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Risk and Rates of Return Stand-alone risk Portfolio risk Risk & return: CAPM / SML 5-1 Investment returns The rate of return on an investment can be calculated as follows: Return = (Amount received – Amount invested) ________________________ Amount invested For example, if $1,000 is invested and $1,100 is returned after one year, the rate of return for this investment is: ($1,100 - $1,000) / $1,000 = 10%. 5-2 What is investment risk? Two types of investment risk Stand-alone risk Portfolio risk Stand-alone risk: The risk an investor would face if he or she held only one asset. Portfolio risk: The riskiness of assets held in portfolios. 5-3 Why is the T-bill return independent of the economy? Do T-bills promise a completely risk-free return? T-bills will return the promised 8%, regardless of the economy. No, T-bills do not provide a risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time. T-bills are risk-free in the default sense of the word. 5-4 Expected Rate of return Company IBM The rate of return expected to be realized from an investment. Expected Rate of Return -22% -2 20 35 50 Probability 10% 20 40 20 10 5-5 Return: Calculating the expected return for each alternative ^ k expected ^ k rate of return n k i Pi i 1 ^ k IBM (-22%) (0.1) (-2%) (0.2) (20%) (0.4) (35%) (0.2) (50%) (0.1) 17.4% 5-6 Summary of expected returns for all alternatives IBM Market USR T-bill Shell Exp return 17.4% 15.0% 13.8% 8.0% 1.7% IBM has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk? 5-7 Risk: Calculating the standard deviation for each alternative Standard deviation Variance n 2 2 ˆ ( k i k ) Pi i 1 5-8 Standard deviation calculation n ^ (k i k ) Pi 2 i 1 (-22.0 - 17.4) (0.1) (-2.0 - 17.4) (0.2) 2 2 (20.0 - 17.4) (0.4) (35.0 - 17.4) (0.2) 2 (50.0 17.4) (0.1) 2 IBM 2 IBM 20.04% T - bills 0.0% 1 2 Shell 13.4% USR 13.8% M 15.3% 5-9 Comments on standard deviation as a measure of risk Standard deviation (σi) measures total, or stand-alone, risk. The larger σi is, the lower the probability that actual returns will be closer to expected returns. Difficult to compare standard deviations, because return has not been accounted for. 5-10 Comparing risk and return Security Expected return 8.0% Risk, σ IBM 17.4% 20.04% Shell 1.7% 13.4% USR 13.8% 13.8% Market 15.0% 15.3% T-bills 0.0% 5-11 Coefficient of Variation (CV) A standardized measure of dispersion about the expected value, that shows the risk per unit of return. Very useful in comparing the risk of assets that have different expected returns. CV Std dev Mean ^ k 5-12 Risk rankings, by coefficient of variation T-bill IBM Shell USR Market CV 0.000 1.152 7.882 1.000 1.020 Shell has the highest degree of risk per unit of return. IBM, despite having the highest standard deviation of returns, has a relatively average CV. 5-13 Investor attitude towards risk Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities. Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities. 5-14 Portfolio construction: Risk and return Assume a two-stock portfolio is created with $50,000 invested in both IBM and Shell. Expected return of a portfolio is a weighted average of each of the component assets of the portfolio. 5-15 Calculating portfolio expected return ^ kp n ^ wi ki i 1 ^ k p 0.5 (17.4%) 0.5 (1.7%) 9.6% 5-16 Calculating portfolio standard deviation Forecasted return Year IBM Shell 2004 2005 2006 2007 2008 8% 10 12 14 16 16% 14 12 10 8 Portfolio Return Calculation (.50*8%) + (.50*16%) (.50*10%) + (.50*14%) (.50*12%) + (.50*12%) (.50*14%) + (.50*10%) (.50*16%) + (.50*8%) Expected Portfolio Return 12% 12% 12% 12% 12% 5-17 Calculating portfolio standard deviation (cont.) Expected value of portfolio return, 2004-2008 12% + 12% + 12% + 12% + 12% KP = 5 = 12% 5-18 Calculating portfolio standard deviation (cont.) n P 2 (k i k ) /n - 1 i 1 P (12% - 12%) 2 (12% - 12%) 2 (12% - 12%) 2 (12% - 12%) 2 (12% - 12%) 2 /( 5 1) 0% 5-19 Alternative Formula for Calculating portfolio standard deviation p 2 W1 1 2 W2 2 2 2 2W1W2 1 2 r 12 W1 Proportion of Asset 1 W2 Proportion of Asset 2 1 Standard Deviation of Asset 1 1 Standard Deviation of Asset 2 r 12 Correlatio n Coefficien t between th e return of assets 1 and 2 5-20 Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0) Stock W Stock M Portfolio WM 25 25 25 15 15 15 0 0 0 -10 -10 -10 5-21 Returns distribution for two perfectly positively correlated stocks (ρ = 1.0) Stock M’ Stock M Portfolio MM’ 25 25 25 15 15 15 0 0 0 -10 -10 -10 5-22 Illustrating diversification effects of a stock portfolio p (%) 35 Company-Specific Risk Stand-Alone Risk, p 20 Market Risk 0 10 20 30 40 2,000+ # Stocks in Portfolio 5-23 Breaking down sources of risk Stand-alone risk = Market risk + Firm-specific risk Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta. (e.g. War, Inflation, High Interest Rates) Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification. 5-24 Capital Asset Pricing Model (CAPM) Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification. CAPM : Ke= Rf + β(Rm – Rf) Rf = Risk free rate of return Rm = Market Return β = Beta Coefficient Ke = Required Return 5-25 Beta Measures a stock’s market risk, and shows a stock’s volatility relative to the market. Indicates how risky a stock is if the stock is held in a well-diversified portfolio. 5-26 Comments on beta If beta = 1.0, the security is just as risky as the average stock. If beta > 1.0, the security is riskier than average. If beta < 1.0, the security is less risky than average. Most stocks have betas in the range of 0.5 to 1.5. The beta coefficient for the market = 1 Betas May be positive or negative. But, positive is the norm. 5-27 The Security Market Line (SML): Calculating required rates of return SML: ki = kRF + (kM – kRF) βi Assume kRF = 8%, kM = 15% and βi =1.3 The market (or equity) risk premium is RPM = kM – kRF = 15% – 8% = 7%. ki = 8.0% + (15.0% - 8.0%)(1.30) = 8.0% + (7.0%)(1.30) = 8.0% + 9.1% = 17.10% 5-28 What is the market risk premium? Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk. Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion. 5-29 An example: Equally-weighted two-stock portfolio Create a portfolio with 50% invested in HT and 50% invested in Collections. The beta of a portfolio is the weighted average of each of the stock’s betas. βP = w1 β1 + w2 β2 βP = 0.5 (1.30) + 0.5 (-0.87) βP = 0.215 5-30 Factor that shifts the SML What if investors raise inflation expectations by 3%, what would happen to the SML? ki (%) D I = 3% SML2 SML1 18 15 11 8 Risk, βi 0 0.5 1.0 1.5 5-31 Factors that change the SML What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML? ki (%) D RPM = 3% SML2 SML1 18 15 11 8 Risk, βi 0 0.5 1.0 1.5 5-32