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

The scenario presents two risky gambles with the same expected value (EV = 40):

Old gamble: 50% chance of 10 and 50% chance of 70.
New gamble: 50% chance of 0 and 50% chance of 80.

Option 1: The certainty equivalent decreases relative to the original gamble.
- This is the key idea. The certainty equivalent (CE) is the guaranteed amount that yields the same utility as the risky gamble. Although both gambles have the same EV, the new gamble is more spread out (it ranges from 0 to 80) than the old gamble (10 to 70). For a risk-averse investor, greater risk (more dispersion) tends to reduce the CE, because the same EV is associated with more downside risk. Therefore, the CE for the new gamble is lower than for the original gamble.

Option 2: Because the expected value remains 40, Emma is indifferent between the old gamble and the new one.
- Indifference would require identical certainty equivalents. However, with increased risk from 0–80 versus 10–70, a risk-averse investor would value them differently, giving a lower CE (and thus not indifference). So this statement is incorrect.

Option 3: The new gamble increases Emma’s expected utility because it has a higher upside.
- Expected utility considers both upside and downside through the utility function and the distribution, not just the potential upside. Even though the upside is higher (80 instead of 70), the downside is worse (0 instead of 10), and for a risk-averse agent this tends to lower expected utility or keep it the same at best, but not guarantee an increase. Therefore this claim is not generally correct.

Option 4: The new belief lowers the risk premium Emma would pay to avoid the gamble.
- The risk premium is EV minus CE. Since the CE decreases when risk increases (as argued for Option 1), the risk premium EV - CE actually increases, not decreases. Thus this statement is false.

In summary, the only statement that aligns with standard risk-return intuition when moving from the old to a more dispersed 0/80 distribution (with the same EV) is that the certainty equivalent decreases relative to the original gamble.

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