# Formula for underdiversification

I'm trying to develop a study which links a person's demographic and social characteristics to a tendency to under diversify their portfolio. So far I was accounting for risk aversion and just going the divide by n method, but is there a more specific way to know if a person is under diversified?

For context, in the study I give participants 3 assets, a low risk bond, moderate risk fund and high risk stock, and ask them to split assets between the three classes. This happens over 3 trading periods, and I have full control over the return of the asset

Sorry if this question isn't very specific.. is there a way?

A variant of the Herfindahl-Hirschman index, specifically its inverse, is probably the most widely used for this sort of thing. It's a measure of portfolio or market concentration, where an equally-weighted portfolio (most diversified) has an effective sample size equal to the number of assets. It's a cousin of the Gini coefficient used a lot in income inequality research.

Calculated as follows, where $$w_i$$ are holding weights:

$$effSS = \frac{1}{\sum_{i}w_i^2}$$

I recommend you to read John Y. Cambell AFA presidential address, namely section III of his paper: https://scholar.harvard.edu/files/campbell/files/householdfinance_jof_2006.pdf

There are few ways of measuring underdiversification, few of them are outlined in the paper above but I guess the most obvious is the first one they mention:

We adopt the perspective that systematic risk is compensated and idiosyncratic risk is not, so that taking idiosyncratic risk is an investment mistake.

So, you can use the amount of idiosyncratic risk as a proxy for underdiversification.