Risk aversion, prudence, and asset allocation: a review and some new developments

Abstract

In this paper, we consider the composition of an optimal portfolio made of two dependent risky assets. The investor is first assumed to be a risk-averse expected utility maximizer, and we recover the existing conditions under which all these investors hold at least some percentage of their portfolio in one of the assets. Then, we assume that the decision maker is not only risk-averse, but also prudent and we obtain new minimum demand conditions as well as intuitively appealing interpretations for them. Finally, we consider the general case of investor’s preferences exhibiting risk apportionment of any order and we derive the corresponding minimum demand conditions. As a byproduct, we obtain conditions such that an investor holds either a positive quantity of one of the assets (positive demand condition) or a proportion greater than 50 % (i.e., the “50 % rule”).

Appendix: A useful expansion formula

Consider two random variables $Z_1$ and $Z_2$ valued in $[a,b]$ and a real valued function $g$ with domain $[a,b]\times [a,b]$. Let $g^{(i,j)}$ denote the $(i,j)$th partial derivative of $g$, i.e., $g^{(i,j)}(z_1,z_2)=\frac{\partial ^{i+j}}{\partial z_1^i\partial z_2^j}g(z_1,z_2)$. Integration by parts shows that

$$\begin{aligned} E[g(Z_1,Z_2)]= & {} E[g(Z_1,b)]-\int _a^bg^{(0,1)}(b,z_2)P[Z_2\le z_2]\mathrm{d}z_2\\&+\int _a^b\int _a^b\Pr [Z_1\le z_1,Z_2\le z_2]g^{(1,1)}(z_1,z_2)\mathrm{d}z_1\mathrm{d}z_2. \end{aligned}$$

Integrating by parts the last double integral gives

$$\begin{aligned}&\int _a^b\int _a^b\Pr [Z_1\le z_1,Z_2\le z_2]g^{(1,1)}(z_1,z_2)\mathrm{d}z_1\mathrm{d}z_2\\&\quad =\int _a^bg^{(1,1)}(b,z_2)\left( \int _a^b\Pr [Z_1\le \xi _1,Z_2\le z_2]\mathrm{d}\xi _1\right) \mathrm{d}z_2\\&\qquad -\int _a^b\int _a^b\left( \int _a^{z_1}\Pr [X_1\le \xi _1,X_2\le z_2]\mathrm{d}\xi _1\right) g^{(2,1)}(z_1,z_2)\mathrm{d}z_1\mathrm{d}z_2. \end{aligned}$$

Now, as

$$\begin{aligned} \int _a^{z_1}\Pr [Z_1\le \xi _1,Z_2\le z_2]\mathrm{d}\xi _1= & {} \int _a^{x_1}E\Big [I[Z_1\le \xi _1]I[Z_2\le z_2]\Big ]\mathrm{d}\xi _1\\= & {} E\left[ \int _a^{z_1}I[Z_1\le \xi _1]\mathrm{d}\xi _1 I[Z_2\le z_2]\right] \\= & {} E\Big [(z_1-Z_1)_+ I[Z_2\le z_2]\Big ] \end{aligned}$$

the expectation $E[g(Z_1,Z_2)]$ can be expanded as follows:

$$\begin{aligned} E[g(Z_1,Z_2)]= & {} E[g(Z_1,b)]-\int _a^bg^{(0,1)}(b,z_2)P[Z_2\le z_2]\mathrm{d}z_2\nonumber \\&+\int _a^bg^{(1,1)}(b,z_2)E\Big [(b-Z_1) I[Z_2\le z_2]\Big ]\mathrm{d}z_2\nonumber \\&-\int _a^b\int _a^bg^{(2,1)}(z_1,z_2)E\Big [(z_1-Z_1)_+ I[Z_2\le z_2]\Big ]\mathrm{d}z_1\mathrm{d}z_2.\nonumber \\ \end{aligned}$$

(5.1)