Variance of an Equal-Weight Portfolio with Uncorrelated and Equicorrelated Assets

A portfolio holds `n` assets. The return of every asset has variance 1, and the portfolio is equally weighted, with weight `1/n` on each asset. Find the variance of the portfolio's return in two settings. ### Clarifying Questions - Should the answer be a closed form in `n` (and later in $\rho$), and should it also describe what happens as `n` grows? - For the first setting, is it enough that the returns are uncorrelated, or should I assume they are fully independent? - Are the weights fixed at exactly `1/n` in both settings, with no short positions or leverage? ### Part 1 — Uncorrelated assets Assume that the returns of different assets are uncorrelated. What is the variance of the equally weighted portfolio? ```hint Expand before you simplify Write the portfolio return as a weighted sum of the asset returns, expand its variance into variance terms and covariance terms, and decide which of them survive. ``` #### What This Part Should Cover - The variance of a weighted sum, written out term by term - The role of the squared weights - How the result depends on `n`, and what it says about diversification ### Part 2 — Equal positive correlation between every pair Now every pair of distinct assets has the same correlation $\rho > 0$. Each asset still has variance 1 and weight `1/n`. What is the variance of the portfolio? ```hint Count the cross terms Count how many covariance terms the expansion contains, and work out what each one equals when every variance is 1. ``` #### What This Part Should Cover - The number and the value of the covariance terms - A simplified closed form, checked against limiting cases such as $\rho = 0$, $\rho = 1$ and a single asset - Which part of the risk diversification removes and which part remains ### What a Strong Answer Covers - Sets the variance up as a double sum, or as the quadratic form of the weights with the covariance matrix, instead of guessing - Treats Part 1 as the special case of Part 2 with zero correlation - Describes the behavior as `n` grows in both settings and explains it as diversifiable versus common risk - Knows that not every correlation value is possible for `n` assets ### Follow-up Questions - As `n` grows, what does the portfolio variance approach in each part? With $\rho = 0.2$, how many assets bring the variance within 10% of that limit? - If $\rho$ were allowed to be negative, how negative could it be for `n` assets, and why? - With identical variances and identical pairwise correlations, is the equally weighted portfolio the fully invested portfolio with the lowest variance? Justify it. - How does the answer change if every pair still has correlation $\rho$ but the assets have different variances $\sigma_i^2$?