power utility function - chunhualiao/public-docs GitHub Wiki
Computes Expected Utility: Calculates the expected Constant Relative Risk Aversion (CRRA) utility of terminal wealth for each simulated path. This utility function quantifies investor satisfaction with wealth, accounting for risk aversion.
The CRRA utility function is defined as: $U(W) = \frac{W^{1-\gamma}}{1-\gamma}$ when gamma is not equal to 1.
And for the special case when gamma equals 1: $U(W) = \log(W)$
where: W = terminal wealth
gamma = coefficient of relative risk aversion
Example: Let's say an investor has a terminal wealth of $100,000 and a risk aversion coefficient (gamma) of 2.
The utility would be calculated as:
U(100000) = (100000^(1-2))/(1-2) = (100000^(-1))/(-1) = -1/100000 = -0.00001
The most common mathematical representation of a CRRA utility function is the power utility function:
$$U(W) = \frac{W^{1-\gamma}}{1-\gamma}$$
Where:
- $U(W)$ is the utility derived from a certain level of wealth, $W$.
- $W$ is the investor's total wealth.
- $\gamma$ (gamma) is the coefficient of relative risk aversion. This is the single most important parameter in the function.
The coefficient $\gamma$ determines an investor's level of risk aversion:
- If gamma is negative: 1-gamma will be >1, it becomes a superlinear or exponential grow: more money, much more utility: this is not the reality
- If $\gamma = 0$, the investor is risk-neutral: it is a straight 45-degree slope, each dollar generates the same utility, not the reality neither
- If $\gamma > 0$ but <1, 1-gama will be a fraction , the utility line will be bending downwards, generating positive utility values. the investor is risk-averse.
- If gamma >1, 1-gama will be negative, the utility line will be bending downwards, generate negative values reaching 0.
- A higher $\gamma$ means higher risk aversion. An investor with $\gamma=5$ is much more conservative than an investor with $\gamma=2$.
Vanguard, in its models, assumes a hypothetical average investor with a specific, proprietary $\gamma$ value that it believes best represents its target client base.
gamma is 1: