# Efficient Portfolios

The graphical approach to Markowitz portfolio analysis presented in Chapter 5 provides an excellent visual introduction. However, for those who wish to be able to solve portfolio problems involving many assets, this chapter is more appropriate. This chapter's mathematical techniques yield the same answers as the graphical technique. However, the mathematical techniques in this chapter are more general, more precise, and offer the tools for solving larger problems.

# 6.1 Risk and Return for Two-Asset Portfolios

A portfolio manager must consider many different securities at once when attempting to delineate an optimal portfolio. In this section, the statistical tools the portfolio manager uses for evaluating securities (and thus portfolios) within the modern portfolio theory framework are developed.

If denotes the return from a portfolio consisting of n securities, that portfolio's expected return, , can be expressed as

2.12 and its variance, , can be expressed as

Equation (series 2.19a) decomposes the first equation into two parts. The first part of ...

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