Consider the risky investment we considered in class: Emma is considering buying a stock that has a 50% chance of being worth 10 and a 50% chance of being worth 70 at the future date on which she would want to sell the stock.

Suppose Emma does some more analysis and changes her mind about the subjective probability distribution. The stock actually has a 50% chance of being worth 0 and a 50% chance of being worth 80. Given this change, which of the following statements is true?

Question

Consider the risky investment we considered in class: Emma is considering buying a stock that has a 50% chance of being worth 10 and a 50% chance of being worth 70 at the future date on which she would want to sell the stock.

Suppose Emma does some more analysis and changes her mind about the subjective probability distribution. The stock actually has a 50% chance of being worth 0 and a 50% chance of being worth 80. Given this change, which of the following statements is true?

BlackTom AI · Accepted Answer

We start by restating the scenario and the options to ensure clarity. Emma compares two gambles:
- Original gamble: 50% chance of 10 and 50% chance of 70. EV = 0.5*10 + 0.5*70 = 40.
- New gamble after updating beliefs: 50% chance of 0 and 50% chance of 80. EV = 0.5*0 + 0.5*80 = 40.

Option 1: The certainty equivalent decreases relative to the original gamble.
- Why this could be true: Certainty equivalent (CE) reflects the guaranteed amount a person considers equally desirable as the gamble, given their risk preferences. Since the two gambles have the same expected value, the comparison hinges on risk. The original gamble (10 or 70) has substantial upside and downside spread (60 units). The new gamble (0 or 80) also has a large spread (80 units) but, for a risk-averse person, the larger or differently distributed risk can affect utility more (or less) than the original, depending on the curvature of the utility function. Because the updated distribution tends to be perceived as riskier (greater dispersion relative to the mean for some risk preferences), a risk-averse Emma would typically require a lower certain amount to be indifferent, i.e., a lower CE. Hence the CE can decrease relative to the original gamble.

Option 2: Because the expected value remains 40, Emma is indifferent between the old gamble and the new one.
- Why this is not correct: Indifference would require identical certainty equivalents. Even with the same EV, changes in the distribution affect utility through risk aversion. A risk-averse person usually does not view two gambles with the same EV as equally desirable when their risk profiles differ. Therefore, indifference is not guaranteed here.

Option 3: The new gamble increases Emma’s expected utility because it has a higher upside.
- Why this is unlikely: Although the new gamble has a high upside (80), the downside is also severe (0). For many individuals with diminishing marginal utility of wealth, the increased risk does not translate into higher expected utility if risk aversion dominates. Unless Emma is risk-seeking in this region, the expected utility would not necessarily rise; it could stay the same or fall depending on the utility function.

Option 4: The new belief lowers the risk premium Emma would pay to avoid the gamble.
- Why this is incorrect: The risk premium is defined as the difference between the certainty equivalent and the expected value (RP = EV − CE). If the CE decreases while EV stays at 40, the risk premium EV − CE increases, not decreases. Thus the statement that the risk premium would be lower contradicts the standard interpretation of a lower CE under the same EV.

In summary, analyzing the impact of the distribution shift under risk aversion indicates that the certainty equivalent tends to decline when the gamble becomes relatively more risky or differently distributed, even if the EV is unchanged. Therefore, option 1 is the consistent choice given typical risk-averse preferences, while the other options misstate indifference, expected utility effects, or risk premium dynamics.

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