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Chapter 12. Risk and Diversification Learning Objectives Risk aversion Investment implications of risk aversion Standard deviation as a measure of risk for individual securities and portfolios Security correlation and portfolio risk Benefits of diversification Efficient diversification and modern portfolio theory 1. Risk aversion Risk is the danger of possible loss People need an incentive to accept risk Utility of wealth For any amount of wealth, the total utility of that wealth is measured by the height (in utils.) of the curve in Exhibit 12.1 The marginal utility of wealth is measured by the slope of the curve at any point on the wealth axis Utility of wealth function for a risk-averse investor Ex 12.1 Risk-neutral investor utility function increases at a constant rate Risk-taker investor utility function increases at an increasing rate Utility of wealth function for risk-neutral investors and risk takers Ex 12.2 Risk aversion and expected returns Higher risk should expect higher returns T-bills average 3.7 %, range 0 to 5% S&P 500 average 12.3 %, range -10% to +30% Relative risk aversion Some investors are more risk averse than others Risk aversion affects choices of securities 2. Measuring risk and return: individual securities Measuring returns - one year return: Investors are generally most concerned with the holding period return that will occur in the future, rather than the holding period return which has just occurred One way to estimate the price expected at the end of a future holding period is to identify the price which will exist under alternative assumptions about the future The return expected for each state of the world is determined by the stock price expected for that state of the world (Dividends are assumed to be constant for all states of the world) The expected return on security i during the holding period which begins today and ends at t+1 is the weighted average of the returns expected in each possible state of the world Ex-ante returns - predicted returns Ex-post returns - historical returns Standard Deviation as a measure of risk statistical measure of dispersion of distribution about a mean 67% fall within one standard deviation 95% lies within two standard deviations Since risk is the difference between the expected return on a security, and each of the possible future returns which may occur, it can be measured by the variance of expected returns Variance is an accurate measure of risk when the possible future returns are normally distributed around the expected future return The range of returns, and the number of negative outcomes are non-parametric risk measures which are not influenced by the distribution of future returns A more straight forward way to measure risk and expected future returns is to compute the mean and the variance from a sample of ex-post returns, and make the assumption that the future will look like the past The coefficient of variation (CV) is usually computed from the ex-post mean and variance of an historical sample of returns The coefficient of variation is a useful estimate of the risk to reward ratio which can be expected to accrue to the owners of a given security 3. Measuring risk and return: security selection Stocks can be grouped on the basis of either the expected return or the average risk Within each group, the stock which has the lowest CV is the stock which provides the highest reward per unit of risk in the group Mean Variance dominance is a property of the stock with the lowest CV in a group of stocks with identical risk or return 4. Portfolio risk and return In general, the purchase of a single stock is a speculation, rather than an investment The owner of single security hopes to profit from an increase in the price of the security during the near term Profits and losses derived from near term changes in the price of a single asset are speculative in nature, and do not constitute investment returns Investing involves buying and holding a portfolio of securities, expecting to profit in the long term from the secular price trend of the market Portfolio diversification can reduce the risk an investor must bear without reducing the return an investor can expect to earn The expected return of a portfolio is a weighted average of the expected returns of each of the securities in the portfolio: The weights are equal to the percentage of the portfolio’s value which is invested in each security The variance of a portfolio is more complex; it is a weighted average of the variances of each security and the correlation between each pair of securities The covariance of historical returns from any pair of securities is measured by the following formula: Notice the similarity between the equation for the COV(A,B) and the equation for the VAR(A) The following conclusion can be drawn: When the holding period returns of two securities move in the same direction, by the same amount at the same time, the pair is perfectly positively correlated When the holding period returns of two securities are totally unrelated to each other, the pair is uncorrelated The risk of a portfolio is the weighted average of the risk of each security in the portfolio, and the covariance between each pair of securities in the portfolio The number of covariance terms increases rapidly as new securities are added to the portfolio 5. Portfolio diversification Diversification can increase the risk/return tradeoff if the correlation coefficient between individual securities in the portfolio is less than 1.0 The benefits of diversification increase as the correlation coefficient gets smaller As the number of securities in a portfolio increases the portfolio risk decreases and approaches the risk of the total market Market risk is inherent from business cycles, inflation, interest rates, and economic factors Firm-specific risk is tied to the company’s labor contracts, new product development and other company related factors Mathematical diversification Increasing the number of stocks reduces the portfolio risk from any individual stock Diversification across time Annualized standard deviations decline as the investment horizon increases Uncertainty also increases and total return is not improved Dollar cost averaging Strategy which invests a set dollar amount at regular intervals Depends upon diversification across time Naive diversification occurs when investors select stocks at random, and purchase and equal dollar amount of each security When N becomes large enough, naive diversification averages out the firm-specific (unsystematic) risk of the stocks in the portfolio, so that only the market (or systematic) risk remains Buy and hold strategies reduce risk by diversifying across time The standard deviation of returns for a naively diversified portfolio held for five years is less than half the standard deviation of a one year holding period Naive diversification, over enough time and enough securities, will eliminate almost all of the firm specific, unsystematic risk of holding each security in the portfolio Naive diversification cannot reduce the risk of a portfolio to a level smaller than the risk of the market portfolio Efficient diversification finds the portfolio with the lowest risk for a given return, or the largest return for a given risk This eliminates all diversifiable risk Modern portfolio theory includes the concept of diversification and measures security risk using the capital asset pricing model (CAPM) 6. Efficient frontier A line graphing the most efficient possible combinations of stocks for maximizing portfolio returns while minimizing portfolio risk - the best risk/return tradeoff From Exhibit 12.19, we can see that each vector of weights results in a different portfolio mean return, and a different portfolio standard deviation Full-market efficient frontier Exhibit 12.20 Implications Grouping many securities together in a portfolio reduces the risk of the portfolio faster than the return is reduced Holding a portfolio over time periods of five years or more reduces risk more than it reduces returns from the portfolio