Measuring Portfolio Underdiversification with Concentration and Idiosyncratic Risk
Summary
The discussion offers two ways to quantify underdiversification in a portfolio allocation study. One is the inverse Herfindahl-Hirschman concentration measure: take the reciprocal of the sum of squared asset weights. It can be interpreted as an effective number of holdings, reaching the number of assets when weights are equal. This provides a compact measure of how concentrated a participant’s allocation is across the available assets.
A second approach uses the amount of idiosyncratic risk in the portfolio as a proxy for underdiversification. This rests on the assumption that systematic risk is compensated while asset-specific risk is not, so bearing avoidable idiosyncratic risk may indicate an investment mistake. The answer points to household finance research for broader measures. These indicators capture different aspects of diversification and should not be treated as interchangeable: concentration depends on weights, while idiosyncratic risk depends on asset exposures and return behavior. The thread does not specify how to adjust either measure for risk preferences or the controlled returns in the proposed experiment.
Key ideas
- The inverse Herfindahl-Hirschman index summarizes portfolio concentration through the effective number of holdings.
- Equal asset weights maximize this effective count for a fixed set of assets.
- Portfolio idiosyncratic risk can serve as a proxy for underdiversification.
- Using idiosyncratic risk as a mistake measure assumes that systematic risk is compensated and asset-specific risk is not.
- Weight concentration and uncompensated risk represent distinct measurement choices.
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# Formula for underdiversification
# 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?
## Answer by Chris (score 1)
https://quant.stackexchange.com/a/49837
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:
$effSS = \frac{1}{\sum_{i}w_i^2}$
## Answer by phdstudent (score 0)
https://quant.stackexchange.com/a/49285
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.