# Herfindahl-Hirsch-Index for FX Portfolios

Assume we have an FX portfolio with $$n$$ currency pairs, such that $$w_i$$ is the weight of currency pair $$i \in \{1, \dots, n\}$$ in the portfolio, and $$\sum_{i=1}^n w_i = 1$$. All pairs have USD as one of the legs (USD is the base currency of the portfolio, and I would like to allocate to other currencies using Markowitz). I would like to compute a concentration measure of the portfolio, and believe that the Herfindahl-Hirsch-Index could be an option (see Chammas, Portfolio Concentration link ). However, Chammas assumes that the portfolio only contains long positions. How should I proceed if one of the portfolio weights is negative?

For example, lets assume that the weight of the USDJPY is $$-0.5$$. Can I simply input that weight in the computation of the HHI?

Is there a more appropriate concentration measure than the HHI for my problem?

• You might consider the Yager Entropy $E_Y = \sum_{i=1}^N | w_i-\frac{1}{N}|$ where the $w_i$ are the portfolio weights. The minimum (i.e. least concentrated portfolio) is achieved when $w_i=1/N$ like for Shannon Entropy but the advantage is that the $w_i$ can be negative. It is a bit obscure and there are only a few papers about using it in a portfolio context AFAIK. Jun 12, 2021 at 23:30
• At the risk of extreme stupidity if you are insisting on long-only positions, why not treat a 0.5 short in USDJPY as a 0.5 long in JPYUSD? Returns measured in the log-price wouldn't be different. Jun 15, 2021 at 1:35