Expected Return and Risk Learning Objectives Explain how

Expected Return and Risk

Learning Objectives • 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.

Investment Decisions • Involve uncertainty • Focus on expected returns – Estimates of future returns needed to consider and manage risk • Goal is to reduce risk without affecting returns – – Accomplished by building a portfolio Diversification is key

Dealing with Uncertainty • Risk that an 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 Can be discrete or continuous

Calculating Expected Return • Expected value – – – The single most likely outcome from a particular probability distribution The weighted average of all possible return outcomes Referred to as an ex ante or expected return

Calculating Risk • Variance - The standard measure of risk is the variance of return Or Its square root: the standard deviation • Measures the spread in the probability distribution – Variance of returns: 2 = (Ri - E(R))2 pri – Standard deviation of returns: =( 2)1/2 • With T observations sample variance is • The standard deviation is

Portfolio Expected Return • 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:

Sample Covariance • The covariance measures the way the returns on two assets vary relative to each other – Positive: the returns on the assets tend to rise and fall together – Negative: the returns tend to change in opposite directions • Covariance has important consequences for portfolios Asset A B Return in 2001 10 2 Return in 2002 2 10

Sample Covariance • Mean return on each stock = 6 • Variances of the returns are • Portfolio: 1/2 of asset A and 1/2 of asset B • Return in 2001: • Return in 2002: • Variance of return on portfolio is 0

Sample Covariance • The covariance of the return is • It is always true that – ii.

Sample Covariance • Example. The table provides the returns on three assets over three years Year 1 Year 2 Year 3 A 10 12 11 B 10 14 12 C 12 6 9 • Mean returns • Covariance between A and B is

Variance-Covariance Matrix • Covariance between A and C is • Covariance between B and C is • The matrix is symmetric

Variance-Covariance Matrix • For the example the variance-covariance matrix is

Population Return and Variance • Expectations: assign probabilities to outcomes • Rolling a dice: any integer between 1 and 6 with probability 1/6 • Outcomes and probabilities are: {1, 1/6}, {2, 1/6}, {3, 1/6}, {4, 1/6}, {5, 1/6}, {6, 1/6} • Expected value: average outcome if experiment repeated

Population Return and Variance • Formally: M possible outcomes • Outcome j is a value xj with probability pj • Expected value of the random variable X is • The sample mean is the best estimate of the expected value

Population Return and Variance • After market analysis of Esso an analyst determines possible returns in 2008 Return Probability 2 6 9 12 0. 3 0. 2 • The expected return on Esso stock using this data is E[r. Esso] =. 2(2) +. 3(6) +. 3(9) +. 2(12) = 7. 3 • The expectation can be applied to functions of X • For the dice example applied to X 2

Population Return and Variance • And to X 3

Population Return and Variance • And to X 3 • The expected value of the square of the deviation from the mean is • This is the population variance

Modelling Returns • States of the world – Provide a summary of the information about future return on an asset • A way of modelling the randomness in asset returns – Not intended as a practical description • Let there be M states of the world • Return on an asset in state j is rj • Probability of state j occurring is pj • Expected return on asset i is

Modelling Returns • Example: The temperature next year may be hot, warm or cold • The return on stock in a food production company in each state State Return Hot 10 Warm Cold 12 18 • If each states occurs with probability 1/3, the expected return on the stock is

Portfolio Expected Return • • • N assets and M states of the world Return on asset i in state j is rij Probability of state j occurring is pj Xi proportion of the portfolio in asset i Return on the portfolio in state j The expected return on the portfolio • Using returns on individual assets • Collecting terms this is • So

Portfolio Expected Return • Example: Portfolio of asset A (20%), asset B (80%) • Returns in the 5 possible states and probabilities are: State Probability Return on A 1 0. 1 2 2 0. 2 6 3 0. 4 9 Return on B 5 1 0 4 5 0. 1 0. 2 1 2 4 • For the two assets the expected returns are • For the portfolio the expected return is 3

Population Variance and Covariance • Population variance • The sample variance is an estimate of this • Population covariance • The sample covariance is an estimate of this

Population Variance and Covariance • M states of the world, return in state j is rij • Probability of state j is pj • Population variance is • Population standard deviation is

Population Variance and Covariance • Example: The table details returns in five possible states and the probabilities State Return Probability 1 5 2 2 3 -1 4 6 5 3 0. 1 0. 2 0. 4 0. 1 0. 2 • The population variance is

Portfolio Variance • Two assets A and B • Proportions XA and XB • Return on the portfolio r. P • Mean return • Portfolio variance

Portfolio Variance • Population covariance between A and B is • For M states with probabilities pj • The portfolio return is • So • Collecting terms

Portfolio Variance • Squaring • Separate the expectations • Hence

Portfolio Variance • Example: Portfolio consisting of – 1/3 asset A – 2/3 asset B • The variances/covariance are • The portfolio variance is

Correlation Coefficient • The correlation coefficient is defined by • Value satisfies • perfect positive correlation r B r. A

Correlation Coefficient • perfect negative correlation r B r. A • Variance of the return of a portfolio

Correlation Coefficient • Example: Portfolio consisting of – 1/4 asset A – 3/4 asset B • The variances/correlation are • The portfolio variance is

General Formula • N assets, proportions Xi • Portfolio variance is • But • so

Effect of Diversification • • Diversification: a means of reducing risk Consider holding N assets Proportions Xi = 1/N Variance of portfolio • N terms in the first summation, N[ N-1] in the second • Gives • Define

Effect of Diversification

Effect of Diversification • Then • Let N tend to infinity (extreme diversification) • Then • Hence • In a well-diversified portfolio only the covariance between assets counts for portfolio variance

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

Risk Reduction in Portfolios • 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” • Only issue is how many securities to hold • Random diversification – Diversifying without looking at relevant investment characteristics – Marginal risk reduction gets smaller and smaller as more securities are added • A large number of securities is not required for significant risk reduction • International diversification is beneficial

Portfolio Risk and Diversification • Random Diversification : ü Act of randomly diversifying without regard to relevant investment characteristics p %ü 15 or 20 stocks provide adequate diversification 35 Total Portfolio Risk 20 Market Risk 0 10 20 30 40 . . . Number of securities in portfolio 100+

International Diversification p % Domestic Stocks only 35 Domestic + International Stocks 20 0 10 20 30 40 . . . Number of securities in portfolio 100+

Markowitz 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

Measuring Co-Movements in Security Returns • Needed to calculate risk of a portfolio: – Weighted individual security risks • Calculated by a weighted variance using the proportion of funds in each security • For security i: (wi i)2 – Weighted co-movements between returns • Return covariances are weighted using the proportion of funds in each security • For securities i, j: 2 wiwj ij

Correlation Coefficient • Statistical measure of relative co-movements between security returns mn = correlation coefficient between securities m & n – mn = +1. 0 = perfect positive correlation – mn = -1. 0 = perfect negative (inverse) correlation – mn = 0. 0 = zero correlation • When does diversification pay? – Combining securities with perfect positive correlation provides no reduction in risk • Risk is simply a weighted average of the individual risks of securities – Combining securities with zero correlation reduces the risk of the portfolio – Combining securities with negative correlation can eliminate risk altogether

Covariance • Absolute measure of association – Not limited to values between -1 and +1 – Sign interpreted the same as correlation – The formulas for calculating covariance and the relationship between the covariance and the correlation coefficient are:

Calculating Portfolio Risk • Encompasses three factors – Variance (risk) of each security – Covariance between each pair of securities – Portfolio weights for each security • Goal: select weights to determine the minimum variance combination for a given level of expected return • Generalizations – – The smaller the positive correlation between securities, the better As the number of securities increases: • The importance of covariance relationships increases • The importance of each individual security’s risk decreases

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

Simplifying Markowitz Calculations • Markowitz full-covariance model – 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

The Single-Index Model • Relates returns on each security to the returns on a common stock index, such as the S&P Composite Index • Expressed by the following equation: • Divides return into two components – a unique part, αi – a market-related part, βi. RMt measures the sensitivity of a stock to stock market If securities are only related in their common response to the market üSecurities covary together only because of their common relationship to the market index üSecurity covariances depend only on market risk and can be written as: