1. The utility function ( U ) of wealth ( W ) can be defined different for different people. The coefficient of risk aversion is defined as
(
A(W):= - \frac{U''(W)}{U'(W)}
)
One example is
(
U(W): = -\frac{exp(-aW)}{a}.
)
and its coefficient of risk aversion can be computed as
(
A=a.
)
2. On the other hand, if the return ( r ) of a portfolio is a random variable with expection (E(r)) and variance ( \sigma^2 ), then the utility of the portfolio return is defined as
(
U(r) = E(r) - A \sigma^2/2
)
where $A$ is also call coefficient of risk aversion.
3. Concerned about the two definitions of utility functions and of coefficients of risk aversion in part 1 and 2, my questions are:
(
A(W):= - \frac{U''(W)}{U'(W)}
)
One example is
(
U(W): = -\frac{exp(-aW)}{a}.
)
and its coefficient of risk aversion can be computed as
(
A=a.
)
2. On the other hand, if the return ( r ) of a portfolio is a random variable with expection (E(r)) and variance ( \sigma^2 ), then the utility of the portfolio return is defined as
(
U(r) = E(r) - A \sigma^2/2
)
where $A$ is also call coefficient of risk aversion.
3. Concerned about the two definitions of utility functions and of coefficients of risk aversion in part 1 and 2, my questions are:
- Are the above two definitions of utility functions related somehow? Is the second one based on the first one?
- How different are the cases where they are intended to use?
- Are their coefficients ( A ) of risk aversion the same thing or differently defined?
