Attilio Meucci. Issues in Quantitative Portfolio Management: Handling Estimation Risk

Transcription

1 Attilio Meucci Lehman Brothers, Inc., Ne York personal ebsite: symmys.com Issues in Quantitative Portfolio Management: Handling Estimation Risk

2 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

3 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

4 ETIMATION vs. MODELING general conceptual frameork Estimation = Invariance (i.i.d.) Detection Projection Modeling & Optimization time series analysis investment horizon P&L investment decision

5 ETIMATION vs. MODELING fixed-income PCA trading recipe 1. consider N series of T observations of homogeneous forard rates X (TxN panel) X'X 2. define (NxN positive definite matrix) EΛE' 3. run PCA (eigenvectors-eigenvalues-eigenvectors)

6 ETIMATION vs. MODELING fixed-income PCA trading recipe 1. consider N series of T observations of homogeneous forard rates X (TxN panel) X'X 2. define (NxN positive definite matrix) EΛE' 3. run PCA (eigenvectors-eigenvalues-eigenvectors) ( N ) 4. analyze the series y Xe of the last factor i z-score: structural bands i juice : b.p. from mean i roll-don/slide-adjusted prospective harpe ratio i reversion timeframe i market events (e.g. Fed, Thursday numbers, ) i relation ith other series (e.g. oil prices) 5. convert basis points to PnL/risk exposure by dv01 small picture big picture variations: transform series, include mean, support series (PCA-regression),

7 ETIMATION vs. MODELING fixed-income PCA trading recipe estimation projection modeling 1. consider N series of T observations of homogeneous forard rates X (TxN panel) X'X 2. define (NxN positive definite matrix) EΛE' 3. run PCA (eigenvectors-eigenvalues-eigenvectors) ( N ) 4. analyze the series y Xe of the last factor i z-score: structural bands i juice : b.p. from mean i roll-don/slide-adjusted prospective harpe ratio i reversion timeframe i market events (e.g. Fed, Thursday numbers, ) i relation ith other series (e.g. oil prices) 5. convert basis points to PnL/risk exposure by dv01 estimation (backard-looking) and projection/modeling (forard-looking) overlap non-linearities not accounted for

8 ETIMATION vs. MODELING fund of funds flaed management recipe 1. consider N series of T observations of fund prices (TxN panel) 2. consider the compounded returns P ( ) ln ( ) C ln P P tn, tn, t 1, n 3. estimate covariance (e.g. the sample non-central) 4. define the expected values (e.g. risk-premium) T 1 Σ CC t T t = 1 µ γ diag( Σ) t '

9 ETIMATION vs. MODELING fund of funds flaed management recipe 1. consider N series of T observations of fund prices (TxN panel) 2. consider the compounded returns P ( ) ln ( ) C ln P P tn, tn, t 1, n 3. estimate covariance (e.g. the sample non-central) 4. define the expected values (e.g. risk-premium) T 1 Σ CC t T t = 1 µ γ diag( Σ) t ' 5. solve mean-variance: argmax { ' µ } C investment constraints ' Σ v grid of significant variances ( i) 6. choose the most suitable combination among according to preferences

10 ETIMATION vs. MODELING fund of funds flaed management recipe estimation 1. consider N series of T observations of fund prices (TxN panel) 2. consider the compounded returns 3. estimate covariance (e.g. the sample non-central) 4. define the expected values (e.g. risk-premium) P ( ) ln ( ) C ln P P tn, tn, t 1, n T 1 Σ CC t T t = 1 µ γ diag( Σ) t ' modeling & optimization 5. solve mean-variance: argmax { ' µ } C investment constraints ' Σ v grid of significant variances ( i) 6. choose the most suitable combination among according to preferences estimation (backard-looking) and modeling (forard-looking) overlap projection (investment horizon) not accounted for non-linearities of compounded returns not accounted for

11 ETIMATION vs. MODELING fund of funds consistent management recipe iestimation: compounded returns ( ) ln ( ) t t t τ τ C ln P P compounded returns are more symmetric, in continuous time they can be modeled (in first approximation) as a Bronian motion estimation interval

12 ETIMATION vs. MODELING fund of funds consistent management recipe iestimation: compounded returns ( ) ln ( ) t t t τ τ C ln P P compounded returns are more symmetric, in continuous time they can be modeled (in first approximation) as a Bronian motion investment horizon iprojection to investment horizon τ τ τ t = τ t J τ t J τ t C C C C ( 1) compounded returns can be easily projected to the investment horizon because they are additive ( accordion expansion)

13 ETIMATION vs. MODELING fund of funds consistent management recipe iestimation: compounded returns ( ) ln ( ) t t t τ τ C ln P P compounded returns are more symmetric, in continuous time they can be modeled (in first approximation) as a Bronian motion iprojection to investment horizon τ = τ + τ + + τ t t J τ t J τ t C C C C ( 1) compounded returns can be easily projected to the investment horizon because they are additive ( accordion expansion) imodeling: linear returns τ L P / P 1 t t t τ linear returns are related to portfolio quantities (P&L): L Π = 'L portfolio return securities returns securities relative eights compounded returns are NOT related to portfolio quantities (P&L): C Π 'C

14 ETIMATION vs. MODELING fund of funds consistent management recipe iestimation: compounded returns sample/risk-premium: iprojection to investment horizon square root rule : ( τ ) τ T 1 τ τ τ Σ ' diag CtCt µ γ Σ T t= 1 ( ) ln ( ) τ C ln P P t t t τ τ τ τ t τ t J τ t J τ t C C C C Σ τ τ τ τ τ τ Σ µ τ µ ( 1) τ imodeling: linear returns τ L P / P 1 t t t τ Black-choles assumption: (log-normal) m n { τ } tn, E L = e τ τ 1 µ n+ Σ τ 2 τ nn τ τ 1 τ τ 1 τ µ { } n+ Σ nn+ µ m+ Σ τ τ mm Σ τ τ 2 2 nm τ τ nm Cov Ltn,, Ltm, = e e 1 the mean - variance optimization can be fed ith the appropriate inputs

15 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

16 CLAICAL OPTIMIZATION mean-variance in theory argmax subject to { ' m} C ' N 1 v vector N N matrix : relative portfolio eights C : set of investment constraints, e.g. v : significant grid of target variances 1 ' = 1, 0 m E { τ L } t +τ { τ } t Cov L +τ

17 CLAICAL OPTIMIZATION mean-variance in practice argmax subject to { ' m} C ' v { ' m} argmax subject to C ' v : relative portfolio eights C v m : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances E { τ L } t +τ Cov { L τ } t+τ m : estimate of e.g. m : estimate of e.g. m 1 T τ lt t= 1 T T 1 ( τ ) τ lt m lt m T t= 1 ( ) '

18 CLAICAL OPTIMIZATION estimation risk The true optimal allocation is determined by a set of parameters that are estimated ith some error: θ ( m, ) θ ( m,) space of possible parameters values true (unknon) θ point estimate θ

19 CLAICAL OPTIMIZATION estimation risk The true optimal allocation is determined by a set of parameters that are estimated ith some error: θ ( m, ) θ ( m,) space of possible parameters values true (unknon) θ point estimate θ The classical optimal allocation based on point estimates sub-optimal θ ( m, ) More importantly, the sub-optimality due to estimation error is large (Jobson & Korkie (1980); Best & Grauer (1991); Chopra & Ziemba (1993)) is

20 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

21 BLACK-LITTERMAN APPROACH inputs: prior linear returns L N ( µ, Σ) unconstrained Markoitz mean-variance optimization 1 ς argmax ' µ ' Σ 2ς ell-diversified portfolio (equilibrium - benchmark) ς 1 ς Σ µ 1 µ Σ ς ς market-implied equilibrium prior expected returns average risk propensity ~ 0.4 exponentially smoothed estimate of covariance

22 BLACK-LITTERMAN APPROACH outputs: posterior official prior on linear returns L, N ( µ Σ) equilibrium-based estimation subjective vies V V ( 2 q ω ) N, p p µ µ ( 2 q ω ) N, K K K K Bayesian posterior: BL 1 ( Σ ) ( ) µ µ + ΣP' P P' + Ω q Pµ p P p N K 1 K matrix of stacked pick ro-vectors

23 BLACK-LITTERMAN APPROACH outputs: portfolios official prior on linear returns L N( µ, Σ) equilibrium-based estimation subjective vies V V ( 2 q ω ) N, p p µ µ ( 2 q ω ) N, K K K K Bayesian posterior: µ µ BL + vies Markoitz mean-variance optimization: ς argmax ' ' µ BL ' Σ 2ς shrinkage to equilibrium

24 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

25 ROBUT OPTIMIZATION the general frameork The point estimate for the parameters must be replaced by an uncertainty region that includes the true, unknon parameters: ( ) Θ θ m, space of possible parameters values true (unknon) θ point estimate θ uncertainty region Θ

26 ROBUT OPTIMIZATION the general frameork The point estimate for the parameters must be replaced by an uncertainty region that includes the true, unknon parameters: ( ) Θ θ m, space of possible parameters values true (unknon) θ point estimate θ uncertainty region Θ The allocation optimization must be performed over all the parameters in the uncertainty region: ( i) argmax... C ( ) ( m, ) m, { } argmax... C ( m, ) Θ { }

27 ROBUT OPTIMIZATION from the standard mean-variance { ' m} argmax subject to C ' v : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : (point) estimate of : (point) estimate of m

28 ROBUT OPTIMIZATION to a conservative mean-variance approach argmax subject to { ' m} C ' v { } m n m m Θ argmax i ' subject to m C { } ' v max Θ : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : (point) estimate of m Θ m : uncertainty set for m : (point) estimate of Θ : uncertainty set for

29 ROBUT OPTIMIZATION uncertainty regions 22 Θ EXAMPLE: 2x2 COVARIANCE MATRIX positivity boundary Trade-off for the choice of the uncertainty regions: Must be as large as possible, in such a ay that the true, unknon parameters (most likely) are captured Must be as small as possible, to avoid trivial and nonsensical results

30 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

31 BAYEIAN OPTIMIZATION Bayesian estimation theory The Bayesian approach to estimation of the generic market parameters differs from the classical approach in to respects: θ ( m,) it blends historical information from time series analysis ith experience the outcome of the estimation process is a (posterior) distribution, instead of a number experience posterior density f po ( θ ) prior density f pr ( θ ) historical information space of possible parameters values θ 0 θ ce θ classical-equivalent

32 BAYEIAN OPTIMIZATION - Bayesian estimation theory m 2 2 less likely estimates m ce EXAMPLE: 2-dim EXPECTED VALUE m m m more likely estimates m in the Bayesian approach the expected values of the returns are a random variable

33 BAYEIAN OPTIMIZATION Bayesian estimation theory 22 less likely estimates more likely estimates ce EXAMPLE: 2x2 COVARIANCE MATRIX positivity boundary 11 in the Bayesian approach the covariance matrix of the returns is a random variable

34 BAYEIAN OPTIMIZATION from the standard mean-variance { ' m} argmax subject to C ' v : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : classical estimate of : classical estimate of m

35 BAYEIAN OPTIMIZATION to the classical-equivalent mean-variance argmax subject to { ' m} C ' v { ' m } ce argmax subject to C ' ce v : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : classical estimate of : classical estimate of m m ce ce : Bayesian classical-equivalent estimate for m : Bayesian classical-equivalent estimate for

36 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

37 ROBUT BAYEIAN OPTIMIZATION the general frameork Robust allocations are guaranteed to perform adequately for all the markets ithin the given uncertainty ranges Bayesian allocations include the practitioner s experience

38 ROBUT BAYEIAN OPTIMIZATION the general frameork Robust allocations are guaranteed to perform adequately for all the markets ithin the given uncertainty ranges. Hoever the uncertainty regions for the market parameters are somehat arbitrary the practitioner s experience, or prior knoledge, is not considered Bayesian allocations include the practitioner s experience. Hoever the approach is not robust to estimation risk Bayesian approach to parameter estimation ithin the robust frameork

39 ROBUT BAYEIAN OPTIMIZATION Bayesian ellipsoids The Bayesian posterior distribution defines naturally a self-adjusting uncertainty region Θ q for the market parameters This region is the location-dispersion ellipsoid defined by a location parameter: the classical-equivalent estimator θ ce a dispersion parameter: the positive symmetric scatter matrix a radius factor q ( ) -1 : θ θ ' ce θ θce q q Θ θ ( ) 2 θ posterior density f po ( θ ) uncertainty region q Θ space of possible parameters values

40 ROBUT BAYEIAN OPTIMIZATION Bayesian ellipsoids tandard choices for the classical equivalent and the scatter matrix respectively: global picture: expected value / covariance matrix ce f ( ) θ θ po θ dθ ( )( ) ' f θ θ θ θ θ po ( θ) dθ ce ce local picture: mode / modal dispersion ce ( θ) θ θ argmax f po θ ln fpo θθ ' { } ( θ) θ ce 1

41 ROBUT BAYEIAN OPTIMIZATION from the standard mean-variance { ' m} argmax C ' v subject to : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : (point) estimate of : (point) estimate of m

42 ROBUT BAYEIAN OPTIMIZATION to the robust mean-variance argmax C { ' m} argmax min C { } ' m m Θ m ' v subject to subject to { } ( i) ' v max Θ : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : (point) estimate of m Θ m : uncertainty set for m : (point) estimate of Θ : uncertainty set for

43 ROBUT BAYEIAN OPTIMIZATION to the robust Bayesian MV argmax C { ' m} ' v subject to, q argmax min ' m q C m Θ m p subject to max { } ( i) ' v p Θ : relative portfolio eights C v : set of investment constraints, e.g. 1 ' = 1, 0 : significant grid of target variances m : (point) estimate of m q Θm : Bayesian ellipsoid of radius q for m : (point) estimate of p Θ : Bayesian ellipsoid of radius for p

44 ROBUT BAYEIAN OPTIMIZATION 3-dim. mean-variance frontier ( i) The robust Bayesian efficient allocations p, q represent a three-dimensional frontier parametrized by: 1. Exposure to market risk represented by the target variance ( i) v 2. Aversion to estimation risk for the expected returns represented by radius m q...indeed, a large ellipsoid corresponds to an investor that is very orried about poor estimates of m q Θm 3. Aversion to estimation risk for the returns covariance represented by radius p Θ p indeed, a large ellipsoid corresponds to an investor that is very orried about poor estimates of

45 RBO EXAMPLE market model We make the folloing assumptions: The market is composed of equity-like securities, for hich the returns are independent and identically distributed across time The estimation interval coincides ith the investment horizon The linear returns are normally distributed: ( ) τ L m, N m, t+ τ We model the investor s prior as a normal-inverse-wishart distribution: here m N m0,, W ν 0, T0 ν 0 (, ): investor s experience on m0 0 ( m, ) ( T, ν ) : investor s confidence on ( m0, 0) 0 0

46 RBO EXAMPLE posterior distribution of market parameters Under the above assumptions, the posterior distribution is normal-inverse-wishart, see e.g. Aitchison and Dunsmore (1975): here m N m1,, W ν1, T1 ν 1 1 T t T τ m l 1 ( τ ) τ t t T l m l m T t= 1 t= 1 ( ) ' T1 T0 + T m 1 1 T 0 0 T T + m m 1 ν 1 ν 0 + T ( )( ) m m m m ' 1 ν 00 + T + ν T0 T

47 RBO EXAMPLE location-dispersion ellipsoids in practice The certainty equivalent and the scatter matrix for the posterior (tudent t) marginal distribution of are computed in Meucci (2005): m 1 ν m = m, = 1 ce 1 m 1 T1ν 1 2 The certainty equivalent and the scatter matrix for the posterior (inverse-wishart) marginal distribution of are computed in Meucci (2005): ν 2ν = = ce 1, 3 ν1 + N + 1 ( ν1 + N + 1) ( ' ( 1 1 D ) ) Ν 1 1 DΝ 1 D Ν here is the duplication matrix (see Magnus and Neudecker, 1999) and is the Kronecker product

48 RBO EXAMPLE efficient frontier Under the above assumptions the robust Bayesian mean-variance problem: subject to simplifies as follos:, q argmax min ' m q C m Θ m p { } ( i) ' v p max Θ ( ) argmax { ' ' } 1 m pq, λ 1 λ C The three-dimensional frontier collapses to a line The efficient frontier is parametrized by the exposure to overall risk, hich includes market risk, estimation risk for m and estimation risk for

49 RBO EXAMPLE efficient frontier

50 RBO EXAMPLE robust Bayesian self-adjusting nature When the number of historical observations is large the uncertainty regions collapse to classical sample point estimates: { } ( λ) argmax ' λ ' m C robust Bayesian frontier = classical sample-based frontier When the confidence in the prior is large the uncertainty regions collapse to the prior parameters: ( λ) argmax { ' } 0 λ ' 0 m C robust Bayesian frontier = a-priori frontier (no information from the market)

51 RBO EXAMPLE robust Bayesian self-adjusting nature 1 robust Bayesian frontier portfolio eights market & estimation risk T 0 T, ν 0 0 T T, ν prior frontier 0 sample-based frontier

52 RBO EXAMPLE robust Bayesian conservative nature (&P 500) ample-based Apr97 ep98 Jan00 May01 Oct02 Feb04 Jul Robust Bayesian Apr97 ep98 Jan00 May01 Oct02 Feb04 Jul05

53 RBO EXAMPLE robust Bayesian conservative nature (&P 500) 4 3 ample-based Apr97 ep98 Jan00 May01 Oct02 Feb04 Jul Robust Bayesian Apr97 ep98 Jan00 May01 Oct02 Feb04 Jul05

54 AGENDA Estimation vs. Modeling Classical Optimization and Estimation Risk Black-Litterman Optimization Robust Optimization Bayesian Optimization Robust Bayesian Optimization References

55 REFERENCE This presentation: symmys.com > Teaching > Talks > Issues in Quantitative Portfolio Management: Handling Estimation Risk implementation code (MATLAB): symmys.com > Book > Donloads > MATLAB Comprehensive discussion of - modeling - estimation - location-dispersion ellipsoid - satisfaction maximization - quantitative portfolio-management - risk-management - estimation risk - Black-Litterman allocation - Bayesian techniques - robust techniques - symmys.com > Book > A. Meucci, Risk and Asset Allocation - pringer (2005)