Value at Risk Sign Conventions and a Portfolio Quantile Example
Summary
The document discusses how to interpret the sign and magnitude of value at risk (VaR). In common practice, VaR is reported as a positive loss amount, so a figure such as 100 denotes the amount at risk rather than a signed portfolio loss. Another answer emphasizes that VaR is a quantile of a loss distribution, so its sign depends on the loss convention and the scenarios: if all outcomes are gains, a quantile can be negative.
A portfolio example explains a VaR of 50: if the probability that at least one of two assets loses money exceeds the chosen confidence tail threshold, and simultaneous losses are treated as unlikely, then a loss equal to half the portfolio value occurs often enough to determine the VaR. The discussion also notes VaR’s failure of subadditivity in some cases, so combined portfolio risk can exceed the sum of individual VaRs. The example relies on simplified probability assumptions, and the sign convention must be stated clearly.
Key ideas
- VaR is commonly presented as a positive amount representing a potential loss.
- As a loss-distribution quantile, VaR can be negative when scenarios imply gains.
- The portfolio example derives VaR from the probability of at least one asset losing value.
- The example simplifies the joint-loss probability by treating simultaneous losses as negligible.
- VaR may fail subadditivity, so diversification behavior can be inconsistent with that property.
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Full text
# Why is Value at Risk non-negative?
# Why is Value at Risk non-negative?
When reading the book of `Financial Risk Forecasting`, I saw the following example. I am not very clear about two points marked with yellow and green respectively.
Regarding the first point marked with yellow color, why $VaR^{1\%}=100$, I think it should equal to $-100$ instead.
Regarding the second point marked with green color, I do not understand how to get $50$.
## Answer by sashkello (score 3, accepted)
https://quant.stackexchange.com/a/12719
Value at risk is quoted by absolute value. This is the amount of money you can lose, so everyone knows the sign by default.
For the second question, the last line explains it. Probability of at least one of the assets losing money is ~9.6%. Probability of both losing money is pretty small and is ignored. So, since 9.6% > 5%, it means that you lose on one of the assets, which is 50% of your portfolio with higher than 5% probability (so, it's your value at risk). This is where 50 comes from.
## Answer by Vincent (score 0)
https://quant.stackexchange.com/a/12720
I think that in theory the VaR can be negative. The VaR is only a given quantile of your loss distribution depending on the confidence level you set and the time horizon you wanna consider (how many loses can I afford next year/quarter/month/week/day etc). Imagine that the loss distribution you are looking at contains only negative values (gains in that case) then whatever the confidence level, your VaR will be negative. In practice, the VaR tends to be always positive because we use high confidence levels such as 95% or 99% and we observe random variables taking their values in R (set of all real values)
Your last example just illustrates one of the VaR shortcomings : it is not subaddative, meaning that the sum of the VaR of two portfolios can be lower than the VaR of the combined two which is not consistent with the theory of diversification.
## Answer by UmaN (score 0)
https://quant.stackexchange.com/a/22023
By convention, a negative number indicates loss. Value at risk gives you the loss at the specific quantile.
However, this number depends entirely on the scenarios on which you run VaR. If your portfolio is a single long position in a stock, and all your scenarios shift the price of the stock upwards different amounts, then you will get a negative number - indicating that the alfa:th quantile worst loss is actually a profit.
Needless to say, this is an indication that your scenarios are poor (or that you hold an incredible portfolio).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.