4106 Advanced Investment Management Asset Pricing Models session 8

1. 4106 Advanced Investment ManagementAsset Pricing Modelssession 8 Andrei Simonov Asset Pricing Models

2. Agenda • CAPM & C-CAPM. • Testing CAPM(s) • Fama & French evidence. • APT & multifactor models Asset Pricing Models

3. Assumptions: • Single holding period • Investors are risk-averse • Investors are ”small” • The information about asset payoffs is common knowledge • Assets are in unlimited supply • Assets are perfectly divisible • No transaction cost • Wealth W is invested in assets Asset Pricing Models

4. E(Ri) - r Reminder on Optimal diversification at individual investor level: condition of optimality • How can you tell whether a portfolio p is well diversified or efficient? • For each security i, E(Ri) - r must be lined up with cov(Ri,Rp) or, equivalently, with: i = cov(Ri,Rp)/var(Rp) i Asset Pricing Models

5. Market level: syllogism port 1 E(Ri) - r avg 1+2 • All weighted averages of efficient portfolios are efficient • Assume each person holds an efficient portfolio • At equilibrium, the market portfolio, m, is an average of individually held portfolios • Therefore, the market portfolio must be efficient port 2 cov(Ri,Rp) Asset Pricing Models

6. Market level: Math (1) • Conditions of optimality (Remember Microeconomics?) Risk Aversion Asset Pricing Models

7. Market level: Math (2) • But Market Portfolio is a legidimate asset, • Market price of risk: Asset Pricing Models

8. E(Ri) - r i Market equilibrium • For each security i, E(Ri) - r must be lined up with cov(Ri,Rm) or, equivalently, with: i = cov(Ri,Rm)/var(Rm) • CAPM can be extended to the case in which there exists no risk-less asset. m = mvar(Rm) = E(Rm) - r Asset Pricing Models

9. Extensions • No risk-free assets • No short sales • Different lending and borrowing rate • Restricted opportunity sets (Hietala, JF89) • Personal taxes • Multi-period extensions • Liquidity The simple form of CAPM is rather robust. Modification of basic assumptions leads to changes in existing terms and appearance of new terms (”induced” factors) However, sumultaneous modification of multiple assumptions leads to SERIOUS departure from standard CAPM. Asset Pricing Models

10. C-CAPM • Individuals have preferences over consumption C described by CRRA u=-C1-g • Certainty case: marginal utility of consumption today =discounted marginal utility of consumption tomorrow times teturn of asset ri: C-g(t)=[(1+ri)/(1+r)] C-g(t+1) • In case of uncertainty C-g(t)=E[[(1+ri)/(1+r)] C-g(t+1)] • Introducing consumption growth g=C(t+1)/C(t)-1 Asset Pricing Models

11. C-CAPM(2) Asset Pricing Models

12. C-CAPM: Equity premium puzzle • Mehra&Prescott(85); Mankiv &Zeldes(91): • 1890-1979[1948-88]: Risk premium=0.06 [.08] • Std of consumption growth =0.036 [0.014] • Std of market returns=0.167 [0.14] • Correlation between consumption growth and market returns = 0.40 [0.45] • 0.06=g*0.40*0.167*.036 => g=25 [90] Asset Pricing Models

13. Equity premium puzzle • Gamble: take 20% paycut if state of the world is ”bad” (prob=1/2) and stay at your current salary in good state, or agree on permanent cut of X%: • 0.5*(0.81-g+1)=x1-g Asset Pricing Models

14. Equity premium puzzle • Gamble: take 20% paycut if state of the world is ”bad” (prob=1/2) and stay at your current salary in good state, or agree on permanent cut of X%: • 0.5*(0.81-g+1)=x1-g • If g=25 then x=17.7% • Realistic estimate for gamma=3 Asset Pricing Models

15. Is CAPM right? • Content: cross-sectional relationship • when comparing securities to each other, linear, positive-slope relationship of mean excess return (risk premium) with beta • zero intercept • no variable, other than beta, matters as a measure of risk Asset Pricing Models

16. Is CAPM right? • How can you tell? • Two-pass approach • for each security, measurement of mean excess return and beta using history of returns (time series) • relate mean excess return to beta (cross section) • First pass has no economic meaning, just a measurement. Second pass is embodiment of CAPM. Asset Pricing Models

17. Example Asset Pricing Models

18. First pass: Security A Asset Pricing Models

19. Second pass: CAPM line Intercept = 3.82%Slope m = 5.21%Adjusted R2 = 0.44 G Asset Pricing Models

20. Discussion: CAPM may not be testable 1. the “market” is not observable (Roll critique) 2. should use time-varying version, based on the information set of the investors. The latter is not observable (Hansen and Richard critique). E(Ri) - r A date 2 B B A B date 1 A i Asset Pricing Models

21. There are deviations from CAPM (or ’s) • Fama and French (1992) investigate 100 NYSE portfolios for the period 1963-1990 • The portfolios are grouped into 10 size classes and 10 beta classes • They find that return differential (risk premium) on  is negative (and non significant) • whereas return differential on size is large and significant. Asset Pricing Models

22. “beta is dead ?” Asset Pricing Models

23. Book/market also Asset Pricing Models

24. Recent thinking • Question: is Fama-French evidence reliable? Return differential may come in “waves”. • Perhaps, CAPM is right at each point in time • But CAPM line moves about • When “indicator variables” are used to track these changes over time (such as variables we listed in lecture on TAA), size and B/M no longer show up in CAPM • So, these variables were showing up in the Fama-French analysis, not because CAPM was wrong, but only because movements in the line had not been properly accounted for Asset Pricing Models

25. Other criticism of CAPM • No account of re-investment risk (multi-period aspects) • inter-temporal hedging • No account of investors’ non traded wealth (similar to Roll critique) • when human capital included, revised CAPM holds up better Asset Pricing Models

26. Conclusion on CAPM • If CAPM were right every mean-variance investor could just hold the market portfolio (“index fund”), adjusting the level of risk by mixing it with riskless asset. • If CAPM is not right, there is room for “tilted” index funds. See Dimensional Fund Advisors case. • If CAPM is not right is it because risk is multi-dimensional? Asset Pricing Models

27. Arbitrage Pricing Theory • Large number of securities, finite number of factors • Residuals become irrelevant. Only exposures b to common factors matter for pricing • Dichotomy of risk variables: • some (factors, in finite number) affect all securities • others (residuals, in large number) affect only one security each • Pricing equation: there must exist premia : • Otherwise, could work out “approximate” arbitrage Asset Pricing Models

28. Math: Derivation of APT(1) • Main requirement: No possibility to create something out of nothing • Certainty case: n+1 assets, K factors, n>K, ”0” asset is risk-free one • Consider portfolio of yj0=(1-Skbjk) units of riskless asset and yjk=bjk of risky asset • Looks similar to returns of asset j! Let us exploit it by buying $1 worth of this portfolio and shorting $1 of asset j. No-arbitrage requirement tells that the return on such portfolio should be 0: Asset Pricing Models

29. Math: Derivation of APT(2) • Uncertainty case: • For any small e, there exists at most N assets, N<n, for which arbitrage condition is violated: • The no-arbitrage condition requires that N/n0 as n. Moreover, as Dybvig(1983) shows, the error also decreases as the number of players in economy increases Asset Pricing Models

30. APT & Ideas that have survived • A statistical concept: exposure • beta is an “exposure” to risk • need to keep track of exposures, when constructing a portfolio • use of “exposures” to classify securities and systematize portfolio construction process • beta measures each asset’s contribution to total portfolio risk • idea useful for risk management: need to develop accounting of risk (or breakdown of risks) • but beta needs a generalization: find more common factors • A pricing concept: • In pricing, only common risk factors matter. Other risks can be diversified away. Asset Pricing Models

31. Statistical models: streamlining the investment process • Covariance matrix is huge (many entries) • Leads to imprecisely computed portfolios • Let us impose structure on the matrix • We may have 10000 securities to choose from • In fact, there may be only 20 basic sources of risks • A particular security can be seen as a portfolio of these basic risks and must be identified as such Asset Pricing Models

32. Example: one factor model • One-factor model ( b’s called “loadings” or “exposures”): where residuals i,t are independent • Then: Asset Pricing Models

33. Example: construct common factor Asset Pricing Models

34. How to build a factor? • Eigenvectors of covariance matrix represent the dimensions of risk. • Procedure: solve det(S-lI)=0 • If there dimension of S is k (=4 in our case), then there are up to k unique solutions. Rank them in descending order: l1>l2>l3>l4 • In our case 0.2>>0.014>>0.007>>0.001 • Find eigenvector  that is the solution of S=l1 • In our case it is  =(0.92,0.40,0.02,0.01) those are non-normalized weights! Asset Pricing Models

35. Example: compare correlations Asset Pricing Models

36. Estimating beta of a security • Use model: (example: “market” = 50% large stocks + 50% small stocks) negligible Asset Pricing Models

37. Extension: multi-factor statistical models • Multi-factor model: • When I hold security i, I am truly holding b1,i of risk #1, b2,i of risk #2 etc.. • Then: Asset Pricing Models

38. Estimating Statistical Factor Models: three approaches Capture the way in which securities returns move together: • Factor analysis • Factor analysis constructs a set of abstract factors that best explain the estimated covariances • Throws no light on underlying economic determinants of the covariances • Use of macroeconomic variables • Use of firm specific variables Asset Pricing Models

39. Estimating Statistical Factor Models:macroeconomic variables • Business cycle risk • unanticipated growth in industrial production • Confidence risk • default spread (Baa - Aaa) which is a proxy for unanticipated changes in risk premia • Term premium risk • return on long bonds minus short bonds, which is a proxy for unanticipated shifts in slope of yield curve • Other: Oil prices • Get exposures (loadings) of each stock by multiple regression Asset Pricing Models

40. Estimating Statistical Factor Models: use of firm-specific information • Form factor-mimicking portfolios which capture factors: • Market: RM - r • B/M: High-minus-low (HML): RHML = RH - RL • Size: Small-minus-big (SMB): RSMB = RS - RB • Estimate exposures by regression Asset Pricing Models

41. Example calculation of multi-factor statistical model Asset Pricing Models

42. BARRA and other “quant” shops • Similar approach: • measure exposures onto any number of risk categories (country, industry, small vs. big etc..). • get value of factor return for each category, each time period, by comparing across firms • interpret this month’s factor return as a way of pinpointing the category of firms that are currently most profitable • Up to 80 factors! Asset Pricing Models

43. Some of BARRA variables (due to Rosenberg-McKibben 73) Asset Pricing Models

44. Predicted BETAlook atwww.barra.com Asset Pricing Models

45. Statistical factor models in the standard CAPM vs. Multi-factor pricing models • Multi-factor statistical models can be used to estimate parameters of the standard CAPM. E.g., securities’ betas from exposures times factor betas • One may still use standard CAPM: still one risk premium (single-factor pricing model) • Opposite: several risk premia. “Multi-factor pricing models” Asset Pricing Models

46. Multi-factor pricing models: state-dependent preferences Investor cares about portfolio variance, and also about performance in a recession: • investors try to buy stocks that do well in a recession • this drives down expected return of those stocks beyond the market beta effect (recession < 0): • Might be a way to price political risk Asset Pricing Models

47. Numerical example • Exposures obtained by multiple regression of individual security on the factors, as in statistical models • Prices of risk obtained by second-pass cross-sectional regression like in CAPM. Asset Pricing Models

48. Example calculation: multi-factor pricing model Asset Pricing Models

49. Example: E(R) on security B Asset Pricing Models

50. “Other” justification for this sort of pricing: Arbitrage Pricing Theory • Large number of securities, finite number of factors • Residuals become irrelevant. Only exposures b to common factors matter for pricing • Dichotomy of risk variables: • some (factors, in finite number) affect all securities • others (residuals, in large number) affect only one security each • Pricing equation: there must exist premia : • Otherwise, could work out “approximate” arbitrage Asset Pricing Models