Factor Model Risk Analysis

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2 Factor Model Risk Analysis Eric Zivot University of Washington BlackRock Alternative Advisors April 29, 2011

3 Outline Factor Model Specification Risk measures Factor Risk Budgeting Portfolio Risk Budgeting Factor Model Monte Carlo

4 Factor Model Specification Factor models for asset returns have the general form R it = α i + β 1i f 1t + β 2i f 2t + + β Ki f Kt + ε it (1) = α i + β 0 i f t + ε it R it is the simple return (real or in excess of the risk-free rate) on asset i (i =1,...,N) in time period t (t =1,...,T), f kt is the k th common factor (k =1,...,K), β ki is the factor loading or factor beta for asset i on the k th factor, ε it is the asset specific factor.

5 Assumptions 1. The factor realizations, f t, are stationary with unconditional moments E[f t ] = μ f cov(f t ) = E[(f t μ f )(f t μ f ) 0 ]=Ω f 2. Asset specific error terms, ε it, are uncorrelated with each of the common factors, f kt, cov(f kt,ε it )=0, for all k, i and t. 3. Error terms ε it are serially uncorrelated and contemporaneously uncorrelated across assets cov(ε it,ε js ) = σ 2 i for all i = j and t = s = 0, otherwise

6 Cross Section Regression The multifactor model (1) may be rewritten as a cross-sectional regression model at time t by stacking the equations for each asset to give R t (N 1) + B f t (N 1) (N K)(K 1) = α β 0 1 B =. (N K) β 0 = N E[ε t ε 0 t f t ] = D = diag(σ 2 1,...,σ2 N ) Note: Cross-sectional heteroskedasticity + ε t (N 1) β 11 β 1K..... β N1 β NK, t =1,...,T (2)

7 Time Series Regression The multifactor model (1) may also be rewritten as a time-series regression model for asset i by stacking observations for a given asset i to give R i (T 1) F (T K) = 1 T = α i (T 1) (1 1) f 0 1. f 0 T = β i (T K) (K 1) + F f 11 f Kt..... f 1T f KT + ε i (T 1), i =1,...,N (3) E[ε i ε 0 i ] = σ2 i I T Note: Time series homoskedasticity

8 Expected Return (α β) Decomposition E[R it ]=α i + β 0 i E[f t] β 0 i E[f t]=explained expected return due to systematic risk factors α i = E[R it ] β 0 i E[f t]=unexplained expected return (abnormal return) Note: Equilibrium asset pricing models impose the restriction α i = 0 (no abnormal return) for all assets i =1,...,N

9 Covariance Structure Using the cross-section regression R t (N 1) + B f t (N 1) (N K)(K 1) = α + ε t (N 1), t =1,...,T and the assumptions of the multifactor model, the (N N) covariance matrix of asset returns has the form Note, (4) implies that cov(r t )=Ω FM = BΩ f B 0 + D (4) var(r it ) = β 0 i Ω fβ i + σ 2 i cov(r it,r jt ) = β 0 i Ω fβ j

10 Portfolio Analysis Let w = (w 1,...,w n ) be a vector of portfolio weights (w i = fraction of wealth in asset i). If R t is the (N 1) vector of simple returns then Portfolio Factor Model R p,t = w 0 R t = NX i=1 w i R it R t = α + Bf t + ε t R p,t = w 0 α + w 0 Bf t + w 0 ε t = α p + β 0 pf t + ε p,t α p = w 0 α, β 0 p = w0 B, ε p,t = w 0 ε t var(r p,t ) = β 0 p Ω fβ p + var(ε p,t )=w 0 BΩ f B 0 w + w 0 Dw

11 Macroeconomic Factor Models Econometric problems: R it = α i + β 0 i f t + ε it f t = observed economic/financial time series Choice of factors Estimate factor betas, β i, and residual variances, σ 2 i, using time series regression techniques. Estimate factor covariance matrix, Ω f, from observed history of factors

12 Risk Measures Let R t be an iid random variable, representing the return on an asset at time t, withpdff, cdff, E[R t ]=μ and var(r t )=σ 2. The most common risk measures associated with R t are 1. Return standard deviation: σ = SD(R t )= q var(r t ) 2. Value-at-Risk: VaR α = q α = F 1 (α), α (0.01, 0.10) 3. Expected tail loss: ETL α = E[R t R t VaR α ],α (0.01, 0.10) Note: VaR α and ETL α are tail-risk measures.

13 Risk Measures: Normal Distribution R t iid N(μ, σ 2 ) R t = μ + σ Z, Z iid N(0, 1) Φ = F Z, φ = f Z Value-at-Risk VaRα N = μ + σ z α, z α = Φ 1 (α) Expected tail loss ETL N α = μ σ 1 α φ(z α)

14 Estimation ˆμ = 1 T TX t=1 d VaR N α = ˆμ +ˆσ z α d ETL N α = ˆμ ˆσ 1 α φ(z α) R t, ˆσ 2 = 1 T 1 TX t=1 (R t ˆμ) 2, ˆσ = q ˆσ 2 d Note: Standard errors are rarely reported for VaR N α and to compute using delta method or bootstrap. d ETL N α, but are easy

15 Risk Measures: Factor Model and Normal Distribution R t = α + β 0 f t + ε t f t iid N(μ f, Ω f ), ε t iid N(0,σ 2 ε), cov(f k,t,ε s )=0for all k, t, s Then E[R t ] = μ FM = α + β 0 μ f var(r t ) = σ 2 FM = β0 Ω f β + σ 2 ε q σ FM = β 0 Ω f β + σ 2 ε VaR N,FM α = μ FM + σ FM z α ETL N,FM 1 α = μ FM σ FM α φ(z α) Note: In practice, α =0is typically imposed so that μ FM = β 0 μ f.

16 Tail Risk Measures: Non-Normal Distributions Stylized fact: The empirical distribution of many asset returns exhibit asymmetry and fat tails Some commonly used non-normal distributions for Skewed Student s t (fat-tailed and asymmetric) Generalized hyperbolic Cornish-Fisher Approximations Extreme value theory: Generalized Pareto

17 Modeling Non-Normal Returns for VaR Calculations R t = μ + σz t, E[R t ]=μ, var(r t )=σ 2 Z t iid (0, 1) with CDF F Z Then VaR q = F 1 (q) =μ + σ FZ 1 (q) normal VaR: Z (q) =N(0,1) quantile Student s t VaR : FZ 1 (q) =Student s t quantile Cornish-Fisher (modified) VaR : FZ 1 (q) =Cornish-Fisher quantile EVT VaR : F 1 (q) =GPD quantile F 1 Z

18 Tail Risk Measures: Cornish-Fisher Approximation Idea: Approximate unknown CDF of Z = (R μ)/σ using 2 term Edgeworth expansion around normal CDF Φ( ) and invert expansion to get quantile estimate: F 1 Z,CF (q) = z q (z2 q 1) skew (z3 q 3z q ) ekurt 1 36 (2z3 q 5z q ) skew z q = Φ 1 (q), skew = E[Z 3 ], ekurt = E[Z 4 ] Note: Very commonly used in industry Reference: Boudt, Peterson and Croux (2008) Estimation and Decomposition of Downside Risk for Portfolios with Nonnormal Returns, Journal of Risk.

19 Tail Risk Measures: Non-parametric estimates Assume R t is iid but make no distributional assumptions: {R 1,...,R T } = observed iid sample Estimate risk measures using sample statistics (aka historical simulation) VaR d HS α = ˆq α = empirical α quantile ETL d HS α = 1 R t 1 {R t ˆq α } [Tα] t=1 1 {R t ˆq α } = 1 if R t ˆq α ; 0 otherwise TX

20 Factor Risk Budgeting Additively decompose (slice and dice) individual asset or portfolio return risk measures into factor contributions Allow portfolio manager to know sources of factor risk for allocation and hedging purposes Allow risk manager to evaluate portfolio from factor risk perspective

21 Factor Risk Decompositions Assume asset or portfolio return R t can be explained by a factor model R t = α + β 0 f t + ε t f t iid (μ f, Ω f ), ε t iid (0,σ 2 ε), cov(f k,t,ε s )=0for all k, t, s Re-write the factor model as R t = α + β 0 f t + ε t = α + β 0 f t + σ ε z t Then = α + β 0 f t β = (β 0,σ ε ) 0, f t =(f t,z t ) 0, z t = ε t σ ε iid (0, 1) σ 2 FM = β 0 Ω f β, Ω f = Ã Ωf 0 0 1!

22 Linearly Homogenous Risk Functions as func- Let RM( β) denote the risk measures σ FM,VaRα FM tions of β and ETL FM α Result 1: RM( β) is a linearly homogenous function of β for RM = σ FM, VaRα FM and ETL FM α. That is, RM(c β) =c RM( β) for any constant c 0 Example: Consider RM( β) =σ FM ( β). Then σ FM (c β) = ³ c β 0 Ω f c β 1/2 = c ³ β0 Ω f β 1/2 = c σ FM ( β)

23 Euler s Theorem and Additive Risk Decompositions Result 2: Because RM( β) is a linearly homogenous function of β, by Euler s Theorem RM( β) = k+1 X j=1 β j RM( β) β j RM( β) = β 1 β 1 RM( β) = β 1 β β k+1 RM( β) β k β k RM( β) β k + σ ε RM( β) σ ε

24 Terminology Factor j marginal contribution to risk RM( β) β j Factor j contribution to risk β j RM( β) β j Factor j percent contribution to risk β j RM( β) β j RM( β)

25 Analytic Results for RM( β) =σ FM ( β) σ FM ( β) = ³ β0 Ω f β 1/2 σ FM ( β) β = 1 σ FM ( β) Ω f β Factor j =1,...,K percent contribution to σ FM ( β) β 1 β j cov(f 1t,f jt )+ + β 2 j var(f jt)+ + β K β j cov(f Kt,f jt ) σ 2 FM ( β), Asset specific factor contribution to risk σ 2 ε σ 2, j = K +1 FM ( β)

26 Results for RM( β) =VaR FM α ( β), ETL FM α ( β) Based on arguments in Scaillet (2002), Meucci (2007) showed that Remarks V arα FM ( β) β j = E[ f jt R t = VaRα FM ( β)], j=1,...,k +1 ETL FM α ( β) β j = E[ f jt R t VaRα FM ( β)], j=1,...,k +1 Intuitive interpretation as stress loss scenario Analytic results are available under normality

27 Marginal Contributions to Tail Risk: Non-Parametric Estimates Assume R t and f t are iid but make no distributional assumptions: {(R 1, f 1 ),...,(R T, f T )} = observed iid sample Estimate marginal contributions to risk using historical simulation Ê HS [ f jt R t = VaR α ]= 1 m TX t=1 f jt 1 Ê HS [ f jt R t VaR α ]= 1 [Tα] ½ VaR d HS TX t=1 α ε R t VaR d HS α f jt 1 ½ VaR d HS α R t ¾ ¾ + ε Problem: Not reliable with small samples or with unequal histories for R t

28 Portfolio Risk Budgeting Additively decompose (slice and dice) portfolio risk measures into asset contributions Allow portfolio manager to know sources of asset risk for allocation and hedging purposes Allow risk manager to evaluate portfolio from asset risk perspective

29 Portfolio Risk Decompositions Portfolio return: R t = (R 1t,...,R Nt ), w t =(w 1,...,w n ) 0 R p,t = w 0 R t = NX i=1 w i R it Let RM(w) denote the risk measures σ, V ar α and ETL α as functions of the portfolio weights w. Result 3: RM(w) is a linearly homogenous function of w for RM = σ, VaR α and ETL α. That is, RM(c w) =c RM(w) for any constant c 0

30 Result 4: Because RM(w) is a linearly homogenous function of w, by Euler s Theorem RM(w) = NX i=1 w i RM(w) w i = w 1 RM(w) w w N RM(w) w N

31 Terminology Asset i marginal contribution to risk RM(w) w i Asset i contribution to risk w i RM(w) w i Asset i percent contribution to risk w i RM(w) w i RM(w)

32 Analytic Results for RM(w) =σ(w) Note Ωw = R p,t = w 0 R t, var(r t )=Ω σ(w) = ³ w 0 Ωw 1/2 σ(w) w = 1 σ(w) Ωw cov(r 1t,R p,t ). cov(r Nt,R p,t ) β i,p = cov(r it,r p,t )/σ 2 (w) = σ (w) β 1,p. β N,p

33 Results for RM(w) =VaR α (w), ETL α (w) Gourieroux (2000) et al and Scalliet (2002) showed that Remarks V ar α (w) w i = E[R it R p,t = VaR α (w)], i=1,...,n ETL α (w) w i = E[R it R p,t VaR α (w)], i=1,...,n Intuitive interpretation as stress loss scenario Analytic results are available under normality and Cornish-Fisher expansion

34 Marginal Contributions to Tail Risk: Non-Parametric Estimates Assume the N 1 vector of returns R t is iid but make no distributional assumptions: {R t,...,r T } = observed iid sample R p,t = w 0 R t Estimate marginal contributions to risk using historical simulation Ê HS [R it R p,t = VaR α ]= 1 m TX t=1 R it 1 Ê HS [R it R p,t VaR α ]= 1 [Tα] ½ VaR d HS TX t=1 α ε R p,t VaR d HS α R it 1 Problem: Very few observations used for estimates ½ VaR d HS α R p,t ¾ ¾ + ε

35 Marginal Contributions to Tail Risk: Cornish-Fisher Expansion Boudt, Peterson and Croux (2008) derived analytic expressions for V ar α (w) w i = E[R it R p,t = VaR α (w)], i=1,...,n ETL α (w) w i = E[R it R p,t VaR α (w)], i=1,...,n based on the Cornish-Fisher quantile expansions for each asset. Results depend on asset specific variance, skewness, kurtosis as well as all pairwise covariances, co-skewnesses and co-kurtosises

36 Factor Model Monte Carlo Problem: Short history and incomplete data limits applicability of historical simulation, and risk budgeting calculations are extremely difficult for nonnormal distributions Solution: FactorModelMonteCarlo(FMMC) Use fitted factor model to simulate pseudo hedge fund return data preserving empirical characteristics Use full history for factors and observed history for asset returns Do not assume full parametric distributions for hedge fund returns and risk factor returns

37 Estimate tail risk and related measures non-parametrically from simulated return data

38 Unequal History f 1T.. f KT. R it. f 1,T Ti +1.. f 1,T Ti +1. R i,t Ti +1 f 11 f 1K Observe full history for factors {f 1,...,f T } Observe partial history for assets (monotone missing data) {R i,t Ti +1,...,R it }, i = 1,...,n; t = T T i +1,...,T

39 Simulation Algorithm Estimate factor models for each asset using partial history for assets and risk factors R it =ˆα i + ˆβ 0 if t +ˆε it, t = T T i +1,...,T Simulate B values of the risk factors by re-sampling with replacement from full history of risk factors {f 1,...,f T }: {f 1,...,f B } Simulate B values of the factor model residuals from empirical distribution or fitted non-normal distribution: {ˆε i1,...,ˆε ib }

40 Create pseudo factor model returns from fitted factor model parameters, simulated factor variables and simulated residuals: {R 1,...,R B } R it = ˆβ 0 if t +ˆε it, t =1,...,B

41 Remarks: 1. Algorithm does not assume normality, but relies on linear factor structure for distribution of returns given factors. 2. Under normality (for risk factors and residuals), FMMC algorithm reduces to MLE with monotone missing data. 3. Use of full history of factors is key for improved efficiency over truncated sample analysis 4. Technical justification is detailed in Jiang (2009).

42 Simulating Factor Realizations: Choices Empirical distribution Filtered historical simulation use local time-varying factor covariance matrices to standardize factors prior to re-sampling and then re-transform with covariance matrices after re-sampling Multivariate non-normal distributions

43 Simulating Residuals: Distribution choices Empirical Normal Skewed Student s t Generalized hyperbolic Cornish-Fisher

44 Reverse Optimization, Implied Returns and Tail Risk Budgeting Standard portfolio optimization begins with a set of expected returns and risk forecasts. These inputs are fed into an optimization routine, which then produces the portfolio weights that maximizes some risk-to-reward ratio (typically subject to some constraints). Reverse optimization, by contrast, begins with a set of portfolio weights and risk forecasts, and then infers what the implied expected returns must be to satisfy optimality.

45 Optimized Portfolios Suppose that the objective is to form a portfolio by maximizing a generalized expected return-to-risk (Sharpe) ratio: μ p (w) max w RM(w) μ p (w) = w 0 μ RM(w) = linearly homogenous risk measure The F.O.C. s of the optimization are (i =1,...,n) 0= Ã! μp (w) 1 μ p (w) = μ p(w) RM(w) w i RM(w) RM(w) w i RM(w) 2 w i

46 Reverse Optimization and Implied Returns Reverse optimization uses the above optimality condition with fixed portfolio weights to determine the optimal fund expected returns. These optimal expected returns are called implied returns. The implied returns satisfy μ implied i (w) = μ p(w) RM(w) RM(w) w i Result: fund i s implied return is proportional to its marginal contribution to risk, with the constant of proportionality being the generalized Sharpe ratio of the portfolio.

47 How to Use Implied Returns For a given generalized portfolio Sharpe ratio, μ implied i (w) is large if RM(w) w i is large. If the actual or forecast expected return for fund i is less than its implied return, then one should reduce one s holdings of that asset If the actual or forecast expected return for fund i is greater than its implied return, then one should increase one s holdings of that asset

48 References [1] Boudt, K., Peterson, B., and Christophe C Estimation and decomposition of downside risk for portfolios with non-normal returns The Journal of Risk, vol. 11, [2] Epperlein, E. and A. Smillie (2006). Cracking VAR with Kernels, Risk. [3] Goldberg, L.R., Yayes, M.Y., Menchero, J., and Mitra, I. (2009). Extreme Risk Analysis, MSCI Barra Research. [4] Goldberg, L.R., Yayes, M.Y., Menchero, J., and Mitra, I. (2009). Extreme Risk Management, MSCI Barra Research.

49 [5] Goodworth,T.andC.Jones(2007). Factor-based,Non-parametricRisk Measurement Framework for Hedge Funds and Fund-of-Funds, The European Journal of Finance. [6] Gourieroux, C., J.P. Laurent, and O. Scaillet (2000). Sensitivity Analysis of Values at Risk, Journal of Empirical Finance. [7] Hallerback, J. (2003). Decomposing Portfolio Value-at-Risk: A General Analysis, The Journal of Risk 5/2. [8] Jiang, Y. (200). Overcoming Data Challenges in Fund-of-Funds Portfolio Management. PhD Thesis, Department of Statistics, University of Washington.

50 [9] Meucci, A. (2007). Risk Contributions from Generic User-Defined Factors, Risk. [10] Scaillet, O. Nonparametric estimation and sensitivity analysis of expected shortfall. Mathematical Finance, 2002, vol. 14, [11] Yamai, Y. and T. Yoshiba (2002). Comparative Analyses of Expected Shortfall and Value-at-Risk: Their Estimation Error, Decomposition, and Optimization, Institute for Monetary and Economic Studies, Bank of Japan.