Risk-Based Portfolios underParameter Uncertainty

R/FinanceMay 20, 2017Lukas Elmiger

Which risk based portfolio strategy offers best out of sample performance …

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Inverse Volatility Minimum Variance Maximum Diversified

1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

1.0

2.5

5.0

10.0

Cum

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etur

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Which risk based portfolio strategy offers best out of sample performance … at smallest sensitivity to parameter estimation.

3

Inverse Volatility Minimum Variance Maximum Diversified

1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

1.0

2.5

5.0

10.0

Cum

ulat

ed R

etur

n

Sensitivity to Calibration• High dispersion of out-of-sample performance of portfolios

calibrated with bootstrapped sample periods indicates elevated sensitivity to parameter estimation.

• Bootstrap does not require distributional assumptions and maintains (nonlinear, higher moment) dependence information.

• Block-bootstrap with monthlong blocks maintains serial dependence.

“Financial variables are intrinsically linked in complicated ways.”Harvey and Liu (2015)

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. . .

. . .

. . .

. . .

. . .

. . .r3 r7 r10 r13

r1 r3 r4 r6 r7 r9 r10 r11 r13 r14 r16

r1 r2 r3 r4 r5 r6 r7 r8 r9 r10 r11 r12 r13 r14 r15 r16

N N

Monthlong blocks of returns:

Bootstrap sample:1. Randomly draw blocks

with replacement from 2. Recycle random sequence over

entire data set ⇒ overlapping parts of calibration

periods remain unchanged. ⇒ Each calibration period

contains N observations.

Calibrate portfolios based on overlapping bootstrap samples

Apply out-of-sample to full return series:

Bootstrap Procedure

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NLength of calibration period

r1 . . . rN

N

Asset UniverseS&P 500 Stocks• Historical index members • Homogeneous correlations • Daily CRSP data • 27 years of data 1990 - 2016

Global Futures• 42 liquid rate, bond, equity,

volatility, commodity futures • Heterogeneous correlations • 17 years of data 1999 - 2016 • Intra-day data from tickdata.com

for concurrent return observations • Define rolling dates and

account for rolling costs

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0

1

2

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−0.5 0.0 0.5 1.0

Asset Universe Global Futures SP500 Stocks

Density of Pairwise Correlation CoefficientsJan 2010 − Dec 2014

Cumulated Returns

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Solid line represents median performance, shaded area spans from 10% to 90% quantile of performances of 50 portfolios calibrated with bootstrapped samples.

S&P500 Stocks Global Futures

Const. Target Volatility

Fully Invested

1995 2000 2005 2010 2015 2005 2010 2015

1.0

2.5

5.0

10.0

1.0

2.5

5.0

10.0

StrategyInverse VolatilityMinimum VarianceMaximum DiversifiedEqual Risk Contribution

Sharpe Ratio Distribution

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S&P500 Stocks Global Futures

Const. Target Volatility

Fully Invested

0.25 0.30 0.35 0.40 0.35 0.40 0.45 0.50 0.55 0.60

0

10

20

30

0

10

20

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StrategyInverse VolatilityMinimum VarianceMaximum DiversifiedEqual Risk Contribution

Density of sharpe ratios of portfolios calibrated with 50 bootstrapped calibration periods.

Take-Out• Portfolios differ both in size and dispersion of sharpe ratios.

• Maximum Diversified portfolio reaches highest sharpe ratio in single US stocks / second highest sharpe ratio in global futures over study period.

• It pays out to use a more complex portfolio strategy in terms of number of parameters. This also leads to higher sensitivity to parameter estimation.

• New question: How to mitigate impact of parameter estimation.

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Thanks to authors of PortfolioAnalytics and FRAPO packages for great R implementations of portfolio strategies!

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Annexes

Portfolio Objectives• Inverse Volatility 1 :

• Minimum Variance 2 :

• Maximum Diversified 3 :

• Equal Risk Contribution 4 :

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ReferencesDavid Ardia, Guido Bolliger, Kris Boudt, and Jean-Philippe Gagnon Fleury. The Impact of Covariance Misspecification in Risk-Based Portfolios. SSRN Electronic Journal, pages 1–17, November 2015.

Denis Chaves, Jason Hsu, Feifei Li, and Omid Shakernia. Risk Parity Portfolio vs. Other Asset Allocation Heuristic Portfolios. The Journal of Investing, 20 (1):108–118, 2011. Yves Choueifaty and Yves Coignard. Toward Maximum Diversification. The Journal of Portfolio Management, 35(1):40–51, 2008. Campbell R Harvey and Yan Liu. Lucky Factors. SSRN Electronic Journal, December 2015.

Robert A Haugen and Nardin L Baker. The efficient market inefficiency of capitalization–weighted stock portfolios. The Journal of Portfolio Management, 17(3):35–40, 1991. Sébastien Maillard, Thierry Roncalli, and Jérôme Teïletche. The Properties of Equally Weighted Risk Contribution Portfolios. The Journal of Portfolio Management, 36(4):60–70, 2010.

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