# Statistics: Value at Risk

1. Oct 25, 2014

### Peter G.

1. The problem statement, all variables and given/known data

Find the VaR for an investment of $500,000 at 1% given that the investment is expected to grow 10% every year with standard deviation of 35% and that the investment is held for two years. 2. Relevant equations E(X + Y) = E(X) + E(Y) E(X*Y) = E(X) * E(Y) (for independent random variables) var (X + Y) = var (X) + var (Y) (for independent random variables) 3. The attempt at a solution So, at first, I thought that the expected return if the investment were held for two years would be: E(X+[(1+X)*Y]) Although I can compute that, if that were the case, then the variance for the two year investment would be given by: var (X + [(1+X)*Y]) = var (X) + var (Y) + var (Y*X) But that cannot be the case I do not know how to calculate that last term. Upon doing some research, it appears that I should be computing E(X + Y) and var (X + Y) instead. However, that does not make much sense to me. For example: if I were to invest 100 dollars on a stock that yielded a return of 10% with SD = 0% every year, then my return on the 100 dollars after two years would be 21%, not 20%, right? Can anyone shed some light on this for me please? Thank you in advance! 2. Oct 25, 2014 ### Ray Vickson What do you mean by "an investment of$500,000 at 1% given that the investment is expected to grow at 10% every year with SD 35% and is held for two years"? The 1% and 10% seem to be contradicting each other. Do they refer to different things? If I invest $500,000 at 1% for onw year I will have$505,000 at the end of the year. So, where do the 10

3. Oct 26, 2014

### Peter G.

I am sorry. The VaR for an investment of \$500,000 at 1% means the value that is at risk in that investment if we exclude the worst 1% of the outcomes.

The core of my problem/difficulty is understanding why one can model the expected value of an investment with 10% yearly returns that is held for two years as simply the sum of the expected returns in each year.

4. Oct 26, 2014

### Ray Vickson

You ask "...why one can model the expected value of an investment with 10% yearly returns that is held for two years as simply the sum of the expected returns in each year". Well, maybe you can't; it depends on some more details. If the investment is left alone for two years it would typically accumulate like "compound interest". That means that if $X_1, X_2$ are the rates of return in years 1 and 2, the total future value (per invested dollar) would be $(1+X_1)(1+X_2)$. This is $1 + X_1 + X_2 + X_1 X_2$. Since expectations add (whether the terms are independent or not), we have
$$E(\text{future value}) = 1 + E\,X_1 + E\,X_2 + E(X_1 X_2).$$
If $X_1$ and $X_2$ are (statistically) independent, we would have $E(X_1 X_2) = (E\,X_1)(E\,X_2)$, and so would lead to
$$E(\text{future value}) = (1+E\,X_1)(1+E\,X_2) \: \Leftarrow \: \text{assumes independence!}$$
This is not just a sum of returns. However, if $E\,X_1 = E\,X_2 = 0.10$, then the exact expected 2-year rate of return for independent yearly returns would be $0.10 + 0.10 + 0.10 \times 0.10 = 0.21,$ which is close to, but not exactly equal to $0.10 + 0.10 = 0.20.$

On the other hand, if the gains accumulate additively (no compounding) then there would be no $X_1 X_2$ term, so adding the expected returns would give the exact answer.

5. Oct 26, 2014

### Peter G.

Hi Ray Vickson,

Thank you very much for your help! Although the question does not specify if the investment is compounded or not, I will have to assume it is not. To answer the Value at Risk component of the question I will need the variance/standard deviation for the return over the two years. If I assume there is compounding, I would end up with a var (X1X2) term, which we have not learned how to deal with.

Thank you once again,

Best,

Peter