the n diagonal elements are estimates of the variances, 2, and the n2 n n(n 1) off-diagonal elements are the estimates of the covariances between each pair of asset returns. (You can verify this from Table 8.2 for the case n 2.) We know that each covariance appears twice in this table, so actually we have n(n 1)/2 different covariances estimates. If our portfolio management unit covers 50 securities, our security analysts need to deliver 50 estimates of expected returns, 50 estimates of variances, and 50 49/2 1,225 different estimates of covariances. This is a daunting task! (We show later how the number of required estimates can be reduced substantially.) Once these estimates are compiled, the expected return and variance of any risky port- folio with weights in each security, wi, can be calculated from the bordered covariance ma- trix or, equivalently, from the following formulas: n E(rp) wi E(ri) i 1 (8.9) n n P   i 1 j 1 wiwj Cov(ri, rj) (8.10) An extended worked example showing you how to do this on a spreadsheet is presented in the next section. We mentioned earlier that the idea of diversification is age-old. The phrase "dont put all your eggs in one basket" existed long before modern finance theory. It was not until 1952, however, that Harry Markowitz published a formal model of portfolio selection embodying diversification principles, thereby paving the way for his 1990 Nobel Prize for economics.9 His model is precisely step one of portfolio management: the identification of the efficient set of portfolios, or, as it is often called, the efficient frontier of risky assets. The principal idea behind the frontier set of risky portfolios is that, for any risk level, we are interested only in that portfolio with the highest expected return. Alternatively, the fron- tier is the set of portfolios that minimize the variance for any target expected return. Indeed, the two methods of computing the efficient set of risky portfolios are equivalent. To see this, consider the graphical representation of these procedures. Figure 8.12 shows the minimum-variance frontier. The points marked by squares are the result of a variance-minimization program. We first draw the constraints, that is, horizontal lines at the level of required expected returns.     9 Harry Markowitz, Portfolio Selection, Journal of Finance, March 1952.