Interpreting Value at Risk When Observed Returns Are All Positive
Summary
The document considers how to report 95% VaR when a return sample contains no losses. Under a historical quantile calculation, the measured lower-tail return may still be positive; the accepted answer recommends flooring reported loss VaR at zero under a loss-only convention. It distinguishes that sample-based result from inference about the unknown return distribution: a finite run of positive observations does not establish that future losses are impossible.
A kernel density example using transformed normal draws illustrates that a fitted distribution can assign probability to negative values even when every observed value is positive, though the example is explicitly uncalibrated. A second answer gives a normal-distribution illustration in which a positive lower-tail return corresponds to negative VaR, meaning the portfolio is expected to profit at that quantile. The discussion shows that sign conventions and estimation method determine the answer; it also notes expected shortfall and Monte Carlo as alternatives or complements, without developing them.
Key ideas
- A historical VaR based on observed returns can be zero when the chosen quantile implies no loss.
- Positive observations alone do not prove that the underlying return distribution excludes losses.
- A modeled lower-tail return above zero can produce negative VaR under a signed convention.
- VaR interpretation depends on whether it is reported as a loss measure or as a signed return quantile.
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# Value at Risk - What if an account has never suffered from a negative return # 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)? ## Answer by rbm (score 4, accepted) https://quant.stackexchange.com/a/32875 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) (note2: also you shouldn't care about VaR, but about expected shortfall) (note3: and you'd also probably want to run MC sims) ## Answer by Chris Degnen (score 1) https://quant.stackexchange.com/a/32876 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
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