Investment Management EffiCient DiversifiCation Efficient-Diversification 3

Summary: Asset Allocation with Many Risky Assets and a Risk-free Asset Specify risk-return characteristics of securities and calculate the portfolio opportunity set Statistical task Find the optimal risky portfolio P (same for all investors) Maximize reward-to-variability ratio Combine the optimal risky portfolio P and the riskless asset This is the optimal complete portfolio C for the given level of risk-aversion Efficient Diversification Fall 2018 25 / 36 Discussion Mean-variance analysis is one of the crown jewels of finance theory (It got Harry Markowitz the Nobel Prize) But there are implementation problems Mean-variance analysis sometimes leads to large short positions in some assets. Is this reasonable/appropriate? Solution: Constrain the weights to be positive Efficient Diversification Fall 2018 26 / 36 Underlying Reason for the Implementation Problems? Precision in estimating the inputs What if you have an asset with interesting risk/return trade-off or correlation properties but these inputs are estimated imprecisely? Also, large covariance matrices are hard to work with. Time variation in the inputs For variances, historical averages are usually okay Covariances are trickier and change over the business cycle. They also change in periods of market turmoil. The hardest part is to estimate expected returns !!! Expected returns are not constant over time, so historical averages are not always helpful. Efficient Diversification Fall 2018 27 / 36 A Single-Index Stock Market Index model is a model that relates stock returns to returns on both a broad market index and firm-specific factors. Ri = β i RM + ei + αi Ri : Excess return of the firm’s stock (stock return−risk-free rate) β i RM : Component of return due to movements in overall market β i : Sensitivity of security’s returns to market factor RM : Excess return of the market index (market index return−risk-free rate) ei : Component attributable to unexpected events relevant only to this security (firm-specific or residual risk) αi : Stock’s expected excess return if the market factor is neutral, i.e. if the market-index excess return is zero Efficient Diversification Fall 2018 28 / 36 Variance of the Excess Return of the Stock Variance(Ri ) = Variance( β i RM + ei + αi ) = Variance( β i RM ) + Variance(ei ) 2 = β2i σM + σ 2 ( ei ) = Systematic risk + Firm-specific risk The total variance of the rate of return of each security is a sum of two components: 1 The variance attributable to the uncertainty of the entire market. 2 , This variance depends on both the variance of RM , denoted by σM and the β of the stock on RM . 2 The variance of the firm-specific return, ei , which is independent of market performance. Efficient Diversification Fall 2018 29 / 36 Statistical and Graphical Representation of Single-Index Model Ri = β i RM + ei + αi may be interpreted as a single-variable regression equation of Ri on the market excess return RM The regression line is called the Security Characteristic Line (SCL) Security Characteristic Line is a plot of a security’s expected excess return over the risk-free rate as a function of the excess return on the market Slope of the SCL is the regression coefficient β and the intercept is the α for the security The algebraic representation of the regression line is E (Ri |RM ) = αi + β i RM Efficient Diversification Fall 2018 30 / 36 Relative Importance of Systematic Risk One way to measure the relative importance of systematic risk is to measure the ratio of systematic variance to total variance Ratio of systematic variance to total variance is the square of the correlation coefficient ρ between Ri and RM Systematic (or explained) variance Total variance 2 2 β2i σM β2i σM = = 2 + σ2 σi2 β2i σM ei ρ2 = A large correlation coefficient (in absolute value terms) means systematic variance dominates the total variance; that is, firm-specific variance is relatively unimportant When the correlation coefficient is small (in absolute value terms), the market factor plays a relatively unimportant part in explaining the variance of the asset, and firm-specific factors dominate Efficient Diversification Fall 2018 31 / 36 Scatter Diagram for Dana Computer Corp. Efficient Diversification Fall 2018 32 / 36 Various Scatter Diagrams Efficient Diversification Fall 2018 33 / 36 Diversification in a Single-Index Security Market Imagine a portfolio P that is divided equally among securities whose returns follow the single-index model The β of the portfolio is a simple average of the individual security βs Hence, there are no diversification effects on systematic risk no matter how many securities are involved Intuition: The systematic component of each security return, β i RM , is driven by the market factor and therefore is perfectly correlated with the systematic part of any other security’s return Efficient Diversification Fall 2018 34 / 36 Diversification in a Single-Index Security Market n Consider a portfolio P of n securities with weights wi ( ∑ wi = 1) i =1 Nonsystematic risk of each security is σe2i Nonsystematic portion of portfolio P’s return is n eP = ∑ w i ei i =1 Portfolio P’s nonsystematic variance is n σe2P = ∑ wi2 σe2 i i =1 The sum is far less than the average firm-specific variance of the stocks in the portfolio Efficient Diversification Fall 2018 35 / 36 Diversification in a Single-Index Security Market The impact of nonsystematic risk becomes negligible as the number of securities grows and the portfolio becomes more diversified The number of securities counts more than the size of their nonsystematic variance Sufficient diversification can virtually eliminate firm-specific risk. Only systematic risk (market risk) remains. For diversified investors, the relevant risk measure for a security is the security’s β, since firms with higher β have greater sensitivity to market risk 2 , will be determined by both the market The systematic risk, β2 σM 2 volatility, σM , and the firm’s β Efficient Diversification Fall 2018 36 / 36