Class 7 Portfolio Analysis

Risk and Uncertainty n Almost all business decisions are made in the face of risk and uncertainty. n So far we have side-stepped the issue of risk and uncertainty, except to say that investments with greater risk should have higher required returns. n A full consideration of risk and uncertainty requires a statistical framework for thinking about these issues.

Random Variables n A random variable is a quantity whose outcome is not yet known.  The high temperature on next July 1st.  The total points scored in the next Super Bowl.  The rate of return on the S&P500 Index over the next year.  The cash flows on an investment project being considered by a firm.

Probability Distributions n A probability distribution summarizes the possible outcomes and their associated probabilities of occurrence.  Probabilities cannot be negative and must sum to 1.0 across all possible outcomes.  Example: Tossing a fair coin. Outcome Probability Heads 50% Tails 50%

Summary Statistics n Mean or Average Value  Measures the expected outcome. n Variance and Standard Deviation  Measures the dispersion of possible outcomes. n Covariance and Correlations  Measures the comovement of two random variables.

Calculating the Mean n Means or expected values are useful for telling us what is likely to happen on average. n The mean is a weighted average.  List the possible outcomes.  For each outcome, find its probability of occurrence.  Weight the outcomes by their probabilities and add them up.

Calculating the Mean n The formula for calculating the mean is:

Calculating the Mean Example n Suppose we flip a coin twice. The possible outcomes are given in the table below.

Properties of Means n E(a) = a where a is constant n E(X+Y) = E(X) + E(Y) n E(aX) = aE(X)

Calculating the Variance and Standard Deviation n The variance and standard deviation measure the dispersion or volatility. n The variance is a weighted average of the squared deviations from the mean.  Subtract the mean from each possible outcome.  Square the difference.  Weight each squared difference by the probability of occurrence and add them up. n The standard deviation is the square root of the variance.

Calculating the Variance and Standard Deviation n The formulas for calculating the variance and standard deviation are:

Calculating Variance and Standard Deviation Example n What is the variance and standard deviation of our earlier coin tossing example? n   

Properties of Variances n var(a) = 0 where a is constant n var(aX) = a 2 var(X) n var(a+X) = var(X) n var(X+Y) = var(X)+var(Y)+2cov(X,Y) n var(aX+bY) = a 2 var(X)+b 2 var(Y) +2ab[cov(X,Y)]

Probability Distribution Graphically Both distributions have the same mean. One distribution has a higher variance.

Covariance and Correlation n The covariance and correlation measure the extent to which two random variables move together.  If X and Y, move up and down together, then they are positively correlated.  If X and Y move in opposite directions, then they are negatively correlated.  If movements in X and Y are unrelated, then they are uncorrelated.

Calculating the Covariance n The formula for calculating the covariance is:

Calculating the Correlation n The correlation of two random variables is equal to the covariance divided by the product of the standard deviations. n Correlations range between -1 and 1.  Perfect positive correlation:  XY = 1.  Perfect negative correlation:  XY = -1.  Uncorrelated:  XY = 0.

Calculating Covariances and Correlations n Consider the following two stocks:

Properties of Covariances n cov(X+Y,Z) = cov(X,Z) + cov(Y,Z) n cov(a,X) = 0 n cov(aX,bY) = ab[cov(X,Y)]

Risk Aversion n An individual is said to be risk averse if he prefers less risk for the same expected return. n Given a choice between $C for sure, or a risky gamble in which the expected payoff is $C, a risk averse individual will choose the sure payoff.

Risk Aversion n Individuals are generally risk averse when it comes to situations in which a large fraction of their wealth is at risk.  Insurance  Investing n What does this imply about the relationship between an individual’s wealth and utility?

Relationship Between Wealth and Utility Utility Function Utility Wealth

Risk Aversion Example n Suppose an individual has current wealth of W 0 and the opportunity to undertake an investment which has a 50% chance of earning x and a 50% chance of earning -x. Is this an investment the individual would voluntarily undertake?

Risk Aversion Example

Implications of Risk Aversion n Individuals who are risk averse will try to avoid “fair bets.” Hedging can be valuable. n Risk averse individuals require higher expected returns on riskier investments. n Whether an individual undertakes a risky investment will depend upon three things:  The individual’s utility function.  The individual’s initial wealth.  The payoffs on the risky investment relative to those on a riskfree investment.

Diversification: The Basic Idea n Construct portfolios of securities that offer the highest expected return for a given level of risk. n The risk of a portfolio will be measured by its standard deviation (or variance). n Diversification plays an important role in designing efficient portfolios.

Measuring Returns n The rate of return on a stock is measured as: n Expected return on stock j = E(r j ) n Standard deviation on stock j =  j

Measuring Portfolio Returns n The rate of return on a portfolio of stocks is: n x j = fraction of the portfolio’s total value invested in stock j.  x j > 0 is a long position.  x j < 0 is a short position.   j x j = 1

Measuring Portfolio Returns n The expected rate of return on a portfolio of stocks is: n The expected rate of return on a portfolio is a weighted average of the expected rates of return on the individual stocks.

Measuring Portfolio Risk n The risk of a portfolio is measured by its standard deviation or variance. n The variance for the two stock case is: or, equivalently,

Minimum Variance Portfolio n Sometimes we are interested in the portfolio that gives the smallest possible variance. We call this the global minimum-variance portfolio. n For the two stock case, the global minimum variance portfolio has the following portfolio weights:

Two Asset Case

n We want to know where the portfolios of stocks 1 and 2 plot in the risk-return diagram. n We shall consider three special cases:   12 = -1   12 = 1   12 < 1

Perfect Negative Correlation n With perfect negative correlation,  12 = -1, it is possible to reduce portfolio risk to zero. n The global minimum variance portfolio has a variance of zero. The portfolio weights for the global minimum variance portfolio are:

Perfect Negative Correlation

Example n Suppose you are considering investing in a portfolio of stocks 1 and 2. n Assume  12 = -1. What is the expected return and standard deviation of a portfolio with equal weights in each stock?

Example n Expected Return n Standard Deviation

Example n What are the portfolio weights, expected return, and standard deviation for the global minimum variance portfolio? n Portfolio Weights

Example n Expected Return n Standard Deviation

Perfect Positive Correlation

n With perfect positive correlation,  12 = 1, there are no benefits to diversification. This means that it is not possible to reduce risk without also sacrificing expected return. n Portfolios of stocks 1 and 2 lie along a straight line running through stocks 1 and 2.

Perfect Positive Correlation n With perfect positive correlation,  12 = 1, it is still possible to reduce portfolio risk to zero, but this requires a short position in one of the assets. n The portfolio weights for the global minimum variance portfolio are:

Example n Consider again stocks 1 and 2. n Assume now that  12 = 1. What is the expected return and standard deviation of an equally- weighted portfolio of stocks 1 and 2?

Example n Expected Return n Standard Deviation

Example n What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio? n Portfolio Weights

Example n Expected Return n Standard Deviation

Non-Perfect Correlation

n With non-perfect correlation, -1<  12 <1, diversification helps reduce risk, but risk cannot be eliminated completely. n Most stocks have positive, but non-perfect correlation with each other. n The global minimum variance portfolio will have a lower variance than either asset 1 or asset 2 if:  <      where    

Example n Consider again stocks 1 and 2. n Assume now that  12 =.25. What is the expected return and standard deviation of an equally-weighted portfolio of stocks 1 and 2?

Example n Expected Return n Standard Deviation

Example n What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio? n Portfolio Weights

Example n Expected Return n Standard Deviation

Multiple Assets n The variance of a portfolio consisting of N risky assets is calculated as follows:

Limits to Diversification n Consider an equally-weighted portfolio. The variance of such a portfolio is:

Limits to Diversification n As the number of stocks gets large, the variance of the portfolio approaches: n The variance of a well-diversified portfolio is equal to the average covariance between the stocks in the portfolio.

Limits to Diversification n What is the expected return and standard deviation of an equally-weighted portfolio, where all stocks have E(r j ) = 15%,  j = 30%, and  ij =.40?

Limits to Diversification Market Risk Total Risk Firm-Specific Risk Portfolio Risk,  Number of Stocks Average Covariance

Examples of Firm-Specific Risk n A firm’s CEO is killed in an auto accident. n A wildcat strike is declared at one of the firm’s plants. n A firm finds oil on its property. n A firm unexpectedly wins a large government contract.

Examples of Market Risk n Long-term interest rates increase unexpectedly. n The Fed follows a more restrictive monetary policy. n The U.S. Congress votes a massive tax cut. n The value of the U.S. dollar unexpectedly declines relative to other currencies.

Efficient Portfolios with Multiple Assets E[r]  0 Asset 1 Asset 2 Portfolios of Asset 1 and Asset 2 Portfolios of other assets Efficient Frontier Minimum-Variance Portfolio

Efficient Portfolios with Multiple Assets n With multiple assets, the set of feasible portfolios is a hyperbola. n Efficient portfolios are those on the thick part of the curve in the figure. They offer the highest expected return for a given level of risk. n Assuming investors want to maximize expected return for a given level of risk, they should hold only efficient portfolios.

Common Sense Procedures n Hold a well-diversified portfolio. n Invest in stocks in different industries. n Invest in both large and small company stocks. n Diversify across asset classes.  Stocks  Bonds  Real Estate n Diversify internationally.