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Part II. Portfolio Theory and Asset Pricing Models 1 1. Risk and Risk Premiums Rates of Return: Single Period HPR P1 P0 D1 P0 HPR = Holding Period Return P0 = Beginning price P1 = Ending price D1 = Dividend during period one 2 Expected Return and Standard Deviation Expected returns E (r ) p( s)r ( s) s p(s) = probability of a state r(s) = return if a state occurs s = state 3 Scenario Returns: Example State Prob. of State r in State 1 .1 -.05 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35 E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35) E(r) = .15 4 Variance or Dispersion of Returns Variance: p( s ) r ( s) E (r ) 2 2 s Standard deviation = [variance]1/2 Using Our Example: Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2…+ .1(.35-.15)2] Var= .01199 S.D.= [ .01199] 1/2 = .1095 5 EXHIBIT 3.14 Geometric Mean rates of Return and Standard Deviation for Sotheby's Indexes, S&P 500, Bond Market Series, One-Year Bonds, and Inflation Chinese Mod Paint Amer Paint Cont Art Ceramic Imp Paint Eng Furn FW Index S&P500 UW Index VW Index Old Master Fr+Cont Furn Cont 19C Euro Amer Fum Ceramic LBGC Eng Silver Cont Silver 1-Year Bond CPI Standard Deviation EXHIBIT 3.17 Alternative Investment Risk and Return Characteristics Futures Art and Antiques Coins and Stamps Warrants and Options US Common Stocks Commercial Real Estate Foreign Common Stock Foreign Corporate Bonds Real Estate (Personal Home) US Corporate Bonds Foreign Government Bonds US Government Bonds T-Bills Covariance of two random variables • Covariance is defined as: n Cov(r1 , r2 ) p( s )[ r1 ( s ) E (r1 )][ r2 ( s ) E (r2 )] i 1 8 Figure 5.4 The Normal Distribution 9 Figure 5.5A Normal and Skewed Distributions (mean = 6% SD = 17%) 10 Figure 5.5B Normal and Fat-Tailed Distributions (mean = .1, SD =.2) 11 Other measures of risks • Value at Risk(Var) is another name for the quantile of a distribution. The quantile (q) of a distribution is the value below which lie q% of the value • Conditional Tail Expectation (CTE) provides the answer to the question, “Assuming the terminal value of the portfolio falls in the bottom 5% of possible outcomes, what is the expected value?” • Lower Partial Standard Deviation (LPSD) is the standard deviation computed solely from values below the expected return 12 Table 5.5 Risk Measures for Non-Normal Distributions 13 Figure 5.6 Frequency Distributions of Rates of Return for 1926-2005 14 Table 5.3 History of Rates of Returns of Asset Classes for Generations, 1926- 2005 15 Arithmetic and Geometric means • Arithmetic mean _ n 1 r rt n i 1 • Geometric mean _ r n Rn Rn (1 r1 )(1 r2 )...( 1 rn ) 16 Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000 17 Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000 18 Risk Aversion measured with Utility Function 1 U E (r ) A 2 2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion 2 = variance of returns 19 Figure 6.2 The Indifference Curve: a curve indicating the same utility level 20 Table 6.3 Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4 21 Indifference curve • The curvature of indifference curve indicates the riskiness of the investor; more steeper means more risk averse • When indifference curve moves to the north- west, it implies higher utility for most investors 22 The Risk-Free Asset • Only the government can issue default-free bonds – Guaranteed real rate only if the duration of the bond is identical to the investor’s desire holding period • T-bills viewed as the risk-free asset – Less sensitive to interest rate fluctuations 23 Figure 6.3 Spread Between 3-Month CD and T-bill Rates 24 2. Portfolio Theory: Portfolios of One Risky Asset and a Risk-Free Asset • It’s possible to split investment funds between safe and risky assets. • Risk free asset: proxy; T-bills • Risky asset: stock (or a portfolio) 25 Example Using Chapter 6.4 Numbers rf = 7% rf = 0% E(rp) = 15% p = 22% y = % in p (1-y) = % in rf 26 Expected Returns for Combinations E (rc ) yE (rp ) (1 y)rf rc = complete or combined portfolio For example, y = .75 E(rc) = .75(.15) + .25(.07) = .13 or 13% 27 Combinations Without Leverage If y = .75, then c = .75(.22) = .165 or 16.5% If y = 1 c = 1(.22) = .22 or 22% If y = 0 c = (.22) = .00 or 0% 28 Capital Allocation Line with Leverage Borrow at the Risk-Free Rate and invest in stock. Using 50% Leverage, rc = (-.5) (.07) + (1.5) (.15) = .19 c = (1.5) (.22) = .33 29 Figure 6.4 The Investment Opportunity Set with a Risky Asset and a Risk-free Asset in the Expected Return-Standard Deviation Plane 30 Risk Tolerance and Asset Allocation • The investor must choose one optimal portfolio, C, from the set of feasible choices – Trade-off between risk and return – Expected return of the complete portfolio is given by: E (rc ) rf y E (rP ) rf – Variance is: y 2 C 2 2 P 31 Table 6.5 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4 32 Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y 33 Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves 34 Mathematically, maximize utility function of investing to get optimum portfolio weights: 1 Maximize U E (rc ) A c2 2 Subject to: E ( rc ) r f y ( E ( rp ) r f ) and c y p 35 The solution is: E (rp ) rf y A p 2 -if risk premium of investing in portfolio p is higher, y is higher -if the investor is more risk averse, less y -if the stock portfolio p is more risky, less y 36 Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates 37 3. Portfolio Theory: Portfolio with Two Risky Securities rp wr D D wEr E rP Portfolio Return wD Bond Weight rD Bond Return wE Equity Weight rE Equity Return E (rp ) wD E (rD ) wE E (rE ) 38 Two-Security Portfolio: Risk w w 2wD wE Cov(rD , rE ) 2 P 2 D 2 D 2 E 2 E D = Variance of Security D 2 2 E = Variance of Security E Cov(rD , rE ) = Covariance of returns for Security D and Security E 39 Covariance Cov(rD,rE) = DEDE D,E = Correlation coefficient of returns D = Standard deviation of returns for Security D E = Standard deviation of returns for Security E Correlation Coefficients: Possible Values Range of values for 1,2 + 1.0 > > -1.0 If = 1.0, the securities would be perfectly positively correlated If = - 1.0, the securities would be perfectly negatively correlated p wE E wD D 2 wE wD DE E D 2 2 2 2 2 if DE 1, then P ( wE E wD D ) 2 2 and ( wE E wD D ) 42 Table 7.1 Descriptive Statistics for Two Mutual Funds Three-Security Portfolio E (rp ) w1E (r1 ) w2 E (r2 ) w3 E (r3 ) 2p = w1212 + w2212 + w3232 + 2w1w2 Cov(r1,r2) + 2w1w3 Cov(r1,r3) + 2w2w3 Cov(r2,r3) Table 7.2 Computation of Portfolio Variance From the Covariance Matrix Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions Minimum Variance Portfolio as Depicted in Figure 7.4 • Standard deviation is smaller than that of either of the individual component assets • Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation Correlation Effects • The relationship depends on the correlation coefficient • -1.0 < < +1.0 • The smaller the correlation, the greater the risk reduction potential • If = +1.0, no risk reduction is possible Figure 7.10 The Minimum-Variance Frontier of Risky Assets 52 Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs The Sharpe Ratio • Maximize the slope of the CAL for any possible portfolio, p • The objective function is the slope: E (rP ) rf SP P 54 Mathematically, we solve for the tangent portfolio weights from maximization the sharpe ratio: E ( rp ) r f Sp p subjectto : E ( rp ) w1 E ( r ) w2 E ( r2 ) 1 p w1 1 w2 2 2 12 w1 w2 1 2 2 2 2 2 The solutions are: E ( RD ) E E ( RE )Cov( RD , RE ) 2 wD E ( RD ) E E ( RE ) D [ E ( RD ) E ( RE )]Cov( RD , RE ) 2 2 wE 1 wD 55 Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio Figure 7.8 Determination of the Optimal Overall Portfolio Efficient Frontier with Lending & Borrowing CAL E(r) B Q M A rf F 58 4. Asset Pricing Models: Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) • It is the equilibrium model that underlies all modern financial theory. • Derived using principles of diversification with simplified assumptions. 59 Assumptions • Individual investors are price takers. • Single-period investment horizon. • Investments are limited to traded financial assets. • No taxes and transaction costs. 60 Assumptions (cont’d) • Information is costless and available to all investors. • Investors are rational mean-variance optimizers. • There are homogeneous expectations. 61 Resulting Equilibrium Conditions • All investors will hold the same portfolio for risky assets – market portfolio. • Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value. 62 Resulting Equilibrium Conditions (cont’d) • Risk premium on the market depends on the average risk aversion of all market participants. • Risk premium on an individual security is a function of its covariance with the market. E (ri ) rf i E (ri ) rf 63 Using GE Text Example Continued • Reward-to-risk ratio for investment in market portfolio: Market risk premium E (rM ) rf Market variance M2 • In equilibrium, reward-to-risk ratios of GE and the market portfolio should be equal to each other: E (r ) r GE E (r (r ) f M f Cov ( rGE , rM ) 2 M • And the risk premium for GE: Cov(rGE , rM ) E (rGE ) rf E (rM ) rf 2 M Expected Return-Beta Relationship • CAPM holds for the overall portfolio because: E (rP ) wk E (rk ) and k P wk k k • This also holds for the market portfolio: E (rM ) rf M E (rM ) rf Figure 9.2 The Security Market Line Security Market Line (SML) i = [COV(ri,rm)] / m2 Slope SML = E(rm) - rf = market risk premium SML = rf + i [E(rm) - rf] Betam = [Cov (ri,rm)] / m2 = m2 / m2 = 1 67 Sample Calculations for SML E(rm) - rf = .08 rf = .03 x = 1.25 E(rx) = .03 + 1.25(.08) = .13 or 13% y = .6 E(ry) = .03 + .6(.08) = .078 or 7.8% 68 Graph of Sample Calculations E(r) SML Rx=13% .08 Rm=11% Ry=7.8% 3% .6 1.0 1.25 y x By 69 Disequilibrium Example E(r) SML 15% Rm=11% rf=3% 1.0 1.25 70 Disequilibrium Example • Suppose a security with a of 1.25 is offering expected return of 15%. • According to SML, it should be 13%. • Under-priced: offering higher rate of return for its level of risk. 71 Active investment with beta • Those who think they are able to time the market can do the followings. – Buy high-beta stocks when they think the market is up – Switch to low-beta stock when they fear the market is down 72 Extensions of the CAPM • Zero-Beta Model – Helps to explain positive alphas on low beta stocks and negative alphas on high beta stocks • Consideration of labor income and non-traded assets • Merton’s Multiperiod Model and hedge portfolios – Incorporation of the effects of changes in the real rate of interest and inflation Black’s Zero Beta Model • Absence of a risk-free asset • Combinations of portfolios on the efficient frontier are efficient. • All frontier portfolios have companion portfolios that are uncorrelated. • Returns on individual assets can be expressed as linear combinations of efficient portfolios. Black’s Zero Beta Model Formulation Cov(ri , rP ) Cov(rP , rQ ) E (ri ) E (rQ ) E (rP ) E (rQ ) P Cov(rP , rQ ) 2 Efficient Portfolios and Zero Companions E(r) Q P E[rz (Q)] Z(Q) E[rz (P)] Z(P) Zero Beta Market Model E (ri ) E (rZ ( M ) ) E (rM ) E (rZ ( M ) ) Cov(ri , rM ) 2 M CAPM with E(rz (m)) replacing rf CAPM with Nontraded Assets -To the extent that risk characteristics of private enterprises differ from those of traded assets, a portfolio of traded assets that best hedges the risk of typical private business would enjoy excess demand from the population of private business owners. The price of assets in this portfolio will bid up relative to the CAPM considerations, and the expected returns on these securities will be lower in relation to their systematic risk. Conversely, securities highly correlated with such risk will have high equilibrium risk premiums and may appear exhibit positive alphas relative to the conventional SML CAPM with Labor Income • An individual seeking diversification should avoid investing in his employer’s stock and limit investments in the same industry. Thus, the demand for stocks of labor-intensive firms may be reduced, and these stocks may require a higher expected return than predicated by the CAPM • Mayer’s model (9.13) Mayers derives the equilibrium expected return-beta for an economy in which individuals are endowed with labor income of varying size relative to their nonlabor capital. Cov( Ri , RM ) PM Cov( Ri , RH ) PH E ( Ri ) E ( RM ) M P Cov( RM , RH ) 2 P H M Where PH =value of aggregate human capital PM =market value of traded assets (market portolio) RH =excess return on aggregate human capital 80 A Multiperiod Model and Hedge Portfolios • Merton relaxes the “single-period” myopic assumptions about investors. He envisions individuals who optimize a lifetime consumption/investment plan, and who continually adapt consumption/investment decisions to current wealth and planned retirement age • One key parameter is the future risk-free rate. If it falls in some future period, one’s level of wealth will now support a lower stream of real consumption. To the extent that returns on some securities are correlated with changes in the risk-free rate, a portfolio can be formed to hedge such risk, and investors will bid up the prices (and bid down the expected return) of those hedge assets. • Another key parameter is inflation risk. For example, investors may bid up share prices of energy companies that will hedge energy price uncertainty K E ( Ri ) iM E ( RM ) ik E ( Rk ) k 1 Extensions of the CAPM Continued • A consumption-based CAPM – Models by Rubinstein, Lucas, and Breeden • Investor must allocate current wealth between today’s consumption and investment for the future • As a general rule, investors will value additional income more highly during difficult times than in affluent times. An asset will therefore be viewed as riskier in terms of consumption if it has positive covariance with consumption path (9.15) E ( Ri ) iC RPC where RP E ( RC ) E (rC ) rf C Liquidity and the CAPM • Liquidity is prefered by investors • Illiquidity Premium • Research supports a premium for illiquidity. – Amihud and Mendelson – Acharya and Pedersen Figure 9.5 The Relationship Between Illiquidity and Average Returns Acharya and Pedersen proposed the following model: E ( Ri ) kE(Ci ) ( L1 L 2 L3 ) Where E (Ci ) =expected cost of illiquidity k =adjustment for average holding period over all securities =market risk premium net of average market illiquidity cost, =measure of systematic market risk L1 , L 2 , L 3 =liquidity betas 85 Three Elements of Liquidity • Sensitivity of security’s illiquidity to market illiquidity: Cov(Ci , CM ) Var ( RM CM ) L1 • Sensitivity of stock’s return to market illiquidity: Cov( Ri , CM ) L2 Var ( RM CM ) • Sensitivity of the security illiquidity to the market rate of return: Cov(Ci , RM ) L3 Var ( RM CM ) How to estimate beta: Single Index Model rit i i rmt eit ßi = index of a securities’ particular return to the factor m = a broad market index like the S&P 500 is the common factor ei = uncertainty about the firm Where Cov(ei , rm ) 0 Cov(ei , e j ) 0 87 Single-Index Model • Regression Equation: ri rf i i (rm rf ) ei • Portfolio n n R p wi Ri wi [ i i Rm ] i 1 i 1 n n wi i ( wi i ) Rm p p Rm i 1 i 1 88 Single-Index Model Continued • Systematic and unsystematic risk – Total risk = Systematic risk + Firm-specific risk (or unsystematic risk): i2 i2 M 2 (ei ) 2 89 Index Model and Diversification • Portfolio’s variance: (eP ) 2 P 2 P 2 M 2 • Variance of the equally weighted portfolio of firm-specific components: 2 n 1 2 1 2 (eP ) (ei ) (e) 2 i 1 n n • When n gets large, 2 (eP ) becomes negligible 90 Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient βp in the Single-Factor Economy 91 Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006 92 Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic Line (SCL) for HP 93 Table 8.1 Excel Output: Regression Statistics for the SCL of Hewlett-Packard 94 Alpha and Security Analysis • Macroeconomic analysis is used to estimate the risk premium and risk of the market index • Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances 95 Optimal Risky Portfolio of the Single-Index Model • Maximize the Sharpe ratio by choosing weights – Expected return, SD, and Sharpe ratio: n 1 n 1 E ( RP ) P E ( RM ) P wi i E ( RM ) wi i i 1 i 1 1 1 2 n 1 2 n 1 2 2 2 P P M (eP ) M wi i wi (ei ) 2 2 2 2 i 1 i 1 E ( RP ) SP P 96 The Information Ratio • The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy): 2 A sM 2 2 sP (e A ) 97 Treynor-Black Allocation CAL E(r) CML P A M Rf Arbitrage Pricing Theory and Multifactor Models of Risk and Return Single Factor Model Equation ri E (ri ) i F ei Or alternatively, ri i i F1 ei ri = Return for security i i = Factor sensitivity or factor loading or factor beta F = Surprise in macro-economic factor (F could be positive, negative or zero) ei = Firm specific events Arbitrage Pricing Theory Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit • Since no investment is required, an investor can create large positions to secure large levels of profit • In efficient markets, profitable arbitrage opportunities will quickly disappear Arbitrage Pricing Theory (APT) • Assume factor model such as ri i i F1 ei • And no arbitrage opportunities exist in equilibrium • Then, we have E (ri ) 0 1 i 102 APT and CAPM Compared • APT applies to almost all individual securities • With APT it is possible for some individual stocks to be mispriced - not lie on the SML • APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio • APT can be extended to multifactor models Multifactor APT • Use of more than a single factor • Requires formation of factor portfolios • What factors? – Factors that are important to performance of the general economy – Fama-French Three Factor Model Two-Factor Model ri E (ri ) i1F1 i 2 F2 ei • The multifactor APR is similar to the one- factor case – But need to think in terms of a factor portfolio Example of the Multifactor Approach • Work of Chen, Roll, and Ross – Chose a set of factors based on the ability of the factors to paint a broad picture of the macro-economy – GDP factor, inflation factor, and interest rate factor Another Example: Fama-French Three-Factor Model • The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums rit i iM RMt iSMB SMBt iHML HMLt eit • Where: – SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess of the return on a portfolio of large stocks – HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book to-market ratio in excess of the return on a portfolio of stocks with a low book- to-market ratio The Multifactor CAPM and the APT • A multi-index CAPM will inherit its risk factors from sources of risk that a broad group of investors deem important enough to hedge • The APT is largely silent on where to look for priced sources of risk Empirical tests of asset pricing models Overview of Investigation • Tests of the single factor CAPM or APT Model • Tests of the Multifactor APT Model • Studies on volatility of returns over time The Index Model and the Single-Factor APT • Test the linear expected return-beta relationship E (ri ) rf i E (rM rf Tests of the CAPM Tests of the expected return beta relationship: • First Pass Regression – Estimate beta, average risk premiums and unsystematic risk • Second Pass: Using estimates from the first pass to determine if model is supported by the data • Most tests do not generally support the single factor model Single Factor Test Results Return % Predicted Actual Beta Roll’s Criticism • The only testable hypothesis is on the efficiency of the market portfolio • CAPM is not testable unless we know the exact composition of the true market portfolio and use it in the tests • Benchmark error Measurement Error in Beta • Statistical property • If beta is measured with error in the first stage, second stage results will be biased in the direction the tests have supported • Test results could result from measurement error Jaganathan and Wang Study • Included factors for cyclical behavior of betas and human capital • When these factors were included the results showed returns were a function of beta • Size is not an important factor when cyclical behavior and human capital are included Table 13.2 Evaluation of Various CAPM Specifications Table 13.4 Determinants of Stockholdings Tests of the Multifactor Model • Chen, Roll and Ross 1986 Study Factors Growth rate in industrial production Changes in expected inflation Unexpected inflation Unexpected Changes in risk premiums on bonds Unexpected changes in term premium on bonds Study Structure & Results • Method: Two -stage regression with portfolios constructed by size based on market value of equity Fidings • Significant factors: industrial production, risk premium on bonds and unanticipated inflation • Market index returns were not statistically significant in the multifactor model Table 13.5 Economic Variables and Pricing (Percent per Month x 10), Multivariate Approach Fama-French Three Factor Model • Size and book-to-market ratios explain returns on securities • Smaller firms experience higher returns • High book to market firms experience higher returns • Returns are explained by size, book to market and by beta Table 13.6 Three Factor Regressions for Portfolios Formed from Sorts on Size and Book-to-Market Ratios (B/M) Interpretation of Three-Factor Model • Size is a proxy for risk that is not captured in CAPM Beta • Premiums are due to investor irrationality or behavioral biases Risk-Based Interpretations • In figure 13.1, it shows that returns on style portfolios (HML or SMB) seem to predict GDP growth, and thus may in fact capture some aspects of business cycle risks • In figure 13.2, it shows the beta of the HML portfolio is lower in good economies while becomes higher in recessions, suggesting also that HML captures some aspects of business cycle risks Zhang (2005) finds that value firms (with high book-to-market ratios) on average have greater amount of tangible capital. Investment irreversibility puts such firms more at risk for economic downturns. In contrast, growth firms are better able to deal with a downturn by deferring investment plans. The greater exposure of high book-to-market firms to recessions will result in higher down-market betas. 126 To quantify this, Petkova and Zhang fit the following model: rHML rMt ei [b0 b1DIVt b2 DEFLT b3TERMt b4TBt ] et t Where: DIV =market dividend yield Default=default spread on corporate bonds (Baa-Ass rates) Term=term structure spread (10-year – 1-year Treasury rates) TB=1-month T-bill rate 127 Figure 13.1 Difference in Return to Factor Portfolios in Year Prior to Above-Average versus Below-Average GDP Growth Figure 13.2 HML Beta in Different Economic States Behavioral Explanations • Market participants are overly optimistic – Analysts extrapolate recent performance too far into the future – Prices on these glamour stocks are overly optimistic – Lower book-to-market on these glamour firms leads to underperformance compared to value stocks • Chan, Karceski and Lakonishok find that B/M ration reflects the past earning growth but not the future, as in Fig. 13.3 La Porta, Lakonishok, Shleifer, and Vishny (2007) demonstrate that growth stocks underperforms value stocks surrounding earnings announcements, suggesting that when news of actual earnings is released to the public, the market is relatively disappointed in stocks is has been predicting as growth firms. 132 Liquidity and Asset Pricing • Acharya and Pedersen – Premiums observed in the three-factor model may be illiquidity premiums, as shown in the first three row of results in Table 13.7 – In table 13.8, it shows that despite that the liquidity adjustments to the market beat are relatively small, accounting for portfolio liquidity materially improves the fit of the model Table 13.7 Properties of Liquidity Portfolios Table 13.8 Estimates of the CAPM With and Without Liquidity Factors Consumption-based Asset Pricing Model (CCAPM) Each individual’s plan is to maximize a utility function of lifetime consumption, and consumption/investment in each period is based on age and current wealth, as well as risk-free rate and the market portfolio’s risk and risk premium E (rM ) rf ACov(rM , rC ) Table 13.10 and Figure 13.7 indicate that Fama-French factors for average returns may in fact reflect the differing consumption risk of those portfolios 137 138 Equity Premium Puzzle • Rewards for bearing risk appear to be excessive • Possible Causes – Predicting returns from realized returns; people underestimated the realized returns in post-war America • Survivorship bias also creates the appearance of abnormal returns in market efficiency studies; see Figure 13.8 Equity Premium Puzzle Period Risk-Free Rate S&P 500 return Equity Premium 1872-1999 4.87 10.97 6.10 1972-1949 4.05 8.67 4.62 1950-1999 6.15 14.56 8.41 Extensions to the CAPM may resolve the equity premium puzzle Constantinides (2008) argues that the standard CAPM can be extended to including habit formation, incomplete markets, the life cycle, borrowing constraints, and other forces of limited stock market participants, to help explain equity premium puzzle 143 Behavioral Explanations of the Equity Premium Puzzle Barberis and Huang (2008) incorporate loss aversion and narrow framing to explain the puzzle. Narrow framing is the idea that investors evaluate every risk they face in isolation. Thus, investors will ignore low correlation of the risk of stock portfolio with other components of wealth, and therefore require a higher risk premium than rational models would predict. Loss aversion also generate larger risk premium 144 Time-Varying Volatility • Stock prices change primarily in reaction to information • New information arrival is time varying • Volatility is therefore not constant through time Stock Volatility Studies and Techniques • Volatility is not constant through time • Improved modeling techniques should improve results of tests of the risk-return relationship • ARCH and GARCH models incorporate time varying volatility Figure 13.5 Estimates of the Monthly Stock Return Variance 1835 - 1987