Impact of Higher Moments on Portfolio Optimisation

By Peter Urbani, CEO KnowRisk Consulting

The standard Mean Variance Optimisation(MVO) of Harry Markowitz's 1959 Modern Portfolio Theory ( MPT) typically considers only the first two moments of both the Portfolio and its underlying assets namely their means and variances. Under this framework maximising your Sharpe ratio, or Return to variability ratio, will produce the 'optimal' set of asset weights relative to your objective function, risk aversion and any other constraints you may have applied.

However, it is now well known that asset returns are not in general perfectly normally distributed and that they can exhibit excess kurtosis and both positive and negative skewness which can manifest itself as Fat Tailed risk where both the magnitude and frequency of Tail Losses are higher than those expected under the assumption of Normality.

Harvey and Siddique amongst others have shown that co-skewness risk tends to be priced and that investors in general have a preference for positive odd moments (Mean and Skew) and a dislike for positive even moments (Variance and Excess Kurtosis).

There is some debate however about the existence of higher moments and their persistence at longer time scales. In this article we look at the impact of different objective functions on the out-of-sample performance of optimised strategy allocations and in particular of the important contribution of co-skewness to the overall skewness of the portfolio for those who care about higher moments.

We show that the inclusion of higher moments in your portfolio construction process is worth +246 bp per annum over and above the performance of a broad passive benchmark  such as the S&P500 (^GSPC) and around +60bp over and above the compound average annual rate or return (CAGR) of an actively managed portfolio of S&P500 sector ETF's. We also show that optimal weights for those who prefer higher odd moments ( Mean, Skew etc ) are generally higher in the direction of those assets exhibiting positive relative skewness or co-skewness with either a benchmark or the portfolio itself.

Methodology

To illustrate we conduct out-of-sample back testing on the 9 iShares S&P500 Sector ETF's  Jan 2004 to Dec 2013. The initial look back window period is 36 months which is then rolled forward by three months and the portfolio is rebalanced at the new optimal weights for the most recent 36 month period. A maximum weight constraint of 50% in any one sector is applied. After each optimisation period we record the weights and the performance of the portfolio over the subsequent three months ( out-of-sample) using the historically optimal weights only. No attempt is made to forecast returns or variances although we believe the returns would be further enhanced by doing so.

In order to compare the performance of a portfolio selected using the first two moments only ( Max Sharpe Ratio ) with one using the first four moments ( Modified ) on a like for like basis we use the STAR ratio as an objective function for both substituting the  Cornish Fisher Modified CVaR for the Normal CVaR  as necessary. We also verify that using the STAR Mean Annual return / CVaR  (Normal ) produces exactly the same results as Maximising the Sharpe ratio. Results were calculated at both the 95th and 99th percentile with most of the illustrations herein using the 99th percentile.

Results ( Out of Sample )

Over the ten year period under review the S&P500 Index generated a CAGR of +5.21%.  Active portfolio rebalancing based on a Mean Variance 2 moment optimisation generated a CAGR of +7.21% whilst actively rebalancing based on the four moment Modified approach generated a CAGR of +7.88%. Whilst the difference of around 59bp may be considered small readers should bear in mind that the assets under consideration generally only have small amounts of skewness and excess kurtosis and that this amount is roughly equivalent to most funds annual costs.

As might be expected using only historical returns does result in some lagged effects and there is relative outperformance in only 4 of the 10 years. However, the main advantage of the four moment Modified version is an almost 10%reduction the large drawdown exhibited by the Normal portfolio in 2008 from -44.84% (Normal) to -40.70% (Modified). We attribute this to the improved Fat Tail risk capture of the four moment method.

To further illustrate we include the in-sample optimal weights for the last 36 month period to end Dec 2013.

As can be seen in the table to the right, the Optimal ( Normal ) Portfolio contains a higher weighting in Consumer Staples of 35.65% versus 3.71% (Modified) and a lower weighting in Utilities of 14.35% versus 36.99% in the Modified portfolio.

There are two reasons why the four moment ( Modified ) method chooses to upweight Utilities at the expense of Consumer Staples. Firstly Healthcare is statistically the favourite asset over the period and would have been upweighted further had we not constrained the maximum in any one ETF to 50%.

Consumer Staples have a higher mean return than both Utilities and Technology which the four moment method prefers.  Consumer Staples also have a lower standard deviation which explains the variance hating Normal method's preference for it. However, Consumer Staples also have a very high average correlation of over 0.80 with all other sectors.  Healthcare is favoured by the Modified method because its lower than Normal Kurtosis reduces it's 'Modified Volatility' from 3.25% to 2.64%.  Similarly, Utilities Modified correlation with Healthcare is almost 22% higher than the 0.488 suggested by the Normal (Pearson) correlation at around 0.598 - Consequently Utilities get upweighted.

In-Sample Optimal Weights for 36 Months to end Dec 2013

It can also be seen that Modified weights generate positive Co-Skewness with Healthcare and Utilities helping to contribute to the Modified portfolio's slightly positive skewness whereas the Normal portfolio has Skewness of  -0.18

Assumed Bi-Variate Normal and Best Fit Bi-Variate relationship between Healthcare and Utilities

The Cornish Fisher Modified approach is not without its problems and more sophisticated approaches such as the joint and marginal density copula approaches offer potentially better solutions. However, as an illustration as to the potential benefits to be had from considering higher moments when building portfolios I believe the Modified method provides a relatively accessible approach.

You can get the Excel demonstration file below:

iShares Four Moment Optimal Portfolio

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