Naive diversification means a portfolio consisting of stocks chosen at random. This is not put all your eggs in one basket approach. Intuitively, as stocks in the portfolio increases, individual fluctuations in asset returns are cancelled out. Thus, the variance of returns on a portfolio should vary inversely with the number of securities in the portfolio. However, ii is not possible to reduce portfolio risk to zero by increasing the number of assets in the portfolio. Equation 3.18 shows that when there are just two securities there are an equal number of variance and co-variance terms. As the number of securities increases, the number of co-variance terms increases much faster. In a portfolio of N securities, there are N variance terms, but N2 – N co-variance terms. If the securities in the Portfolio have equal weights, the portfolio variance is given by Equation 3.18.
Portfolio variance: = 1/N * average variance + (1 – 1/N) average co-variance.
As N increases, the portfolio variance steadily approaches to average co-variance. This is the limit-the level of systematic risk–below which portfolio risk cannot be reduced through naive diversification. Systematic risk refers to the overall market risk that affects all securities and cannot be diversified away. Empirical studies have shown that this limit is reached at a relatively low level of diversification; say 10 to 15 securities can eliminate most of the non-systematic risk of the portfolio. non-systematic risk is firm or specific and can be avoided by diversification.