Sparse and stable Markowitz portfolios

Joshua Brodie1 Ingrid Daubechies1 Christine De Mol2

Domenico Giannone3 Ignace Loris2

1Princeton University, PACM

2Université Libre de Bruxelles, ECARES and Math DeptVrije Universiteit Brussel, Math Dept and CAMP

3European Central Bank (DG-R) and ECARES

Workshop "Stats in the Château", HEC, Jouy-en-Josas,Paris, September 2009

Portfolio optimization

• N securities with (stationary) returns ri t at time t .• rt = N × 1 vector of returns at time t .• Vector of expected returns : µµµ = E[rt ]

Covariance matrix of returns:

E[(rt −µµµ)(rt −µµµ)>] = CCC

• A portfolio is defined by a N × 1 vector of weights wisumming to one (unit of capital): w>1N = 1,where 1N denotes the N × 1 vector of ones.

• Expected return of the portfolio: w>µµµVariance of the portfolio: w>CCC w

Markowitz portfolios

• Find a portfolio w which has minimal variance for a givenexpected return ρ = w>µµµ, i.e.

w = arg minw

w>CCC w

s. t. w>µµµ = ρ

w>1N = 1

• Since CCC = E[rtr>t ] − µµµµµµ>, this is equivalent to

w = arg minw

E[|ρ−w>rt |2

]s. t. w>µµµ = ρ

w>1N = 1

Our proposal

• For empirical implementation, replace expectations withsample averages and solve the following regressionproblem

w = arg minw

1T‖ρ1T −RRRw‖22

s. t. w>µµµ = ρ

w>1N = 1,

where µµµ = 1T

∑Tt=1 rt and RRR is the T × N matrix of the

available returns.

• Add a L1-norm penalty to ensure sparsity and stability.

Our sparse and stable portfolios

• Find the weights

w[τ ] = arg minw

[||ρ1T −RRRw||22 + τ ||w||1

](1)

s. t. w>µµµ = ρ (2)w>1N = 1. (3)

where ||w||1 =∑

i |wi | and τ is a positive parameter tuningthe balance between the two terms.

• “Lasso” regression (Tibshirani 1996), but with extraconstraints.

Why stability?

• We suspect ill-conditioning due to high collinearity is amajor problem for practical implementations of theMarkowitz framework and may also account for thefollowing fact.

• It was recently shown that many portfolio constructionsproposed in the literature fail to outperform the naiveequally-weighted (“1/N”) portfolio.(DeMiguel, Garlappi and Uppal 2007)

• The L1-norm penalty is known to regularize (stabilize) theproblem.(Daubechies, Defrise and De Mol 2004)

Sparsity

• The L1-norm penalty enforces sparsity of the portfolio,i.e. favors the presence of many zero weights(↔ few active assets).

• This allows for variable (asset) selection.

• Why?

Lasso regression and sparsity

Lasso regression and sparsity

Lasso regression and sparsity

Lasso regression and sparsity

Why sparsity?

• Difficulty of managing several hundreds of assets.

• The traditional Markowitz framework does not take intoaccount transaction and monitoring costs.

• The L1-norm penalty allows to cope with the transactioncosts, modelling linear transaction costs and/or serving asa proxy to the L0-norm (↔ sparsity, to keep control on thefixed costs)

Constrained LARS Algorithm

• The recursive LARS/homotopy algorithm allows tocompute efficiently the Lasso regression solutions for allvalues of τ , starting from the largest ones.(Osborne et al. 2000, Efron et al. 2004)

• We devised a modification of LARS able to enforce thelinear constraints.

• Varying τ allows to tune the number of active positions.

Special case: No-short portfolios

• Notice∑

i wi = 1 =⇒∑

i |wi | ≥ 1.

• Limit case for τ large:∑

i |wi | = 1⇐⇒ wi ≥ 0 for all i(no short positions).

• Positive portfolios known for their good performances(Jagannathan and Ma 2003) .

• But the fact that no-short portfolios are sparse seems tohave gone unnoticed !

Empirical application

• We used as assets the Fama and French 48 industryportfolios (FF48) and 100 portfolios formed on size andbook-to-market (FF100).

• We constructed our portfolios in June of each year from1976 to 2006 using 5 years of historical (monthly) returnsand a target return equal to the historical return of theequally-weighted portfolio.

• Performance is evaluated by out-of-sample monthly meanreturn m, standard deviation σ and Sharpe ratio S = m/σ.

• Benchmark (tough!) is the evenly-weighted portfolio.

Empirical results

Performance of sparse portfolio with no short-selling (FF48)

Evaluation wi > 0 Equalperiod for all i weight

m σ S m σ S06/76-06/06 17 41 41 17 61 2707/76-06/81 23 48 49 29 66 4407/81-06/86 23 41 57 18 58 3107/86-06/91 9 45 20 5 72 707/91-06/96 16 26 62 18 41 4407/96-06/01 16 40 40 11 67 1707/01-06/06 13 43 30 18 60 30

Empirical results

Performance of sparse portfolio with no short-selling (FF100)

Evaluation wi > 0 Equalperiod for all i weight

m σ S m σ S06/76-06/06 16 53 30 17 59 2807/76-06/81 12 59 21 23 61 3807/81-06/86 24 49 49 20 53 3807/86-06/91 10 65 15 9 71 1307/91-06/96 19 31 61 18 34 5307/96-06/01 18 52 35 16 63 2607/01-06/06 11 55 21 12 64 19

Empirical results FF48Sh

arpe

Rat

io (i

n %

) wpos17−24

25−32

33−40

41−48

number of active positions5 10 15 20 25 30 35 40 4515

20

30

25

35

40 8−16

w

w

wpos

bin

K

Empirical results FF100Sh

arpe

Rat

io (i

n %

)

wpos

5 10 15 20 25 30 35 40 45 50 55 60

40

35

30

25

number of active positions

11−20

21−30

31−4041−50

51−60

1

w

w

wpos

bin

K

Generalizations

• Weighted L1-norm penalty:∑

i si |wi |.

• Index tracking (replace target return ρ by index yt ).

• Portfolio Adjustment (Rebalancing).

Full Paper

• Sparse and stable Markowitz portfoliosJoshua Brodie, Ingrid Daubechies, Christine De Mol,Domenico Giannone and Ignace LorisWorking paper (July 2007), ECORE DP2007/61, CEPRDP6474,Working Paper European Central Bank No 936 (2008)Downloadable from http://arxiv.org/abs/0708.0046Published in PNAS 2009 106:12267-12272(No 30, 28 July 2009)