I recently saw a VaR calculation, and I was wondering whether that calculation made sense. Here the details: 1. Construction of a total return bond portfolio index. By total return I mean that the index takes on account capital gains/losses from bond price movements; and that accrued interests are also taken on account as part of the return. 2. Calculation of this TRI over a certain period of time gave only positive returns. 3. In conclusion, the Historical VaR is "positive", meaning that you never lose money when taking on account accrued interest. Guys, will that make sense to you? Basel 3 and Solvency 2 propose internal models based on price movements; do you think it makes sense making the VaR calculations over TRI portfolios (equities o bonds)?
Of course this does not make sense. But the problem is not the total return index but (most likely) the range of historical values used in the calibration. In a Solvency II setting we are talking about annual VaR on the 99.5% level. As a quick reality check assume you are a well versed extreme event modeller. In fact you just need at least 5 events. Furthermore you are an optimistic individual and agree that you are fine with a time series which has a 90% chance of observing those 5 events.
Question: How many years of data do you need for the calibration of your model? Observing the event is Bernoulli with a probability of 0.005. So the probability to observe more than 5 "successes" in n years is 1 - B(n,0.005,5) where B(n, p, k) is the Binomial Distribution. A quick look into Excel shows n=1853 years are necessary to reach 90%. So unless those guys had bond prices roughly since the birth of Christ they do not have enough data for a VaR based on historic data alone.
At these VaR levels historic testing just does not work! You always need to augment historic data by a parametric assumption (such as Pareto) and by additional "artificial" stress scenarios.
You need to take in to account the variation in return ( i.e. standard deviation) of the returns to get the VaR. Alternatively,you can simulate the total return using Excel as well and find the VaR accordingly.
Statistics is about comparing like with like to come up with a conclusion.
First thing in any analysis is to demean the data.
After demeaning, absolute VaR may still be less than the mean return, but if the mean is a fat risk premium which you might not achieve - that can be serious also.
Lastly, VaR figures are only as good as their backtest. If they don't come with a pvalue, they're worthleworthless. Whether positive or negative.