On the Out-of-Sample Importance of Skewness and Asymmetric DependenceSkewness and Asymmetric Dependence

for Asset Allocation

Andrew Patton

London School of Economics.

1

Outline of talkOutline of talk

MotivationDefinition of asymmetric dependenceSet-up of the problem

DataInvestor’s utility function and optimisation problemDensity models: mean variance skewness and copulaDensity models: mean, variance, skewness and copulaInvestment strategiesPortfolio performance measuresp

ResultsUnconstrained versus short sales constrained

f

2

Economic significanceStatistical significance

Motivation: stock returns are non-normalMotivation: stock returns are non normal

The distribution of stock returns are widely reported as be being skewed, see Kraus and Litzenberger g , g(1976), Harvey and Siddique (1999,2000), inter alia.

Recent studies report that stock returns are more higly correlated in bear markets than bull markets –a form of asymmetric dependence, see Erb et al.y p ,(1994), Longin and Solnik (2001), Ang and Chen (2002).

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Describing asymmetric dependenceDescribing asymmetric dependence

There are a number of ways of trying to measure andThere are a number of ways of trying to measure and present asymmetric dependence

One simple way is to look at exceedence correlations , see Longin and Solnik (2001) and Ang and Chen (2002):(2002):

Correl [ X , Y | Quantile(X) < q , Quantile(Y) < q ], for q ≤ 0.5

Correl [ X , Y | Quantile(X) > q , Quantile(Y) > q ], for q ≥ 0.5

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[ I don’t use this measure in the modelling stage, but it is useful for preliminary analysis of the data. ]

Asymmetric dependenceAsymmetric dependence

Exceedence correlations between raw excess returns

0.9

1

Empirical Bivariate normal

0.6

0.7

0.8Unconditional correlation = 0.72

atio

n

0 3

0.4

0.5

Cor

rela

0.1

0.2

0.3

5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantile, q

Asymmetric dependenceAsymmetric dependence

Exceedence correlations between transformed residuals

0.9

11

Empirical Bivariate normal

0.6

0.7

0.8

atio

n Unconditional correlation = 0.73

Rotated gumbel copula

0 3

0.4

0.5

Cor

rela

0.1

0.2

0.3

6

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Quantile, q

Goal of this researchGoal of this research

The presence of skewness and/or asymmetricThe presence of skewness and/or asymmetric dependence violates the assumption that stock returns are normally distributed

I attempt to determine the economic and statistical significance of these non normalities for a particularsignificance of these non-normalities for a particular pair of indices, in the context of out-of-sample asset allocation

I find substantial economic significance, and d t t ti ti l i ifi

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moderate statistical significance

Investor’s optimisation problemInvestor s optimisation problem

The investor’s optimisation problem is:The investor s optimisation problem is:

ˆ*

dydxyxhyxU

YXUE

tyx

tytxtt

.).,(ˆ.)(maxarg

})]([maxarg 1*

∫∫ +≡

+= −

ωω

ωωωω

dydxyGxFcygxfyxU tttttyx

tyx

.))(ˆ),(ˆ(ˆ).(ˆ).(ˆ).(maxarg ∫∫∫∫

+= ωωω

ω

where U is a CRRA utility function with RRA of 1, 3, 7 10 and 20

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7, 10 and 20.

Data and EstimationData and Estimation

Monthly data from Jan 1954 to Dec 1999 on a U.S. risk-free asset, a small cap and a big cap stock indexindex.

In-sample period: Jan 1954 – Dec 1989, 420 obsOut-of-sample period: Jan 1990 – Dec 1999, 120 obsOut of sample period: Jan 1990 Dec 1999, 120 obs

Model selection is done only once, using the in-sample datasample data.

Parameters of the model are estimated recursively throughout the out of sample period

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throughout the out-of-sample period.

Copulas and Sklar’s theoremCopulas and Sklar s theorem

Sklar (1959) showed that we may decompose theSklar (1959) showed that we may decompose the distribution of (X,Y) into three parts:

H( x , y ) ⇔ C( F(x) , G(y) ) ∀ x , y

Joint dist’n of X and Yof X and Y

Marginal Copula of X and Y

Marginal dist’n of Y

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gdist’n of X

X and Y

All of these distributions have N(0,1) marginal distributions and ρ=0 50Gaussian

marginal distributions and ρ=0.50Student’s tMixed Normal

Clayton GumbelJoe-Clayton

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The density modelsThe density models

I compare the performance of three density models.p p y

All have AR models for the mean, and TARCH models for the varianceAll use DIV, RF and SPR as explanatory variables

1. The first assumes a bivariate normal densityy

2. The second allows for time-varying skewness, via Hansen’s (1994) skewed t, but imposes a normal ( ) , pcopula

3. The third allows for time-varying skewness and

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chooses the optimal copula model from a set of 9 possible copulas (selects the ‘rotated Gumbel’ copula)

The asset allocation decision rulesThe asset allocation decision rules

1 100% i ht i ll1. 100% weight in small caps

2. 100% weight in big caps

3. 50% weight in each stock index

4 Optimise using unconditional distribution4. Optimise using unconditional distribution

5. Optimise using a bivariate normal

6. Optimise using a skewed t – Normal copula

7. Optimise using a skewed t –flexible copula

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7. Optimise using a skewed t flexible copula

Portfolio performance measuresPortfolio performance measures

I use four measures of portfolio performance:I use four measures of portfolio performance:

1-3. Mean to risk ratios:Mean / standa d de iation (Sha pe atio)Mean / standard deviation (Sharpe ratio)

Mean / 5% Value-at-Risk

Mean / 5% Expected ShortfallMean / 5% Expected Shortfall

4. Management feeA more interpretable value than average realisedA more interpretable value than average realised utilityThis is a fee, expressed in basis points per year, that

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a particular investor would be willing to pay to switch from a 50:50 portfolio to another portfolio.

Short sales constraintsShort sales constraints

Short sales constraints have two interpretations inShort sales constraints have two interpretations in this context:

1 E i ll th fl t th t i t th t1. Economically they reflect the constraints that many market participants face, and so possibly make the study more realistic

1. Econometrically they can be interpreted as an ‘insanity filter’, preventing the hypothetical investor f ki i i i h kfrom taking extreme positions in the market.

→ Stock and Watson (1999), for example, find that such filters improve forecast accuracy from non-linear

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filters improve forecast accuracy from non linear models.

Economic significanceEconomic significance

Gumbel model out-performs the normal model 16 outGumbel model out-performs the normal model 16 out of 20 comparisons

Overall average out-performance is 16.7%Overall average out performance is 16.7%Average out-performance in management fee is 41 (1) basis points for unconstrained (constrained) investors.investors.

Gumbel model out-performs the ‘intermediate’ model in all 20 comparisonsp

Overall average out-performance is 52.3%Average out-performance in management fee is

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21 (1.5) basis points for unconstrained (constrained) investors

Management feeUnconstrained Normal Unconstrained intermediateManagement fee Unconstrained intermediateUnconstrained Gumbel

A t i t ld b illi t t it h f th b d h ld tf li

2030

r

Amount investor would be willing to pay to switch from the buy and hold portfolio

-100

10

s pe

r yea

r

-40-30-20

sis

poin

ts

-70-60-5040

Bas

17

-8070

1 3 7 10 20

Relative risk aversion

Management feeConstrained Normal Constrained intermediateManagement fee Constrained intermediateConstrained Gumbel

A t i t ld b illi t t it h f th b d h ld tf li

25

30

ar

Amount investor would be willing to pay to switch from the buy and hold portfolio

20

25

nts

per y

ea

10

15

asis

poi

n

0

5

B

18

1 3 7 10 20Relative risk aversion

Pair-wise comparison bootstrap testsPair wise comparison bootstrap tests

Focussing on results using realised utility:Focussing on results using realised utility:

Unconstrained investors:Unconstrained investors:

Gumbel model significantly outperformed both the Normal and intermediate models for all levels of riskNormal and intermediate models for all levels of risk aversion

Normal and intermediate models were not distinguishable

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Pair-wise comparison bootstrap testsPair wise comparison bootstrap tests

Short sales constrained investors:Short sales constrained investors:

Gumbel out-performed Normal model for high risk i (RRA 10 d 20) hil N l taversion (RRA=10 and 20) while Normal out-

performed Gumbel for RRA=1

Gumbel outperformed the intermediate model for all levels of risk aversion

Normal and intermediate models were again indistinguishable

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indistinguishable

Bootstrap reality check resultsBootstrap reality check results

Reject benchmark portfolio if ‘consistent’ p-value is less j p pthan 0.10

Benchmark portfolio: Normal

Unconstrained Short sales constrained

RRA Lower Consistent Upper Lower Consistent Upper

1 N/A N/A N/A 0.316 0.316 0.896

3 N/A N/A N/A 0.586 0.667 0.792/ / /

7 0.042 0.042 0.042 0.746 0.792 0.842

10 0 034 0 034 0 034 0 373 0 384 0 593

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10 0.034 0.034 0.034 0.373 0.384 0.593

20 0.117 0.185 0.309 0.082 0.082 0.535

Bootstrap reality check resultsBootstrap reality check results

Reject benchmark portfolio if ‘consistent’ p-value is less j p pthan 0.10

Benchmark portfolio: Intermediate

Unconstrained Short sales constrained

RRA Lower Consistent Upper Lower Consistent Upper

1 0.126 0.126 0.126 0.556 0.556 0.932

3 0.066 0.066 0.317 0.319 0.368 0.470

7 0.067 0.067 0.305 0.349 0.394 0.493

10 0 023 0 023 0 224 0 380 0 511 0 579

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10 0.023 0.023 0.224 0.380 0.511 0.579

20 0.238 0.380 0.380 0.151 0.161 0.611

Summary of ResultsSummary of Results

Capturing skewness and asymmetric dependenceCapturing skewness and asymmetric dependence leads to better portfolio performance:

Noteworthy as in many cases simpler models do bestNoteworthy, as in many cases simpler models do best in out-of-sample comparisons

For these assets, it seems that asymmetric dependence is more important than skewness

Statistical significance of improvement is moderate

Short sales constraints improve portfolio decisions made using out-of-sample density forecasts

E i i ifi i t t f t i d

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Economic significance is greatest for unconstrained investors, eg: hedge funds.

Future workFuture work

1. Impact of parameter estimation uncertainty on all of these results

2. Compare flexible parametric methods, like mine or those of Ang and Bekaert (2001), with nonparametric g ( ), pmethods like those of Brandt (1999) and Aït-Sahalia and Brandt (2001)?

3. Extensions to higher dimensions: are the improvements even greater, or does estimation error dominate?

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dominate?

Management fee U t i d N lManagement fee Unconstrained Normal Unconstrained Gumbel

A t i t ld b illi t t it h f th I t di t tf liAmount investor would be willing to pay to switch from the Intermediate portfolio

15

25

ar

5

15

nts

per y

ea

-15

-5

Bas

is p

oin

-35

-25

1 3 7 10 20

25

-451 3 7 10 20

Relative risk aversion

Management fee U t i d N lManagement fee Unconstrained Normal Unconstrained Gumbel

A t i t ld b illi t t it h f th I t di t tf li

2

2.5

Amount investor would be willing to pay to switch from the Intermediate portfolio

1

1.5

per y

ear

0 5

0

0.5

s po

ints

p

-1.5

-1

-0.5

Bas

i

26

-21 3 7 10 20

Relative risk aversion