Calculating Return Risk from Scenario Probabilities

Summary

The document explains how to calculate the standard deviation of project returns when outcomes depend on discrete market scenarios with given probabilities. For each project, first compute the probability-weighted expected return, then take the square root of the probability-weighted squared deviations from that mean. The examples show that a project with identical returns in both scenarios has zero standard deviation, while another project’s scenario-dependent returns produce a positive risk estimate.

The table provides two scenarios and return figures for three projects. The answer works through the first two calculations and directs the reader to apply the same procedure to the third, but does not show that final value. Although the question mentions weights, the explanation treats the scenarios’ probabilities as weights and does not calculate portfolio risk for a weighted combination of projects. It also gives no discussion of covariance or dependence across projects.

Key ideas

• Compute expected return as the probability-weighted average across scenarios.
• Measure each project’s risk as the square root of its probability-weighted squared deviations from expected return.
• A project with the same return in every scenario has zero standard deviation.
• Scenario probabilities act as weights for each individual project’s return distribution.

Tags

# Answer by Neeraj (score 1, accepted)


# Compute the risk measured by the standard deviations $\sigma K_1, \sigma K_2, \sigma K_3$, does this have to do with weights?












Compute the risk measured by the standard deviations $\sigma K_1, \sigma K_2, \sigma K_3$ for each of the investment projects, where the returns $K_1, K_2$, and $K_3$ depend on the market scenario:

$$ \begin{matrix} Scenario & Probability & Return K_1 & Return K_2 & Return K_3 \\ \omega_1 & 0.3 & 12\% & 11\% & 2\% \\ \omega_2 & 0.7 & 12\% & 15\% & 22\% \\ \end{matrix} $$

I am not sure what this question is asking me to do, I think it has something to do with weights?

## Answer by Neeraj (score 1, accepted)

https://quant.stackexchange.com/a/24307

This is very basic question. You just need to compute the standard deviation of three projects $K_1$, $K_2$ and $K_3$.

## $$\text{Standard deviation}= \sqrt{E[(X-\mu)^2]}$$

## For the first project $K_1$:

Expected return ($\mu_{K_1}$) = $.3*.12 + .7*.12 = .12$

Standard Deviation ($\sigma K_1$) = $\sqrt{.3*(.12-.12)^2+.7*(.12-.12)^2}=0 $

## For the second project $K_2$:

Expected return ($\mu_{K_2}$) = $.3*.11 + .7*.15 = .138$

Standard Deviation ($\sigma K_1$) = $\sqrt{.3*(.11-.138)^2+.7*(.15-.138)^2}=0.018330=1.83\% $

## You may follow the same procedure for the third project.

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.