A diversification benefit exists when a portfolio’s standard deviation can be reduced without reducing expected return.

The diversification benefit is possible when return correlations between portfolio assets is less than perfect positive correlation (<+1.0).

If assets have less than a +1.0 correlation, then some of the random fluctuation around the expected trend rates of return will cancel each other out and lower the portfolio’s standard deviation (risk).

If assets have a perfect negative correlation (-1.0), then some combination of asset weights will eliminate all of the portfolio’s expected standard deviation (risk).

As the number of assets grows large, the variance of the portfolio will approach the average covariance of the asset pairs comprising the portfolio; this is the upper limit of portfolio diversification benefits.

Covavg = (ρavg * σavg2)

An analyst can assess how the variance (and standard deviation) of a portfolio will decline by adding more assets with following formula:

σport new2 = σcurrent2 ((1-ρavg)/n + ρavg)

Where average correlation, ρavg  = Covavg / σcurrent2

Remember that the above formula is for variance, so if the question asks for standard deviation, the square root will need to be taken.

An exercise can be performed that shows where the initial benefits of adding assets is dramatic but the benefits increase at a decreasing rate as more and more assets are added.

• This indicates that the benefits of diversification can be realized from a small number of well-chosen assets.
• For example, a stock portfolio may be able to achieve strong diversification from 30 stocks; however the incremental diversification benefits from adding another 1,000 stocks may not be incredibly significant.