# Diversification Score

A measure of how diversified a portfolio is. Northstar Risk uses the following formula to compute diversification scores:

The diversification score ranges from 0 to 1. If the diversification score is equal to 0, the portfolio has no diversification. It is as if the entire portfolio were in only one security. If the diversification score is equal to 1, the portfolio has maximum diversification. In this case, all the positions appear to be perfectly hedged and the portfolio standard deviation is zero.

As a measure of risk, diversification score has some shortcomings. For example, replacing an ETF with the underlying equities in the ETF will increase the diversification score, even though there is no impact on actual market risk. For long/short equity portfolios the diversification score can be very sensitive to the net exposure. For long/short portfolios, looking at the diversification score of the long and short books separately may be more meaningful.

In certain situations you may want to look at the diversification not by position, but by strategy or book. In this case, you would replace “positioni” in the above formula with “strategyi” or “booki”. Northstar Risk has a version of the diversification score, “Diversification score (group)”, which calculates diversification at the highest level, given the current grouping. “Diversification score (leaf)” calculates the diversification score based on the lowest level. The default diversification score, uses the “leaf” method with the portfolio grouped so that the lowest level is positions. For the same portfolio, the leaf method will always produce a higher diversification score.

The definition of diversification score provided above is far from universal. One common variation simply calculates the ratio of the portfolio standard deviation to the sum of the position standard deviations, in effect, removing the “1 −” from the equation given above. In this case 0 would be high diversification and 1 would be low diversification. This can be confusing because a higher score is associated with lower diversification. For the definition used by Northstar Risk a higher diversification score is always associated with higher diversification and a lower score with lower diversification. Other variations have unbounded ranges (unlike the Northstar Risk definition which always ranges from 0 to 1), or use other risk measures. Substituting VaR for standard deviation in the above definition is common. For most portfolios the results will be very similar, and using standard deviation is more computationally efficient.