Choosing Portfolio Weights for Historical Risk Contributions

Summary

The document asks how to calculate asset-level contributions to portfolio risk when asset weights have drifted over the historical sample. It presents the familiar two-asset portfolio variance formula and describes risk contribution as portfolio weight multiplied by the asset’s marginal contribution to portfolio volatility. The central issue is that the covariance matrix is estimated from historical returns, while the holdings may have changed through performance. The question considers whether to use beginning weights, ending weights, or an average, and recognizes that the formula may be an ex-ante calculation for a specified portfolio rather than a direct summary of realized historical risk.

The document provides no answer, worked calculation, or academic reference, so it should be read as a framing of the measurement problem rather than a settled method. In practice, the appropriate weights depend on the question being asked: risk of a current portfolio, risk at a historical point, or attribution over a period with changing exposures. The sample alone does not determine one uniquely correct choice.

Key ideas

• Portfolio variance depends on asset weights, individual variances, and cross-asset covariance.
• A marginal risk contribution must be combined with the portfolio weight to obtain an asset contribution.
• Changing holdings make historical covariance estimates and portfolio weights refer to potentially different exposures.
• The choice of weights depends on whether the goal is current risk estimation or historical risk attribution.

Tags

# Portfolio risk decomposition using historical data: which weights to use for assets?


# Portfolio risk decomposition using historical data: which weights to use for assets?












I am trying to decompose portfolio risk given historical returns of each asset in the portfolio. For a basic 2 asset portfolio, the portfolio risk is given as

$$σ_p^2 = w_x^2 \cdot σ_x^2+ w_y^2 \cdot σ_y^2 + 2\cdot w_x\cdot w_y\cdot σ_{xy}$$

Now I know that the risk contribution of a single asset can be estimated as the product of its weight in the portfolio and the marginal contribution of that asset to the portfolio volatility.

So I calculated the variance-covariance matrix for the 2 assets using historical returns and started calculating the risk contribution.

The question which has me stumped is this: since the weight of the assets changes with time (because of difference in performance), which weight should I use to calculate the risk contribution? Is it the starting weight or the end weight or is it some sort of average? Or is this question meaningless because this formula makes sense only when the asset variance and covariances are a given and this will only give you an ex-ante estimation of risk contribution.

Thanks in advance and I'd be really glad if someone can guide me to any academic reference on this.

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