# How to calculate a position's contribution to its portfolio's tracking error?

Say we have assets X (with weight $$w_a$$) and Y (with weight $$w_y$$) in a portfolio. X and B returns are correlated: $$Cov(R_x, R_y)\neq 0$$.

The portfolio's tracking error is: $$std(R_p - R_b) = std((w_x*(R_x-R_b)+w_y*(R_y -R_b))$$.

How can I calculate, based on the asset's tracking error ($$std(R_i-R_p)$$) and its normalised weight $$w_i$$, this asset's contribution to the portfolio's tracking error?

Remarks:

• I saw this Quant question but I don't think it answers my question.
• Bloomberg has something called "tracking error contribution", but I don't know which formula they are using.

Hi: You can calculate the weights in the index of the two stocks. $$w_{A}$$ and $$w_{B}$$ and the weights of the stocks in the portfolio, $$w^{\prime}_A$$ and $$w^{\prime}_B$$. Then, the return contribution due to the mis-weighting, is $$(w_{A} - w^{\prime}_{A}) R_{A} + (w_{B} - w^{\prime}_{B}) R_{B}$$.

Then, assuming you don't have a risk model such as Barra, you can use brute force in order to obtain the variance of the return contribution above. You get

$$(w_{A} - w^{\prime}_{A})^2 \times Var(R_{A})$$ +

$$(w_{B} - w^{\prime}_{B})^2 \times Var(R_{B})$$ +

$$2 \times (w_{A} - w^{\prime}_{A})( w_{B} - w^{\prime}_{B}) \times Cov(R_{A}, R_{B})$$

Getting estimates of Var and Cov without a risk model is not straightforward. One way is to just use estimates from the time period that you are concerned with.

• Thanks. In this case, what is the contribution of asset A, say, to the portfolio's total tracking error? Aug 6 '20 at 7:50
• hi: you can't know unless you know the total tracking error. if it's just the two stocks in the whole index, then its first term over the three terms. Aug 6 '20 at 18:42
• Note that I had to edit above because I left out the multiplicative factor of two on the third term. Aug 6 '20 at 18:58
• Note that when you calculate tracking error, its always with respect to some index or atleast some benchmark. So, when you write that equation for the portfolio's tracking error, that doesn't seem correct. unless maybe the benchmark is cash ? Aug 6 '20 at 21:00
• thx. You are right: the formula for the portfolio wasn't correct. I edited it now. So what is the formula for the general case then? Aug 7 '20 at 14:26