Copyright © 2000 by Harcourt, Inc. All rights reserved. Introduction In the next three chapters (and part of Ch. 22, together with Chs. 2 – 5 of Haugen), we will briefly examine different aspects of quantitative investment management, including: Bringing risk and return into the picture of investment management – Markowitz optimization and enhanced index portfolios Modeling risk and return – CAPM, APT, and Behavioural models Estimating risk and return – the Single-Index Model (SIM) and risk and expected return factor models
Version 1.2 Copyright © 2000 by Harcourt, Inc. All rights reserved. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department Harcourt, Inc Sea Harbor Drive Orlando, Florida Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Sixth Edition by Frank K. Reilly & Keith C. Brown Chapter 8
Copyright © 2000 by Harcourt, Inc. All rights reserved. Background Assumptions As an investor you want to maximize the returns for a given level of risk. Your portfolio includes all of your assets and liabilities The relationship between the returns for assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments.
Copyright © 2000 by Harcourt, Inc. All rights reserved. Risk Aversion Given a choice between two assets with equal rates of return, risk averse investors will select the asset with the lower level of risk. –Investors prefer a sure thing to a “fair game” –$1 preferred to 50:50 chance of $2 or nothing Risk aversion is a consequence of decreasing marginal utility (e.g., 2 nd BMW is less valuable to you than the 1 st one; next two cheeseburgers give you less value than the prior two did). All investors are assumed to be risk-averse.
Copyright © 2000 by Harcourt, Inc. All rights reserved. Evidence That Investors are Risk-Averse Many investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of loss Yield on bonds increases with risk classifications from AAA to AA to A….
Copyright © 2000 by Harcourt, Inc. All rights reserved. Are all investors risk-averse? Risk preference may have to do with amount of money involved - risking small amounts with expected, though small, losses (e.g., lottery tickets), but insuring against large losses But, outcomes aren’t comparable – winning lottery tickets can put purchaser into a whole new class of consumption (e.g., not just a 2 nd car, but 1 st Lamborghini vs. 1 st Ford Focus) Also, risk aversion doesn’t mean you don’t take on risks, but that you are willing to take on risks only with the expectation of a potential higher return
Copyright © 2000 by Harcourt, Inc. All rights reserved. Definition of Risk 1. Uncertainty of future outcomes or 2. Probability of an adverse outcome We will consider several measures of risk that are used in developing portfolio theory
Copyright © 2000 by Harcourt, Inc. All rights reserved. Alternative Measures of Risk Range of returns Returns below expectations –Semivariance - measure expected returns below some target –Intended to minimize the damage Variance or standard deviation of returns –Generally works as well as the other measures, but mathematically more convenient to use –Gives us information about the uncertainty of returns, and can be used to determine the probability of adverse outcomes
Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Portfolio Theory Quantifies risk Derives the expected rate of return for a portfolio of assets and an expected risk measure Shows the variance of the rate of return is a meaningful measure of portfolio risk Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio Provides both: –the foundation for Modern Finance –a key tool for Haugen’s New Finance
Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 1.Investors view potential investments in terms of the distributions of their returns over some given holding period
Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 2. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth. –I.e., investors like higher returns, but they are risk-averse in seeking those returns
Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 3. Investors measure the risk of their portfolios in terms of the variance of the portfolio’s returns. –I.e., variance is the key measure of risk
Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 4. Investors base decisions solely on expected return and risk, due to either: –Investors’ utility curves are functions of only expected return and the variance (or standard deviation) of returns. –Stocks’ returns are normally distributed or follow some other distribution that is fully described by mean and variance.
Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory 5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk.
Copyright © 2000 by Harcourt, Inc. All rights reserved. Assumptions of Markowitz Portfolio Theory Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.
Copyright © 2000 by Harcourt, Inc. All rights reserved. Expected Rates of Return Individual risky asset –Sum of probability times possible rate of return Portfolio –Weighted average of expected rates of return for the individual investments in the portfolio
Copyright © 2000 by Harcourt, Inc. All rights reserved. Computation of Expected Return for an Individual Risky Investment Table 8.1
Copyright © 2000 by Harcourt, Inc. All rights reserved. Computation of the Expected Return for a Portfolio of Risky Assets Table 8.2
Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment Standard deviation is the square root of the variance Variance is a measure of the variation of possible rates of return R i, from the expected rate of return [E(R i )]
Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment where P i is the probability of the possible rate of return, R i
Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment Standard Deviation
Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for an Individual Investment Table 8.3 Variance ( 2 ) =.0050 Standard Deviation ( ) =.02236
Copyright © 2000 by Harcourt, Inc. All rights reserved. But what if you don’t know the true probability distribution (including the true probabilities for each of the possible outcomes) for the stocks you are studying? One alternative – use the ex post, time series distribution –Assumes past is representative of the future, with each past period’s outcome an equally likely event (probability = 1 / T) –Naïve, limited information approach –Nonetheless, can be complicated – many choices and tradeoffs must be made to implement this approach (will be discussed more later)
Copyright © 2000 by Harcourt, Inc. All rights reserved. Ex-Post Estimation of Mean, Variance, and Covariance Computation of Monthly Rates of Return Table 8.4
Copyright © 2000 by Harcourt, Inc. All rights reserved. Time Series Returns for Coca-Cola: 1998 Figure 8.1
Copyright © 2000 by Harcourt, Inc. All rights reserved. Times Series Returns for Exxon: 1998 Figure 8.2
Copyright © 2000 by Harcourt, Inc. All rights reserved. Times Series Returns for Coca-Cola and Exxon: 1998
Copyright © 2000 by Harcourt, Inc. All rights reserved. Juxtaposing these two plots on one graph helps to illustrate why portfolio management isn’t simply about selecting a collection of individually good stocks Co-movements between the stocks in the portfolio can have important implications for the portfolio as a whole One measure of such co-movements is covariance, or its standardized analogue, correlation …
Copyright © 2000 by Harcourt, Inc. All rights reserved. Variance (Standard Deviation) of Returns for a Portfolio For two assets, i and j, the covariance of rates of return is defined as: Cov ij = E{[R i - E(R i )][R j - E(R j )]} = s [(R is - E(R i ))(R js - E(R j ))] P s, where s refers to the possible state of the economy
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation of Covariance of Returns for Coca-Cola and Exxon: 1998 Table 6.5
Copyright © 2000 by Harcourt, Inc. All rights reserved. Scatter Plot of Monthly Returns for Coca-Cola and Exxon: 1998 Figure 8.3
Copyright © 2000 by Harcourt, Inc. All rights reserved. Covariance and Correlation Covariance is an absolute measure of co- movement, correlation is a relative or standardized measure of co-movement Correlation coefficient varies from -1 to +1
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation of Standard Deviation of Returns for Coca-Cola and Exxon: 1998 Table 8.6
Copyright © 2000 by Harcourt, Inc. All rights reserved. Parameters vs. Estimates No one knows the true values for the expected return and variance and covariance of returns These must be estimated from the available data The most basic way to estimate these is the naïve or unconditional estimate using the sample mean, sample variance, and sample covariance from a time series sample of stock returns Typical time series used: –Last 60 months’ worth of monthly returns –Last 3 years’ worth of weekly returns
Copyright © 2000 by Harcourt, Inc. All rights reserved. Sample Estimates Sample Mean: Sample Variance: Sample Covariance: =
Copyright © 2000 by Harcourt, Inc. All rights reserved. Now, the key equation for Markowitz portfolio optimization Takes us up from the level of the individual stock to the level of the portfolio as a whole
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Standard Deviation Formula
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Standard Deviation Calculation Any asset of a portfolio may be described by two characteristics: –The expected rate of return –The expected standard deviations of returns The correlation, measured by covariance, affects the portfolio standard deviation Low correlation reduces portfolio risk while not affecting the expected return
Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining Stocks with Different Returns and Risk Case Correlation Coefficient Covariance a b c d e
Copyright © 2000 by Harcourt, Inc. All rights reserved. Combining Stocks with Different Returns and Risk Assets may differ in expected rates of return and individual standard deviations Negative correlation reduces portfolio risk If correlation = -1.0 between two stocks, portfolio can be constructed with zero risk (s.d. = 0) and certain return
Copyright © 2000 by Harcourt, Inc. All rights reserved. Constant Correlation with Changing Weights 1.10 r ij =
Copyright © 2000 by Harcourt, Inc. All rights reserved. Constant Correlation with Changing Weights
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk- return along a line between either single asset
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = f g h i j k 1 2 With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = R ij = f g h i j k 1 2 With correlated assets it is possible to create a two asset portfolio between the first two curves
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = R ij = R ij = f g h i j k 1 2 With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset
Copyright © 2000 by Harcourt, Inc. All rights reserved. Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = R ij = R ij = f g h i j k 1 2 With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk R ij = Figure 8.7
Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Optimization and Efficient Diversification Use info. on E(R i )’s and ij ’s to obtain maximum benefits from diversification Q: What proportion to invest in each asset?
Copyright © 2000 by Harcourt, Inc. All rights reserved. Markowitz Model Modern Portfolio Theory (MPT) Minimize portfolio risk: subject to (a) minimum required return and (b) full investment: yields the “Efficient Frontier”
Copyright © 2000 by Harcourt, Inc. All rights reserved. The Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Frontier will be portfolios of investments rather than individual securities –Exceptions being the asset with the highest return and the asset with the lowest risk
Copyright © 2000 by Harcourt, Inc. All rights reserved. Efficient Frontier for Alternative Portfolios Efficient Frontier A B C Figure 8.9 E(R) Standard Deviation of Return
Copyright © 2000 by Harcourt, Inc. All rights reserved. The Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The slope of the efficient frontier curve decreases steadily as you move upward These two interactions will determine the particular portfolio selected by an individual investor
Copyright © 2000 by Harcourt, Inc. All rights reserved. The Efficient Frontier and Investor Utility The optimal portfolio has the highest utility for a given investor It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility
Copyright © 2000 by Harcourt, Inc. All rights reserved. Selecting an Optimal Risky Portfolio X Y U3U3 U2U2 U1U1 U 3’ U 2’ U 1’ Figure 8.10
Copyright © 2000 by Harcourt, Inc. All rights reserved. Use of MPT in Practice The most common use of Markowitz optimization is in conjunction with Passive (Index or Benchmark) Portfolio Management Goal is to optimize the portfolio in order to beat the benchmark by enough basis points on a risk- adjusted basis to cover the fund’s administrative and trading expenses (see Ch. 22)
Copyright © 2000 by Harcourt, Inc. All rights reserved. An Overview of Passive Equity Portfolio Management Strategies Replicate the performance of an index May slightly underperform the target index due to fees and commissions Costs of active management (1 to 2 percent) are hard to overcome in risk-adjusted performance Many different market indexes are used for tracking portfolios
Copyright © 2000 by Harcourt, Inc. All rights reserved. Passive Equity Portfolio Management Techniques Full replication Sampling Quadratic optimization or programming
Copyright © 2000 by Harcourt, Inc. All rights reserved. Full Replication All securities in the index are purchased in proportion to weights in the index This helps ensure close tracking Increases transaction costs, particularly with dividend reinvestment
Copyright © 2000 by Harcourt, Inc. All rights reserved. Sampling Buys a representative sample of stocks in the benchmark index according to their weights in the index Fewer stocks means lower commissions Reinvestment of dividends is less difficult Will not track the index as closely, so there will be some tracking error Frequently used in conjunction with quadratic optimization (see below)
Copyright © 2000 by Harcourt, Inc. All rights reserved. Expected Tracking Error Between the S&P 500 Index and Portfolio Samples of Less Than 500 Stocks Figure Expected Tracking Error (Percent) Number of Stocks
Copyright © 2000 by Harcourt, Inc. All rights reserved. Quadratic Optimization (or programming techniques) Historical information on price changes and correlations between securities are input into a computer program (an optimization program) that determines the composition of a portfolio that will maximize the portfolio’s excess return over the benchmark ( ) while minimizing its tracking error with the benchmark ( ) –Variation of Markowitz Portfolio Theory, but … –Rather than maximize E(R) while minimizing , –Maximize while minimizing
Copyright © 2000 by Harcourt, Inc. All rights reserved. Efficient Frontier for Enhanced/Optimized Index Funds Efficient Frontier pp Tracking Error ( ) pp 0
Copyright © 2000 by Harcourt, Inc. All rights reserved. Quadratic Optimization (or programming techniques) This is the application for which Markowitz optimization is most frequently used in practice Suffers from the same problems as other applications of Markowitz optimization, such as: –Relies on historical correlations, which may change over time, leading to failure to track the index –Also, still need to use some type of factor model to provide structure to the correlations and thereby reduce the number of elements that must be estimated
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation Issues Results of portfolio allocation depend on accurate statistical inputs Estimates of –Expected returns (n) –Standard deviation (n) –Correlation coefficient (n(n-1)/2) Among entire set of stocks (assuming 60 observations per stock): With 100 stocks, –6,000 data points & 4,950 correlation estimates With 500 stocks, –30,000 data points & 124,950 correlation estimates
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation Issues Large number of quantities to be estimated means that estimation error is a significant potential problem, unless valid simplifying assumptions can be applied Typical simplifying assumption is the use of a risk-factor model to describe the variances and covariances of the the stock returns (see also Ch. 3 of Haugen) Most basic model is the single-index model (SIM) –Time series analogue of CAPM –Also used in the estimation of beta
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation Issues With the assumption that stock returns can be described by a single index model, such as the Market Model, the number of correlations required reduces to one plus the number of assets, or (n+1) Single index market model: b i = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market R m = the returns for the aggregate stock market
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation Issues Under the assumptions of the Market Model: Covariance of returns of different stocks is a function of their betas Total variance of the returns for a single risky security is equal to market + idiosyncratic variance
Copyright © 2000 by Harcourt, Inc. All rights reserved. Estimation Issues In practice, quantitative portfolio managers generally use risk-factor models that include multiple risk factors (generally APT factors – see chapters 9 & 10) to model the correlations between the stocks in the portfolio.
Copyright © 2000 by Harcourt, Inc. All rights reserved. The Internet Investments Online
Copyright © 2000 by Harcourt, Inc. All rights reserved. Future topics Modeling risk and return CAPM & APT – models, testing, and extensions Estimating risk and return References: –Reilly, Chs. 9 – 10 –Haugen, Chs. 2 – 5