Diversification (finance) - Effect of Diversification On Variance

Effect of Diversification On Variance

One simple measure of financial risk is variance. Diversification can lower the variance of a portfolio's return below what it would be if the entire portfolio were invested in the asset with the lowest variance of return, even if the assets' returns are uncorrelated. For example, let asset X have stochastic return and asset Y have stochastic return, with respective return variances and . If the fraction of a one-unit (e.g. one-million-dollar) portfolio is placed in asset X and the fraction is placed in Y, the stochastic portfolio return is . If and are uncorrelated, the variance of portfolio return is . The variance-minimizing value of is, which is strictly between and . Using this value of in the expression for the variance of portfolio return gives the latter as, which is less than what it would be at either of the undiversified values and (which respectively give portfolio return variance of and ). Note that the favorable effect of diversification on portfolio variance would be enhanced if and were negatively correlated but diminished (though not necessarily eliminated) if they were positively correlated.

In general, the presence of more assets in a portfolio leads to greater diversification benefits, as can be seen by considering portfolio variance as a function of, the number of assets. For example, if all assets' returns are mutually uncorrelated and have identical variances, portfolio variance is minimized by holding all assets in the equal proportions . Then the portfolio return's variance equals = =, which is monotonically decreasing in .

The latter analysis can be adapted to show why adding uncorrelated risky assets to a portfolio, thereby increasing the portfolio's size, is not diversification, which involves subdividing the portfolio among many smaller investments. In the case of adding investments, the portfolio's return is instead of and the variance of the portfolio return if the assets are uncorrelated is which is increasing in n rather than decreasing. Thus, for example, when an insurance company adds more and more uncorrelated policies to its portfolio, this expansion does not itself represent diversification—the diversification occurs in the spreading of the insurance company's risks over a large number of part-owners of the company.