The Effect of Estimation in High dimensional Portfolios

The Effect of Estimation in High–dimensional Portfolios

Luitgard A. M. Veraart

Joint work with Axel Gandy, Imperial College London

Analysis, Stochastics, and ApplicationsVienna 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


Optimal Portfolio SelectionThe 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 fullrank (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)XtdSi (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 →∞?


Optimal Portfolio SelectionThe 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 fullrank (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)XtdSi (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 →∞?


Plug-in Merton Strategy with Estimated µ

General Unbiased Plug-in Estimator

Estimate µ by µ and define the plug-in strategy π = Σ−1(µ−r1).

Assume π ∼ N(Σ−1(µ− r1),V 20 ), V0 ∈ Rd×d , then

V (π) = V (π∗)− T

2trace(ΣV 2

0 ).

Specific Plug-in Estimator

Observation period [−test , 0] for test > 0.

Set

µi =log(Si (0))− log(Si (−test))

test+

1

2

d∑j=1

σ2ij .

Then π ∼ N(Σ−1(µ− r1),Σ−1/test).

V (π) = V (π∗)−d T2test

.

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


James-Stein-Type Shrinkage of the Strategy

The James-Stein-Strategy

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

πJS ,π0

=

(1− a

(π − π0)TΣ(π − π0)

)(π − π0) + π0.

The Expected Utility for the JS-Strategy

Let µ ∼ N(µ,Σ/test), K ∼ Poisson(λ), λ = (π∗ − π0)TΣ(π∗ − π0)/2:

V (πJS ,π0) = V (π) +

T

2a

[2d − 2

test− a

]E[

testd − 2 + 2K

].

πJS ,π0

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

Special Choices for π0 and Optimal a

π0 = π∗: V (πJS ,π0) = V (π∗)− T

test.

π0 = βd 1, β ∈ R: In some situations V (πJS ,π

0)→∞ as d →∞.


L1–constrained Strategies - LASSO

General Idea

Require that π satisfies ‖π‖1 =∑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) 6→ −∞ 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 →∞:I the distribution of #{i : π∗i 6= 0}, if ρ = 0,I an upper bound on limd→∞ P(#{i : π∗i 6= 0} > k), if ρ > 0.


Other Strategies and Performance for d →∞

Other Norm Constraints

L0-restricted strategies: no degeneration of expected utility asd →∞.

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):

πEWEki

=c

βdsign(aki )I(i ≤ βd), i = 1, . . . , d ,

where ai = µi−rΣii

, c > 0, β ∈ (0, 1) constants, ki are such that|ak1 | > |ak2 | > · · · > |akd |.


Example - Trading S&P500

Stocks in S&P 500 index on 01/01/2006 having daily returns for alltrading days between 2001 and 2008 (373 stocks, n=2011 tradingdays, daily returns).

Specific random ordering of stocks. Allow the strategies to invest inthe first d stocks of this ordering.

X0 = 1, r = 0.02, roughly T = 1.

Use of unbiased estimators based on observed stock prices at timepoints 0,∆, 2∆, . . . , (n − 1)∆:

µdata =1

∆ξ +

1

2diag(Σdata),

Σdataµ,ν =

1

∆(n − 2)

n−2∑i=0

[Rµ(i)− ξµ

] [Rν(i)− ξν

]for µ, ν = 1, . . . , d , where Rµ(i) = log

(Sµ((i+1)∆)

Sµ(i∆)

),

ξµ = 1n−1

∑n−2i=0 Rµ(i).


Analytic and Simulation Results−2

0−1

00

1020

d

n == 504

expe

cted u

tility

V((ππ|µµ))

Merton−Strategies

µµ,ΣΣ knownµµ estimated, ΣΣ knownµµ,ΣΣ estimated

d

n == 1008

d

n == 2016−2

0−1

00

1020

d

expe

cted u

tility

V((ππ|µµ))

James−Stein Strategies

JS, ππ0 == ππ*JS, ππ0 == ππ*, ΣΣ est.JS, ππ0 == 0JS, ππ0 == 0, ΣΣ est.

d d

0 100 200 300

0.00.1

0.20.3

0.40.5

dd

expe

cted u

tility

V((ππ|µµ))

L1−restricted Strategies

L1L1,µµ est.L1,µµ,ΣΣ est1 dEWE, ββ == 0.1

0 100 200 300dd

0 100 200 300dd

Expected utility plotted against the number d of available stocks.

Out-of-Sample Performance2005

−1

−0.5

0

0.5

1

log((

XT))

0 100 200 300

−− ∞∞

−20

−10

0

10

log((

XT))

d

2006

0 100 200 300

d

2007

0 100 200 300

d

2008Strategy

1/dL1EWE, ββ == 0.1

0 100 200 300

Strategy

MertonJS

d

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


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-constrainedstrategies.

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/dstrategy perform well also in out-of-sample tests.


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 riskassessment with gross exposure constraints on vast portfolios.Preprint, Princeton University.

James, W. & Stein, C. (1961). Estimation with quadratic loss. InProc. 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 acontinuous-time model. J. Econ. Theory 3, 373–413.

Stein, C. M. (1981). Estimation of the mean of a multivariate normaldistribution. Ann. Statist. 9, 1135–1151.

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso.J. Roy. Stat. Soc. B 58, 267–288.


Analytic Results for LASSO with Σ = η2I

Optimal L1-constrained strategy with known µ

Suppose |µ1 − r | > |µ2 − r | > . . . > |µd − r |. Then π† = 1η2 (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

∑li=1(|µi − r | − |µl+1 − r |)

},

a = 1k

[η2c −

∑ki=1 |µi − r |

]and µd+1 = r .

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

Let c = αcd , cd > 0 norming constants from extreme value theory forfolded normal distribution FN(E(µ1 − r),Var(µ1)). Then

#{i : π∗i 6= 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

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