Statistics: Value at Risk

[Peter G.]

Homework Statement

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.

Homework 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)

The Attempt at a Solution

[/B]
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!

 

[Ray Vickson]

Homework Statement

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.

Homework 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)

The Attempt at a Solution

[/B]
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!

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

 

[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.

Sorry about the confusion!

 

[Ray Vickson]

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.

Sorry about the confusion!

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.

 

[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,
Peter