I find the way value at risk is discussed in the notes and tested in previous examinations to be a bit strange. VaR is intuitively simple. Take say a £1000 portfolio, have a distribution for the value of the portfolio value in 1 year, then compute the 5% (lower) tail value of this distribution, which gives a 'worst case' portfolio value. Then subtract this value from the initial £1000, which in this case gives the 1-year 95% VaR. The first thing that doesn't seem quite right in the notes is the definition. We are given a formula the VaR 'of a continuous random variable' (which I don't think really makes sense in itself) as VaR(X) = -t where P(X < t) = p But if we are to take X (going back to the example above) as equal to the portfolio value in 1 year, t in this case would come out being the negative of the worst case portfolio value, and not the actual VaR value we want. We have to take X to be the return on the portfolio (in £), not the value of the portfolio, for this definition to make sense. The mathematical definition above is applied in previous exam questions (e.g. April 2015 Q2) to a random variable representing a return distribution, giving a VaR that is a percentage. This also seems strange, because the VaR is then not a monetary value. I also found a question (September 2016 Q1) where we are asked for the VaR of a Poisson distribution. The probabilities are simple to look up in the tables, the answer comes out to be -5. While this is consistent with the above mathematical definition, this again uses the idea of the 'VaR of a random variable', which I find a bit weird. I think my gripe here is to do with how VaR is defined and used. The portfolio value example seems to make sense and is, I believe, the standard definition of VaR, but everything else seems to be various degrees of muddled.
The good news is that you clearly understand how to calculate a VaR and what it represents As you have identified, in the definition, X represents the change in the portfolio value (if working in monetary amounts), or the percentage return on the portfolio (if we're given a distribution of investment returns). I'm not sure I see your point that X is a random variable is weird - the future value of a portfolio is unknown, so would ordinarily be represented by a random variable in a probability expression, as it is in the definition provided. But I'm not sure this is important in the context of answering exam questions.... Where possible, I'd advise quoting the VaR as a monetary amount (in line with the third bullet point on page 7 of the chapter). So if you're given the distribution of percentage investment returns, then having found the lower 5% tail of this distribution, you should multiply by the size of the portfolio to obtain a monetary amount. Clearly, April 2015 Question 2, does not provide a portfolio value, so there's not really anything you can do there. If that's the case in an exam question, you'd just leave your answer in percentage terms.
Hi Mark, thanks for the response. I still think the VaR of a random variable where the variable has nothing to do with financial investments - e.g. in one past exam question the number of apples falling from a tree - makes no sense. But at least I'll know how to answer such a question in the exam, should it occur.
I agree that it's a little unusual, but if you think of the farmer's portfolio of assets as the apple tree, and the apples harvested as the return or yield from his assets, then it is at least a comparable situation to the investment scenario we ordinarily consider. After all, the farmer will be able to sell the apples for money.
