Continuing Example E in Section 4.3, suppose there are n securities, each with the same...

Continuing Example E in Section 4.3, suppose there are n securities, each with the same expected return, that all the returns have the same standard deviations, and that the returns are uncorrelated. What is the optimal portfolio vector? Plot the risk of the optimal portfolio versus n. How does this risk compare to that incurred by putting all your money in one security?

Reference

Investment Portfolio

We are now in a position to further develop the investment theory discussed in Section 4.1.2, Example E, and Section 4.2, Example D. Please review those examples before continuing.We first consider the simple example of two securities, assuming that they have the same expected returns μ₁ = μ₂ = μ and their returns are uncorrelated: σi j = Cov(R_i , R_j ) = 0. For a portfolio (π, 1 − π), the expected return is

The investment decision, the choice of the portfolio vector π, is often couched as that of maximizing expected return subject to the risk being less than some value the individual investor is willing to tolerate. Some investors are more risk averse than others, so the portfolio vectors will differ from investor to investor. Equivalently, the decision may be phrased as that of finding the portfolio vector with the minimum risk subject to a desired return; there may well be many portfolio choices that give the

As a general rule, risk is reduced by diversification and can be decreased with only a small sacrifice of returns. Figure 4.6 from Bernstein (1996, p. 254) illustrates this point empirically. The point labeled “Index” shows the monthly average versus standard deviation for an investment that was equally weighted across all the markets.

A reasonably high return with relatively little risk would thus have been obtained by spreading investments equally over the 13 stock markets. In fact, the risk is less than that of any of the individual markets. Note that these emerging markets were riskier than the U.S. market, but that they were more profitable.