# Utility function for avoiding investment

An investor has initial wealth $30000$ and utility function $\ln{x}$. He is planning to invest in a project where he has $60%$ chance of gaining $\alpha%$ and $40%$ chance of losing $\beta%$. Express the certainty equivalent of this investment in terms of $\alpha$ and $\beta$. If $\beta=20$, find the range of values of $\alpha$ for which the investor will avoid this investment.

$$\begin{array}{c|lcr} \text{p} & \text{x} & \text{U(W_0+x)} \\ \hline 0.6 & 30000(1+0.0\alpha ) & U[30000(1+0.0\alpha )] \\ 0.4 & 30000(1+0.0\beta ) & U[30000(1+0.0\beta )] \\ \end{array}$$

Certainty equivalent

$=0.6\ln30000[(1+0.0\alpha )]+0.4\ln[30000(1+0.0\beta )]$

Introducing $\beta=20$ and since avoid investment

$=0.6\ln[30000(1+0.0\alpha )]+0.4\ln(24000) \le \ln{30000}$

Solving this I get $\alpha$ to be negative. There exist some error somewhere.

Assuming that $W_0$ is the initial wealth and that $\alpha$ and $\beta$ are yields of return, the final wealth is a discrete random variable \begin{align} W_T = \begin{cases} W_0(1+\alpha) &\text{ with probability }\quad p=0.6\\ W_0(1-\beta) &\text{ with probability }\quad 1-p=0.4 \end{cases} \end{align} The investment should be avoided iff the expected utility of the terminal wealth is smaller than or equal to that of the initial fortune: $$\Bbb{E}[ U(W_T) ] \leq U(W_0)$$ this gives \begin{align} & 0.6 \ln(W_0(1+\alpha)) + 0.4\ln(W_0(1-\beta)) \leq \ln(W_0) \\ \iff& 0.6 \ln(W_0) + 0.6\ln(1+\alpha) + 0.4\ln(W_0) + 0.4\ln(1-\beta) \leq \ln(W_0) \\ \iff& \ln(1+\alpha) \leq -\frac{2}{3}\ln(1-\beta) \end{align} Further assuming $\beta = 20\%$ yields the following range of values of $\alpha$ for which the investor will avoid this investment. $$\alpha \leq \exp(-2/3\ln(1-\beta)) - 1 \approx 16.04\%$$