QuantPort India

Cokurtosis

A fourth-moment measure that quantifies how an asset's returns interact with extreme market movements, extending portfolio optimization beyond mean-variance-skewness frameworks.

Why Cokurtosis?

Traditional portfolio theory focuses on the first two moments of return distributions, while more advanced approaches incorporate the third moment (skewness). Cokurtosis extends this further by capturing the fourth moment, which measures the propensity for extreme outcomes—both positive and negative—and the "fatness" of the return distribution tails.

Assets with high positive cokurtosis tend to amplify extreme market movements, potentially exacerbating portfolio losses during market crashes. Conversely, assets with low or negative cokurtosis can provide a cushioning effect during extreme market events, making them valuable for tail risk management and crisis-resilient portfolio construction.

1. Mathematical Definition

1.1 Kurtosis: The Fourth Moment

Kurtosis measures the "tailedness" of a probability distribution. For a return series $r$ , kurtosis is defined as:

Kurt(r) = \frac{\mathbb{E}[(r - \mu)^4]}{[\mathbb{E}[(r - \mu)^2]]^{2}} = \frac{\mathbb{E}[(r - \mu)^4]}{\sigma^4}

Where $\mu$ is the mean return and $\sigma$ is the standard deviation. A distribution with kurtosis greater than 3 (the kurtosis of a normal distribution) is called leptokurtic and has fatter tails, indicating a higher probability of extreme outcomes.

1.2 Cokurtosis Definition

Cokurtosis extends this concept to measure the co-movement between an asset's returns and cubed market returns:

k_{i,m} = \frac{\mathbb{E}[(r_i - \mu_i)(r_m - \mu_m)^3]}{\sigma_i \sigma_m^3}

Where $r_i$ and $r_m$ are the returns of asset $i$ and the market, $\mu_i$ and $\mu_m$ are their respective means, and $\sigma_i$ and $\sigma_m$ are their standard deviations.

In an unstandardized form, cokurtosis can be expressed as:

Cokurt(r_i, r_m) = \mathbb{E}[(r_i - \mu_i)(r_m - \mu_m)^3]

1.3 Matrix Representation

For a portfolio of $n$ assets, cokurtosis can be represented using a four-dimensional matrix:

K_{i,j,k,l} = \mathbb{E}[(r_i - \mu_i)(r_j - \mu_j)(r_k - \mu_k)(r_l - \mu_l)]

The cokurtosis of a portfolio with weights $w$ can then be calculated as:

k_p = \sum_{i=1}^{n}\sum_{j=1}^{n}\sum_{k=1}^{n}\sum_{l=1}^{n} w_i w_j w_k w_l K_{i,j,k,l}

In practice, we often focus on the cokurtosis between each asset and the market portfolio for computational tractability.

1.4 Estimation via Regression

Similar to coskewness, cokurtosis can be estimated using regression analysis:

r_i = \alpha + \beta r_m + \gamma r_m^2 + \delta r_m^3 + \epsilon

Where $\delta$ captures the asset's sensitivity to cubed market returns (cokurtosis), alongside the traditional beta $\beta$ and coskewness $\gamma$ coefficients.

1.5 Financial Significance

Cokurtosis is financially significant for several reasons:

Tail risk management: Cokurtosis directly measures an asset's contribution to portfolio tail risk, helping identify securities that might amplify losses during market crashes.
Pricing impact: Assets with high positive cokurtosis (which tend to exacerbate extreme market movements) may command higher risk premiums in a four-moment asset pricing framework.
Crisis resilience: Building portfolios with controlled cokurtosis can enhance resilience to market crashes and extreme events beyond what traditional diversification achieves.
Options pricing: Cokurtosis helps explain the volatility smile observed in options markets, as it captures the non-normality of return distributions that affects option values.

2. Default Parameters

Parameter	Default	Description
window	`252`	Rolling window length in trading days (~1 year)
standardized	`True`	Whether to use standardized or raw cokurtosis
method	`"dittmar"`	Estimation method (dittmar, fang-lai)
min_periods	`100`	Minimum observations required for estimation
rf	`0.0`	Risk-free rate subtracted from returns before estimation

3. Implementation Considerations

When implementing cokurtosis in portfolio optimization:

Data requirements: Fourth-moment statistics require substantial historical data for reliable estimation—typically two to three years of daily returns at minimum. Longer periods provide more stable estimates but may include outdated market regimes.
Estimation precision: Cokurtosis is even more sensitive to outliers and estimation error than coskewness. Shrinkage estimators or robust statistics can improve reliability.
Computational complexity: The full cokurtosis tensor for a portfolio of $n$ assets has $n^4$ elements, making it computationally intensive for large portfolios. Market-based simplifications reduce this to $n$ calculations.
Optimization challenges: Including cokurtosis in portfolio optimization leads to quartic programming problems, which are more complex than quadratic (mean-variance) or cubic (mean-variance-skewness) problems.
Empirical relevance: While theoretically important, the empirical significance of cokurtosis varies across markets and time periods. Test its relevance in your specific investment universe before implementation.

4. Advantages and Limitations

Advantages

Captures tail risk beyond what variance and skewness measures.
Helps identify assets that amplify or dampen extreme market movements.
Enhances crisis-period portfolio management and tail hedging.
Aligns with investor preferences for avoiding extreme negative outcomes.
Improves model fit for non-normal return distributions common in financial markets.

Limitations

Requires substantially more data than lower-moment statistics for reliable estimation.
Highly sensitive to outliers and estimation error.
Computationally intensive for large portfolios.
Challenging to implement in standard optimization frameworks.
Benefits may be marginal in some markets compared to simpler three-moment approaches.

5. References

Fang, H., & Lai, T. Y. (1997). Co-kurtosis and capital asset pricing. The Financial Review, 32(2), 293-307.
Dittmar, R. F. (2002). Nonlinear pricing kernels, kurtosis preference, and evidence from the cross section of equity returns. Journal of Finance, 57(1), 369-403.
Christoffersen, P., Feunou, B., Jacobs, K., & Turnbull, S. (2021). Option-Based Estimation of the Price of Coskewness and Cokurtosis Risk. Journal of Financial and Quantitative Analysis, 56(1), 65-91.
Jondeau, E., & Rockinger, M. (2006). Optimal portfolio allocation under higher moments. European Financial Management, 12(1), 29-55.
Martellini, L., & Ziemann, V. (2010). Improved estimates of higher-order comoments and implications for portfolio selection. The Review of Financial Studies, 23(4), 1467-1502.
Guidolin, M., & Timmermann, A. (2008). International asset allocation under regime switching, skew, and kurtosis preferences. The Review of Financial Studies, 21(2), 889-935.