Wilmott's Two Vars (Question 6 in Quant A)
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Sathya,
I appreciate that observation b/c in writing the notes, I pinged him to say I think it's a minor typo. He shows two VaRs (relative and absolute). Each formula itself is fine, but the "relative" VaR gives a positive [i.e., negative of a negative NORMSINV()] which creates a technical error in his absolute VaR. Strictly speaking, I agree with you (and so does Jorion, as his formula is positive).
But in practical terms it does not matter. This is a good example of where we might not let the formulas block our intuition. It doesn't matter b/c you see it both ways vis a vis relative VaR: "a loss of -X" or a VaR of X."
Where it may matter is that relative VaR is just a special case of the full "absolute" VaR (where absolute offsets the loss by the expected gain) but where we are assuming the expected return is zero (as VaR started as a daily trading metric, the distinction matter less, but it has become more of an economic capital metric). So, Wilmott shows two VaR formulas but a way to think of it is that there is only one:
VaR = (mean return) - (volatility)(alpha)(square root of delta time), or
VaR = - (mean return) + (volatility)(alpha)(square root of delta time)
The latter, for example, is the way Kevin Dowd writes it.
For what it's worth, the way that i think about this is that parametric VaR is merely scaled volatility:
VaR = (mean return) - (volatility)[scaled by time and confidence]
VaR = - (mean return) + (volatility)[scaled by time and confidence]
And the special case, relative VaR, is when we can assume mean return = 0, such that
VaR = - (volatility)(alpha)(square root of delta time), or
VaR = (volatility)(alpha)(square root of delta time), or
And wilmott's alpha =NORMSINV() which is negative (e.g., -2.33 or -1.65), so that will give a +VaR in his case. But, so far, it does not matter because we understand we are talking about losses to the left of mean.
Where it matters is the full "absolute" VaR. In the Culp's liquidity reading, he gives liquidity VaR with an error:
VaR = (mean return) - (volatility)(alpha)(square root of delta time) + (1/2)(bid ask spread)
and now an error has been introduced! And, strictly speaking, so is Willmott's absolute VaR in error for it adds the return to the VaR.
If you are still with me (and this is why I agree with you, where his alpha = NORMSINV()), this is why I like Kevin Dowd's best. But, again, I think the most important thing is to keep an eye on the intuition. I hope that helps..
David
Append: after all that, I forgot your example. Note that if volatility = 20%, then 95% VaR under Wilmott's formula will be a positive: (negative)(20%)NORMSIMV(1-95%) = about +33%. So, far no problem. The problem is assume expected return of +10%, then his absolute formula is wrong: 10% - [(20%)[-1.645]).
I appreciate that observation b/c in writing the notes, I pinged him to say I think it's a minor typo. He shows two VaRs (relative and absolute). Each formula itself is fine, but the "relative" VaR gives a positive [i.e., negative of a negative NORMSINV()] which creates a technical error in his absolute VaR. Strictly speaking, I agree with you (and so does Jorion, as his formula is positive).
But in practical terms it does not matter. This is a good example of where we might not let the formulas block our intuition. It doesn't matter b/c you see it both ways vis a vis relative VaR: "a loss of -X" or a VaR of X."
Where it may matter is that relative VaR is just a special case of the full "absolute" VaR (where absolute offsets the loss by the expected gain) but where we are assuming the expected return is zero (as VaR started as a daily trading metric, the distinction matter less, but it has become more of an economic capital metric). So, Wilmott shows two VaR formulas but a way to think of it is that there is only one:
VaR = (mean return) - (volatility)(alpha)(square root of delta time), or
VaR = - (mean return) + (volatility)(alpha)(square root of delta time)
The latter, for example, is the way Kevin Dowd writes it.
For what it's worth, the way that i think about this is that parametric VaR is merely scaled volatility:
VaR = (mean return) - (volatility)[scaled by time and confidence]
VaR = - (mean return) + (volatility)[scaled by time and confidence]
And the special case, relative VaR, is when we can assume mean return = 0, such that
VaR = - (volatility)(alpha)(square root of delta time), or
VaR = (volatility)(alpha)(square root of delta time), or
And wilmott's alpha =NORMSINV() which is negative (e.g., -2.33 or -1.65), so that will give a +VaR in his case. But, so far, it does not matter because we understand we are talking about losses to the left of mean.
Where it matters is the full "absolute" VaR. In the Culp's liquidity reading, he gives liquidity VaR with an error:
VaR = (mean return) - (volatility)(alpha)(square root of delta time) + (1/2)(bid ask spread)
and now an error has been introduced! And, strictly speaking, so is Willmott's absolute VaR in error for it adds the return to the VaR.
If you are still with me (and this is why I agree with you, where his alpha = NORMSINV()), this is why I like Kevin Dowd's best. But, again, I think the most important thing is to keep an eye on the intuition. I hope that helps..
David
Append: after all that, I forgot your example. Note that if volatility = 20%, then 95% VaR under Wilmott's formula will be a positive: (negative)(20%)NORMSIMV(1-95%) = about +33%. So, far no problem. The problem is assume expected return of +10%, then his absolute formula is wrong: 10% - [(20%)[-1.645]).
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