Rethinking the Fundamental Law

Transcription

1 Rethinking the Fundamental Law Jose Menchero

2 Simplicity as a Theore@cal Objec@ve Everything should be made as simple as possible, but no simpler - Albert Einstein 2

3 Outline Fundamental Law of Management (FLAM) Overview of standard Problems with standard of FLAM New of FLAM skill and breadth Examples Transfer Coefficient Summary Technical Appendix the alphas 3

4 Fundamental Law of Management 4

5 Residual Returns and Alpha Residual returns r n = β n R B + e n r = βr B + e Vector Form ˆr E[r] Expected Return Residual returns are uncorrelated with the benchmark ( ) = 0 cov e,r B Alphas represent the expected value of the residuals α = E[e] Alphas Only constraint on alpha: benchmark must have zero alpha: h! B α = 0 5

6 Let Ω T be the asset covariance matrix for total returns Ω T = cov( r,r) σ 2 T = h# A Ω T h A Total Variance Let Ω denote the covariance matrix of residual returns Ω R = cov( e,e) 2 = Ω T β β$ σ B Residual Covariance Matrix Informa@on Ra@o using residual risk and returns: σ 2 R = h" A Ω R h A Residual Variance IR R = h" α A σ R Residual IR Informa@on Ra@o using total risk and returns IR T = h! Aˆr σ T Total IR Both IR defini@ons are equally valid Total IR includes market risk 6

7 Standard Form of the Fundamental Law Grinold (1989, 1994) introduced the of the fundamental law FLAM was later codified by Grinold and Kahn (2000) Essence of FLAM is to explain por`olio in terms of skill (IC) and breadth (BR) IR = IC BR To obtain high IR, play well (high IC) and play oben (large BR) Grinold defines IC for stock n as series correla@on between the forecasts α nt and the realiza@ons e nt Grinold describes BR as the number of independent bets 7

8 of Standard FLAM Find a predic@ve variable (e.g., earnings yield or momentum) Standardize the raw values as z- scores Next, scale the alphas using Grinold (1994) rule: α n = σ n IC z n Compute IC Chicken and egg: alpha depends on IC, but IC depends on alpha Solu@on: IC is a constant, so use score (σ n z n ) as the forecast Compute IR: IR = IC BR Many prac@@oners use the number of stocks N as the breadth 8

9 the Coefficient The IC is the between the forecasts and the r nt = β nt R Bt + e nt α nt = E! " e nt # $ Correla@ons can be computed series, then averaged IC n = corr( α n,e ) n IC = 1 N n IC n Time- Series Correla@on Correla@ons can be computed on cross- sec@on, then averaged IC t = corr( α t,e ) t IC = 1 T t IC t Cross- Sec@on Correla@on 9

10 Problems with the Standard Makes that all assets have the same IC Not clear how to compute breadth (except for uncorrelated assets) Many incorrectly assume BR=N, even for correlated assets This typically results in overly Even for uncorrelated assets, standard FLAM is not exact This would imply a maximum aeainable IR of In reality, there is no theore@cal bound on the informa@on ra@o Alpha scaling rule (α n = σ n IC z n ) is not always valid Implies std(α/σ) 1 (no theore@cal basis for such a restric@on) Implies asset IR s are mean zero (not true for fund- of- funds applica@on) Key point: alpha is whatever the PM thinks that it is (subject to α B =0) N 10

11 Buckle (2004) Buckle (2004) derives an expression for breadth: BR = ij ρ ij P ij 1 ρ = correlation matrix of return forecasts P = correlation matrix of asset returns Deriva@on rests on several assump@ons: Return forecasts and the errors are independent Return forecasts are normally distributed Stock- level IC s are small and equal in magnitude Shortcomings of Buckle result: Assump@ons restrict the applicability of FLAM and may be violated in prac@ce Not clear how to compute the correla@on matrix of return forecasts ρ for the general case 11

12 of Qian, Hua and Sorensen (2007) rests on several key Por`olio has zero factor risk Stock- level are mean zero Risk- adjusted residual returns are mean zero Coefficient is computed as a cross- sec@onal correla@on! α IC t = ρ nt t, e nt # " σ nt σ nt Informa@on Ra@o is given by: $ & % Single Period IC = 1 T t IC t Mul@ple Periods IR = IC = IC N IC N σ IC 1 IC 2 Informa@on Ra@o 12

13 of Ding and (2015) Ding and explicitly take factor risk into account They regress risk- adjusted realized returns against z- scores!e nt e nt σ nt!e nt = f t Z nt +ε nt One- factor model Factor returns represent informa@on coefficients: ( ) ( ) f t = cov!e,z nt nt var Z nt = corr (!e nt,z ) nt f t = IC t They derive expression for IR in terms of IC Factor Returns IC = avg( IC ) t 2 σ IC = var( IC ) t IR = IC N 1 IC Nσ IC Informa@on Ra@o 13

14 of Ding and (cont) In the limit of large N (i.e., por`olio dominated by factor risk): IR = IC σ IC Represents an upper bound on IR In the limit of zero factor vola@lity: IR = IC N IC N 1 IC 2 Replicates result of Qian et al. Ding and Mar@n approach generalizes previous work on FLAM: They relax the assump@on of zero factor risk Provides insight on rela@onship between IR and IC Shortcoming: Does not provide a way to compute breadth 14

15 New of FLAM 15

16 Exact Expression for the Required inputs: stock alphas and asset covariance matrix Let Y denote the maximum IR por`olio with fixed vola@lity σ Y h Y = σ Y Ω 1 α α$ Ω 1 α Op@mal Holdings Vector Expected return of por`olio Y E! " R Y # $ = α% h = σ α% Ω 1 α Y Y Expected Returns The IR of por`olio Y immediately follows IR Y = α! Ω 1 α Informa@on Ra@o 16

17 New of FLAM As Grinold & Kahn, decompose IR into a product of skill and breadth However, we now assign a different interpreta@on to these quan@@es IR Y = Q D New Formula@on of FLAM Q denotes the InformaBon Quality; Q is representa@ve of skill Q specifies the strength of the underlying alpha signals D denotes the DiversificaBon Benefit D is akin to the square root of breadth (i.e., number of independent bets) If the assets are uncorrelated, then D = N D quan@fies the gain in IR achieved through op@mal alloca@on of risk budget 17

18 the Quality Q Skill is independent of asset correla@ons Finding skill for one set of correla@ons represents the general solu@on For uncorrelated assets, we know that n D = For uncorrelated assets, the covariance matrix Ω is diagonal: IR Y = α! Ω 1 α IR Diag Y = α n σ n Compute Informa@on Quality (skill) directly N ( ) 2 Q = IR Diag Y N = 1 N ( α n σ ) 2 n n Q = RMS α n ( σ ) n Informa@on Quality Unlike the IC, the Informa@on Quality Q may exceed 1 18

19 Benefit D (General Case) For general case, asset correla@ons are non- zero (Ω is not diagonal) IR Y = α! Ω 1 α Informa@on Ra@o Q = RMS α n ( σ ) Informa@on n Quality Diversifica@on Benefit represents closed- form solu@on for breadth D = α! Ω -1 α RMS α n ( σ ) n Diversifica@on Benefit Proper@es of the Diversifica@on Benefit: D is scale invariant (e.g., doubling the alphas doesn t change D) D 1 (i.e., diversifica@on benefit is always posi@ve): For uncorrelated assets, D = N " IR Y max α n $ # σ n % ' & RMS " α % n $ ' # & σ n 19

20 The Link Between Q and IC Skill is measured by Informa@on Quality Q = RMS α n ( σ ) n Assuming that the stock- level Informa@on Ra@os are mean zero Q = STD α n ( σ ) n Assuming that the Grinold alpha scaling rule applies: ( ) = IC α n = σ n IC z n STD α n σ n Hence, under these two assump@ons, IC is equivalent to Q However, these assump@ons are oben invalid in prac@ce: Informa@on Ra@os need not be mean zero Grinold scaling rule does not always apply 20

21 Example 1: Two Correlated Assets (Fund of Funds) Consider a manager alloca@ng between two hedge funds, A and B Five parameters fully specify the op@miza@on problem: The expected returns of each fund (α A and α B ) The vola@li@es of each fund (σ A and σ B ) The correla@on between the funds (ρ) Suppose both funds have same risk (10%) Suppose E[R A ]=8.5%, E[R B ]=11.3% " Ω = $ $ # σ A 2 ρσ A σ B ρσ A σ B σ B 2 Covariance Matrix % ' ' & IR A = 0.85, IR B =1.13 Q =1.00 Informa@on Quality Study por`olio characteris@cs versus correla@on ρ 21

22 Asset Weights For large the asset weights are 50/50 since the two funds are perfect hedges For large por`olio levers in Fund A, and hedges by Fund B Fund Weights 22

23 Expected Return, Risk and Note: Since Q=1, the Benefit is the same as IR Expected return is fairly constant, except for large Risk goes to zero for large ρ, and diverges to infinity for large ρ Info diverges for extreme or Return, Risk and IR 23

24 Example 2: Factor Risk and the Use a factor model with two major assump@ons: Assume only one factor drives all systema@c risk, with vola@lity σ F Assume all N stocks have the same specific risk, σ S The factor covariance matrix is given by: Ω = XF X " + Δ, where F = σ 2 F, Δ = σ 2 S I Risk factor X is standardized in usual way: i.e., µ=0 and σ=1 To solve op@miza@on problems, we need the inverse of Ω Inverse can be solved analy@cally using the Woodbury formula: Ω 1 = I σ S 2 1 σ S 2! 2 σ $ F # Nσ 2 2 " F +σ & X X' S % 24

25 Closed- Form Assume alpha factor lies in same as the risk factor ( ) α = ax a = STD α Op@mal por`olio Y is given by usual expression: Informa@on Ra@o IR Y and Informa@on Quality Q: h Y = Ω 1 α IR Y = a N Nσ F 2 +σ S 2 Q = a 1 N n! # " X n 2 X n2 σ F 2 +σ S 2 $ & % Diversifica@on Benefit is the ra@o of the two: D = IR Y Q = N Nσ F 2 +σ S 2 ( * )* 1 N n! # " X n 2 X n2 σ F 2 +σ S 2 $ + & - %,- 1/2 25

26 The Effect of Factor If factor is zero, result is consistent with standard D increases linearly with the square root of N Even for modest levels of factor vola@lity, D hits a ceiling As the factor vola@lity goes to infinity, D approaches 1, i.e., Diversifica@on benefit disappears Diversifica@on Benefit 26

27 The Effect of Number of Stocks For low factor (e.g., σ F < 0.05σ S ), there is a significant benefit to holding more stocks For high factor vola@lity (e.g., σ F > 0.20σ S ), there is almost no benefit to holding more stocks Note: A weak factor may have vola@lity σ F 0.05σ S A strong factor may have vola@lity σ F 0.20σ S Diversifica@on Benefit 27

28 Example 3: Risk Model Style Factors D for some factors is greater than naïve FLAM Rela@on between D and factor vola@lity: D tends to decrease as factor vola@lity rises Rela@on between D and the number of stocks: D tends to increase as the size of the index grows Excep@on is the size factor Diversifica@on Benefit for Major US Stock Indices Axioma US Equity Risk Model April 17,

29 Drilling Deeper into the Construct unit- exposure por`olios for each style factor k: Ω 1 X k h k = X# k Ω 1 X k Compute stock alphas and expected por`olio returns: α k = a X k E$ % R k & ' = h( α = a k k Expected Return Compute por`olio risk: σ 2 k = h" k Ωh k σ k = a " α k Ω 1 α k Vola@lity Compute informa@on ra@o: IR k = E! " R k # $ σ k IR k = a ' X k Ω 1 X k Informa@on Ra@o 29

30 Benefit: Taking a Deeper Look Consider size and value factors (select a=1% to produce reasonable IR) In all cases, IR increases with the size of the universe (as required) For value factor, Q is fairly constant across indices Drop in D for size factor due to sharp increase in Informa@on Quality Reflec@on of the distribu@on of size factor exposures 30

31 The Transfer Coefficient 31

32 Risk Budget Approach to Derive Transfer Coefficient Let por`olio Y represent the unconstrained op@mal por`olio Aeribute the return of por`olio Y to source por`olios m R Y = m x m Y g m x m Y = source exposure; g m = source return Aeribute the risk of por`olio Y using x- sigma- rho formula σ Y = m x my σ m ρ m Y Por`olio Vola@lity Informa@on Ra@o of por`olio Y given by IR Y = E[R Y ] σ Y = m! x Y m E[g m ] $ # & " % σ Y Informa@on Ra@o 32

33 Exact Form of TC and FLAM Trick: each term by x- sigma- rho IR Y = E[R Y ] σ Y = m! # " x my ρ my σ m σ Y $! x Y m E[g m ] $ & # %" x my ρ my σ & m % First term represents risk weight; second term is the marginal IR For op@mal por`olio Y, all sources must have the same marginal IR IR Y = E[g ] m IR m = ρ Y m IR Y ρ my σ m Since this holds for any por`olio P TC P = h! Ωh P Y σ P σ Y Exact TC Risk- Budget Equa@on IR P = TC P Q D Exact FLAM 33

34 Expression for the Transfer Coefficient Clarke et al. (2006) derived a different expression for the exact TC TC =! α h P α! Ω 1 α h! P Ωh P Alterna@ve Defini@on of the Transfer Coefficient In the diagonal approxima@on, this becomes: " TC corr α n $,h σ n # n P σ n % ' & Diagonal Approxima@on Aber a few lines of algebra, Clarke s exact TC may be rewrieen TC = h! Ωh P Y σ P σ Y (iden@cal to our previous result) 34

35 Summary We introduced a new formula@on of FLAM Reinterpret skill as the RMS Informa@on Ra@o of underlying assets Only required inputs are alphas and the asset covariance matrix Ω New formula@on offers important advantages over standard FLAM: The formula@on is exact (free of approxima@ons) Provides simple closed- form solu@ons for skill and breadth Broadens applicability of FLAM (e.g., fund- of- funds use case) Impact of constraints measured by the Transfer Coefficient TC represents the predicted correla@on between unconstrained op@mal por`olio and the actual por`olio 35

36 Technical Appendix 36

37 Alphas with One Signal (Time Series) Regress residual returns for stock n against series z- score: e nt = z nt b n +ε nt b n = cov e nt, z nt ( ) var( z ) nt Z- scores have unit standard devia@on by construc@on: b n = corr( e nt, z nt )std( e ) nt = IC n σ n The alphas are given by the expected values of the residual returns: α nt = σ n IC n z nt Grinold Alpha Scaling Rule Implies Apple would have a nega@ve exposure to the size factor if the market cap were below its trailing average (not very intui@ve) Some prac@@oners blindly apply the Grinold scaling rule even when it is inappropriate (e.g., cross- sec@onal model) 37

38 Alphas with One Signal (Cross Regress residual returns for stock n against a cross- sec@onal z- score: e nt = X nt f t + u nt f t = cov e nt, X nt ( ) var( X ) nt Factor exposures have unit standard devia@on by construc@on: f t = corr( e nt, X nt )std( e ) nt = IC t σ t The alphas are given by the expected values of the residual returns: α nt = σ t IC t X nt Alpha Scaling Rule (Cross Sec@on) Alpha is s@ll score, except now: Vola@lity is cross- sec@onal for one period IC is measured as a cross- sec@onal correla@on The score is standardized on the cross sec@on 38

39 Alphas with Signals (Cross Define alpha factors as factors with non- zero expected return Define risk factor as factors with zero expected return Regress stock returns against the combined factor set: r n = X (α ) f (α ) nk + (σ X ) (σ ) k nk f k k (α ) k (σ ) + u n If predicted beta is included as a risk factor, all alpha factors are uncorrelated with the benchmark (i.e., strictly residual) α n = E! r n " # $ = X (α ) E! (α ) f # nk " k $ k (α ) Expected factor returns can be wrieen as product of IR and vola@lity E! (α ) f # " k $ = IR k (α ) σ k (α ) Stock Alphas 39