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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 $50$.

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Just to add a comment regarding the title, not as much the question, $VaR\geq 0$ whenever you are implementing a true arbitrage strategy. – user25064 Jun 19 '14 at 17:53
up vote 3 down vote accepted

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.

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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.

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I think you meant a portfolio with two assets. – Student T Jun 19 '14 at 10:41
@vincent 's post is my understanding, VaR can be negative which means there's likely no loss. – athos Jun 25 '14 at 9:59

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).

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