Optimal Portfolio, Special Cases

The optimal portfolio depends on the utility function $u(y)$. Consider, for example, the optimal portfolio for three different utility functions.

The first utility function is linear

$$u(y)= c y.$$ (11)

This function is for "rich" persons. Rich persons want to maximize the average wealth. They are not emotional about accidental losses or gains. In the linear case (11), the optimal portfolio is to invest all the capital in an object with the highest product $p_ic_i$.

The second utility function is for "prudent" persons which averse risk

$$u(y)= \left\{ \begin{array}{ll} 0 & \mbox{if $0 \le y < a$} \\ 1 & \mbox{if $a \le y \le c$} \end{array}. \right.$$ (12)

Here $a$ is a risk threshold. $c=\max_i c_i x_i$ denotes the maximal return of invested capital (see expression (4)). If $a= 1/m\ min_i\ c_i x_i$ then, in the risk-averse case the optimal decision is $x_i^*=1/m,\ i=1,...,m$. Here one divides the capital equally between all the objects¹.

The third utility function is for "risky" persons. Risky persons are ready to risk for the great win $c$.

$$u(y)= \left\{ \begin{array}{ll} 0 & \mbox{if $0 \le y < c$} \\ 1 & \mbox{if $ y = c$} \end{array}. \right.$$ (13)

Here one invests all the capital in the object with highest wealth return. Therefore, $x_i=1$, if $c_i=\max_j c_j=c$.

These examples are abstract. An average person behaves "risky," if only a small part of his resources is involved. The same person behaves prudently, if all his wealth is at stake. There is a point $r$ between areas of risky and prudent behavior. At this point an average person behaves like the "rich" one. Here is an example

$$u(y) < y,\ if\ 0 \le y < r,$$ (14)

$$u(y)=y,\ if\ y= r,$$

$$u(y) > y,\ if\ r < y \le c.$$

Here $r$ is a boundary point between risky and prudent areas.