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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.

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    $\begingroup$ I strongly disagree with the fact that the equally-weighted portfolio is perfectly diversified. If you assume that, then indeed your diversification measure should be the distance to that portfolio. But that would just be wrong in my opinion, and if you choose another measure you should expect the most-diversified portfolio, according to that measure, not to be the equally-weighted portfolio. $\endgroup$ – SRKX Mar 3 '15 at 7:23
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    $\begingroup$ I agree with SRKX but would even go one step further. It does not make a lot of sense (at least to me) to discuss diversification unless you specify a risk measure. Then the perfectly diversified portfolio is one which minimises your particular risk measure. $\endgroup$ – g g Mar 3 '15 at 8:34
  • $\begingroup$ True there are many ways to measure diversity. I merely suggest this as one possible way. One draw back with a risk parity approach is that it assumes that your moment estimates are accurate. Perhaps a good risk control policy on top of some risk parity approach is to say that your investment in any asset is between 1% and 5% of total portfolio value. This puts in place a risk control that is independent of your estimates and perhaps sheds some light on why we may want to look at equal weight portfolios. $\endgroup$ – Wintermute Mar 3 '15 at 14:57
  • $\begingroup$ I would disagree with this measure because it does not control for number of assets as well. Bu this definition, putting all your eggs in one basket (n=1) will a 'perfectly' diversified portfolio $\endgroup$ – ChinG Nov 11 '15 at 16:14
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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.

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  • $\begingroup$ I've seen this approach before. I always get a little leery when PC analysis is used in risk estimation. PC analysis assumes the existence of zero variance portfolios. $\endgroup$ – Wintermute Mar 3 '15 at 15:03
  • $\begingroup$ "PC analysis assumes the existence of zero variance portfolios" - no, it does not. It is rather the opposite ... where do you have this from? Any proof or reference? $\endgroup$ – Richard Mar 3 '15 at 15:48
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    $\begingroup$ @Richard Meucci seems to have moved on past that paper to his one on minimum torsion bets. I've been playing around with it the past couple of days and have found it to be MUCH better. $\endgroup$ – John Mar 4 '15 at 5:06
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    $\begingroup$ @mtiano No, if your assets are linearly independent the PCA will not give you eigenvalues of zero. The covariance matrix will have full rank and all eigenvalues will be bigger than zero. If you have more assets than observation times then you have to use another estimator for the covariance matrix (e.g. shrinkage) $\endgroup$ – Richard Mar 4 '15 at 7:13
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    $\begingroup$ @Richard I can fix you up in the meantime: papers.ssrn.com/sol3/papers.cfm?abstract_id=2276632 $\endgroup$ – vanguard2k Mar 4 '15 at 7:47
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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 $N$.
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    $\begingroup$ You could add the title and the name of the authors of the paper, it would be nice for them... $\endgroup$ – SRKX Mar 3 '15 at 7:25
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I recommend you this paper Measuring Portfolio Diversification Ulrich Kirchner & Caroline Zunckel http://arxiv.org/pdf/1102.4722.pdf

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  • $\begingroup$ The 20 word summary of this paper "We propose the negative information entropy of the probability distribution of the final portfolio value as a suitable diversification measure". $\endgroup$ – noob2 Apr 1 '16 at 16:55
  • $\begingroup$ Could you consider describing why you think this paper contains any nice insights instead of just posting a link? This would improve overall quality of your answer by far=) $\endgroup$ – muffin1974 Apr 13 '16 at 6:56
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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.

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