2 + ? ?wiwj?iji = 1 i = 1 j = 1 i ?j Where ?2p ?i2 ?ijwnin the variance of the return on the portfolio the variance of return for security = the covariance between the returns for securities i and j . = the portfolio weights or percentage of investable funds invested in security i ? ? = a double summation sign indicating that n2 numbers are to be added i =1 j =1 together (i.e., all possible pairs of values for i and j) It states exactly the same messages for the two-stock portfolio. This message is portfolio risk is a function of; The weighted risk of each individual security (as measured by its variance) The weighted covariance among all pairs of securities Note that three variables actually determine portfolio risk: variances, covariances, and weights. Simplifying the Markowitz Calculations: In the case of two securities, there are, two covariances, and we multiply the weighted covariance term by two,' since the covariance of A with B is the same as the covariance of B with A. In the case of three securities, there are six covariances; with four securities, 12 covariances; and so forth, based on the fact that the total number of covariances in the Markowitz model is calculated as n (n - 1), where n is the number of securities. For the case of two securities, there are n2, or four,: total terms in the matrix—two variances and two covariances. For the case of four securities, there are n2, or 16 total terms in the matrix—four variances and 12 covariances. The variance terms are on the diagonal of the matrix; in effect represent the covariance of a security with itself. Efficient Portfolios: Markowitz's approach to portfolio selection is that an investor should evaluate portfolios on the basis of their expected returns and risk as measured by the standard deviation. He was 021018400 210 Investment Analysis & Portfolio Management -635-444600 the first to derive die concept of an efficient portfolio, which is defined as one that has the smallest portfolio risk for a given level of expected return or the largest expected return for a given level of risk. Investors can identify efficient portfolios by specifying an expected portfolio return and minimizing the portfolio-risk at this level of return. Alternatively, they can specify a portfolio risk level they are willing to assume and maximize the expected return on the portfolio for this level of risk. Rational investors will seek efficient portfolios, because these portfolios are optimized on the two dimensions of most importance to investors, expected return and risk. To begin our analysis, we must first determine the risk-return opportunities available to an investor from a given set of securities. A large number of possible portfolios exist when we realize that varying-percentages of an investor's wealth can be invested in each of the assets under consideration investors should be interested in only that subset of the available portfolios known as the efficient set. The assets generate the attainable set of portfolios, or the opportunity set. The attainable set is the entire set of all portfolios that could be found from a group of n securities. However, risk-averse investors should be interested only in those portfolios with the lowest possible risk for any given level of return. All other portfolios in the attainable set are dominated. Using the inputs described earlier—expected returns, variances, and covariances—-we can calculate the portfolio with the smallest variance, or risk, for a given level of expected return based on these inputs. Given the minimum-portfolios, we can plot the minimum-variance frontier. Point A represents, the global minimum-variance portfolio, because no other minimum-variance portfolio has a smaller risk. The bottom segment of tile minimum variance frontier, AC, is dominated by portfolios on the upper segment, AB. For example, since portfolio X has a larger return than portfolio Y for the same level of risk, investors would not want to own portfolio Y. The segment of the minimum-variance frontier above the global minimum variance portfolio, AB, offers the best risk-return combinations available to investors from this particular set of inputs, this segment is referred to as the efficient set of portfolios. This efficient set is determined by the principle of dominance—portfolio X dominates portfolio Y if it has the same level of risk but a larger expected return or the same expected return but a lower risk. The solution to the Markowitz model revolves around the portfolio weights, or percentages of investable funds to be invested in each security. Because the expected returns, standard deviations, and correlation coefficients for the securities being considered are inputs in, the Markowitz analysis, the portfolio weights are the only variable that can be manipulated to solve the portfolio problem of determining efficient portfolios. Think of efficient portfolios as being derived in the following manner. The inputs are obtained and a level of desired expected return for a portfolio is specified; for example, 10 percent. Then all combinations of securities that can be combined to form a portfolio with an expected return of I0 percent are determined, and the one with the smallest variance of return is selected as the efficient portfolio. Next, a new level of portfolio expected return is specified— for example; 11 percent — and the process is repeated. This continues until the feasible range of expected returns is processed. Of .course, the problem could be solved by specifying levels of portfolio risk and choosing that portfolio with the largest expected return for the specified level of risk. -63559245400 211 Investment Analysis & Portfolio Management 0-444600 Lecture 10 PORTFOLIO SELECTION Building a Portfolio Using Markowitz Principles: To select an optimal portfolio of financial assets using the Markowitz analysis; investors should; Identify optimal risk-return combinations available from the set of risky assets being considered by using the Markowitz efficient frontier analysis. This step; uses the inputs from, the expected returns, variances, and covariances for a set of securities. Choose the final portfolio from among those in the efficient set based on an investor's preferences. Using the Markowitz Portfolio Selection Model: Even if portfolios are selected arbitrarily, some diversification benefits are gained. This results in a reduction of portfolio risk. However, to take; the full information set into account, we use portfolio theory as developed by Markowitz, Portfolio theory is nonnative, meaning that it tells investors; how they should act to diversify optimally. It is based on a small set of assumptions, including; A single investment period; for example, one year. Liquidity of positions; for example, there are no transaction costs. Investor preferences based only on a portfolio's expected return and risk; as measured by variance or standard deviation. Efficient Portfolios: Markowitz's Approach to portfolio selection is that an investor should evaluate portfolios on the basis of their expected returns and risk as measured by the standard deviation. He was the first to derive the concept of an efficient portfolio, defined as one that, has the smallest portfolio risk for a given level of expected return or the largest expected return for a given level of risk. Rational investors will seek ethcient portfolios, because these portfolios are optimized on the two dimensions of most importance to investors, expected return and risk. The basic details of how to derive efficient portfolios? In brief, based on inputs consisting of estimates of expected return and risk for each security being considered of well as the correlation between pairs of securities, an optimization program varies the weights for each security until an efficient portfolio is determined. This portfolio will have the maximum expected return for a given level of risk or the minimum risk for a given level of expected return. Selecting an Optimal Portfolio of Risky Assets: Once the efficient set of portfolios is determined using the Markowitz model, investors must select from this set the portfolio most appropriate for them. The Markowitz model does not specify one optimum portfolio. Rather it generates the efficient set of portfolios, all of which, by definition; are optimal portfolios (for a given level pf expected return or risk). In economics in general and finance in particular, we assume investors are risk averse. This 012382400 212 Investment Analysis & Portfolio Management -635-444600 means that investors, if given a choice, will not take a "fair gamble," defined as one with an expected payoff of zero and equal probabilities of a gain or a loss. In effect, with a fair gamble, the disutility from the potential loss is greater than the utility from the potential gain. The greater the risk aversion; the greater the disutility from the potential loss. ALTERNATIVE METHODS OF OBTAINING THE EFFICIENT FRONTIER: The single-index model provides an alternative expression- for portfolio variance, which is easier to calculate than in the case of the Markowitz analysis. This alternative approach can be used to solve the portfolio problem as formulated by Markowitz—determining the efficient set of portfolios. It requires considerably fewer calculations. The Single - Index Model: William Sharpe, following Markowitz, developed the single-index model, which relates returns on each security to the returns on a common index. A broad market index of common stock returns is generally used for this purpose. Think of the S&P 500 as this index. The single-index model can be expressed by the following equation: Ri = ?i+ ?iRM + ?i Where; Ri = the return (TR) on security i RM = the return (TR) on the market index: ?i = that part of security i’s .return independent of market performance ?i = a constant measuring the expected change in the dependent variable, Ri given a change in the independent variable, RM?i = the random residual error; The single-index model divides a security's return into two components: a unique part, represented by ai and a market related part represented by ?iRM. The unique part is a micro event, affecting an individual company but not all companies in general. Examples include the discovery of new ore reserve, a fire, a strike, or the resignation of a key company figure. The market related part, on the other hand, is a macro event that is broad based and affects all (or; most) firms. Examples include a Federal Reserve announcement about the discount rate, a change in the prime rate, or an unexpected announcement about the money supply. Given these values, the error term is the difference between the left-hand side of the equation, the return on security i, arid the right-hand side of the equation, the sum of the two components of return. Since the single-index model is, by definition, equality, the two sides must be the same. Selecting Optimal Asset Classes—the Asset Allocation Decision: The Markowitz model is typically thought of in terms of selecting portfolios of individual securities; indeed, that is how Markowitz expected his model to be used. As we know, however, it is a cumbersome model to employ because of the number of covariance estimates needed when dealing with a large number of individual securities. An alternative way to use the Markowitz model as a selection technique is to think in terms of asset classes, such as domestic stocks, foreign stocks of industrialized countries, the -6358318400 213 Investment Analysis & Portfolio Management 0-444600 stocks of emerging markets, bonds, and so forth. Using, the model in this manner, investors decide what asset classes to own and what proportions of the asset classes to hold. The allocation of a portfolio's funds to classes of assets, such as cash equivalents, bonds, and equities The asset allocation decision refers to the allocation of portfolio assets to broad asset markets; in other words, how much of the portfolio's funds are to be, invested in stocks, in bonds, money market assets, and so forth. Each weight can range from zero percent to 100 percent. Asset allocation is one of the most; widely used applications of modern portfolio theory (MPT). Examining the asset allocation decision globally leads us to ask the following questions: What percentage of portfolio funds is to be invested in each of the countries for which financial markets are available to-investors? Within each country, what percentage of portfolio funds is to be invested in stocks, bonds, bills, and other assets? Within each of the major asset .classes, what percentage of portfolio funds is to be invested in various individual securities? Many knowledgeable market observers agree that the asset allocation decision is the most important decision made by an investor. According to some studies, for example, the asset allocation decision accounts for more than 90 percent of the variance in quarterly returns for a typical large pension fund. The rationale behind this approach is that different asset classes offer various potential returns and various levels of risk, and the correlation coefficients between some of these asset classes may be quite tow, thereby providing beneficial diversification effects. As with the Markowitz analysis applied to individual securities, inputs remain a problem, because they must be estimated. However, this" will always ¥e a problem in investing, because we are selecting assets to be held over the uncertain future. The Impact of Diversification on Risk: The Markowitz analysis demonstrates that the standard deviation of a portfolio is typically less than the weighted average of the standard deviations of the securities in the portfolio. Thus, diversification typically reduces the risk of a portfolio—as "the number of portfolio holdings increases, portfolio risk declines. . Systematic and Nonsystematic Risk: The riskiness of the portfolio generally declines as more stocks are added, because we are eliminating the nonsystematic risk, or company- specific risk. This is unique risk related to a particular company. However, the extent of the risk reduction depends upon the degree of correlation among the stocks. As a general rule, correlations among stocks, at least domestic stocks and particularly large domestic stocks, are positive, although less than 1.0. Adding more stocks will reduce risk at first, but no matter how many partially correlated stocks we add to the portfolio, we can riot eliminate all of the risk. Variability in a security's total returns that is directly associated with overall movements in the general market or economy is called systematic risk, or market risk, or non diversifiable risk. Virtually all securities have some systematic risk, whether bonds or stocks, because systematic risk directly 214 Investment Analysis & Portfolio Management encompasses interest rate risk, market risk, and inflation risk. After the non systematic risk is eliminated, what is left is the non diversifiable portion, or the market risk (systematic part). This part of the risk is inescapable, because no matter how well an investor diversifies, the risk of the overall market cannot he avoided. Investors can construct a diversified portfolio and eliminate part of the total risk, the diversifiable or non market, part. As more securities are added, the non systematic risk becomes smaller and smaller, and the total risk for the portfolio approaches its systematic risk. Since diversification cannot reduce systematic risk, total portfolio risk can be reduced no lower than the total risk of the market portfolio. Diversification can substantially reduce the unique risk of a portfolio. However, we cannot eliminate systematic risk. Clearly, market risk is critical to all investors. It plays a central role in asset-pricing, because it is the risk that investors can expect to be rewarded for taking. The Implications of the Markowitz Portfolio Model: The construction of optimal portfolios and the selection of the best portfolio for, an investor have implications for the pricing of financial assets. Part of the riskiness of the, average stock can be eliminated by holding a well-diversified portfolio. This means that part of the risk of the average stock can be eliminated and part cannot. Investors need to focus on that part of 'the risk that cannot be eliminated by diversification, because this is the risk that should be priced in the financial markets. The relevant risk of an individual stock is its contribution to the-riskiness of a well-diversified portfolio. The return that should be expected on the basis of this contribution can be estimated by the capital asset pricing model. Investment Analysis & Portfolio Management 0-444600 ASSET PRICING MODEL Capital Market Theory: Capital market theory is a positive theory in that it hypothesis how investors do behave rather than, how investors should behave, as, in the case of Modem Portfolio Theory (MPT). It is reasonable "to view capital market” theory; as an extension of portfolio theory, but it is important to understand that MPT is not based on the validity, or lack thereof, of capital market theory. The equilibrium model of interest to many investors is known as the capital asset pricing model, typically referred to as the CAPM. It allows us to measure the relevant risk of an individual security as well as to assess the relationship between relevant risk of and the returns expected from investing. The CAPM is attractive as an equilibrium model because of its simplicity and its implications. Because of serious challenges, to the model, however, alternatives have been developed. The primary alternative to the CAPM is arbitrage pricing theory, or APT, which allows for multiple sources of risk. Capital Theory Assumptions: Capital market theory involves a set of predictions concerning equilibrium expected return on risky assets. It typically is derived by making some simplifying assumptions in order to facilitate the analysis and help us to understand the arguments without fundamentally changing the predictions of asset pricing theory. Capital market theory builds on Markowitz portfolio theory to diversify his or; her portfolio, according to the Markowitz model, choosing a location on the efficient frontier that matches his or her return-risk references. Because of the complexity of the real world, additional assumptions; are made to make individual more alike. All investors can borrow or lend money at the risk-free rate of return. All investors have identical probability distributions for future rates of return; they have homogeneous expectations with respect to the three inputs of the portfolio model i.e. expected returns, the variance of returns, and the correlation matrix. Therefore, given a set of security prices and a risk-free rate, all investors use the same information to generate an efficient frontier. All investors have the same one-period time horizon. There are no transaction costs. There are no personal income taxes---investors are indifferent between capital gains and dividends. There is no inflation. There are many investors, and no single investor can affect the price of a stock through his or her buying and selling decisions. Investors are price takers and act as if prices are unaffected by their own trades. Capital markets are in equilibrium. These assumptions appear to be unrealistic and often disturb investors encountering capital market theory for the first time. However, the important issue is how well the theory predicts or describes reality, and not the realism of its assumptions. If CMT does a good job of explaining the returns on risky .assets, it is very useful, and the assumptions made in deriving the theory are of less importance. 09651900 216 Investment Analysis & Portfolio Management -635-444600 Most of these assumptions can be relaxed without significant effects on the CAPM or its implications in other words, the CAPM is robust. Although the results from such a relaxation of the assumptions may be less clear-cut and precise no significant damage is done. Many conclusions of the basic model still hold. Finally, most investors recognize that all of the assumptions of CMT are not unrealistic. For example, some institutional investors are tax exempt, and brokerage costs today, as percentage of the transaction, are quite small. Nor is it too unreasonable to as that for the, one-period horizon of the model, inflation may be fully (or mostly) anticipated and, therefore, not a major factor. The Equilibrium Return-Risk Tradeoff: Given the previous analysis, we can now derive some predictions concerning equilibrium expected returns and risk. The CAPM is an equilibrium model that encompasses two important relationships. The first, the capital market line specifies the equilibrium relationship between expected return and risk for efficient portfolios. The second, the security market line specifies the equilibrium relationship between expected return and systematic risk. It applies to individual securities as well as portfolios. The Capital Market Line: We now know that portfolio M is die tangency point to a straight line drawn front RF to the efficient frontier and that this straight line is the best obtainable efficient set line. All investors will hold portfolio M as their optimal risky portfolio, and all investors will be somewhere on this steepest; trade-off line between expected return and risk, because it represents, those combinations of risk-free investing / borrowing and portfolio M that yield the highest return obtainable for a given level of risk. The straight line usually referred to as the capital market line (CML, depicts the equilibrium conditions that prevail in the market for efficient portfolios consisting of the optimal portfolio of risky assets and die risk-free assets. All combinations of the risk-free asset and the risk portfolio M are on the CML, and, in equilibrium, all investors will end up ;with portfolios somewhere on the CML. The Market Portfolio: Portfolio M is called the market portfolio of risky securities. It is the highest point of tangency between RF and the efficient frontier and is the optimal risky portfolio. All investors would want to be on the optimal line RF-M-L, and, unless they invested 100 percent of their wealth in the risk-free asset, they would own portfolio M with some portion of their investable wealth or they would invest their own wealth plus borrowed funds in portfolio M. This portfolio is the optimal portfolio of risky assets. Why do all investors hold identical risky portfolios? Based on our assumptions above, all investors use the same Markowitz analysis on the same set of securities, have the same expected returns and covariance’s and have an identical time horizon Therefore, they will, arrive at the same optimal risky portfolio, arid it will be the market portfolio, designated M. -63590995400 217 Investment Analysis & Portfolio Management 0-444600 The Separation Theorem: We have established that each investor will hold combinations of the risk-free asset (either lending or borrowing) and the tangency portfolio from the efficient frontier, which is the market portfolio. Because we are assuming homogeneous expectations, in equilibrium all investors will determine the same tangency portfolio. Further, under the assumptions of CMT all investors agree on the risk-free rate. Borrowing and lending possibilities, combined with one portfolio of risky assets M, offer an investor whatever risk-expected return combination he or she seeks; that is, investors can be anywhere they choose on this line depending on their risk-return preferences. An investor could: Invest 100 percent of investable funds in the risk-free asset, providing an expected return of RF and zero risk. Invest 100 percent of investable funds in risky-asset portfolio-M, offering E (RM), with its risk ?M. Invest in any combination of return and risk between these two points; obtained by varying the proportion wRF invested in the risk-free asset. Invest more than 100 percent of investable funds in the risky-asset portfolio M by borrowing money at the rate RF, thereby increasing both the expected return and the risk beyond that offered by portfolio M. Different investors will choose different portfolios because of their risk preferences (they have different indifference curves), but they will choose the same combination of risky securities as denoted by the tangency point M. Investors will then borrow or lend to achieve various positions on the linear trade-off, between expected return and risk. Unlike the Markowitz analysis; it is not necessary to match each client's indifference curves with a particular efficient portfolio, because only one efficient portfolio is held by all investors. Rather each client will use his or her indifference curves to determine where along the new efficient frontier RF-M-L he or she should be; In effect, each client must determine how much of investable funds should be lend or borrowed at RF and how much should be invested in portfolio M. This result is referred to as a separation property. The Security Market Line: The capital market line depicts the risk-return trade-off in the financial markets in equilibrium. However, it applies only to efficient portfolios and cannot be used to-assess the equilibrium expected return for a single security. What about individual securities or inefficient portfolios? Under the CAPM all investors will hold the market portfolio, which is the bench-mark portfolio against which other portfolios are measured. How does an individual security contribute to the risk of the market portfolio? Investors should expect a risk premium for buying a risky asset such as a, stock. The greater the riskiness of that stock, the higher the risk premium should be. If investors hold well-diversified portfolios, they should be interested in portfolio risk rather than individual security risk. Different stocks will affect a well-diversified portfolio differently. The relevant risk for an individual stock is its contribution to the riskiness of a well-diversified portfolio. And the risk of a well-diversified portfolio is market risk, or systematic risk, 026161900 218 Investment Analysis & Portfolio Management which is non-diversifiable. Beta: We now know that investors should hold diversified portfolios to reduce the portfolio risk. When an investor adds a security to a portfolio what matters is the security's average covariance with the other securities in the portfolio. We also now know that under CMT all investors will hold the same portfolio of risky assets, the market portfolio. Therefore, the risk that matters when we consider any security is its covariance with the market portfolio. We could relate the expected return on a stock to its covariance with the market portfolio. However, it is more convenient to use a standardized measure of the systematic risk that cannot be avoided through diversification. Beta is a relative measure of risk-—the risk of an individual stock relative to the market portfolio of all stocks. If the security’s returns move, more (less) than, the market's returns as the latter changes, the security's returns have more (less) volatility (fluctuations in price) than those of the market. For example, a security whose returns rise or fall on average 15 percent when the market return rises or falls 10 percent is said to be an aggressive or volatile security. CAPM's Expected Return-Beta Relationship: The security market line (SML) is the CAPM specification of how risk and required rate of return for any asset security, or portfolio are related. This theory posits a linear relationship between an asset's risk and its required rate of return. This linear relationship, called the security market line (SML). Required rate of return is on the vertical axis and beta, the measure of risk, is on the horizontal axis. The slope of the line is the difference between the required rate of return on the market index and RF, the risk-free rate. The capital asset pricing model (CAPM) formally relates the expected rate of return for any security or portfolio with .the relevant risk measure. The CAPM's expected return-beta relationship is the most-often cited form of the relationship. Beta is the relevant measure of risk that cannot be diversified away in a portfolio of securities and, as such, is the measure that investors should consider in their portfolio management decision process. The CAPM in its expected return-beta relationship form is a simple but elegant statement. It says that the expected rate of return on an asset is a function of the two components of the required rate of return—the risk-free rate and the risk premium. Thus; ki = Risk-free rate + Risk premium = RF + ?i[E (R M) - RF] Where; ki = the required rate of return on asset i E (RM) = the expected rate of return on the market portfolio ?i = the beta coefficient for asset i This relationship provides an explicit measure of the risk premium. It is the product of the beta for a particular security i and the market risk premium, E (RM) - RF. Thus, Risk premium for security i = ?i(market risk premium) = ?i[E (RM) - RF] -63542227400 219 Investment Analysis & Portfolio Management 0-444600 The CAPM’s expected return-beta relationship is a simple but elegant statement about expected (required) return and risk for any security or portfolio. It formalizes the basis of investments which is that the greater the risk assumed, the greater the expected (required) return should be. This relationship states that an investor require (expects) a return on arisky asset equal to the return on a risk-free asset plus a risk premium and the greater the risk assumed, the greater the risk premium. 0784605900 ASSET PRICING MODEL Contd… Introduction of the Risk-Free Asset: The first assumption of CMT listed above is that investors can borrow and lend at the risk-free rate. Although the introduction of a risk-free asset appears to be a simple step to take in the evolution of portfolio and CMT, it is a very significant step. In-fact, it is the introduction of a risk-free asset that allows us to develop CMT from portfolio theory. With the introduction of risk-free asset, investors can now invest part of their wealth; in this asset and the remainder in any of the risky portfolios in the Markowitz efficient set. Lt allows Markowitz portfolio theory to be extended in such a way that the efficient frontier is completely changed, which in turn leads to a general theory for pricing assets under uncertainty. A risk-free asset can be defined as one, with a certain-to-be-earned expected return and a variance of return of zero. Since variance = 0, the nominal risk-free rate in each period will be equal to its expected value. Furthermore, the covariance between the risk-free asset and any risky asset i will be zero. The true risk-free asset is best thought of as a Treasury Security, which has no risk of default, with a maturity matching /the holding period of the investor. In this case, the amount of money to be received at the end of, the holding period is known with certainty at the beginning of the period. The Treasury bill typically is taken to be the risk-free asset, and its rate of return is referred to here as RF. Risk-Free Borrowing and Lending: Assume; that the efficient frontier, has been derived by an investor. The arc AB delineates the efficient set of portfolios of risky assets. We now introduce a risk-free asset with return RF and ?= 0. What if we extend this analysis to allow investors to borrow money? The investors no longer restricted to his or, her wealth when investing in risky assets. Technically, we are show selling the riskless asset. One way to accomplish this borrowing is to buy stocks on margin, which has a current initial margin requirement of 50 percent. We will assume that investors can also borrow at the risk-free rate RF. This assumption can be removed without changing the basic arguments. Borrowing additional investable funds and investing them together with the investor's own wealth allows investors to seek higher, expected returns while assuming greater risk. These borrowed funds can be used to lever the portfolio position beyond point M, the point of tangency between the straight line emanating from RF and the efficient frontier AB. Estimating the SML: To implement the SML approach described here an investor needs estimates of the return on the risk-free asset (RF), the expected return on the market index, and the beta for an individual security. How difficult are these to obtain? The RF should, be the easiest of the three variables, to obtain. In estimating RF, the investor -63512382400 221 Investment Analysis & Portfolio Management can use as a proxy the return on Treasury bills for the coining period (e.g., a year). Estimating the market return is more difficult, because the expected return for the market index is not observable. Furthermore, several different market indexes could be used. Estimates of the market return could be derived from a study of previous market returns. Alternatively, probability estimates of market returns could be made, and the expected value calculated. This would provide an estimate of both the expected return and the standard deviation for the market. Finally, it is necessary to estimate the betas for individual securities. This is a crucial part of the CAPM estimation process. The estimates of RF and the expected return on the market are the same for each security being evaluated. Only beta is unique, bringing together the investor's expectations of returns for the stock with those for the market. Beta is the only company-specific factor in the CAPM; therefore, risk is the only asset-specific forecast that must be made in the CAPM. Estimating Beta: A less restrictive form of the single-index model is known as the market model. This model is identical to the Single-index model except that the assumption of the error terms for different securities being uncorrelated is not made. The market model equation is the same as for the single-index model: Ri = ?i+ ?iRM + ?i Where; Ri = the return (TR) on security i RM = the return (TR) on the market index: ?i = the intercept term ?i = the slope term ?i = the random residual error; The market model produces an estimate of return for any stock. To estimate the market model, the TRs for stock i can be regressed on the corresponding TRs, for the market index. Estimates will be obtained of ?i (the constant return on security i that is earned regardless of the level of market returns) and ?i, (the slope coefficient that indicates the expected increase in a security's return for a 1 -percent, increase in market return). This is how the estimate of a stock's beta is often derived. Arbitrage Pricing Theory: An equilibrium theory of expected returns for securities involving few assumptions about investor preferences The CAPM is not the only model of security pricing. Another model that has received attention is based on arbitrage pricing theory (APT) as developed by Ross and enhanced by others. In recent years, APT has emerged as an alternative theory-of asset pricing to the CAPM. Its appeal is that it is more general than the1 CAPM, with less restrictive assumptions. However, like the CAPM, it has limitations, and like the CAPM, it is not the final word in asset pricing. 026415900 222 Investment Analysis & Portfolio Management -635-444600 Similar to the CAPM, or any other asset-pricing model, APT posits a relationship between expected return and risk. It does so, however, using different assumptions and procedures. Very importantly, APT is not critically dependent on an underlying market portfolio as is the CAPM, which predicts that only market risk influences expected returns. Instead, APT recognizes that several types of risk may affect security returns. APT is based on the law of one price, which states that two otherwise identical assets cannot sell at different prices. APT assumes that asset returns are linearly related to a set of indexes, where each index represents a factor that influences the return on an asset. Market participants develop expectations about the sensitivities of assets to the factor. They buy and sell securities so that, given, the law of one price, securities affected equal by the same factors will have equal expected returns. This buying and selling is the arbitrage process, which determines the prices of securities. APT states that equilibrium market prices will adjust to eliminate any arbitrage opportunities, which refer to situations where a zero investment portfolio can be constructing that, will yield a risk-free profit. If arbitrage opportunities arise, a relatively few investors can act to restore equilibrium. Unlike the CAPM, APT does not assume: A single-period investment horizon The absence of taxes Borrowing and lending at the rate RF Investors select portfolios on the basis of expected return and variance APT, like the CAPM, does assume: Investors have homogeneous beliefs Investors are risk-averse utility maximizers Markets are perfect Returns are generated by a factor model Factor Model used to depict the behavior of security prices by identifying major factors in the economy that affect large numbers of securities A factor model is based on the view that there are underlying risk factor's that affect realized and expected security returns. These risk factors represent broad economic forces and hot company-specific characteristics, and by definition they represent the element of surprise in the risk factor—the difference between the actual value for the factor and its expected value. The factors must possess three characteristics: Each risk factor must have a pervasive influence on stock returns. Firm-specific events are not APT risk factors. These risk factors must influence expected return, which means they must have nonzero prices. This issue must be determined empirically, by statistically analyzing stock returns to see which factors pervasively affect returns. At the beginning of each period, the risk factors must be unpredictable to the market as a whole, this raises an important point. In our example above, we used in flatten and the economy's output as the two factors affecting portfolio returns. The rate of inflation is not an APT risk factor, because it is at least partially predictable. In an -6359016900 223 Investment Analysis & Portfolio Management 0-444600 economy with reasonable growth where the quarterly rate of inflation has averaged 3 percent on an annual basis, we can reasonably assume that next quarter inflation rate is not going to be 10 percent. On the other hand, unexpected inflation—the difference between actual inflation, and expected inflation—is an APT risk factor. By definition, it cannot be predicted, since it is unexpected. What really matters are the deviations of the factors from their expected values. For example, if the expected value of inflation is 5 percent and the actual rate of inflation for a period is only 4 percent, this, 1-percent deviation will affect the actual return for the period. Portfolio Management: Portfolio management involves a series of decisions and actions that must be made by every investor whether an individual or institution. Portfolios must be managed whether investors follow a passive approach or ail active approach to selecting and holding their financial assets. As we saw when we examined portfolio theory, the relationships among the various investment alternatives that are held as a portfolio must be considered if an investor is to hold an optimal portfolio, and achieve his or her investment objectives. Portfolio management can be thought of as a process. Having the process clearly in mind is very important, allowing investors to proceed in an orderly manner. In this chapter, we outline the portfolio management process, making it clear that a logical and orderly flow does exist. This process can be applied to each investor and by any investment manager. Details may vary from client to client, but the process remains the same. . Portfolio Management as a Process: The portfolio management process has been described by Maginn and Tuttle in a book that forms the basis for portfolio management as envisioned by the Association for Investment Management and Research (AIMR), and advocated in its curriculum for the Chartered Financial Analyst (CFA) designation. This is an important development because of its contrast with the past, where portfolio management was treated on an ad hoc basis, matching investors with portfolios on an individual basis. Portfolio management should be structured so that any investment organization can carry it out in an effective and timely manner without serious omissions. Maginn and Tuttle emphasize that portfolio management is a process, integrating a set of activities in a logical and orderly manner. Given the feedback loops and monitoring that is included; the process is both continuous and systematic. It is a dynamic and flexible concept, and extends to all portfolio investments, including real estate, gold, and other real assets. The portfolio management process extends to all types of investment organizations and investment styles. In fact, Maginn and Tuttle specifically avoid advocating how the process should be organized by money management companies or others, who should make the decisions, and so forth. Each investment management organization, should decide for itself how best to carry out its activities consistent with viewing portfolio management as a process. Having structured portfolio management as a process, any portfolio manager can 028066900 224 Investment Analysis & Portfolio Management -635-444600 execute the necessary decisions for an investor. The process provides a framework and a control over the diverse activities involved, and allows every investor, an individual or institution, to be accommodated in a systematic, orderly manner. As outlined by Maginn and Tuttle, portfolio management is an ongoing process by which: Objectives, constraints, and preferences are identified for each investor. This leads to the development of an explicit investment policy statement which is used to guide the money management process. Capital market expectations for the economy, industries and sectors, and individual securities are considered arid quantified. Strategies are developed arid implemented. This involves asset allocation, portfolio optimization, and selection of securities. Portfolio factors are monitored and responses are; made as investor objectives and constraints and/or market expectations change. The portfolio is rebalanced as necessary by repeating the asset allocation, portfolio strategy, and security selection steps. Portfolio performance is measured and evaluated to ensure attainment of the investor objectives. Individual Investors Vs Institutional Investors: Significant differences exist among investors as to objectives, constraints, and preferences. We are primarily interested here in the viewpoint of the individual investor, but the basic investment management process applies to all investors, individuals, and institutions. Furthermore, individuals are often the beneficiaries of the activities of institutional investors, and an understanding of how institutional investors fit into the investment management process is desirable. A major difference between the two occurs with regard to time horizon, because institutional investors are often thought of on a perpetual basis, but this concept has no meaning when applied to individual investors. For individual investors, it is often useful to think of a life- cycle approach, as people go from the beginning of their careers to retirement. This approach is less useful for institutional investors, because they typically maintain a relatively constant profile across time. Kaiser has summarized the differences between individual investors and institutional investors as follows: Individuals define risk as ''losing money”, whereas institutions use approach, typically defining risk in terms of standard deviation. Individuals can be characterized by their personalities, whereas for institutions, we consider the investment characteristics of those with a beneficial interest in the portfolios managed by the institutions. Goals are a key part of what individual investing is all about, along with their assets, whereas for institutions, we can be more precise as to their total package of assets and liabilities. Individuals have great freedom in what they can do with regard to investing whereas institutions are subject to numerous legal and regulatory constraints. Taxes often are a very important consideration for individual investors, whereas many institutions, such as pension funds, are free of such considerations. -63526225400 Investment Analysis & Portfolio Management The implications of all of this for the investment management process are as follows: For individual investors: Because each individual's financial profile is different, aninvestment policy for an individual investor must incorporate that investor's unique factors. In effect, preferences are self-imposed constraints. For institutional investors: Given the increased complexity in managing institu-tional portfolios, it is critical to establish a well defined and effective policy. Such a policy must clearly delineate the objectives being sought, the institutional investor's risk tolerance, and the investment Constraints and preferences under which it must operate. The primary reason for establishing a long term investment policy for institutional investors is two fold: It prevents arbitrary revisions of a soundly designed investment policy. It helps the portfolio manager to plan and execute on a long term basis and resist short term pressures that could derail the plan.