Infiniti Capital Four Moment Risk Decomposition

By Peter Urbani and Mitchell Bristow

Introduction to the Infiniti Capital Four Moment Risk Decomposition

Higher Moments and RiskClassical Investment Theory assumes that investment returns are normally distributed.

The Normal distribution can be full described by its first two moments, Mean (Mu) and Standard Deviation (Sigma).

The empirical evidence indicates that only about 12% of Hedge Funds have returns that are normally distributed.

Hedge Fund returns exhibit significant amounts of skewness and excess ( > 3 ) kurtosis (third and fourth statistical moments).

Investors generally have a preference for positive skewness and moderate to low levels of kurtosis.

Consequently the impact of higher moments is of great interest to people building portfolios of hedge funds or managing their risk.

One way to include the impact of higher moments into the calculation of Value at Risk (VaR) for such portfolios is to use the Cornish Fisher modification to the Normal VaR calculation.

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Normal versus Modified VaR

In the case of a standard normal distribution (Mean=0, Std Dev=1, Skew=0, Kurt=3) both the Normal Var and Cornish Fisher Modified VaR give the same answer.

Note this assumes Raw Kurtosis – Excels formula assumes Excess Kurtosis > 3 and subtracts 3 automatically. This is why Kurtosis is not 0 in the above example.

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Normal versus Modified VaR

In the case of a slightly positive skewness and slightly higher than normal kurtosis (Mean=0, Std Dev=1, Skew=0.5, Kurt=4) the Cornish Fisher Modified VaR is lower (less negative) than that given by the Normal VaR calculation.

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Normal versus Modified VaR

In the case of a slightly negative skewness and slightly higher than normal kurtosis (Mean=0, Std Dev=1, Skew=-0.5, Kurt=4) the Cornish Fisher Modified VaR is higher (more negative) than that given by the Normal VaR calculation.

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The Cornish Fisher Modification

In Excel

for use with Excess Kurtosis = (Skew*(ZScore^2-1)/6)+(Kurt*(ZScore^3-3*ZScore)/24)-((Skew^2)*(2*ZScore^3-5*ZScore)/36)

for use with Raw Kurtosis=(1/6)*(ZScore^2-1)*Skew+(1/24)*((ZScore^3)-3*ZScore)*(Kurt-3)-(1/36)*(2*Zscore^3-5*ZScore)*Skew^2

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Moving from the Univariate to the Multivariate

(normal)Variance

Covariance Matrix

Std Devs

(normal)Correlation

Matrix

Weights

(normal)VaR

(normal)Variance

Covariance Matrix

Std Devs

Co-Skewness

Co-Kurtosis

(modified)Correlation

Matrix

Weights

(modified)VaR

Mod Std Devs

-> -> ->+

-> ->

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Infiniti Capital Modified VaR Decomposition

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Normal & Modified VaR for a portfolio

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Basic Portfolio Statistics

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Covariance Matrix

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‘Modified’ Covariance Matrix

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Correlation Matrix

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‘Modified’ Correlation Matrix

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‘Modified’ Volatility

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Normal & Modified VaR from these Matrices

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Infiniti Capital Four Moment Risk Decomposition

http://www.infiniti-analytics.com17

What can we learn from this Analysis ?

First we can see that the portfolio’s negative Skew of -1.28 and excess Kurtosis of +4.51 contribute to the 95% Modified VaR being higher than the Normal VaR at -2.28% versus -1.90%.

This is even more evident when we look at the Modified Conditional VaR (Modified CVaR) versus the Normal CVaR of -4.09% versus -2.56%.

Whilst the differences between the Normal and Modified VaR for individual positions may be small in some cases, those for the Normal and Modified CVaR (average tail expectation beyond the VaR a.k.a expected shortfall (ES)) tend to be far more pronounced.

Thus we can see that largest % Contributions to the portfolio’s Modified CVaR came from Emerging Markets, Convertible Arbitrage, Event Driven and Distressed.

However when we look at ratio of % Contribution to Weight we can see that although Distressed contributed 18.0% to the Risk (Modified CVaR), it also contributed 18.4% to the return. Comparatively Emerging Markets was far riskier as although it contributed 19.3% to return, it also contributed 39.6% to the Risk.

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Disclaimer

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