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Suppose that the probability that determines the state of the economy is unknown. That is, you do not know whether the booms or recessions are more likely. Calculate the expected return and the volatility of the stock under the following payoff table.

I believe the expected return is 0, but how do you calculate the standard deviation? Which probability should be used for the $ P_i$? 0.5 for both or 0.25 and 0.75?

$ σ^2 = sqrtΣ(r_i-E(r))^2cdot P_i$

Edit:

Can I also confirm my solution for the final section?

"Would a typical mean-variance utility maximizer prefer the top or the bottom table? Intuitively, would you prefer the recession probability to be uncertain as in the top table?"

Since the$ E(r) and Var(r)$ are the same in both tables, the investor is indifferent towards both. However, in reality most investors would prefer the recession probability to be certain, as they are risk-averse.

Seems like the total law of variance problem:

$Vleft[Yright]=Eleft[ Vleft[Y mid X right] right]+Vleft[ Eleft[Y mid X right] right]$

Mean on the other hand will be just the iterated expectation problem:

$Eleft[Yright]=Eleft[ Eleft[Y mid X right]right]$

This one's not too difficult. Because the p() of the boom and bust regimes are a 50:50, the vol remains 10%

Where you vary the regime probabilities, life gets only a little more complicated. You have four scenarios, as per above. The mean is the sum of the scenario probability * payoff. The variance is sum of the scenario probability * (scenario payoff - mean)^2. The sigma is the root of the variance. Simples.

Where you run into trouble is trying to calculate a vol from Markov regimes. 30% chance of -10% +/- 20% Gaussian, versus 70% chance of 5% +/- 10% Gaussian. That's what breaks the models here, when you want model the return distribution "normally" rather than approximating this as a 50:50 of +/-1 sigma.

all the best...