Portfolio Theory Chapter 7 Charles P. Jones, Investments: Analysis and Management, Twelfth Edition, John Wiley & Sons

Investment Decisions Involve uncertainty Focus on expected returns Estimates of future returns needed to consider and manage risk Investors often overly optimistic about expected returns 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 Probabilities weight outcomes Assigned to each possible outcome to create a distribution History provides guide but must be modified for expected future changes Distributions can be discrete or continuous

Calculating Expected Return Expected value The weighted average of all possible return outcomes included in the probability distribution Each outcome weighted by probability of occurrence Referred to as expected return

Calculating Risk Variance and standard deviation used to quantify and measure risk Measure the spread or dispersion around the mean of the probability distribution Variance of returns: σ² = (Ri - E(R))²pri Standard deviation of returns: σ =(σ²)1/2 σ is expected (ex ante) Actual (ex post) σ useful but not perfect estimate of future

Modern Portfolio Theory Provides framework for selection of portfolios based on expected return and risk Used, to varying degrees, by financial managers Shows benefits of diversification The risk of a portfolio does not equal the sum of the risks of its individual securities Must account for correlations among individual risks

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

Portfolio Risk Portfolio risk not simply the sum or the weighted average of individual security risks Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio Diversification almost always lowers risk Measured by the variance or standard deviation of the portfolio’s return

Risk Reduction in Portfolios Assume all risk sources for a portfolio of securities are independent This assumption is unrealistic when investing Market risk affects all firms, cannot be diversified away If risks 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

Risk Reduction in Portfolios Random (or naïve) diversification Diversifying without looking at how security returns are related to each other Marginal risk reduction gets smaller and smaller as more securities are added Beneficial but not optimal Risk reduction kicks in as soon as additional securities added Research suggests it takes a large number of securities for significant risk reduction

Measuring Portfolio Risk 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: 2wiwj × ij

Correlation Coefficient Statistical measure of relative association ij = correlation coefficient between securities i and j ij = +1.0 = perfect positive correlation ij = -1.0 = perfect negative (inverse) correlation ij = 0.0 = zero correlation

Correlation Coefficient When does diversification pay? With perfectly positive correlation, risk is a weighted average, therefore there is no risk reduction With perfectly negative correlation, diversification assures the expected return With zero correlation If many securities, provides significant risk reduction Cannot eliminate risk Negative correlation or low positive correlation ideal but unlikely

Covariance Absolute, not relative, measure of association Not limited to values between -1 and +1 Sign interpreted the same as correlation Size depends on units of variables Related to correlation coefficient

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

Calculating Portfolio Risk Generalizations the smaller the positive correlation between securities, the better Covariance calculations grow quickly As the number of securities increases: The importance of covariance relationships increases The importance of each individual security’s risk decreases

Simplifying Markowitz Calculations Markowitz full-covariance model Requires a covariance between the returns of all securities in order to calculate portfolio variance [n(n-1)]/2 set of covariances for n securities Markowitz suggests using an index to which all securities are related to simplify

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