Let $E(R)_X$ denote the expected return of asset $X$.
Given a market with only 3 assets; $A$, $B$ and $C$, the following three things can happen at the next timepoint:
With probability $ p = 0.5$: $$E(R)_A = 0.4, E(R)_B = 0.02, E(R)_C = 0.3$$
With probability $p = 0.4$: $$E(R)_A = 0.2, E(R)_B = 0.02, E(R)_C = 0.25$$
With probability $p = 0.1$: $$E(R)_A = 0.1, E(R)_B = 0.02, E(R)_C = 0.15$$
Now, I'm trying to find the standard deviation for each asset.
Hence, I've found the expected return for each asset, via the following (for example, asset $A$): $$E(R)_A = 0.5(0.4) + 0.4(0.2) + 0.1(0.1) = 0.29$$
Now, how do I find the standard deviation?
Do I simply do: $$ \sigma = \sqrt{\frac{(0.4-0.29)^2 + (0.2-0.29)^2 + (0.1-0.29)^2}{3}}$$
or do I need to somehow incorporate the probabilities in to this formula?
Best Answer
You need to incorporate the probabilities so
$$\sigma = \sqrt{0.5(0.4-0.29)^2 + 0.4(0.2-0.29)^2 + 0.1(0.1-0.29)^2}$$
This is also
$$\sigma = \sqrt{0.5(0.4)^2 + 0.4(0.2)^2 + 0.1(0.1)^2 -0.29^2}$$