Calculating Return Variance with Uncertain Regime Probabilities
Summary
The document considers expected return and volatility when returns depend on economic regimes, such as boom and recession, and the regime probabilities are uncertain. It points to the law of total expectation for the mean and the law of total variance for variance across conditional regimes. For a finite set of outcomes, the practical calculation is to assign each scenario its probability, compute the probability-weighted mean, then take the probability-weighted squared deviations from that mean; standard deviation is the square root of variance.
The response distinguishes a simple discrete scenario table from a richer regime model in which returns within each regime also vary, for example through Gaussian distributions. In that setting, conditional variance and variation in conditional means both contribute to overall variance. A mean-variance investor is indifferent between distributions with the same mean and variance under that criterion, but other preferences may differ. The discussion is explanatory and does not supply the missing payoff table or resolve every ambiguity in the original setup.
Key ideas
- Use iterated expectations to calculate the unconditional mean from conditional regime means.
- Overall variance includes expected within-regime variance and variance across regime means.
- For discrete outcomes, weight squared deviations from the overall mean by scenario probabilities.
- Standard deviation is the square root of variance.
- Mean-variance preferences treat distributions with the same mean and variance as equivalent.
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Full text
# Stock Volatility with Uncertain Probability
# Stock Volatility with Uncertain Probability
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?
$\ σ^2 = \sqrt{Σ(r_i-E(r))^2\cdot P_i}$
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.
## Answer by Magic is in the chain (score 2)
https://quant.stackexchange.com/a/47418
Seems like the total law of variance problem:
$V\left[Y\right]=E\left[ V\left[Y \mid X \right] \right]+V\left[ E\left[Y \mid X \right] \right]$
Mean on the other hand will be just the iterated expectation problem:
$E\left[Y\right]=E\left[ E\left[Y \mid X \right]\right]$
## Answer by demully (score 2)
https://quant.stackexchange.com/a/47419
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...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.