Teaching Portfolio Theory: A tool for demonstrating the diversification effect and related issues

Analysis of risk and return for portfolios is a key component of many undergraduate and postgraduate Principles of Finance and Corporate Finance modules. While the depth of the analysis of risk and return may vary across modules and levels, it is (almost certainly always) introduced via consideration of a two asset example. To do this the two relevant risky assets need to be defined along with their expected returns, the variance (or standard deviation) of these returns, the proportion of the portfolio accounted for by each asset and, importantly, the correlation between the returns on the assets. These can be denoted as follows:

• The two assets: x and y.
• The expected returns on the two assets: E[r_x], E[r_y].
• The variance of the returns on the two assets: σ_x², σ_y².
• The proportion of the portfolio accounted for by each asset: λ_x, λ_y.  (Note λ_x + λ_y = 1).
• The correlation between the returns on the assets: ρ_x,y.

With these terms defined, calculation of the expected return on the portfolio, denoted as E[r_p], is straightforward as it is simply the weighted sum of the expected return on the individual assets:

E[r_p] = λ_xE[r_x] + λ_yE[r_y]

Derivation of the variance of the returns on the portfolio, denoted as σ_p², is more involved, although it is quite intuitive: the variance depends the variances of the individual assets, the amount of each asset in the portfolio and the co-movement of the returns on the assets. As with the expected return, the variance of the return of the portfolio is a familiar feature in the presentation of this material and using the above notation is given as:

σ_p² = λ_x² σ_x² + λ_y² σ_y² + 2λ_x λ_y ρ_x,y σ_x σ_y

The expected return for the portfolio is then plotted against the standard deviation of the return for the portfolio to provide a standard risk-return diagram. This diagram is central to the analysis of risk and return and can be used to discuss a range of issues including:

• Identification of the opportunity set of alternative portfolios of the two assets;
• Identification of the efficient set of alternative portfolios of the two assets;
• ‘Dominated’ portfolios;
• The diversification effect.

The use of risk-return diagrams is clearly essential to the analysis of this material given the manner in which it assists the understanding of a variety of underlying concepts. As a consequence, I have constructed a simple Excel spreadsheet to allow the calculation of risk-return schedules for hypothetical assets with hypothetical returns, variances of returns and correlations between returns. I feel this is a very useful tool to help demonstrate all of the above issues, particularly the diversification effect and its dependence upon ρ_x,y. With the spreadsheet provided, students can experiment with different values of ρ_x,y to see how the diversification effect increases as ρ_x,y moves from its upper limit of +1 towards its lower limit of -1. In the version of the spreadsheet available here, the values for the relevant returns, variances and correlation are those presented in an example provided in a particular popular Corporate Finance text (Hillier et al., 2003). Obviously, the flexibility of the spreadsheet is such that it can be used to replicate examples in textbooks, or extend such examples, or provide completely new examples to complement those seen elsewhere. During lectures and tutorials use can be made to the spreadsheet to generate alternative risk-return diagrams to illustrate and discuss a variety of issues central to portfolio theory, including those listed above.

Download the Excel spreadsheet

References

Hiller, D., Ross, S., Westerfield, R. and Jaffe, J. (2013) Corporate Finance (Second European Edition), London: McGraw-Hill. ISBN 9780077139148