Variance and diversification of a portfolio

Question

Suppose I have a portfolio composed by $n$ assets and fixed total size, with stochastic returns. I'm looking for a result stating that as $n$ increases the variance (or any other measure of riskiness) of the portfolio return decreases. If this result exists, could you state the result and the required assumptions? I would prefer not to restrict the portfolio to the case of an equally weighted portfolio.

Answer 1

Let $R_i$ denote the return and $w_i$ denote the portfolio weight of asset $i$ for $i = 1,2, \ldots, n$.

The portfolio return is

$$R_P = \sum_{i=1}^n w_iR_i$$

and the expectation and variance of portfolio return are, respectively,

$$\mu_P = \mathbf{w}^T\mathbf{\mu}, \\\sigma_P^2 = \mathbf{w}^T\mathbf{\Sigma}\mathbf{w}.$$

Here $\mathbf{w}$ and $\mathbf{\mu}$ are vectors of weights and expected returns, respectively, and $\mathbf{\Sigma}$ is the covariance matrix of the asset returns.

There are, obviously, infinitely many possible portfolio constuctions with risk-return characteristics represented by points in the $(\sigma_P,\mu_P)$-plane. Assuming all assets have positive expected returns, these points lie in the first quadrant. As an exercise in quadratic programming, we can find the optimal portfolios that minimize the variance for a given expected return subject to other constraints. These additional constraints typically include $\mathbf{w}^T\mathbf{1}=1$ (full investment) and $\mathbf{0} \leqslant \mathbf{w} \leqslant \mathbf{1}$ (no leverage). This optimization yields the efficient frontier. For a given expected return there is no portfolio with a lower variance than the portfolio on the efficient frontier.

It also can be shown that there is a minimum variance portfolio on the effficient frontier which globally has the least variance of all possible portfolio constructions.

Let $\sigma_{min,S_n,n}^2$ denote this minimum variance for a set of assets $S_n$ with cardinality $n$.

With this background, lets consider the stated objective:

I'm looking for a result stating that as $n$ increases the variance (or any other measure of riskiness) of the portfolio return decreases.

In full generality, there is no simple definitive result. If we expand the asset universe, then new portfolio constructions (with more assets) do not necessarily have lower variance (risk) than portfolios constructed in the smaller universe -- simply because $n$ is larger.

However, we can ask if the number of assets is increased to $m > n$ such that $S_n \subset S_m$ is it the case that the efficient frontier moves to the left and

$$\sigma_{min,S_m,m}^2 \leqslant \sigma_{min,S_n,n}^2.$$

This is clearly true since opimization over the subset $S_n$ must find a minimum that is no smaller then the global minimum for $S_m$. With some further specification of the properties of the additional assets we can show that this inequality is strict.