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Modern Portfolio Theory 1 Page 1 of 129 MODERN PORTFOLIO THEORY 1. Modern Portfolio Theory § Background o The Primary principle upon which Modern Portfolio Theory is based (MPT) is the RANDOM WALK HYPOTHESIS which sates that the movement of asset prices follows an Unpredictable path: the path as a TREND that is based on the long-run nominal growth of corporate earnings per share, but fluctuations around the trend are random. There are 3 Forms of the Hypothesis: § Weak Form: Security Prices reflect ALL information about price & trading behavior in the market. Thus, analyzing the security’s or market’s data contains NO information that enables predictions on future price to be made. Thus, CHARTING & TECHNICAL ANALYSIS do NOT Work § Semi-strong Form: The Markets react quickly to new public information, whether it relates to trading activity (weak form) or fundamental earnings. Studying historical information, thus, is not too relevant and won’t enable superior results.. § Strong Form: All relevant information knowable about a company is already imbedded in the price of a security Only new information produces systematic (non-random) price changes. Since new information enters the marketplace randomly, asset price movements are random. § Efficient Pricing Structure o The EFFICIENT MARKET THEORY states that asset prices are set in the market by the MARGINAL Buyer & Seller. These buyers & sellers are motivated by various factors; both rational & irrational. The market is not efficient in the sense that it prices securities correctly, but it is efficient in the sense that the market is a reasonable speculation. Efficiency means there is even odds on winning or losing. § Return as a Random Variable o When Security prices are determined within an efficient market structure, a PROBABILITY DISTRIBUTION can be used to describe them. If the normal probability distribution is assumed as an appropriate description of the return function, then one needs to know 2 parameters § Expected Return: the return around which the probability distribution is centered; the expected value or mean of the probability distribution of returns § Standard Deviation: The parameter which describes the width & shape of the distribution of possible returns o Measuring Risk: Risk exists when more than one outcome is possible from an investment. It can be defined as the probability that the ACTUAL RETURN will be SIGNIFICANTLY DIFFERENT from the EXPECTED RETURN. With small standard deviations, there is little chance that the actual return will be significantly different from the expected return. With large standard deviations, there is a good chance that the actual return will be significantly different from the expected return. The SOURCES of Risk are Business risk, financial risk, Liquidity risk, and exchange rate/country risk (for foreign stocks). The Variance and Standard Deviations of Returns are MEASURES of Risk. In reality, the distribution of returns is probably NOT NORMAL (probably log-normal, and Modern Portfolio Theory CFA Level III © Gillsie June, 1999 Page 2 of 129 should be stated on a continuously compounded basis rather than on annualized compounded basis) § Alternative Definitions of Risk o While σ is used mostly in the CFA, there are other ways to Measure Risk § The RANGE of Returns, which is naïve as only extreme values are considered § The SEMI-VARIANCE of returns, which is the variance of ONLY those returns below zero (or some min. target return) § RELATIVE LOWER PARTIAL MOMENTS of the 2nd Order or higher. These count only the probability that an asset’s return will fall BELOW some benchmark return as a risk event and penalize Large return shortfalls from the benchmark return proportionately more than small return shortfalls § Alternative Measures of Investment Risk o σ is the conventional way of measuring risk. But there are several problems with this measure § Variance Measures UNCERTAINTY, but that is not the same thing as risk. For example, if 2 investments have the same level of variance, however one has a variance that is always significantly above the expected return (positive bias), then that should not be considered as risky as the equally variable security that’s returns vary equally around the expected return. § Variance is a SQUARED TERM. Thus, it treats any deviation above the mean return as being as risky as any deviation below the mean return. This says that Outperforming the expected return is just as risky as under- performing it § Using variance as a measure of risk is only applicable to distributions that are NORMAL. When distributions are SKEWED, more parameters should be used. Whenever a portfolio contains options, it’s returns will be skewed § For Variance to be a meaningful measure of risk, it must be assumed that the distributions of returns is STATIONARY, meaning that the mean & variance of the returns remain constant over time. This is not probable o Due to these shortcomings MARKOWITZ suggested that the variance not be used to measure the risk of a portfolio. He suggested SEMI-VARIANCE be employed. But, when he wrote his seminal work, computing was not available, so he ASSUMED investment returns were normally distributed and that variance could be used as the measure of risk. But, this is not realistic today. § Characteristics of a Good Measure of Investment Risk o Should Define Risk as the PROBABILITY of Producing a Return that is LESS than SOME Minimum Objective which the investor wishes to obtain. Variance does not do this. Variance measure doing differently from expected à different can be better OR worse, and the expected return could be higher or lower than the investor’s minimum objective o Should assess both the PROBABILITY that the actual return will be less than the min. return objective and also the SEVERITY of the Shortfall (like insurance risk: Frequency & Severity) Modern Portfolio Theory CFA Level III © Gillsie June, 1999 Page 3 of 129 o Should recognize that Investors are RISK AVERSE. Thus, their utility functions are NTO linear, they are quadratic or logarithmic. (investors are not twice as unhappy to lose 20% as 10%; probably 4 times as unhappy) § Devising a Good Measure of Investment Risk o The FIRST Criterion of a good measure of investment risk is that it should be a RELATIVE measure of risk (i.e., it should define risk as the probability that a portfolio’s return will fall below some BENCHMARK RETURN RB. This Benchmark return is the minimum return objective of the investor. Some possible Benchmarks include à § RB = 0. Risk is when a portfolio’s return is zero or less. § RB = I. Risk that the portfolio will grow as fast as inflation. If not, the investor loses purchasing power & wealth § RB = RM. Risk that the portfolio under-performs the Market § RB = RAvg. Portfolio Manager. Risk of under-performing peers § RB = Actuarial Assumptions (like Pensions & Insurance) o When Risk is Defined that the Portfolio’s Actual Return (RP) will fall below the Benchmark (RB), There are a few ways to QUANTIFY that risk in a Single Summary Measure § VAR – Value At Risk Analysis. When disaster strikes, what’s the most that can be lost. (but, this measure does not consider the probability of the worst possible outcome) § Relative First Order Lower Partial Moment – measures the expected shortfall below the Benchmark. RLPM1 = Σ(RP – RB) * P(RP - RB) This works for both normal & non-normal distributions. But, it does assume that the investor utility functions are linear, rather than quadratic or logarithmic. § Relative Semivariance – aka the RELATIVE SECOND-ORDER LOWER PARTIAL MOMENT. This formulation measures risk as a shortfall from the benchmark return with ONLY UNDERPERFORMANCE being construed as Risk and this causes the disutility of the portfolio to rise with the square of the shortfall 2 RLPB2 = Σ(RP – RB) * P(RP – RB) Note: only use for levels of RP where (RP – RB ≤ 0) Van Harlow’s Study comparing stock/bond portfolios generated using this measure of risk shows that this measure of risk produces Allocations that are slightly more concentrated in bonds than portfolios constructed in the traditional manner. Thus, conventionally generated portfolios seem to have more built-in risk than assumed, and in times of crises, perform worse than expected. § Higher Order Relative Lower Partial Moments – These may produce even better results than relative semivariance. This is because relative semivariance measures assume that investor utility functions are quadratic. But, there are some indications that Investor Utility functions are not: • A quadratic utility function implies that the second derivative of the utility function will always be negative. This means wealthier Modern Portfolio Theory CFA Level III © Gillsie June, 1999 Page 4 of 129 investors prefer less risk than poor investors. This has not been proved empirically. • Some empirical evidence show investor utility functions are DISCONTINUOUS; i.e., risk aversion increases sharply & discontinuously at retirement. • If the utility function is NOT quadratic, the relative semivariance risk measure might be too simplistic. Perhaps a skewed distribution can be described using MEAN, VARIANCE, & SKEWNESS (which is the 3rd Moment). • RELATIVE SEMISKEWNESS à RELATIVE 3rd-ORDER LOWER PARITAL MOMENT 3 RLPM3 = Σ(RP – RB) * P(RP – RB) Note: only use for levels of RP where (RP – RB ≤ 0) This measure of Risk is similar to semivariance, but it makes adverse outcomes even more unfavorable (due to the cubing). • In addition to being Skewed, return distributions could be PLATYKURTIC or LEPTOKURTIC or MESOKURTIC (appear symmetrical and normal, but not). • When try to use Relative 4th Order Lower Partial Moment, will find it almost impossible to achieve an Optimal Portfolio Mix § Non-Constant Benchmark Returns o In the above analysis, RB, was assumed to be constant. But, benchmark returns are usually dynamic. But, when RB is fluid, the modeling process becomes more complex because both RP & RB would be probability distributions.
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