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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?

idknuttin
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1 Answers1

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There are many ways to measure 'risk' like variance, standard deviation, coefficient of variation, etc.

Here, we have investments $K_i$ whose values are given by Return$(K_i)(\omega)$. Each investment $K_i$ has expected return $E[Return(K_i)]$ and variance $Var[Return(K_i)]$. The standard deviation of investment $K_i$ is denoted by $\sigma K_i$ which is the square root of $Var[Return(K_i)]$.

The weights could refer to:

  1. The probabilities of the $\omega$'s

  2. hypothetical weights we could assign to the investments.

Suppose we invest 1 unit in investment $K_1$, x times as much in $K_2$ as we do in $K_1$ and y times as much in $K_3$ as we do in $K_1$.

Then we expect to get:

$E[Return(K_1) + xReturn(K_2) + yReturn(K_3)]$

The variance of our portfolio is:

$Var[Return(K_1) + xReturn(K_2) + yReturn(K_3)] = Var[Return(K_1)] + x^2Var[Return(K_2)] + y^2Var[Return(K_3)]$

I guess the weights would be $1$, $x$ and $y$ or $\frac{1}{1+x+y}, \frac{x}{1+x+y}, \frac{y}{1+x+y}$

Read more: https://quant.stackexchange.com/questions/1356/diversification-rebalancing-and-different-means

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