Calculating Portfolio Volatility Across Normal and Crisis States

Summary

The document explains how to calculate expected return and volatility when returns come from two possible states: a normal environment and a crisis environment. Each state has its own probability, conditional mean return, and conditional standard deviation. The expected return is the probability-weighted average of the state means.

For total variance, the answer applies the law of total expectation to squared returns: weight each state’s conditional second moment, calculated as the squared mean plus the conditional variance, then subtract the square of the overall expected return. This accounts for both within-state variability and the separation between state means. The example yields a 6% expected return and a standard deviation of about 19.21%, showing why using only a weighted average of the two conditional volatilities misses the added dispersion from regime changes. The calculation assumes the stated state probabilities and conditional return moments; it does not address how those inputs should be estimated or whether returns are normally distributed.

Key ideas

• Expected return is the probability-weighted average of conditional state means.
• Overall variance equals the weighted average of conditional second moments minus the squared overall mean.
• Each conditional second moment combines the squared conditional mean and conditional variance.
• Regime shifts can add substantial total volatility beyond the within-state standard deviations.

Tags

# Portfolio Volatility Calculation


# Portfolio Volatility Calculation












The question: you have a portfolio of risky assets that with a 90% probability (normal state of the world) has an expected annual return of 10% plus a random variable with a standard deviation of 15%. With a 10% probability (crisis state) the annual return would be -30% plus a random variable with a standard deviation of 15%. What is the expected return and volatility of the portfolio?

I know the answer is 6% expected return and 19.21% volatility. I understand the annual return but get 12% when calculating the volatility. What am I doing wrong? Thanks!

## Answer by stans (score 1, accepted)

https://quant.stackexchange.com/a/60715

$$ Var[R] = E[R^2] - E[R]^2 = $$ $$ = 0.9 * E[R^2|\text{normal state}] + 0.1 * E[R^2|\text{crisis state}] - E[R]^2 = $$ $$ = 0.9 * (E[R|\text{normal state}]^2 + Var[R|\text{normal state}]) + 0.1 * (E[R|\text{crisis state}]^2 + Var[R|\text{crisis state}]) - E[R]^2 = $$ $$ = 0.9 * (10^2 + 15^2) + 0.1 * (30^2 + 15^2) - 6^2 = 369 $$ $$ \Longrightarrow $$ $$ SD[R] = \sqrt{369} = 19.20937. $$

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.