McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Single Index Models Chapter 6 1

6-2 WARNING! DANGER! To date we have been mostly solving problems We are now beginning the VERY CONCEPTUAL component of the class  Instead of simple being able to solve the problems you need to understand the theoretical concepts and their implications  In general this will require a much deeper understanding of the material  Please be prepared 2

6-3 Learning Objectives Describe the advantages of a single-factor model Define systematic risk and firm-specific risk and estimate the contribution of each to a firm’s total risk Describe the relationship between firm- specific risk and portfolio diversification Identify the inputs of the single index model and describe the security characteristic line

6-4 4 return PP Where we left off Efficient Portfolios CAL Optimal Risky Portfolio Risk Free

6-5 Single Index Stock Market In the real world constructing the efficient frontier is practically impossible  Too many interactions However, we can simplify by assuming that all co-movement in returns results from a single risk factor  Idea is that a single common risk factor (systematic) is responsible for all the co-movement in returns 5

6-6 Assumptions: Single Index Model Returns are driven by a single, common systematic factor  Stock returns are joint normally distributed The factor is systematic (Macroeconomic) affects everything   measures a securities sensitivity to the factor Varies across securities Firms are correlated with each other through their correlation to the systematic (macroeconomic) factor  Macroeconomic surprises and firm-specific surprises are not correlated  Firm specific surprises are not correlated across firms

6-7 Single Index Stock Market r i = β i r M + e i + α i A stocks excess return (r i )has three parts 1. Return due to movements in the risk factor: β i r M r M is the market’s excess return β i is the sensitivity of the security’s returns to the market factor; Systematic risk measure  Β i > 1: Cyclical stocks  Β i < 1: Defensive stocks 2. Return due to firm specific events: e i ; Residual risk 3. Return beyond that induced by the market: α i Under or Over priced Remember excess return → net of risk free 7

6-8 Where Does α i Come From? Stocks have two values:  Intrinsic Value (IV) the present value of the expected future cash flows “True” Price according to a valuation model  Market Value (MV) is the consensus value of all market participants α i is postive when IV > MV, Under-Priced α i is negative when IV < MV, Over-Priced α i is 0 when IV = MV, Correctly Priced 8

6-9 Single Index Stock Market r i = β i r M + e i + α i r i is the security’s excess return r M is the market’s excess return β i is the sensitivity of the security’s returns to the market factor; Systematic risk measure  Β i > 1: Cyclical stocks  Β i < 1: Defensive stocks e i is firm-specific or residual risk  Surprises, return independent of market factor α i is the stock’s expected return beyond that induced by market index; Under or Over priced 9

6-10 Breaking Up Returns A stocks excess return has three parts 1. Return due to movements in the risk factor: β i r M 2. Firm Specific unexpected events: e i 3. Stock expected excess return: α i Is the stock under or over priced 10

6-11 Single Index Graph 11 Security Characteristic Line r i = β i r M + α i

6-12 Security Characteristic Line Does NOT depict actual returns Does represent average tendencies  Provides the security’s expected return given r m r D = β D R M + α D  e i is assumed to be 0 12

6-13 Example If we expect the market to return to be 15%, what return do we expect from a stock with a β of 1.2? If its α is 3%? What about if the market only returned 13%? If e is -1%? 13

6-14 Expect versus Realized Returns Actual return = Expected return + the effect of surprises r i = Actual return earn on the security E(r i )= Expected return on the security β i = Factor sensitivity or factor loading or factor beta F = Surprise in macro-economic factor (+/-) e i = Firm specific events 14 E(r i ) riri

6-15 Expect v Actual Return Example If the market is expected to return 12% over the next year, what is the expected return for a stock with a β of 1.2? The risk free rate is 3%. If the actual market return was 9%:  What is the market surprise?  What was the actual return earned over the year? 15

6-16 Breaking Down Variance Variance is a measure of TOTAL risk  Variance = Systematic risk + Firm-specific risk  Systematic risk = β i 2 σ m 2  Firm-specific risk = σ(e i ) 2 σ i 2 = β i 2 σ m 2 + σ(e i ) 2 For a well diversified portfolio, what does σ(e i ) 2 equal? 16

6-17 Variance Example What is the variance of a stock with a beta of 0.9, if the standard deviation of the market is 25%, and its residual standard deviation is 30%? What if the market standard deviation increases to 28%? 17

6-18 How Important is the Market? To determine the importance of systematic risk we measure the ratio of systematic variance to total variance This is the correlation coefficient squared As ρ 2 increases → the market is more important for explaining firm returns σ 2 (e D ): variance of firm specific surprises Determines spread of actual returns around SCL Influences the importance of the market 18

6-19 Single Index Graph 19 β: Systematic risk -Steepness r i = β i r M + e i + α i Spread for the SCL is idiosyncratic risk

6-20 Diversification in a Single Index World All securities have systematic risk exposure, β  Can’t get rid of this Portfolio β is just a weighted average of the stock in the portfolio 20

Diversification Continued Variance of the equally weighted portfolio of firm-specific components: When n gets large, σ 2 (e p ) becomes negligible and firm specific risk is diversified away.

6-22 Questions Which offers more diversification benefit? Which is riskier for an undiversified investor? Which is riskier for a diversified investor? 22

6-23 Does investment horizon matter? This is hotly debate Many belief that long term investors should hold more stock (riskier assets) because they become less risky over the long run  Time Diversification The book and many academics argues against this position 23