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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!

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  • $\begingroup$ There is also volatility in the normal state and volatility in the crisis state. Something that you did not tell us about. Something you are not accounting for in your calculations. $\endgroup$
    – stans
    Commented Jan 26, 2021 at 6:12
  • $\begingroup$ @stans sorry I did forget to put something into the question. I’ve updated the question. Thanks. $\endgroup$ Commented Jan 26, 2021 at 10:37
  • $\begingroup$ Thank you. I have posted an answer. $\endgroup$
    – stans
    Commented Jan 26, 2021 at 10:50

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

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  • $\begingroup$ Thank you very much! $\endgroup$ Commented Jan 26, 2021 at 20:07

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