We will demon- strate the method using Microsoft Excel. Excel is far from the best program for this pur- pose and is limited in the number of assets it can handle, but working through a simple portfolio optimizer in Excel can illustrate concretely the nature of the calculations used in more sophisticated "black-box" programs. You will find that even in Excel, the computa- tion of the efficient frontier is fairly easy. We will apply the Markowitz portfolio optimizer to the problem of international diver- sification. Table 8.4A is taken from Chapter 25, "International Diversification," and shows average returns, standard deviations, and the correlation matrix for the rates of return on the stock indexes of seven countries over the period 1980-1993. Suppose that toward the end of 1979, the analysts of International Capital Management (ICM) had produced an input list that anticipated these results. As portfolio manager of ICM, what set of efficient port- folios would you have considered as investment candidates? After we input Table 8.4A into our spreadsheet as shown, we create the covariance ma- trix in Table 8.4B using the relationship Cov(ri, rj) ij i j. The table shows both cell for- mulas (upper panel) and numerical results (lower panel). Next we prepare the data for the computation of the efficient frontier. To establish a benchmark against which to evaluate our efficient portfolios, we use an equally weighted portfolio, that is, the weights for each of the seven countries is equal to 1/7 .1429. To compute the equally weighted portfolios mean and variance, these weights are entered in the border column A53-A59 and border row B52-H52.10 We calculate the variance of this portfolio in cell B77 in Table 8.4C. The entry in this cell equals the sum of all elements in the border-multiplied covariance matrix where each element is first multiplied by the port- folio weights given in both the row and column borders.11 We also include two cells to     10 You should not enter the portfolio weights in these rows and columns independently, since if a weight in the row changes, the weight in the corresponding column must change to the same value for consistency. Thus you should copy each entry from col- umn A to the corresponding element of row 52. 11 We need the sum of each element of the covariance matrix, where each term has first been multiplied by the product of the port- folio weights from its row and column. These values appear in Panel C of Table 8.4. We will first sum these elements for each col- umn and then add up the column sums. Row 60 contains the appropriate column sums. Therefore, the sum of cells B60-H60, which appears in cell B61, is the variance of the portfolio formed using the weights appearing in the borders of the covariance matrix. II. Portfolio Theory 8. Optimal Risky Portfolio The McGraw−Hill Companies, 2001           230 PART II Portfolio Theory