Value at Risk - What if an account has never suffered from a negative return

I want to implement an algorithm that calculates an account's 95% value at risk on a monthly return base. The case I want to describe in this question is rather academic and will probably never happen in real life.

I am wondering what happens if an account has never suffered a negative return, i.e. has always made money instead of loosing. Is the VaR 0 in this edge case (since the account never lost money, hence there is NO value at risk), or is the VaR negative (since the account always won money, hence the account's capital being at risk is negative and therefore is expected to grow, even in the worst case)?

By definition, your loss cannot be positive, so you'd set the VaR to zero. But it really depends, on how you calculate your VaR.

If you calculate your returns, sort them and look at the 5% quantile (which, as you say, may be positive), then you'd simply set your VaR to zero.

But if you treat your returns as realizations of some (unknown) random variable, then just because you don't have any negative returns, that doesn't strictly mean that the random variable cannot be negative.

A simple example follows: we get 250 N(0,1) samples and make them all positive (via abs) and try to estimate the density of the data and plot the density curve:

set.seed(11)
returns <- abs(rnorm(250))
# no negative returns - all are positive
min(returns)
kde <- density(returns)
plot(density(returns))
abline(v=0, col=2)


gives

> # no negative returns - all are positive
> min(returns)
[1] 0.006010746


which as you can see also has data for negative values, and you'd probably want to use that to calculate VaR.

(note1: this is a very simply example, the density is not calibrated in any way)

(note3: and you'd also probably want to run MC sims)

• Thanks rbm for your excellent response. As you guessed correctly, I am using the order-and-look-at-the-5%-quantile-method :-) Mar 7 '17 at 12:13
• Wiki on Var says: "A negative VaR would imply the portfolio has a high probability of making a profit." Mar 7 '17 at 12:16
• Hey Chris. That's a great hint. But it kinda conflicts with ram's answer, doesn't it? Mar 7 '17 at 12:38
• Yes, a purely technical interpretation could be 'worst case you'll make a profit', but that would go against the 'risk' idea of what VaR should represent. It's a question of how you (your risk manager) want to look at the numbers, but again, no proper model would ever give you only positive returns.
– rbm
Mar 7 '17 at 12:50
• @rbm The negative returns could be negligible, as illustrated in my post on this question. I think Christian has a good question here. Mar 7 '17 at 13:21

Further to my comment on rbm's answer regarding negative VaR, consider a distribution with mean return = 7% and standard deviation 2%.

In a typical sample the returns are all positive, as the OP describes.

Taking the normally distributed 5% quantile as -1.645, the 5% tail extends to

0.07 + (-1.645 * 0.02) = 0.0371

The 5% quantile return is 3.71% so the VaR is -3.71% of the total asset value.

Quoting Wikipedia

A negative VaR would imply the portfolio has a high probability of making a profit.

5% VaR on sample and ideal distributions with μ = 0.07 & σ = 0.02