# ESG score for shorted stocks and for long-short portfolio

I was wondering how to compute an extra-financial score of a portfolio like, for instance, the ESG score. This score can is typical bounded between 0 and 10 (or 100) (see for example IVA methodology of MSCI. How would one get the score of a portfolio from its constituents? What is the score of a shorted stock?

More generally I am interested in the construction of portfolios with a minimum score but that is trivial after the above question is solved.

• Can you not just take the weighted average as long as the weights add up to one? – Bob Jansen Nov 24 '18 at 23:54
• I tried that (and this is what is MSCI says it does msci.com/esg-ratings) but I am getting some meaningless results. For instance suppose I have a good stock A with a score $s_A=10$ and a bad one B with a score $s_B=2$. If I go long A and short B with (say) weights $w_A=2$ and $w_B=-1$ then the score is $2 s_A - 1 s_B$. Note that the score of shorting B reduces the portfolio score instead of increasing it contrary to what one would have expected by shorting a bad stock. – Borun Chowdhury Nov 25 '18 at 16:24
• Why does your bad stock have a positive score? ;) in any case, shorting stocks with lower scores and using the proceeds for buying stocks with higher scores increases the overall score. – Bob Jansen Nov 25 '18 at 16:47
• It shouldn't according to me but this is what the standard scores (MSCI, MorningStar etc) use. While it is true that the overall score is increasing in my example the fact that the contribution of the bad stock is to decrease the store is counterintuitive and doesn't make sense. Interestingly if the scores are shifted by -5 then this problem is not there. – Borun Chowdhury Nov 25 '18 at 16:52
• But I guess your point is correct in that is that the effect of the shorting the bad stock is indeed in the right direction compared to the mean. Indeed this can be clearly seen by defining the scores to be be wrt to the mean score 5 i.e. scores get shifted by -5. But the effect does not depend on the origin chosen as it is a linear problem. – Borun Chowdhury Nov 25 '18 at 17:02

For example, if shorting a "bad ESG" stock is a goal of the portfolio, then the taking a weighted average, i.e. sum(position_size * IVA), where position sizes are allowed to be negative, will work as intended. This will allow the opt. engine to attempt to short as many "bad esg" stocks and long "good esg" stocks as possible within the additional constraints that you provide.

This can be thought of similar to calculating the beta of a portfolio. Beta of a portfolio can be calculated by taking the weighted sum, i.e. sum(position_size * beta), and they all add up nice and linearly. Beta of a portfolio can for sure flex to negative (i.e. short 100% S&P 500 ETF (SPY) has a market beta of -1).

As the answer by mperlow states the score may be simply the weighted average of scores where the weights are the portfolio weights. This is indeed the method used by MorningStar with an extra effect added based on other consideration. At zeroth order this answer would do.

This in hindsight is obvious and was what I thought of in the beginning but it was a bit confusing in that the contribution of a bad stock (with say a score of 2) seemed to reduced the score of the portfolio

$$s_{portfolio} = \dots - w_B s_B \dots$$

but this is an illusion because the scores are bounded in $$[0,10]$$. If we think of the scores as the deviation from the mean $$\tilde s_i = s_i \to s_i - 5$$ then its clear that the effect of shorting a bad stock is actually to increase the score of the portfolio

$$\begin{eqnarray} \tilde s_{portfolio}-5 &=& s_{portfolio}-5 \\ &=& \dots - w_B \tilde s_B \dots \\ &=& \dots - w_B (s_B-5) \dots \end{eqnarray}$$

This doesn't change anything (because of the linear nature of the problem) but does suggest that using scores that are deviations from the mean may be a bit more intuitive. Note also that the score of the portfolio is not bound to be in the same interval as that of the individual companies but that is not a surprise given that we are leveraging.

• I would add that you can find the "gross normalized score" of your portfolio, which will be comparable to individual stocks, by dividing your portfolio ESG score by gross leverage, (long_market_value + abs(short_market_value)) / aum, effectively converting it to "unit" ESG score. This will allow simpler comparison across portfolios with different leverage and against stocks. – mperlow Nov 26 '18 at 2:50