The Effect of Estimation in Highdimensional Portfolios Luitgard A. - - PowerPoint PPT Presentation

The Effect of Estimation in High–dimensional Portfolios

Luitgard A. M. Veraart Joint work with Axel Gandy, Imperial College London Analysis, Stochastics, and Applications Vienna University, July 2010

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 1 / 12

Outline

1 Classical Portfolio Optimisation 2 Plug-In Strategies with Estimated Parameters 3 James-Stein-Shrinkage Applied to Strategies 4 L1–Constrained Strategies - LASSO 5 Other Strategies 6 Application to Empirical Data

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 2 / 12

Optimal Portfolio Selection

The asset prices: 1 bond S0(t) = ert, d risky assets dSi(t) = Si(t)[µidt +

d

• j=1

σijdWj(t)], Si(0) > 0, i = 1, . . . , d, r > 0 interest rate, µ ∈ Rd drift, σ ∈ Rd×d volatility matrix of full rank (all constant), W d-variate Brownian motion. Investor has T > 0 fixed time horizon, X0 > 0 constant initial wealth, chooses πi(t) fraction of the wealth invested in the ith asset at time t, resulting in time-t-wealth Xt with dXt = d

i=0 πi(t)Xt dSi(t) Si(t) .

Investor seeks π to maximise V (π) := E[log(XT)]. Optimal solution π∗ = Σ−1(µ−r1), where π0(t) = 1 − d

i=1 πi(t), π = (π1, . . . , πd)T, Σ = σσT.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 3 / 12

Optimal Portfolio Selection

The asset prices: 1 bond S0(t) = ert, d risky assets dSi(t) = Si(t)[µidt +

d

• j=1

σijdWj(t)], Si(0) > 0, i = 1, . . . , d, r > 0 interest rate, µ ∈ Rd drift, σ ∈ Rd×d volatility matrix of full rank (all constant), W d-variate Brownian motion. Investor has T > 0 fixed time horizon, X0 > 0 constant initial wealth, chooses πi(t) fraction of the wealth invested in the ith asset at time t, resulting in time-t-wealth Xt with dXt = d

i=0 πi(t)Xt dSi(t) Si(t) .

Investor seeks π to maximise V (π) := E[log(XT)]. Optimal solution π∗ = Σ−1(µ−r1), where π0(t) = 1 − d

i=1 πi(t), π = (π1, . . . , πd)T, Σ = σσT.

The Problem

What if we need to estimate µ? What if the number of risky assets d → ∞?

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 3 / 12

Plug-in Merton Strategy with Estimated µ

General Unbiased Plug-in Estimator

Estimate µ by ˆ µ and define the plug-in strategy ˆ π = Σ−1(ˆ µ−r1). Assume ˆ π ∼ N(Σ−1(µ − r1), V 2

0 ), V0 ∈ Rd×d, then

V (ˆ π) = V (π∗) − T 2 trace(ΣV 2

0 ).

Specific Plug-in Estimator

Observation period [−test, 0] for test > 0. Set ˆ µi = log(Si(0)) − log(Si(−test)) test + 1 2

d

• j=1

σ2

ij.

Then ˆ π ∼ N(Σ−1(µ − r1), Σ−1/test). V (ˆ π) = V (π∗)−d

T 2test .

There are realistic scenarios in which even V (ˆ π) → −∞ as d → ∞.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 4 / 12

James-Stein-Type Shrinkage of the Strategy

The James-Stein-Strategy

Let ˆ π = Σ−1(ˆ µ − r1), π0 ∈ Rd, a > 0 fixed constants. Consider ˆ πJS,π0 =

• 1 −

a (ˆ π − π0)TΣ(ˆ π − π0)

• (ˆ

π − π0) + π0.

The Expected Utility for the JS-Strategy

Let ˆ µ ∼ N(µ, Σ/test), K ∼ Poisson(λ), λ = (π∗ − π0)TΣ(π∗ − π0)/2: V (ˆ πJS,π0) = V (ˆ π) + T 2 a

• 2d − 2

test − a

• E
• test

d − 2 + 2K

• .

ˆ πJS,π0 dominates ˆ π for 0 < a < 2(d − 2)/test; optimal a = (d − 2)/test.

Special Choices for π0 and Optimal a

π0 = π∗: V (ˆ πJS,π0) = V (π∗)− T

test .

π0 = β

d 1, β ∈ R: In some situations V (ˆ

πJS,π0) → ∞ as d → ∞.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 5 / 12

L1–constrained Strategies - LASSO

General Idea

Require that π satisfies π1 = d

i=1 |πi| ≤ c for a constant c ≥ 0.

V (π) ≥ log(X0) + rT − TE

• c maxi |µi − r| + c2

2 maxi,j |Σij|

• .

If maxi |µi − r|, maxi,j |Σij| bounded, V (π) → −∞ as d → ∞.

Specific Results

For Σ = η2(ρ11T + (1 − ρ)I), η > 0, 0 ≤ ρ ≤ 1 analytic results for the optimal L1-constrained strategies, if µ known. for the L1-constrained plug-in strategy as d → ∞:

◮ the distribution of #{i : π∗

i = 0}, if ρ = 0,

◮ an upper bound on limd→∞ P(#{i : π∗

i = 0} > k), if ρ > 0.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 6 / 12

Other Strategies and Performance for d → ∞

Other Norm Constraints

L0-restricted strategies: no degeneration of expected utility as d → ∞. L2-restricted strategies: degeneration possible.

Special L1-Constraints

1/d-strategy: Strategy that invests the same amount into all stocks, i.e. πc/d = c

d 1 for some c > 0.

Equal Weighting of the most Extreme stocks (EWE): πEWE

ki

= c βd sign(ˆ aki)I(i ≤ βd), i = 1, . . . , d, where ˆ ai = ˆ

µi−r Σii , c > 0, β ∈ (0, 1) constants, ki are such that

|ˆ ak1| > |ˆ ak2| > · · · > |ˆ akd|.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 7 / 12

Example - Trading S&P500

Stocks in S&P 500 index on 01/01/2006 having daily returns for all trading days between 2001 and 2008 (373 stocks, n=2011 trading days, daily returns). Specific random ordering of stocks. Allow the strategies to invest in the first d stocks of this ordering. X0 = 1, r = 0.02, roughly T = 1. Use of unbiased estimators based on observed stock prices at time points 0, ∆, 2∆, . . . , (n − 1)∆: ˆ µdata = 1 ∆ ˆ ξ + 1 2 diag( Σdata),

• Σdata

µ,ν =

1 ∆(n − 2)

n−2

• i=0
• Rµ(i) − ˆ

ξµ Rν(i) − ˆ ξν

• for µ, ν = 1, . . . , d, where Rµ(i) = log
• Sµ((i+1)∆)

Sµ(i∆)

• ,

ˆ ξµ =

1 n−1

n−2

i=0 Rµ(i).

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 8 / 12

Analytic and Simulation Results

−20 −10 10 20

n = 504 expected utility V(

(π

π|µ µ)

)

Merton−Strategies µ,Σ known µ estimated, Σ known µ,Σ estimated

n = 1008 n = 2016

−20 −10 10 20

expected utility V(π|µ)

James−Stein Strategies JS, π0 = π* JS, π0 = π*, Σ est. JS, π0 = 0 JS, π0 = 0, Σ est. 100 200 300 0.0 0.1 0.2 0.3 0.4 0.5

d expected utility V(π|µ)

L1−restricted Strategies L1 L1,µ est. L1,µ,Σ Σ est 1 d EWE, β = 0.1 100 200 300

d

100 200 300

d

Expected utility plotted against the number d of available stocks.

Out-of-Sample Performance

2005

−1 −0.5 0.5 1

log(XT)

100 200 300 −∞ −20 −10 10

log(XT) d 2006

100 200 300

d 2007

100 200 300

d 2008

Strategy 1/d L1 EWE, β = 0.1 100 200 300 Strategy Merton JS

d

log(XT) with T = 1 year plotted against the number d of available stocks.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 10 / 12

Summary

Main Contributions

Quantification of the effect of estimation in vast portfolios (unknown µ and large d). Analysis of strategies which are less affected by estimation. Analytic formulae for James-Stein and optimal L1-constrained strategies.

Specific Conclusions

Estimation effects must not be ignored in vast portfolios! Simple plug in strategies have a loss through estimation linear in d. James-Stein shrinkage performs better than simple plug in strategies. L1-constrained strategies cannot degenerate. L1-constrained strategies and particularly the EWE-strategy and 1/d strategy perform well also in out-of-sample tests.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 11 / 12

References

Brodie, J., Daubechies, I., De Mol, C., Giannone, D. & Loris, I. (2009). Sparse and stable Markowitz portfolios. P. Natl. Acad. Sci. 106, 12267–12272. Fan, J., Zhang, J. & Yu, K. (2009). Asset allocation and risk assessment with gross exposure constraints on vast portfolios. Preprint, Princeton University. James, W. & Stein, C. (1961). Estimation with quadratic loss. In

• Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1

(Univ. of Calif. Press), 361–379. Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413. Stein, C. M. (1981). Estimation of the mean of a multivariate normal

• distribution. Ann. Statist. 9, 1135–1151.

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso.

• J. Roy. Stat. Soc. B 58, 267–288.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 12 / 12

Analytic Results for LASSO with Σ = η2I

Optimal L1-constrained strategy with known µ

Suppose |µ1 − r| > |µ2 − r| > . . . > |µd − r|. Then π† = 1

η2 (µ − r1) is a

solution to the L1-constrained optimisation problem if π†1 ≤ c. Otherwise, the unique solution is π∗ := 1 η2 (sign(µ1 − r)(|µ1 − r|−a), . . . , sign(µk − r)(|µk − r|−a), 0, . . . , 0)T, k = min

• l ∈ {1, . . . , d} : c ≤ 1

η2

l

i=1(|µi − r| − |µl+1 − r|)

• ,

a = 1

k

• η2c − k

i=1 |µi − r|

• and µd+1 = r.

Plug-in strategy with iid normally distributed estimators ˆ µ1, . . . , ˆ µd

Let c = αcd, cd > 0 norming constants from extreme value theory for folded normal distribution FN(E(ˆ µ1 − r), Var(ˆ µ1)). Then #{i : π∗

i = 0} L

→ K + 1 (d → ∞), where K is a Poisson distribution with expected value αη2.

Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 13 / 12