Standard deviation/E@R over time with changing exposure

#1
Hello,

I need some help with this statistics/finance problem:

Imagine you own an asset over four periods and the value of it is affected by market price movements. You know the standard deviation and the position that will be held during each period. To complicate the problem, these are different in each period. All of this information is known at inception.

(Standard deviation is the standard price change for one period, not the entire four)

How do you calculate the standard deviation and the Earnings@Risk across the sum of all four periods?

If the StDev and position never change I believe you can do StDev*sqrt(4) to get the StDev for 4 periods. Multiply this by some Zstat (i.e. 1.96) and by the size of the position to get an Earnings@Risk (95%) in Euros.

However, I do not know how to correct for the fact that StDev and position changes in each period. I have attached an excel with a clear example, and two guesses in G12 and J12.

Any ideas on how to do this properly? Thank you
 

BGM

TS Contributor
#2
Seems that you want to calculate the overall Value at Risk over the four periods.

Let \( W_1, W_2, W_3, W_4 \) be the amount invested in the four periods (or you called the exposures) respectively.

Let \( R_1, R_2, R_3, R_4 \) be the normally distributed random return of the four periods respectively, with different variance as you stated.

The total random earnings can be expressed as the sum

\( W_1R_1 + W_2R_2 + W_3R_3 + W_4R_4 \)

Now the questions are:

1. Are those random returns independent?
2. Are those invested amount deterministic (non-random)? It maybe random here because it maybe dependent on the return of the previous period. I am not sure about your setup.

If yes to both, you have a relatively simple case as the sum is also normally distributed. Now you can calculate the overall variance and standard deviation, and then calculate your normal VaR again.