Summary of factor model analysis by Fan at al.

Transcription

1 Summary of factor model analysis by Fan at al. W. Evan Durno July 11, 2015 W. Evan Durno Factoring Porfolios July 11, / 15

2 Brief summary Factor models for high-dimensional data are analysed [Fan et al., 2008]. Majority of work emphasizes O(n) O(p). 1 Convergence is studied from the perspective of matrix norms. 2 Convergence speed is theorized in the O( )-style, emphasizing how increasing samples, dimensions, and factors influence error. 3 Simulations demonstrate adequacy of a proposed norm. 4 Analysis is described as theoretical W. Evan Durno Factoring Porfolios July 11, / 15

3 Factor model review Our n data, Y, are in p dimensions with k factors. 1 Ψ R p p 0, diagonal, not random; L = [l i,j] R p k, not random 2 F = [f j ] N(0, I k ); E = [e i ] N(0, Ψ) 3 E does not depend on F 4 Y = µ + LF + E Y N(µ, Σ) s.t. Cov[Y ] = Σ = LL + Ψ Estimation for L can be done in a few ways, of which MLE and Eigenvectors are popular. The authors use least squares given Y & F. 1 ˆΣ = ˆB ˆ Cov[F ]ˆB + ˆΨ, ˆΨ = diag[e v E v /n], ˆB = YX (XX ) 1 2 such that E v = [E 1,..., E n ], X = [F 1,..., F n ] 3 Criticism: I don t get to observe F! So it is a theoretical exercise. W. Evan Durno Factoring Porfolios July 11, / 15

4 Matrix norms This paper primarily uses the following norms. 1 Frobenious [Horn & Johnson, 1990]: A = [tr(aa )] 1/2 = [ p j=1 λ(a)2 ] 1/2 2 Entropy loss: [James &) Stein, 1961] L 1 (ˆΣ, Σ) = tr (ˆΣΣ 1 log ˆΣΣ 1 p 3 Custom A Σ = p 1 Σ 1/2 AΣ 1/2 Their custom norm is a scaled quadratic norm L 2. Precisely, A Σ Σ = p 1 L 2 (ˆΣ, Σ) = p 1 tr [ AΣ 1 I ]. Convention: Eigenvalues of A are λ 1 (A),..., λ p (A) decreasing. W. Evan Durno Factoring Porfolios July 11, / 15

5 Assumptions The following assumptions are employed partially or entirely. Let b = E Y 2, c = max 1 i k E[f 4 j ], d = max 1 i k E[e 4 i ] 1 (A) (Y i, F i ) iid, E[E i F i ] = 0, Cov[E F ] = Ψ, distribution of F is continuous, k p. 2 (B) b = O(p), c & d bounded. σ 1 > 0 : λ k (LL ) σ 1 n 3 (C) σ 2 > 0 : λ p (Ψ) > σ 2 n 4 (D) The k factors F = [f j ] are fixed accross n. W. Evan Durno Factoring Porfolios July 11, / 15

6 Frobenious is a poor descriptor of convergence Theorem (1) Under assumptions (A) - (C), 1 ˆΣ Σ = O P (pk/ n) 2 ˆΣ s Σ = O P (pk/ n) λi 3 max 1 i p (ˆΣ) λ i (Σ) = O P (pk/ n) λi 4 max 1 i p (ˆΣ s ) λ i (Σ) = O P (pk/ n) Notation 1 Let ˆΣ denote the factor model estimate of Σ. 2 Let ˆΣ s denote the sample covariance estimate of Σ. W. Evan Durno Factoring Porfolios July 11, / 15

7 Quadratic norm better describes convergence Theorem (2) Under assumptions (A) - (C), if K = O(n α 1 ), p = O(n α ), then 1 ˆΣ Σ Σ = O P (n β/2 ) : β = min{1 2α 1, 2 α α 1 } 2 ˆΣ s Σ Σ = O P (n β1/2 ) : β 1 = 1 max{α, 3α 1 /2, 3α 1 α} Remark: α > 2α 1 & α 1 < 1 β > β 1 So the factor estimate converges faster when the number of dimensions grows faster than the number of factors. Also note that α 1 ˆΣ is root-n-consistent under Σ. W. Evan Durno Factoring Porfolios July 11, / 15

8 More theory Theorem (3) Under assumptions (A) - (C), 1 ˆΣ 1 Σ 1 = o P ([p 2 k 4 log n/n] 1/2 ) 2 ˆΣ 1 s Σ 1 = o P ([p 4 k 2 log n/n] 1/2 ) Remark: ˆΣ 1 & ˆΣ 1 s do refer matrix inverses of the estimates. Theorem (4) Under assumptions (A), (B), and (D), ˆΣ Σ converges in distribution to normal errors. Language is purposefully vague for brevity. W. Evan Durno Factoring Porfolios July 11, / 15

9 Markowitz portfolio optimization Define the following variables. 1 There are p risky assets with rate of return µ and covariance Σ. 2 ξ is the variance-minimum allocation vector with rate of return γ. 3 ξ g is the minimum variance porfolio (no desired rate of return). Theorem (Markowitz [Markowitz, H.M., 1952]) The solution to the problem: min ξ Σξ : ξ 1 = 1 & ξµ = γ ξ is this: ξ = φ γψ ρφ ψ 2 Σ γρ ψ ρφ ψ 2 Σ 1 µ such that ρ = 1 Σ 1 1, ψ = 1 Σ 1 µ, φ = µ Σ 1 µ with variance ξ Σξ = ργ2 2φγ+φ ρφ ψ 2. To get ˆξ g, replace γ with ψ/φ. W. Evan Durno Factoring Porfolios July 11, / 15

10 Factor estimates have less porfolio variance If we simply plug in estimates ˆΣ, ˆΣ n, ˆµ we get allocation estimates ˆξ, ˆξ g. Theorem (5, global minimum variance convergence) Under (A) - (C) and ρ > 0, 1 ˆξ g ˆΣˆξ g ξ g Σξ g = o P ([p 4 k 4 log n/n] 1/2 ) 2 ˆξ g ˆΣ s ˆξ g ξ g Σξ g = o P ([p 6 k 2 log n/n] 1/2 ) Theorem (5, optimal variance convergence) Under (A) - (C) and ρφ ψ 2 > max{0, ρ/(ρφ ψ 2 ), ψ/(ρφ ψ 2 ), φ/(ρφ ψ 2 )}, 1 ˆξ ˆΣˆξ ξ Σξ = o P ([p 4 k 4 log n/n] 1/2 ) 2 ˆξ ˆΣ s ˆξ ξ Σξ = o P ([p 6 k 2 log n/n] 1/2 ) W. Evan Durno Factoring Porfolios July 11, / 15

11 Weak convergence of variance Require ξ = O(1)1 to avoid extreme short positions. Theorem Under (A) and (B), 1 ξ ˆΣξ ξ Σξ = o P ([p 4 k 2 log n/n] 1/2 ) 2 ξ ˆΣ s ξ ξ Σξ = o P ([p 4 k 2 log n/n] 1/2 ) and requiring NO short positions results in 1 ξ ˆΣξ ξ Σξ = o P ([p 2 k 2 log n/n] 1/2 ) 2 ξ ˆΣ s ξ ξ Σξ = o P ([p 2 k 2 log n/n] 1/2 ) Notice the equivalence. W. Evan Durno Factoring Porfolios July 11, / 15

12 A simulation study Under fixed n and k, p was allowed to increase. The study revealed the following. 1 The Frobenious norm was insufficient to demonstrate covariance estimate inefficiency. 2 The scaled quadratic norm was sufficient to demonstrate covariance estimate efficiency. 3 When data follow a factor model, observing the factors and including them in the model greatly increases covariance modelling efficiency 4 Estimated portfolio variance MSEs greatly reduce with the factor model when data follow a factor model, even when p > n. W. Evan Durno Factoring Porfolios July 11, / 15

13 Recap 1 Estimation is easy when the model is known and latent variables are observable. 2 Results suggest that correct usage of a factor model may result in reduce variance in covariance estimation and porfolio variance. 3 Results suggest that a simple porfolio (ξ = O(1)1) without shorts does not enjoy a better variance reduction, so standard covariance estimation is adequate. 4 The Frobenious norm may is not always adequate for understanding covariance estimate inefficiency. W. Evan Durno Factoring Porfolios July 11, / 15

14 References Jianquing Fan, Yingying Fan, Jinchi Lv (2008) High dimensional covariance matrix estimation using a factor model Journal of Econometrics 147, Horn, R.A. & Johnson, C.R. (1990) Matrix Analysis Cambridge Press James, W. & Stein, C. (1961) Estimation iwth quadratic loss Proc. Fourth Berkeley Symp. Math. Stat. Prob. vol 1, California Press, Berkeley, pp Markowitz (1952) Portfolio selection Journal of Finance 7, W. Evan Durno Factoring Porfolios July 11, / 15

15 The End W. Evan Durno Factoring Porfolios July 11, / 15