Chapter 7
Advanced Mathematical Portfolio Analysis
This chapter presents a more advanced mathematical solution for the same portfolio problem that was discussed in Chapter 6. A specific solution for the optimal weights for efficient portfolios is presented with and without a risk-free asset. Several numerical examples are provided. An equation specifying the relationship between the expected returns and variances of efficient portfolios is presented: this equation represents the efficient frontier. Although this chapter employs more concise vector and matrix notations to represent the same problem used in Chapter 6, no new financial insights will be presented and this book's reading continuity will not be lost if this chapter is passed over.
7.1 Efficient Portfolios without a Risk-Free Asset
As shown in Chapters 5 and 6, if a risk-free asset is included in the list of risky investment candidates, a linear efficient frontier results. This section omits the risk-free asset from the list of risky investment candidates. As a result of this omission, this section derives a convex efficient frontier.
7.1.1 A General Formulation
Consider how a portfolio on the efficient frontier can be identified. What is desired are weights for the portfolio that minimize the portfolio variance, 
, at a given level of the expected return, . As mentioned in the previous paragraph, this section's optimization ...
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