Diversification

We showed that the variance and covariances of returns to a portfolio is given by

I want to get at the intuition behind asset diversification, the idea of not putting all your eggs in one basket, as they say. The intuition is that diversification spreads risk. Let's show this intuition. We can illuminate this concept more easily if we look at a portfolio in which the assets have identical weights—an equally weighted portfolio of N assets with each . With no loss in generality, also assume their expected returns are all the same. Then the variance-covariance for this portfolio can be written as:

The bracketed terms are just the cross-products of all the returns with themselves and the other assets’ returns. The expectation operator takes the sum of these squares and divides by the number of terms (we'll get to this later but for now understand that if there are N assets, then there are N² cross-products in total). We can parse this out into the variances and covariances as follows:

The first term in the square brackets is the sum of the individual asset variances. This second term is the sum of the covariances among assets. The total sum is the risk on the portfolio. Clearly, it is larger if there is positive covariance between the returns. If, on the other hand, the covariances are zero, meaning the returns are independent, then portfolio variance (volatility) is just a function of the individual variances that each asset contributes to the portfolio. If the covariances are negative, on sum, then the overall risk (variance) on the portfolio can be reduced even further. Separately, we state without proof that under fairly general circumstances, as the number of assets in the portfolio increases, the variance of the portfolio falls (this result depends on the denominator N² increasing at a faster rate than the sum of covariances). The point we make here is showing how diversification of assets in the portfolio affects the risk on the portfolio. Since covariances are generally weaker than variances, then N² increases faster than the sum of the covariances and variances demonstrating that diversification (rising N) can help reduce portfolio risk. How large does N have to be to capitalize on the risk reduction benefits of diversification? The answer depends on the pool of assets we are selecting from; if the pool consists of similar assets (say, investment grade bonds), then N may have to be quite large. On the other hand, N may be as small as 15 or 20 stocks randomly chosen across the Russell 3000 index. It depends on how heterogeneous the population of assets one is selecting from.