# Chapter 11 Introduction to Portfolio Theory

Updated: May 12, 2021

Copyright © Eric Zivot 2015, 2016, 2021

Chapters 7 through 9
focused on the development and estimation of, and the statistical
inference for, the GWN model for asset returns. In this chapter, we
use the GWN model for asset returns as the basis for the quantitative
analysis of portfolios and asset allocation. Specifically, we introduce
the mean-variance portfolio analysis framework originally developed
by Harry Markowitz.^{59} This framework is the widely used in academia and industry and is
covered in most standard textbooks on investment analysis such as
(Bodie, Kane, and Marcus 2013), (Elton et al. 2014), and (Sharpe 1999). In this
framework, asset returns are assumed to be normally distributed and
so investor preferences over portfolios of assets depend only on the
expected return and variance of the portfolio returns. Investors are
assumed to like portfolio with high mean returns but dislike portfolio
with high return variances. Since portfolios that have high (low)
expected returns tend to have high (low) variances investors face
a mean-variance tradeoff between different portfolios. Markowitz’s
mean-variance portfolio theory quantifies this mean-variance tradeoff
and provides a methodology that investors can use to determine optimal
portfolio choice.

This chapter introduces Markowitz’s mean-variance portfolio theory in a simplified setting where there are only two risky assets and a single risk-free asset. This allows us to present the main results of mean-variance portfolio theory graphically and with simple algebra. However, as we shall see in the next chapter, many of the analytic results we derive for portfolios of two risky assets, and for portfolios of two risky assets and a single risk-free asset, can be applied to situations in which there are many risky assets and a single risk-free asset. Therefore, it is highly beneficial to study and understand well the portfolio theory results in the simplified framework of this chapter.

This chapter is organized as follows. Section 11.2 reviews expected return, variance and value-at-risk calculations for portfolios of two risky assets. Section 11.3 introduces mean-variance portfolio analysis for portfolios of two risky assets. The portfolio risk-return frontier is introduced, efficient portfolios are defined, and the global minimum variance portfolio is defined. Section 11.4 presents the analysis of portfolios with a single risky asset and a risk-free asset. The Sharpe ratio/slope of a risky asset is defined and illustrated graphically. Portfolios of two risky assets and a risk-free asset are discussed in Section 11.5. The tangency portfolio is defined and the Mutual Fund Separation Theorem is presented. Section 11.6 provides a brief illustration of the theory to a real world two asset portfolio of stocks and bonds. The Appendix reviews some basic results on constrained optimization of functions.

The R packages used in this chapter are **IntroCompFinR**, **kableExtra**, **knitr**, and **PerformanceAnalytics**. Make sure these packages are installed
and loaded in R before replicating the chapter examples.

### References

Bodie, Z., A. Kane, and A. J. Marcus. 2013. *Investments. 10th Edition*. McGraw-Hill Education.

Elton, E., G. Gruber, S. J. Brown, and W. N. Goetzmann. 2014. *Modern Portfolio Theory and Investment Analysis. 9th Edition*. New York: Wiley.

Markowitz, H. 1959. *Portfolio Selection: Efficient Diversification of Investments*. New York: Wiley.

Markowitz, H. 1987. *Mean-Variance Analysis in Portfolio Choice and Capital Markets*. Cambridge, MA: Basil Blackwell.

Sharpe, W. 1999. *Investments. 6th Edition*. Prentice Hall.