The question asks for calculation of regulatory capital. I understand the theory of VaR but find it tricky in use. In the question, investment returns are expected to be 4%. Std Dev is 25% of mean. And 0.5th percentile is 2.2 Std Devs from mean. Firstly, why is VaR not Min (0; L - Z) where Z = L[1+(25% x 4% x 2.2)] L = lump sum invested at start of year. My reasoning is 0.5th percentile in this case is 1.8% returns (4% x 2.2 x 25%), hence does not result in a loss of any of the initial investment and therefore should be reserved for. Alternatively, why is it not (1.04 x L x 25% x 2.2) - 0.04 x L My reasoning here is expected value at end of period is 1.04 x L. 0.5th percentile is 2.2 x 25% of this. But since this reduction includes the returns gained on the initial investment, then x L should be deducted from the result to find the value that is at risk of loss within the year. Why is the VaR not lump-sum at start x 1.04 x 2.2 x 25% The model answer says VaR = 4% x 25% x 2.2 x L. In my mind, this is a positive return of 1.8%, not a loss, although lower than the 4% we expected. Help find the lost brother.
Correction, this should read: ... does not result in a loss of any of the initial investment and therefore should NOT be reserved for.
I'm a bit confused by your argument Firstly, why is VaR not Min (0; L - Z) where Z = L[1+(25% x 4% x 2.2)] L = lump sum invested at start of year. L - Z does give L x (25% x 4% x 2.2) which is the model answer The VaR is not measuring absolute losses, but losses relative to that expected. If the 4% return has been priced into a premium say then a lower return than expected can be considered a loss
Thanks a lot, Simon. The premium assumption argument makes sense. Some follow-up questions to help complete my understanding: 1. What about if we assume this is not for regulatory capital but for an investment portfolio estimating its VaR. If we assume that 'investment returns are expected to be 4%. Std Dev is 25% of mean. And 0.5th percentile is 2.2 Std Devs from mean.'; would the VaR still be 4% x 25% x 2.2 x L? This is where my intuitive sense of VaR here breaks down because the L would still be wholly intact so none of it is at risk. 2. In what 'state of the world' would we have a lower value than L at the end of the period. What would need to be the Std Dev as a % of mean and the .5th percentile distance from the mean? I'm thinking the Std Dev. as % of mean may be quoted as negative in that case? But how would we interret the negative result for VaR in that case?
Hi 1. You have still suffered a "loss" relative to what you have expected. If you are investing to meet some liabilities (or simply saving for retirement etc) the value of your assets has fallen. VaR is simply a measure of the extent of the possible fall in value of the assets. 2. The VaR above is a loss/negative. The formula is L-Z. This means the result is negative. It is being expressed as a positive. VaRs might be equivalently expressed as "-100" or "a loss of 100"
