# 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.

• What is the question you're trying to answer? Why are you trying to decompose portfolio variance? – Matthew Gunn Dec 1 '17 at 16:47
• I want to trim position in assets which have historically contributed more than x% of portfolio risk. That is my end-game. – ragster Dec 2 '17 at 21:12
• Should historical risk contributions matter at all? If your portfolio used to be 100% IBM, and now it's .0001% IBM, should you trim IBM because it historically contributed 100% of the risk? – Matthew Gunn Dec 2 '17 at 21:46
• You're right. But let's say that I keep my portfolio composition roughly constant (+-10%) by rebalancing. – ragster Dec 3 '17 at 9:07