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Diversification and the CAPM The relationship between risk and expected returns 1 ©1999 Thomas A. Rietz Introduction Investors are concerned with – Risk – Returns What determines the required compensation for risk? It will depend on – The risks faced by investors – The tradeoff between risk and return they face 2 ©1999 Thomas A. Rietz Agenda Concepts of risk for – A single stock – Portfolios of stocks Risk for the diversified investor: Beta – Calculating Beta The relationship between Beta and Return: The Capital Asset Pricing Model (CAPM) 3 ©1999 Thomas A. Rietz Overview Investors demand compensation for risk – If investors hold “diversified” portfolios, risk can be defined through the interaction of a single investment with the rest of the portfolios through a concept called “beta” The CAPM gives the required relationship between “beta” and the return demanded on the investment! 4 ©1999 Thomas A. Rietz Vocabulary Expected return: – What we expect to receive on average Standard deviation of returns: – A measure of risk appropriate for diversified investors Correlation – The tendency for two returns to fall above or below the expected return a the same or different times Diversified investors – Investors who hold a portfolio of many investments – A measure of dispersion of actual returns Beta The Capital Asset Pricing Model (CAPM) – The relationship between risk and return for diversified investors 5 ©1999 Thomas A. Rietz Measuring Expected Return We describe what we expect to receive or the expected return: E (r ) pi ri i – Often estimated using historical averages (excel function: “average”). Example: Die Throw Suppose you pay $300 to throw a fair die. You will be paid $100x(The Number rolled) The probability of each outcome is 1/6. The returns are: – (100-300)/300 = -66.67% – (200-300)/300 = -33.33% …etc. The expected return E(r) is: – 1/6x(-66.67%) + 1/6x(-33.33%) + 1/6x0% + 1/6x33.33% + 1/6x66.67% + 1/6x100% = 16.67%! Example: IEM Suppose – You buy and AAPLi contract on the IEM for $0.85 – You think the probability of a $1 payoff is 90% The returns are: – (1-0.85)/0.85 = 17.65% – (0-0.85)/0.85 = -100% The expected return E(r) is: – 0.9x17.65% - 0.1x100% = 5.88% Example: Market Returns Recent data from the IEM shows the following average monthly returns from 5/95 to 10/99: – (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html) AAPL Average Return 2.42% IBM MSFT SP500 T-Bills 3.64% 4.72% 1.75% 0.35% Month Oct-99 Jul-99 Apr-99 Jan-99 Oct-98 Jul-98 Apr-98 Jan-98 Oct-97 Jul-97 Apr-97 Jan-97 Oct-96 Jul-96 Apr-96 Jan-96 Oct-95 Jul-95 Apr-95 Value of Investment Growth of $1000 Investments $14,000 $12,000 $10,000 $8,000 $6,000 AAPL IBM MSFT SP500 T-Bill(2) $4,000 $2,000 $- Measuring Risk: Standard Deviation and Variance Standard Deviation in Returns: pi ri E (r ) 2 i 2 2 2 p r E ( r ) Var ii i i Often estimated using historical averages (excel function: “stddev”) Example: Die Throw Recall the dice roll example: – – – – You pay $300 to throw a fair die. You will be paid $100x(The Number rolled) The probability of each outcome is 1/6. The expected return E(r) is 16.67%. The standard deviation is: 1 1 2 (66.67% ) (33.33% )2 6 6 1 1 2 (0% ) (33.33% )2 56.93% 6 6 1 1 2 (66.67% ) (100% )2 16.67%2 6 6 Example: IEM Suppose – You buy and AAPLi contract on the IEM for $0.85 – You think the probability of a $1 payoff is 90% The returns are: – (1-0.85)/0.85 = 17.65% – (0-0.85)/0.85 = -100% The expected return E(r) is: – 0.9x17.65% - 0.1x100% = 5.88% The standard deviation is: – [0.9x(17.65%)2 + 0.1x(-100%)2 - 5.88%2]0.5 = 35.29% Example: Market Returns Recent data from the IEM shows the following average monthly returns & standard deviations from 5/95 to 10/99: – (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html) AAPL IBM MSFT SP500 T-Bills Average Return 2.42% 3.64% 4.72% 1.75% 0.35% Std. Dev 14.84% 10.31% 8.22% 3.82% 0.06% Month Oct-99 Jul-99 Apr-99 Jan-99 Oct-98 Jul-98 Apr-98 Jan-98 Oct-97 Jul-97 Apr-97 Jan-97 Oct-96 Jul-96 Apr-96 Jan-96 Oct-95 Jul-95 Apr-95 Value of Investment Growth of $1000 Investments $14,000 $12,000 $10,000 $8,000 $6,000 AAPL IBM MSFT SP500 T-Bill(2) $4,000 $2,000 $- Risk and Average Return 5.0% MSFT 4.5% Average Return 4.0% IBM 3.5% 3.0% AAPL 2.5% 2.0% S&P500 1.5% 1.0% 0.5% T-Bill 0.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Standard Deviation 12.0% 14.0% 16.0% Measures of Association Correlation shows the association across random variables Variables with – Positive correlation: tend to move in the same direction – Negative correlation: tend to move in opposite directions – Zero correlation: no particular tendencies to move in particular directions relative to each other Covariance and Correlation Covariance in returns, AB, is defined as: AB pi rAi E (rA )rBi E (rB ) pi rAi rBi E (rA )E (rB ) i i The correlation, rAB, is defined as: r AB AB A B – rAB is in the range [-1,1] – Often estimated using historical averages (excel function: “correl”) Notation for Two Asset and Portfolio Returns Item Actual Return Expected Return Variance Std. Dev. Asset A rAi E(rA) A 2 A Correlation in Returns Covariance in Returns Asset B rBi E(rB) B2 B rAB AB = ABrAB Portfolio rPi E(rP) P2 P Example: IEM Suppose – You buy an MSFT090iH for $0.85 and a MSFT090iL contract for $0.15. – You think the probability of $1 payoffs are 90% & 10% The expected returns are: – 0.9x17.65% + 0.1x(-100%) = 5.88% – 0.1x566.67% + 0.9x (-100%) = -33.33% The standard deviations are: – [0.9x(17.65%)2 + 0.1x(-100%)2 - 5.88%2]0.5 = 35.29% – [0.1x(566.67%)2 + 0.9x(-100%)2 - (-33.33%)2]0.5 = 200% The correlation is: 0.9 17.65% (-100%) 0.1 566.67% (-100%) - 5.88% (-33.33)% -1 35.29% 200% Example: Market Returns Recent data from the IEM shows the following monthly return correlations from 5/95 to 10/99: – (http://www.biz.uiowa.edu/iem/markets/compdata/compfund.html) AAPL IBM AAPL 1.000 0.262 IBM 1.000 MSFT SP500 T-Bills MSFT 0.102 0.240 1.000 SP500 0.046 0.362 0.550 1.000 T-Bills -0.103 -0.169 -0.073 -0.003 1.000 Correlation of AAPL & IBM $1 y = 0.3777x + 0.0105 Correl = 0.262 $0 $0 IBM Return $0 $0 $-20.00% -10.00% 0.00% $(0) 10.00% $(0) $(0) $(0) AAPL Return 20.00% 30.00% 40.00% Risk and Average Return 5.0% MSFT 4.5% Average Return 4.0% IBM 3.5% 3.0% AAPL 2.5% 2.0% S&P500 1.5% 1.0% 0.5% T-Bill 0.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Standard Deviation 12.0% 14.0% 16.0% Two Asset Portfolios: Risk The standard deviation is not a linear combination of the individual asset standard deviations Instead, it is given by: p w 2A A2 + (1 w A )2 B2 2w A (1 w A ) A B r AB The standard deviation a the 50%/50%, AAPL & IBM portfolio is: p 2 0.52 0.14842 0.52 0.10312 10.08% 2x0.5x0.5 0.1484 0.1031 0.262 The portfolio risk is lower than either individual asset’s because of diversification. Correlations and Diversification Suppose – E(r)A = 16% and A = 30% – E(r)B = 10% and B = 16% Consider the E(r)P and P of securities A and B as wA and r vary... Exp. Ret. Case 1: Perfect positive correlation between securities, i.e., rAB = +1 17% 16% 15% 14% 13% 12% 11% 10% 9% 8% (16%,30%) (10%,16%) 0% 10% 20% Std. Dev. 30% 40% Exp. Ret. Case 2: Zero correlation between securities, i.e., rAB = 0. 17% 16% 15% 14% 13% 12% 11% 10% 9% 8% (16%,30%) Min. Var. (11.33%,14.12%) (10%,16%) 0% 10% 20% Std. Dev. 30% 40% Exp. Ret. Case 3: Perfect negative correlation between securities, i.e., rAB = -1 17% 16% 15% 14% 13% 12% 11% 10% 9% 8% Zero Var. (11.33%,14.12%) (16%,30%) (10%,16%) 0% 10% 20% Std. Dev. 30% 40% Exp. Ret. Comparison 17% 16% 15% 14% 13% 12% 11% 10% 9% 8% (16%,30%) r=1 r=0 r=-1 (10%,16%) 0% 10% 20% Std. Dev. 30% 40% 3 Asset Portfolios: Expected Returns and Standard Deviations Suppose the fractions of the portfolio are given by wAAPL, wIBM and wMSFT. The expected return is: – E(rP) = wAAPLE(rAAPL) + wIBME(rIBM) + wMSFTE(rMSFT) The standard deviation is: w AAPL AAPL w 2 p 2 w MSFT MSFT IBM r AAPL, IBM 2 IBM + 2w AAPL w IBM AAPL 2 IBM 2 + 2w AAPL w MSFT AAPL MSFT r AAPL, MSFT + 2w IBMw MSFT IBM MSFT rIBM,MSFT 2 For the Naively Diversified Portfolio, this gives: 1 1 1 E (RP ) 0.0242 0.0364 0.0472 3.59% 3 3 3 2 p 2 2 2 1 1 1 0.14842 0.10312 0.08222 3 3 3 1 1 + 2 0.1484 0.1031 0.262 3 3 1 1 + 2 0.1484 0.0822 0.102 3 3 1 1 + 2 0.1031 0.0822 0.240 3 3 7.75% For the Naively Diversified Portfolio, this gives: 5.0% MSFT 4.5% Average Return 4.0% Naive Portfolio 3.5% IBM 3.0% AAPL 2.5% 2.0% S&P500 1.5% 1.0% 0.5% T-Bill 0.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% Standard Deviation 12.0% 14.0% 16.0% The Concept of Risk With N Risky Assets As you increase the number of assets in a portfolio: – the variance rapidly approaches a limit, – the variance of the individual assets contributes less and less to the portfolio variance, and – the interaction terms contribute more and more. Eventually, an asset contributes to the risk of a portfolio not through its standard deviation but through its correlation with other assets in the portfolio. This will form the basis for CAPM. Variance of a naively diversified portfolio of N assets Portfolio variance consists of two parts: – 1. Non-systematic (or idiosyncratic) risk and – 2. Systematic (or covariance) risk p 2 1 2 i n Non systematic risk 1 1 ij n Systematic risk The market rewards only systematic risk because diversification can get rid of nonsystematic risk Naive Diversification 80% r.5 r.2 r0 60% 40% 20% Number of Assets 100 91 82 73 64 55 46 37 28 19 10 0% 1 Var. of Portfolio 100% 26 Risky Assets Over a 10 Year Period 6% 5% MSFT Expected Return 4% IBM U 3% 2% 1% 0% 0.00% -1% JNJ XRX HWP GE WMT VIA AAPL F CAT QUIZ S OAT EK BA T NOVL P K DE LE YES R 5.00% 10.00% 15.00% Z -2% Standard Deviation in Return 20.00% Consider Naive Portfolios of 1 through all 26 of these Assets (Added in Alphabetical Order) 14% 12% 10% 8% Standard Devaition 6% 4% 2% Number of Stocks in Portfolio 25 23 21 19 17 15 13 11 7 5 3 9 Average Monthly Return 0% 1 Expected Portfolio Return and Standard Deviation 16% The Capital Asset Pricing Model CAPM Characteristics: – bi = imrim/m2 Asset Pricing Equation: – E(ri) = rf + bi[E(rm)-rf] CAPM is a model of what expected returns should be if everyone solves the same passive portfolio problem CAPM serves as a benchmark – Against which actual returns are compared – Against which other asset pricing models are compared CAPM Assumptions No transactions costs No taxes Infinitely divisible assets Perfect competition – No individual can affect prices Only expected returns and variances matter – Quadratic utility or – Normally distributed returns Unlimited short sales and borrowing and lending at the risk free rate of return Homogeneous expectations Feasible portfolios with N risky assets Feasible Set Expected return (Ei) Efficient frontier Std dev (i) Dominated and Efficient Portfolios Expected return (Ei) B C A Std dev (i) How would you find the efficient frontier? 1. Find all asset expected returns and standard deviations. 2. Pick one expected return and minimize portfolio risk. 3. Pick another expected return and minimize portfolio risk. 4. Use these two portfolios to map out the efficient frontier. Utility Maximization Expected return (Ei) Utility maximizing risky-asset portfolio D Std dev (i) Utility maximization with a riskfree asset Expected return (Ei) E M D Std dev (i) Three Important Funds The riskless asset has a standard deviation of zero The minimum variance portfolio lies on the boundary of the feasible set at a point where variance is minimum The market portfolio lies on the feasible set and on a tangent from the riskfree asset A world with one riskless asset and N risky assets Expected return (Ei) Riskless asset All risky assets and portfolios Market Portfolio Minimum Variance Portfolio Efficient frontier Std dev (i) Tobin’s Two-Fund Separation When the riskfree asset is introduced, All investors prefer a combination of 1) The riskfree asset and 2) The market portfolio Such combinations dominate all other assets and portfolios The Capital Market Line All investors face the same Capital Market Line (CML) given by: E (rm ) rf E (re ) rf e m Equilibrium Portfolio Returns The CML gives the expected return-risk combinations for efficient portfolios. What about inefficient portfolios? – Changing the expected return and/or risk of an individual security will effect the expected return and standard deviation of the market! In equilibrium, what a security adds to the risk of a portfolio must be offset by what it adds in terms of expected return – Equivalent increases in risk must result in equivalent increases in returns. How is Risk Priced? Consider the variance of the market portfolio: Lim N 2 m N X i i 1 i m r im It is the covariance with the market portfolio and not the variance of a security that matters Therefore, the CAPM prices the covariance with the market and not variance per se The CAPM Pricing Equation! The expected return on any asset can be written as: E(Ri ) R f E(Rm ) R f b i i m r im im where b i 2 2 m m This is simply the no arbitrage condition! This is also known as the Security Market Line (SML). Notes on Estimating b’s Let rit, rmt and rft denote historical returns for the time period t=1,2,...,T. The are two standard ways to estimate historical b’s using regressions: – Use the Market Model: rit-rft = ai + bi(rmt-rft) + eit – Use the Characteristic Line: rit = ai + birmt + eit ai = ai + (1-bi)rft and bi = bi Typical regression estimates: – Value Line (Market Model): 5 Yrs, Weekly Data, VW NYSE as Market – Merrill Lynch (Characteristic Line): 5 Yrs, Monthly Data, S&P500 as Market Example Characteristic Line: AAPL vs S&P500 (IEM Data) 50% 40% AAPL Premium 30% 20% y = 0.1844x + 0.0182 R2 = 0.0022 10% 0% -15% -10% -5% 0% -10% -20% -30% -40% S&P500 Premium 5% 10% 15% Example Characteristic Line: IBM vs S&P500 (IEM Data) 40% y = 0.9837x + 0.0191 R2 = 0.1325 30% IBM Premium 20% 10% 0% -15% -10% -5% 0% -10% -20% -30% S&P500 Premium 5% 10% 15% Example Characteristic Line: MSFT vs S&P500 (IEM Data) 30% y = 1.1867x + 0.027 R2 = 0.3032 25% MSFT Premium 20% 15% 10% 5% 0% -15% -10% -5% 0% -5% -10% -15% -20% S&P500 Premium 5% 10% 15% Notes on Estimating b’s Betas for our companies Raw: Adjusted: Avg. R: AAPL 0.1844 0.4563 2.42% IBM 0.9838 0.9891 3.64% MSFT 1.1867 1.1245 4.72% SP500 1 1 1.75% Average Returns vs (Adjusted) Betas 5.00% MSFT 4.50% Average Return 4.00% IBM 3.50% 3.00% 2.50% AAPl 2.00% S&P500 1.50% 1.00% 0.50% T-Bills 0.00% - 0.20 0.40 0.60 Beta 0.80 1.00 1.20 Summary State what has been learned Define ways to apply training Request feedback of training session 64 ©1999 Thomas A. Rietz Where to get more information Other training sessions List books, articles, electronic sources Consulting services, other sources 65 ©1999 Thomas A. Rietz