Certainty Equivalent and Avoidance Threshold for a Risky Investment
Summary
The document derives an expected-utility rule for deciding whether to accept a two-outcome investment. An investor with logarithmic utility receives one return with probability 0.6 and incurs a loss with probability 0.4. The investment is avoided when the expected utility of terminal wealth is no greater than the utility of keeping initial wealth. Expressing gains and losses as return fractions makes the initial-wealth terms cancel, leaving a condition relating the gain and loss rates.
For a loss rate of 20%, the response gives an upper bound of about 16.04% on the gain rate for which the investor avoids the investment. This corrects the question’s apparent sign and percentage notation confusion: a loss of 20% multiplies wealth by 1 minus 0.20, while a gain multiplies it by 1 plus the gain rate. The result assumes logarithmic utility, the stated probabilities, and no other costs or outcomes; it is an expected-utility threshold, not a market estimate.
Key ideas
- Compare expected utility from the risky investment with utility from retaining initial wealth.
- With logarithmic utility, common initial-wealth terms cancel from the acceptance inequality.
- Represent a gain as one plus its return rate and a loss as one minus its loss rate.
- At a 20% loss rate under the stated probabilities, gains up to about 16.04% meet the avoidance condition.
- The threshold depends on the assumed utility function and outcome probabilities.
Tags
Full text
# Utility function for avoiding investment
# 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.
## Answer by Quantuple (score 2, accepted)
https://quant.stackexchange.com/a/30413
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. $$\alpha \leq \exp(-2/3\ln(1-\beta)) - 1 \approx 16.04\% $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.