Measuring Portfolio Diversification with Risk Contributions and Variance

# What to use as portfolio diversification measure?


# What to use as portfolio diversification measure?












Suppose that we have a portfolio of $n$ assets.

A perfectly diversified portfolio is one in which each asset has equal weights, i.e. each asset has weight $\frac{1}{n}$. Of course this is usually not the case.

What are some of the ways we can measure how well diversified our portfolio is?

We could measure how far our portfolio is from the equally-weighted portfolio.

This of course will depend on the geometry of the space which is not euclidean since the sum of the weights must be one.

## Answer by Richi Wa (score 9, accepted)

https://quant.stackexchange.com/a/16834

If you measure risk by the standard deviation of the portfolio return $$ \sigma = \sqrt{w^T \Sigma w}, $$ then it is usual to define risk contributions for each asset by $$ \sigma_i = w_i (\Sigma w)_i/\sigma, $$ then diversified could mean that these $\sigma_i$ are evenly spread over the assets in the portfolio.

You find this approach and more in this paper by Meucci

There you also find the variance concentration curve that uses principle components (PCs) of the asset universe and the weighting of the assets to analyze how much the PCs contribute.

Ad good place to read about the application of PCA to portfolio analysis is Regularization of Portfolio Allocation by B. Bruder, N. Gaussel, J-C. Richard and T. Roncalli.

## Answer by jaamor (score 4)

https://quant.stackexchange.com/a/16832

This paper, Equity Portfolio Diversification by W. Goetzmann and A. Kumar, uses the following diversification measures to measure the diversification of retail investors:

- Normalized portfolio variance: $$ NV = \frac{\sigma_p ^2}{\bar{\sigma} ^2} $$

- Sum of Squared Portfolio Weights (SSPW). Since the weight in the market portfolio is very small diversification could be approximated by the sum of squared portfolio weights: $$ SSPW = \sum w_i ^2 $$

- A very crude diversification measure would be the number of assets $N$.

## Answer by Jorge Sabat (score 0)

https://quant.stackexchange.com/a/25181

I recommend you this paper Measuring Portfolio Diversification Ulrich Kirchner & Caroline Zunckel http://arxiv.org/pdf/1102.4722.pdf

## Answer by lebelinoz (score 0)

https://quant.stackexchange.com/a/34492

I've been struggling to quantify and explain diversity for a while and I think I've found something which captures the essence of a portfolio manager's ability to diversify away risk.

Say you have a portfolio where each stock has volatility $\sigma_i$, weight $w_i$, and pairwise correlation $\rho_{ij}$. Then the portfolio's volatility $\sigma$ can be computed as:

$$\sigma^2 = \underbrace{\sum_i w_i^2 \sigma_i^2}_A + \underbrace{\sum_i\sum_{j \neq i}w_i w_j \sigma_i \sigma_j \rho_{ij}}_{B}$$

Part $A$ of the equation is a pure volatility part, and part $B$ is correlation-dependent part which portfolio managers are forever trying to minimise my choosing stocks which are not too correlated to each other.

Surprisingly, the $A$ part is usually much smaller than the $B$ part. For example, when I calculate the MSCI AC World's volatility in this way (using a five years of monthly returns of stocks currently in the index), I find $A = 0.01\%$ and $B = 1.04\%$. When I split my fund's portfolio risks in this manner, I always find $B$ to be much bigger than $A$ (four to ten times bigger).

Since the goal is to minimize $B$, I propose $\sigma^2/A$, or $$\frac{\sigma^2}{\sum w_i^2 \sigma_i^2}$$ as a ratio for measuring diversity: the lower the better. It's similar to Goetzmann & Kumar's Normalized portfolio variance which @jaamor mentioned in his answer, but makes the denominator more relevant to the portfolio you're trying to measure (a lot of information is lost in a straight average $\bar{\sigma}$ of the $\sigma_i$'s).

Edit: I think it makes even more sense to scale the denominator somehow, so portfolios with low numbers of stocks don't get a misleadingly bad score (or high numbers of stocks good scores). Maybe divide by $\sum w_i^2$ or multiply by the number of stocks.

## Answer by varan (score 0)

https://quant.stackexchange.com/a/63449

The problem with the measure used by lebelinoz above is that the denominator, which is the square of the standard deviation of the hypothetical portfolio of the same assets wherein the cross-correlations between the assets are all zero, is not necessarily greater than the numerator for all possible values of the cross-correlations.

A better quantity to use in the denominator is the square of the standard deviation of the hypothetical portfolio of the same assets wherein the cross-correlations between the assets are all one. In that case it can be guaranteed that the ratio is always between zero and one.