I already asked this yesterday at "Economics Stack Exchange" but think this question might be better suited here. In the meantime i really tried to solve it by myself, but couldn't find anything what might help me. It's not just about the solutions, i do really want to understand how to solve problems like this.
"I am currently studying for my upcomming exams. There is an exercise i can't solve or even understand properly.
The full exercise is: "You bought 100 shares of company A and 200 shares of company B. The shares of A are bid \$50 and ask \$60, while the shares of B are bid \$25 and ask \$35. The bid-ask spreads of both A and B are normally distributed with mean \$10 and standard deviation \$3.
Determine the distributions of the proportional bid-ask spreads for A and B."
I already got the proportional bid-ask-spread for A and B by the formula $s_{p}(X) = \frac{ASK - BID}{MEAN}$. Therefore $s_{p}(A) \approx 0.18$ and $s_{p}(B) \approx 0.33$.
Now i need to calculate the distributions of those spreads. (The actual aim of this exercise is to calculate the cost of liquidation in a stressed market.)
I'm not quite sure what is meant by "distribution", so i assume it's the mean and standard deviation of those spreads. I just can't get my head around the standard deviation, since i need at least two values to calculate the standard deviation. (as far as i undersand) But i don't have more than one value for each spread.
How do i have to solve this exercise? Like, is there a general way of doing it?"
UPDATE: I need these results to calculate the "cost of liquidation in a stressed market". I read that one needs to use the following formula for this.
$\sum_{i=1}^{n} \frac{1}{2}(\mu_i + \lambda_i \sigma_i)\alpha_i$, where
$n=2$,
$\mu_i = s_p(X_i)$, so in my case $\mu_1 = 0.18$ and $\mu_2 = 0.33$,
$\alpha_i = \text{volume } X_i$, in my case $\alpha_1 = 100$ and $\alpha_2 = 200$,
$\lambda_i = \text{confidence-level}$, like $\lambda_1 = \lambda_2 = 2.33$ for $99\%$ confidence-level and finally
$\sigma_i = \text{that "distribution" (standard-deviation?) value i can't calculate}$.
Maybe this can describe my problem in more detail.