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4106 Advanced Investment Management Asset Pricing Models session 3 Andrei Simonov Asset Pricing Models 1 3/23/2016 Agenda CAPM & C-CAPM. Testing CAPM(s) Fama & French evidence. APT & multifactor models Asset Pricing Models 2 3/23/2016 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 3 3/23/2016 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: E(Ri) - r i = cov(Ri,Rp)/var(Rp) i Asset Pricing Models 4 3/23/2016 Market level: syllogism port 1 All weighted averages of E(Ri) - r avg 1+2 efficient portfolios port 2 are efficient Assume each person holds an efficient cov(Ri,Rp) portfolio At equilibrium, the market portfolio, m, is an average of individually held portfolios Therefore, the market portfolio must be efficient Asset Pricing Models 5 3/23/2016 Market level: Math (1) Conditions of optimality (Remember Microeconomics?) ~ ~ W W0 1 rf j w j ~ rj rf , ~ ~ ~ ~ ~ ~ ~ E u ( W ) E u ' ( W ) r r 0 , j E u ' ( W ) E r r Cov u ' ( W ), rj j f j f max a j ~ ~ bi ~ 2 ai ~ ui (W ) aiW W E Wi 2 bi i i i j a ~ E~ rj rf i E M i bi 1 1 Wm 0Cov~ rM , ~ rj i1 Wm 0Cov~ rM , ~ rj i Risk Aversion Asset Pricing Models ~ ~ E rj rf Cov Wi , ~ rj ~ ai ~ ~ E rj rf E M E rj rf i bi Cov ij wij ~ rj rf , ~ rj Wm 0Cov~ rM , ~ rj ai ~ LHS E W i i b i i ~ RHS Cov W , ~ r 6 3/23/2016 Market level: Math (2) But Market Portfolio is a legidimate asset, 1 1 ~ ~ E rM rf i Wm 0Var rM Cov~ rM , ~ rj ~ i ~ E r r E rM rf j f 1 ~ Var rM 1 ~ ~ ~ E rj rf i Wm 0CovrM , rj i ~ ~ E r r E r r j f j M f Market price of risk: E~ r r E ~ r r M f M f ~ , , E r r ~ ~ M f Var r Std r M M Asset Pricing Models 7 3/23/2016 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) E(Ri) - r m = m var(Rm) = E(Rm) - r i CAPM can be extended to the case in which there exists no risk-less asset. Asset Pricing Models 8 3/23/2016 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 9 3/23/2016 What is expected risk premium? Expected risk premium= Var(rM)/E[W/] Plays central role in any discussion about the market What is that? How to measure it? What will it tell us about mankind and economy (or Asset Pricing Model)? – Historical perspective – Equilibrium perspective Asset Pricing Models 10 3/23/2016 Asset Classes Returns: US History Asset Pricing Models 11 3/23/2016 Asset Classes Returns: Swedish History 1 SEK invested in 1900 1000000 100000 10000 DMS Global Sweden Bill TR DMS Global Sweden Inflation DMS Global Sweden Equity TR DMS Global Sweden Bond TR 1000 100 10 1 1900 0.1 Asset Pricing Models 1920 1940 1960 1980 2000 12 3/23/2016 Goetzmann&Jorion: International Evidence Asset Pricing Models 13 3/23/2016 Goetzmann&Jorion: International Evidence Median Market RR 0.75% GDP-weighted RR 4% Asset Pricing Models 14 3/23/2016 Estimates from historical data Ibbotson & Sinquefield (76): Real ERP= 5% Ibbotson & Chen (2000) 4% Fama & French (2002) longer period 4.4% Jagannathan, McGrattan,Scherbina (2000) – 1926-70 ERP=7% – 1971-99 ERP=0.7% Asset Pricing Models 15 3/23/2016 Equilibrium Approach: 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-gt[(1+ri)/(1+r)] C-gt+1 In case of uncertainty C-gtE[(1+ri)/(1+r)] C-gt+1 Introducing consumption growth g=C(t+1)/C(t)-1 Asset Pricing Models 16 3/23/2016 C-CAPM(2) E 1 r 1 g 1 r g i Taylor expansion : 1 ri 1 g 1 ri gg ggri g g (1 g ) 2 g2 Applying Expectatio ns operator : r Eri gE g g E ri E g covri , g g (1 g ) Eg Varg 2 2 If t is small, then quadratic terms can be disregarde d : r Eri gE g g covri , g g (1 g ) Varg 2 This eq. should be true for both risky and riskless assets : g (1 g ) r rf gE g Varg Eri rf g covri , g 2 Asset Pricing Models 17 3/23/2016 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 18 3/23/2016 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*(0.81g+1)=x1g Asset Pricing Models 19 3/23/2016 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*(0.81g+1)=x1g If g=25 then x=17.7% Realistic estimate for gamma=3 Asset Pricing Models 20 3/23/2016 How can we solve it? Habit formation u=-(Ct-Ct-1)1-g – Increases demand for bonds, lower Rf – “Keeping up with the Joneses”: instead C(t-1) there is AVERAGE consumption in the reference group. Idiosynchratic labor risk Disaster states and survivorship bias. Liquidity premium Limited Participation Asset Pricing Models 21 3/23/2016 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 22 3/23/2016 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 23 3/23/2016 Example Excess returns % Market Year index A B C D E F G H I 1 2 3 4 5 6 7 8 9 10 11 12 29.65% -11.91% 14.73% 27.68% 5.18% 25.97% 10.64% 1.02% 18.82% 23.92% -41.61% -6.64% 33.88% -49.87% 65.14% 14.46% 15.67% -32.17% -31.55% -23.79% -4.59% -8.03% 78.22% 4.75% -25.20% 24.70% -25.04% -38.64% 61.93% 44.94% -74.65% 47.02% 28.69% 48.61% -85.02% 42.95% 36.48% -25.11% 18.91% -23.31% 63.95% -19.56% 50.18% -42.28% -0.54% 23.65% -0.79% -48.60% 42.89% -54.39% -39.86% -0.72% -32.82% 69.42% 74.52% 28.61% 2.32% 26.26% -68.70% 26.27% -39.89% 44.92% -3.91% -3.21% 44.26% 90.43% 15.38% -17.64% 42.36% -3.65% -85.71% 13.24% 39.67% -54.33% -5.69% 92.39% -42.96% 76.72% 21.95% 28.83% 18.93% 23.31% -45.64% -34.34% 74.57% -79.76% 26.73% -3.82% 101.67% 1.72% -43.95% 98.01% -2.45% 15.36% 2.27% -54.47% 40.22% -71.58% 14.49% 13.74% 24.24% 77.22% -13.40% 28.12% 37.65% 80.59% -72.47% -1.50% 90.19% -26.64% 18.14% 0.09% 8.98% 72.38% 28.95% 39.41% 94.67% 52.51% -80.26% -24.46% First pass Mean 8.12% 5.18% 4.19% 2.75% 6.15% 8.05% 9.90% 11.32% 13.11% 22.83% 1 -0.470 0.594 0.417 1.379 0.901 1.777 0.663 1.911 2.083 8.997% -0.631% -0.636% -5.051% 0.729% -4.528% 5.936% -2.410% 5.918% a Asset Pricing Models 24 3/23/2016 First pass: Security A 100.00% = -0.4704 a= 8.997% 80.00% excess return on security A 60.00% 40.00% 20.00% -50.00% -40.00% -30.00% -20.00% -10.00% 0.00% 0.00% 10.00% 20.00% 30.00% 40.00% -20.00% -40.00% -60.00% excess return on market Asset Pricing Models 25 3/23/2016 Second pass: CAPM line 25.00% Intercept = 3.82% Slope m = 5.21% Adjusted R2 = 0.44 mean excess return of each security 20.00% I 15.00% H aG 10.00% G F E market A D 5.00% B C 0.00% -1 -0.5 0 0.5 1 1.5 2 2.5 beta of each security Asset Pricing Models 26 3/23/2016 Discussion: CAPM may not be testable 1. the “market” is not observable (Roll critique) 2. should use timevarying version, based on the information set of the investors. The latter is not observable (Hansen and Richard critique). 30.00% E(Ri) - r date 2 20.00% B 15.00% A B B 10.00% date 1 A 5.00% i 0.00% 0 Asset Pricing Models A 25.00% 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 27 3/23/2016 There are deviations from CAPM (or a’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 28 3/23/2016 “beta is dead ?” Average Returns (in %/year) for portfolios Sorted on Size and Betas All firms Low Beta High Beta All firms 15.0% 16.1% 13.7% Small cap 18.2% 20.5% 17.0% Large cap 10.7% 12.1% 6.7% Asset Pricing Models 29 3/23/2016 Book/market also Average Returns (in %/year) for portfolios Sorted on Size and B/M Ratios All firms Low B/M High B/M (“Growth”) (“Value”) All firms 14.8% 7.7% 19.6% Small cap 17.6% 8.4% 23.0% Large cap 10.7% 11.2% 14.2% Asset Pricing Models 30 3/23/2016 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 FamaFrench analysis, not because CAPM was wrong, but only because movements in the line had not been properly accounted for Asset Pricing Models 31 3/23/2016 Other criticism of CAPM No account of re-investment risk (multiperiod 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 32 3/23/2016 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 multidimensional? Asset Pricing Models 33 3/23/2016 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 E ( Ri ) 0 b1,i 1b2,i 2.. Asset Pricing Models 34 3/23/2016 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 K ~ rj a j k 1 jk Fk , j 1,2,..., n Consider portfolio of yj0=(1-Skjk) units of riskless asset and yjk=jk of risky asset K K ~ rp 1 k 1 jk rf k 1 jk Fk 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: 1 K K K ~ a j 1 k 1 jk rf Asset Pricing Models ~ r k 1 jk Fk a j k 1 jk Fk 0 k 1 jk f K 35 3/23/2016 Math: Derivation of APT(2) Uncertainty case: ~ rj a j k 1 jk Fk j , j 1,2,..., n K For any small e, there exists at most N assets, N<n, for which arbitrage condition is violated: a j 1 k 1 jk rf e , j 1,2,..., N (n) K The no-arbitrage condition requires that N/n0 as n. Moreover, as Dybvig(1983) shows, the error also decreases as the number of players in economy increases Asset Pricing Models 36 3/23/2016 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 37 3/23/2016 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 38 3/23/2016 Example: one factor model One-factor model ( b’s called “loadings” or “exposures”): Ri,t ai bi Ft e i,t where residuals ei,t are independent Then: cov( Ri , R j ) bi b j var F ; i j var( Ri ) bi 2 var F vare i Asset Pricing Models 39 3/23/2016 Example: construct common factor 10000 Small stocks Large stocks 1000 Long-term T-Bonds Intermediate-term T-Bonds T-Bills Inflation 100 Re-scaled First factor 10 1 1920 1930 1940 1950 1960 1970 1980 1990 2000 0.1 Asset Pricing Models 40 3/23/2016 How to build a factor? Eigenvectors of covariance matrix represent the dimensions of risk. Procedure: solve det(S-I)=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: 1>2>3>4 In our case 0.2>>0.014>>0.007>>0.001 Find eigenvector that is the solution of S1 In our case it is =(0.92,0.40,0.02,0.01) those are non-normalized weights! Asset Pricing Models 41 3/23/2016 Example: compare correlations Correlation original Small stocks - Tbills Large stocks - Tbills Long-term T-Bonds - Tbills Intermediate-term T-Bonds - Tbills Small stock s - Tbills Large stock s - Tbills Long-term T-Bonds - Tbills Intermediate-term T-Bonds - Tbills 1.0000000 0.8091950 1.0000000 0.1052384 0.2364890 1.0000000 0.0885737 0.1944899 0.7678739 1.0000000 Correlation reconstructed Small stocks - Tbills Large stocks - Tbills Long-term T-Bonds - Tbills Intermediate-term T-Bonds - Tbills Small stock s - Tbills Large stock s - Tbills Long-term T-Bonds - Tbills Intermediate-term T-Bonds - Tbills 1.0000000 0.8644608 1.0000000 0.1378009 0.1206442 1.0000000 0.1155091 0.1011278 0.0161205 1.0000000 Variances of securities 0.169297376 Asset Pricing Models 0.043319362 0.006016743 0.003450979 42 3/23/2016 Estimating beta of a security Use model: cov( Ri , Rm ) i bi F e i var( Rm ) negligible original betas beta of factor loadings beta from model IntermediateSmall stocks Large stocks Long-term Tterm T-Bonds Tbills Tbills Bonds - Tbills Tbills 1.3587 0.6413 0.0409 0.0257 x = 0.9954 1.3798 0.9954 0.6111 0.9954 0.0363 0.9954 0.0230 1.3735 0.6083 0.0361 0.0229 (example: “market” = 50% large stocks + 50% small stocks) Asset Pricing Models 43 3/23/2016 Extension: multi-factor statistical models Multi-factor model: Ri,t ai b1,i F1,t b2,i F2,t .. e i,t When I hold security i, I am truly holding b1,i of risk #1, b2,i of risk #2 etc.. Then: cov( Ri , Rm ) i b1,i F1 b2,i F2 .. e i var( Rm ) Asset Pricing Models 44 3/23/2016 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 45 3/23/2016 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 46 3/23/2016 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 47 3/23/2016 Example calculation of multifactor statistical model Excess returns % Market index Year 1 2 3 4 5 6 7 8 9 10 11 12 First pass (time series) Mean Exposure to market risk Exposure to I - A risk 29.65% -11.91% 14.73% 27.68% 5.18% 25.97% 10.64% 1.02% 18.82% 23.92% -41.61% -6.64% 8.12% 1 0 Asset Pricing Models I - A risk 56.31% 23.23% -47.00% -14.37% -6.69% 104.55% 60.50% 63.20% 99.26% 60.54% -158.48% -29.21% A B 33.88% -49.87% 65.14% 14.46% 15.67% -32.17% -31.55% -23.79% -4.59% -8.03% 78.22% 4.75% C -25.20% 24.70% -25.04% -38.64% 61.93% 44.94% -74.65% 47.02% 28.69% 48.61% -85.02% 42.95% D 36.48% -25.11% 18.91% -23.31% 63.95% -19.56% 50.18% -42.28% -0.54% 23.65% -0.79% -48.60% E 42.89% -54.39% -39.86% -0.72% -32.82% 69.42% 74.52% 28.61% 2.32% 26.26% -68.70% 26.27% F -39.89% 44.92% -3.91% -3.21% 44.26% 90.43% 15.38% -17.64% 42.36% -3.65% -85.71% 13.24% G 39.67% -54.33% -5.69% 92.39% -42.96% 76.72% 21.95% 28.83% 18.93% 23.31% -45.64% -34.34% 74.57% -79.76% 26.73% -3.82% 101.67% 1.72% -43.95% 98.01% -2.45% 15.36% 2.27% -54.47% H I 40.22% -71.58% 14.49% 13.74% 24.24% 77.22% -13.40% 28.12% 37.65% 80.59% -72.47% -1.50% 90.19% -26.64% 18.14% 0.09% 8.98% 72.38% 28.95% 39.41% 94.67% 52.51% -80.26% -24.46% 17.65% 5.18% 4.19% 2.75% 6.15% 8.05% 9.90% 11.32% 13.11% 22.83% 0 1.1033751 -0.57441 0.750926 0.449096 -0.18206 1.833844 1.034249 1.662516 1.103375 1 -0.616466 0.457581 -0.13089 0.364342 0.424359 -0.02226 -0.14529 0.097397 0.383534 48 3/23/2016 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 49 3/23/2016 Some of BARRA variables (due to RosenbergMcKibben 73) Asset Pricing Models 50 3/23/2016 Predicted BETA look at www.barra.com Asset Pricing Models 51 3/23/2016 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. “Multifactor pricing models” Asset Pricing Models 52 3/23/2016 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): E Ri r m bi,m recession bi,recession – Might be a way to price political risk Asset Pricing Models 53 3/23/2016 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 54 3/23/2016 Example calculation: multifactor pricing model Excess returns % Market index Year First pass (time series) cf. slide 15 Mean 8.12% Exposure to market risk 1 Exposure to I - A risk 0 I - A risk A B C D E F G H I 17.65% 5.18% 4.19% 2.75% 6.15% 8.05% 9.90% 11.32% 13.11% 22.83% 0 1.1033751 -0.57441 0.750926 0.449096 -0.18206 1.833844 1.034249 1.662516 1.103375 1 -0.616466 0.457581 -0.13089 0.364342 0.424359 -0.02226 -0.14529 0.097397 0.383534 Second pass (cross-section) Adjusted R Square 0.50857 Intercept lambda /mkt lambda /I-A 3.07% 6.29% 13.26% Line 9.36% Compare to: CAPM line 9.03% Asset Pricing Models 16.33% 1.84% 5.53% 6.06% 10.73% 7.55% 14.31% 7.65% 14.82% 15.09% 1.37% 6.91% 5.99% 11.00% 8.51% 13.08% 7.27% 13.77% 14.67% 55 3/23/2016 Example: E(R) on security B Risk factor Exposures b Contribution Prices of risk Intercept (should be 3.07% zero really) Market risk -0.57441 6.29% = -3.61% I - A risk 0.457581 13.26% = 6.07% Expected excess return = 5.53% Asset Pricing Models 56 3/23/2016 “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 E ( Ri ) 0 b1,i 1b2,i 2.. Asset Pricing Models 57 3/23/2016 Conclusions on Asset Pricing Models: Securities are analyzed as portfolios of underlying risks Make clear distinction between risk and exposure to risk! Less restrictive approach to asset pricing: several dimensions of risk are priced – general multi-factor model based on incompletely specified investor behavior – APT relies on absence of “approximate” arbitrage Asset Pricing Models 58 3/23/2016 Practicality: Estimation Risk Óptimization results are usually suffering from: – Huge short positions in many assets in no-constraint case. – “Corner” solutions with zero positions in may assets if constraints are imposed (most portfolios have < 3 assets). – Huge positions in obscure markets with small cap – Large shifts in positions when exp. returns or covariances changes just a bit… All of those are coming from one common cause: difficulties in estimation of expected returns. Asset Pricing Models 59 3/23/2016 Another example (GS 2003): Here are forecasts given by “Wall Street Protagonist” (Columns 1 and 2) and set of portfolio weights that are generated by it. RETURN STD DEV Japanese Gov Bonds 4.7% 4.2% EU Gov Bonds 5.1% 3.6% US Gov Bonds 5.2% 4.6% US Equities 5.4% 15.5% Global Fixed income 6.0% 3.6% EU Equities 6.1% 16.6% US Inv Grade Corp Bonds 6.3% 5.4% EAFE 8.0% 15.3% Hedge Fund Portfolio 8.0% 5.2% US High Yield 8.9% 7.3% Private Equity 9.0% 28.9% Emerging Debt 9.0% 17.6% REITs 9.0% 13.0% Japanese Equity 9.5% 19.6% Emerging Equities 11.8% 23.4% Portfolio ER Portfolio Volatility Asset Pricing Models Unconstrained No Short weights Sales -2.02 0.00 -3.21 0.00 -4.84 0.00 -0.11 0.00 14.93 0.00 -2.58 0.00 -3.86 0.00 3.14 0.00 0.58 0.55 -0.10 0.36 0.01 0.00 -0.29 0.00 0.04 0.08 -0.72 0.01 0.03 0.00 4.90% 18.20% 5.10% 8.40% 60 3/23/2016 Equilibrium and individual asset’ expected return From previous section one can expect that ERP is between 4 and 6% and is fairly stable with time One can make forecast for individual assets that are different from long term premium. But by forecasting one asset class, we are implicitly making forecast for other asset classes as well. Asset Pricing Models 61 3/23/2016 Practicality: How to express views? Method is due to Black & Litterman (Goldman Sachs). The core themes: equilibrium returns and views. Investors normally have views/preferences. They are NOT incorporated into optimization process. Views=mathematically expressed preferences of individual investors. Step Action 1 2 3 4 5 6 Calculate equilibrium returns Define weight for news Set target tracking error Set target market exposure Get portfolio weight Examine risk distribution Asset Pricing Models Purpose Set neutral reference point Dampen impact of aggressive news Control risk wrt benchmark control directional effect Find allocation that maximize performance Is risk diversifies? 62 3/23/2016 Equilibrium optimal portfolio Imagine that the investor thin that US is still in recession. Thus, stocks will perform badly, and bonds will perform OK. Mathematically, it is equivalent to assuming that bonds will go up 0.8%, and stocks will drop 2.5% Result: see Table 8: Asset Pricing Models 63 3/23/2016 Updating of discrete probabilities 1. We have a probability estimate for event H: prior probability P(H) 2. New information D is gained 3. Update the estimate using Bayes’ theorem: posterior probability P(H|D) Asset Pricing Models 64 3/23/2016 The Bayes’ theorem The updating is done using the Bayes’ theorem: P( D | H ) P( H ) P( H | D) P( D) Asset Pricing Models 65 3/23/2016 Example: Using Bayes’ theorem 1,5 % of the population suffer from schizophrenia P(S) = 0.015 (prior probability) Brain atrophy is found in – 30 % of the schizophrenic P(A|S) = 0.3 – 2 % of normal people P(A|S) = 0.02 If a person has brain atrophy, the probability that he is schizophrenic (posterior probability) is: P( A | S ) P( S ) P( S | A) P( A | S ) P( S ) P( A | S ) P( S ) 0.3 0.015 0.186 0.3 0.015 0.02 0.985 Picture: Clemen s. 250 Figure: Posterior probability with different prior probabilities. Asset Pricing Models 66 3/23/2016 Updating of continuous distributions Choose a theoretical distribution, P(X=x|), for the physical process of interest. Assess uncertainty about parameter : prior distribution, f() Note: Uncertainty about X has two parts: Observe data x1 Update using Bayes’ theorem: posterior distribution of , f(|x1) Asset Pricing Models 1. Due to the process itself, P(X=x|). 2. Uncertainty about , f(), later updated to f(|x1). 67 3/23/2016 Updating of continuous distributions Bayes’ theorem for continuous : f ( | x1 ) f ( x1 | ) f ( ) f ( x1 | ) f ( )d f(x1|) is called the likelihood function of with a given observed data x1. In most cases the posterior distribution can not be calculated analytically, but must be solved numerically. Asset Pricing Models 68 3/23/2016 Normal distribution 1. The physical process of interest is normal distributed: X ~ N(, 2) ( is assumed to be known) 2. Prior distribution for : ~ N(m0, 20) (notation: 20 = 2 / n0) 3. Observe a sample of the physical process: – sample size: n1 – sample mean: x1 4. The posterior distribution, calculated using the Bayes’ theorem, gets reduced to: m0 2 n1 x1 02 n0 m0 n1 x1 m* 2 2 ~ N(m*, 2*), where n1 0 n0 n1 2 2 n1 2 2 0 * 2 2 n1 0 n0 n1 Asset Pricing Models 69 3/23/2016 Expressing Views Thus, one need to specify set of “views” and “precisions” of views for each asset f(x|). “No views” is equivalent to having x For this case posterior=prior. Model will deviate further for assets where views are stronger. All assets are affected: Asset Pricing Models 70 3/23/2016 Further risk control & results: Minimize tracking error Mkt. Exposure of the portfolio () (neutral should be 1) Look at diversification (are all eggs in one basket?) Results (according to GS, 95-97): (+103 bp, +83 bp, -26bp) Asset Pricing Models 71 3/23/2016 Other uses: Black-Litterman model is essentially Tactical Asset Allocation model (provided that algorithm of selecting “views” is specified). But it can be used effectively in updating priors on the distribution of the signals. It can be used to bring in new asset classes for which the recorded history is short or unreliable (venture capital funds, hedge funds, emerging markets, etc.) Asset Pricing Models 72 3/23/2016