chapter 3 - Bentham Science

142 Applications of Spreadsheets in Education The Amazing Power of a Simple Tool Clarence C.Y. Kwan
the use of multivariate calculus for optimization and matrix algebra. For students who have learned
normative portfolio analysis, however, such a task is not burdensome. Notice that the equilibrium
analysis to be presented in this chapter is not intended to replace the textbook version. Rather, the
two versions complement each other, so that students can better appreciate the various implications
and limitations of the model.
The matrix operations as required for the equilibrium analysis here are confined to matrix addition,
multiplication, transposition, and inversion, as well as finding the determinant of a matrix. As
spreadsheet tools for matrix operations are easy for students to learn, using them can significantly
reduce the computational burden. With the relevant input parameters arranged as blocks of data in a
spreadsheet, to perform matrix operations is straightforward. With computational issues no longer
a concern, students can focus their attention on conceptual issues pertaining to the model.
To facilitate the analysis that follows, Section 8.2 first provides some essential material on risk
aversion. Section 8.3 starts with a basic equilibrium analysis for asset price determination. An
extension of the analysis, by relaxing a simplifying assumption in the CAPM, is also considered.
Implications of this extension are discussed as well. Section 8.4 presents an Excel example to
illustrate the required computations. Section 8.5 offers some concluding remarks. For readers who
are unfamiliar with the CAPM, the appendix at the end of this chapter provides an abbreviated
version of its derivation in textbooks. 4
8.2 Preliminaries
Let U(W) be the utility function of an investor with wealth W. The investor is rational if the first
derivative of U(W), written as U ′ (W), is positive. The idea is that, if wealth is the only factor in
the determination of satisfaction, a rational investor must prefer more wealth to less wealth. The
investor is also risk averse if the second derivative, U ′′ (W), is negative. The idea is that a risk
averse investor considers the increase in satisfaction for having an incremental wealth ΔW less than
the decrease in satisfaction for losing the same ΔW. To illustrate, if an investment has equal chances
of gaining $h or losing $k, a risk averse investor will not find the investment worthwhile if h≤k;
exactly how high h−k has to be for the investment to appeal to the investor depends on the function
U(W).
For a risk averse investor with an initial wealth W0, suppose that an investment will result
in W0+z as the final wealth for the investor, with the outcome of the investment z being random.
To assess the investment, the investor considers the expected utility of all potential outcomes,
E[U(W0+z)]. Here, E[·] represents the expected value of the variable[·] in question. To the investor,
this risky investment is equivalent to a risk-free investment with a certain outcome of E[z]−π,
where π > 0 can be viewed as a premium that the investor is willing to pay in order to avoid risk.
To determine π, we can start with
U(W0+ E[z]−π)=E[U(W0+z)] (8.1)
where W0+ E[z]−π is called the certainty equivalent wealth, CEQ.
4 See, for example, Chapter 6 in [8].