Chapter 7 Expected Return and Risk

Explain how expected return and risk for securities are determined. Explain how expected return and risk for portfolios are determined. Describe the Markowitz diversification model for calculating portfolio risk. Simplify Markowitz’s calculations by using the single-index model. Learning Objectives

Involve uncertainty  Investors are trading a known present value for some expected future value that is not known with certainty Focus on expected returns  To estimate the returns from various securities, investors must estimate the cash flows these securities are likely to provide Goal is to reduce risk without affecting returns  Accomplished by building a portfolio  Diversification is key to effective risk management Investment Decisions

Risk that the expected return will not be realized Investors must think about return distributions, not just a single return Use probability distributions  A probability should be assigned to each possible outcome to create a distribution  A probability represents the likelihood of various outcomes and is expressed as a decimal or fraction  Sum of the probabilities of all possible outcomes must be 1.0  Can be discrete or continuous (eg., normal distribution) (Fig.7.1 pg 188) Dealing with Uncertainty

Expected value (Expected rate of return)  Calculate the expected value in order to describe the single most likely outcome from a particular probability distribution  The weighted average of all possible return outcomes, where each is weighted by its respective probability of occurrence  Referred to as an ex ante or expected return Calculating Expected Return

E(R) = the expected return on a security R_i = the ith possible return pr_i = the probability of the ith return m = the number of possible returns

Variance and standard deviation are used to quantify and measure risk  Measure the spread or dispersion in the probability distribution  The larger the dispersion the larger the variance and standard deviation  Variance of returns:  2 =  (R i - E(R)) 2 pr i  Standard deviation of returns:  =(  2 ) 1/2 Calculating Risk

 Calculating a standard deviation using probability distribution involves making subjective estimates of the probabilities and the likely returns  We cannot avoid making estimates because future returns are uncertain  The relevant  in this situation is the ex ante standard deviation and not the ex post standard deviation based on realized returns  In this chapter we are interested in the variability associated with future expected returns

Weighted average of the individual security expected returns  Each portfolio asset has a weight, w, which represents the percent of the total portfolio value  The expected return on any portfolio can be calculated as: Portfolio Expected Return

Example: Portfolio Expected Return (Pg 191) Consider a three stock portfolio consisting of stocks G, H, and I with expected returns of 12%, 20%, and 17% respectively. Assume that 50% of investable funds is invested in security G, 30% in H, and 20% in I. Calculate the expected return on the portfolio

Portfolio risk is not simply the sum of individual security risks Emphasis is on the risk of the entire portfolio and not on the risk of individual securities in the portfolio Individual stocks are risky only if they add risk to the total portfolio Portfolio Risk

Measured by the variance or standard deviation of the portfolio’s return  Portfolio risk is not a weighted average of the risk of the individual securities in the portfolio Portfolio Risk

Although the expected return of a portfolio is a weighted average of its expected returns, portfolio risk is less than the weighted average of the risk of the individual securities in a portfolio of risky securities

Assume all risk sources for a portfolio of securities are independent The larger the number of securities, the smaller the exposure to any particular risk  “Insurance principle” the insurance company reduces its risk by writing many policies against many independent sources of risk Risk Reduction in Portfolios

Random (naïve) diversification  Diversifying without looking at relevant investment characteristics such as expected return or industry classification  An investor simply selects a relatively large number of securities randomly  Marginal risk reduction gets smaller and smaller as more securities are added (Fig. 7.2 pg 192)  Most finance textbooks contain similar diagrams, with the number of stocks required to achieve diversification varying depending upon the market and the particular empirical study referred to in the diagram (eg., 25 to 30 or 15 to 20) Risk Reduction in Portfolios

How many securities are enough to diversify properly?  A recent study by Campbell, Lettau, Malkiel, and Xu showed that, between 1962 and 1997 the market’s overall volatility did not change whereas the volatility of individual stocks increased sharply. As a result, investors need more stocks today to adequately diversity (40 rather than 20)  Another recent study by Vladimir de Vassal examined the period from 1993 to 1999 and found that with a portfolio of 15 stocks, the probability of underperforming the market benchmark by 100% or more was 13.5%, a substantial risk. With a portfolio of 40 stocks the probability declines to only 2.4%

Risk Reduction in Portfolios International diversification  Ignoring the hazards of foreign investing, such as currency risk, we can conclude that if domestic diversification is good, international diversification is better (Fig 7.3 pg 195)  Traditional thinking focused on diversifying across countries, but the current trend is to diversify across industries and across countries simultaneously (Fig. 7.4 pg 196)  Since recent research suggests that industry factors play as big a role (if not bigger) in obtaining diversification benefits

 p % Number of securities in portfolio Total Portfolio Risk Market Risk Portfolio Risk and Diversification

 p % Number of securities in portfolio Domestic Stocks only Domestic + International Stocks International Diversification

Non-random diversification  Active measurement and management of portfolio risk  Investigate relationships between portfolio securities before making a decision to invest  Takes advantage of expected return and risk for individual securities and how security returns move together Markowitz Diversification

We need to consider two factors in order to calculate risk of a portfolio as measured by the variance or standard deviation:  Weighted individual security risks Calculated by a weighted variance using the proportion of funds in each security For security i: (w i   i ) 2  Weighted co-movements between returns as measured by the covariance between returns Return covariances are weighted using the proportion of funds in each security For securities i, j: 2w i w j   ij Measuring Co-Movements in Security Returns

Statistical measure of relative co-movements between security returns (it measures how security returns move in relation to one another) Limited to values between -1 and +1  mn = correlation coefficient between securities m and n   mn = +1.0 = perfect positive correlation   mn = -1.0 = perfect negative (inverse) correlation   mn = 0.0 = zero correlation Correlation Coefficient

With perfect positive correlation, the returns have a perfect direct linear relationship. Knowing what the return on one security will do allows an investor to forecast perfectly what the other will do (Fig. 7.5 pg198) With perfect negative correlation, the securities’ returns have an inverse linear relationship to each other. Therefore, knowing the return on one security provides full knowledge about the return on the other (Fig. 7.6 pg 198) With zero correlation, there is no relationship between the returns on the two securities. Knowledge of the return on one security is of no value in predicting the return of the second security

When does diversification pay?  Combining securities with perfect positive correlation provides no reduction in risk Risk of the resulting portfolio is simply a weighted average of the individual risks of securities (Fig. 7.5 pg 198)  Combining securities with zero correlation reduces the risk of the portfolio If more securities with uncorrelated returns are added to the portfolio, significant risk reduction can be achieved The portfolio risk cannot be eliminated in this case Correlation Coefficient

 Combining securities with negative correlation can eliminate risk altogether If the correct portfolio weights are chosen  In the real world, securities typically have some positive correlation with each other since all security prices tend to move with changes in the overall economy (Fig. 7.7 pg 199,  = +0.55) As a result, risk can be reduced it cannot be eliminated Any reduction in risk that does not adversely affect return has to considered beneficial

Example: Correlation Coefficient (Pg 200) Over the period the average monthly return on Abitibi Consolidated (A) common stock was 0.05%, and the standard deviation of monthly returns was 10.69% During the same period, the average monthly return for Air Canada’s (AC) common stock was -0.29%, and the standard deviation of monthly returns was 22.42% The correlation between the returns on A and AC was 0.12 over this period. Calculate the return and standard deviation for an equally weighted portfolio of these two securities.

Absolute measure of (the degree of association) the extent to which two random variables, such as the return on two securities, tend to covary, or move together over time  Not limited to values between -1 and +1  Sign interpreted the same as correlation (+, -, 0)  The formulas for calculating covariance and the relationship between the covariance and the correlation coefficient are: Covariance

Covariance The covariance allows us to measure the amount of co- movement and incorporate it into any measure of portfolio risk (covariance formula is similar to the variance formula) σ_AB = the covariance between securities A and B R_A,i = one estimated possible return on security A E(R_A) = expected return for security A m = the number of likely outcomes for a security for the period pr_i = the probability of attaining a given return R_A,i

Encompasses three factors  Variance (risk) of each security  Covariance between each pair of securities (σ_AB = ρ_AB σ_A σ_B)  Portfolio weights for each security Goal: select weights to determine the minimum variance combination for a given level of expected return Calculating Portfolio Risk

Generalizations  The smaller the positive correlation between securities, the better The only case where there are no risk reduction benefits obtained from two-security diversification occurs when the correlation coefficient is +1  As the number of securities increases: The importance of covariance relationships increases The importance of each individual security’s risk (variance) decreases Calculating Portfolio Risk

The number of relevant covariances for an n-security portfolio equals n(n-1) For example, the number of relevant covariances in a 100-security portfolio would equal 100(100-1) = 9,900. On the other hand, the number of relevant variances will be 100

Two-Security Case: Calculating Portfolio Risk N-Security Case:

Example: Portfolio Risk Prove that in the two-security case, the portfolio standard deviation will be the weighted average of the standard deviations of the individual securities when the correlation coefficient is equal to +1

Example: Portfolio Risk ( Pg 202) We have an equally weighted portfolio that is compromised of stock A (TR = 26.3%, σ = 37.3%) and stock B (TR = 11.6%, σ = 23.3%) Calculate the standard deviation of the portfolio if the correlation between A and B is: +1, +0.5, +0.15, 0, -0.5, and -1)

Markowitz full-covariance model  Allows us to determine the portfolio expected return and risk  Can be used to determine the optimal portfolio combinations (Ch. 8)  Main problem is its complexity  Requires a covariance between the returns of all securities in order to calculate portfolio variance  Full-covariance model becomes burdensome as number of securities in a portfolio grows n(n-1)/2 unique covariances for n securities Therefore, Markowitz suggests using an index to simplify calculations Simplifying Markowitz Calculations

Developed by William Sharpe Relates returns on each security to the returns on a common stock index, such as the S&P/TSX Composite Index Expressed by the following equation: The Single-Index Model

R_i = the total return on security i α_i = the part of security i’s return independent of the market performance (intercept coefficient) β_i = a coefficient that measures the expected change in the dependent variable R_i given a change in the independent variable R_M R_M = the total return on the market index e_i = the random residual error

The Single-Index Model Divides return into two components  a unique part, α_i Is a micro-event, affecting an individual company, but not all companies in general (e.g., strike or resignation of CEO)  a market-related part, β_iR_M Is a macro-event, affecting all (or most) firms (e.g., inflation or oil prices)

Example: The Single-Index Model (Pg 206) Assume that the return for the market index for period t is 12%, with α_i = 3%, and β_i = 1.5 Use the single index model to calculate the return for stock i for time t Assuming that the actual return on stock i for period t in the previous example is 19%, calculate the error term

 (slope coefficient) measures the sensitivity of a stock to the market movements The single-index model assumes that  Residuals for different securities are uncorrelated  Securities are only related in their common response to the market Securities covary together only because of their common relationship to the market index Security covariances depend only on market risk and can be written as: The Single-Index Model