Hugo has a concave utility function of U(W)=W^0.5 . His only...

Hugo has a concave utility function of $U(W)=W^{0.5} .$ His only asset is shares in an Internet start-up company. Tomorrow he will learn the stock's value. He believes that it is worth $\$ 225$ with probability $20 \%$ and $\$ 256$ with probability $80 \%$ What is his expected utility? What risk premium would he pay to avoid bearing this risk? (Hint: See Solved Problem $17.2 .)$

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Key Concepts

Expected Utility

The concept of expected utility involves calculating the weighted average of utilities associated with all possible outcomes, where the weights are the probabilities of each outcome. This is fundamental in decision theory to assess the desirability of risky prospects by taking into account the decision maker’s preferences and risk attitudes.

Concave Utility Function

A concave utility function, such as the square root function, implies risk aversion. This means that the decision maker values the increment of utility less as wealth increases, reflecting diminishing marginal utility. In a concave utility function, the decision maker prefers a certain outcome over a risky one with an equal expected wealth.

Certainty Equivalent

The certainty equivalent is the guaranteed amount of wealth that provides the same utility as the expected utility of a risky prospect. It is a measure used to evaluate how much a decision maker would accept, instead of facing a risky situation, reflecting their risk tolerance.

Risk Premium

The risk premium is the difference between the expected value of a risky prospect and its certainty equivalent. It represents the amount of wealth a risk-averse individual would be willing to pay to avoid the uncertainty associated with the risky prospect, thus quantifying the cost of risk.

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