VAR, UL and EL
- Thread starter rajaar
- Start date
Hi Rajaar,
Yes, agreed, one of my high-priority feedbacks to GARP concerns this exact relationship and definitional conflicts in the assignments, which has often been discussed in the forum; e.g., here
VaR can be either "relative" or "absolute" and this explains why (strictly speaking) it can include or exclude EL. In regard to each of the three major risk buckets, here are the two variations and the "default" assumption:
Credit Risk:
Relative credit risk VaR = UL = Crouhy's P(c) - EV = EC = denominator of RAROC (b/c the numerator similarly deducts EL)
Absolute credit risk VaR = UL + EL
Default assumption: following Basel, as credit portfolios provision for EL, we assume EC = UL (as Basel, unless EL is not provisioned, charges capital for UL)
note: but even Basel in the IRB assignment defines CVaR = EL + UL. So, there is (IMO) no point is getting attached to either CVaR = UL or CVaR = UL + EL.
Market Risk:
Relative market risk VaR = Volatility * Deviate * SQRT(time delta); i.e., loss relative to future expected value
Absolute market risk VaR = -Drift * (time delta) + Volatility * Deviate * SQRT(time delta); i.e., loss relative to initial/zero
Default assumption: note, unlike credit, the drift is positive! Depends on time frame, but following Basel, under daily and since it is conservative, assume drift = 0, and therefore EC = UL which is the same as EC = UL + (EL = 0)
...but in regard to market risk VaR, my recommendation is to memorize Dowd's Absolute VaR:
-Drift * (time delta) + Volatility * Deviate * SQRT(time delta)
...because then your bases are covered; e.g., if the question does not include an expected return, just assume zero
Operational Risk:
Absolute oprisk VaR = UL + EL
Relative oprisk VaR = UL
Default assumption: drift is negative, but unlike credit, banks often do not provision for operational EL. Therefore, following Basel, default is EC = UL + EL
A brief note about a classic confusion:
There is a confusion here, often and yesterday in my comments section, because (e.g) Crouhy shows relative VaR (and Jorion has a similar variation) as Expected profit - Worst case loss @ 99%. So, this confuses because the "absolute" include expected profit. But please note: e.g., if initial portfolio value (w0) = $100, and expected return = +5% with return volatility = 10%, then
relative VaR = 10%* 1.645 @ 95% * 100 = $16.45
and this is the same as Crouhy's relative VaR = +$5 - (- 11.45)... because the loss cutoff is 100 - 11.45 = 88.55, which i find personally confusing
absolute VaR = -5% + 10%*1.645 = 11.45% = $11.45
in market risk VaR, where the drift is always positive (expected return > 0), the absolute VaR must be less than (<) the relative VaR
If the relative/absolute distinction is still confusing, please maybe keep in mind we can have two loss perspectives, starting with today (T0)
* absolute: what is the worst expected loss from my current (today's initial) position?
* relative: what is the worst expected loss from my expected future position; i.e., which includes the positive drift in the case of market risk and negative drift (EL) in the case of OpRisk and Credit risk?
Finally,
* Although CVaR can include/exclude EL, UL is straightforward. This is why I would prefer to move to Crouhy's unambiguous: UL = WCL(confidence) - EL
* If you are not sure, your best guess is EC = UL (b/c that will be the case except maybe for OpRisk)
* However, the treatment of EL should be given!
sorry for length, i hope collecting the summarized view is helpful...David
Yes, agreed, one of my high-priority feedbacks to GARP concerns this exact relationship and definitional conflicts in the assignments, which has often been discussed in the forum; e.g., here
VaR can be either "relative" or "absolute" and this explains why (strictly speaking) it can include or exclude EL. In regard to each of the three major risk buckets, here are the two variations and the "default" assumption:
Credit Risk:
Relative credit risk VaR = UL = Crouhy's P(c) - EV = EC = denominator of RAROC (b/c the numerator similarly deducts EL)
Absolute credit risk VaR = UL + EL
Default assumption: following Basel, as credit portfolios provision for EL, we assume EC = UL (as Basel, unless EL is not provisioned, charges capital for UL)
note: but even Basel in the IRB assignment defines CVaR = EL + UL. So, there is (IMO) no point is getting attached to either CVaR = UL or CVaR = UL + EL.
Market Risk:
Relative market risk VaR = Volatility * Deviate * SQRT(time delta); i.e., loss relative to future expected value
Absolute market risk VaR = -Drift * (time delta) + Volatility * Deviate * SQRT(time delta); i.e., loss relative to initial/zero
Default assumption: note, unlike credit, the drift is positive! Depends on time frame, but following Basel, under daily and since it is conservative, assume drift = 0, and therefore EC = UL which is the same as EC = UL + (EL = 0)
...but in regard to market risk VaR, my recommendation is to memorize Dowd's Absolute VaR:
-Drift * (time delta) + Volatility * Deviate * SQRT(time delta)
...because then your bases are covered; e.g., if the question does not include an expected return, just assume zero
Operational Risk:
Absolute oprisk VaR = UL + EL
Relative oprisk VaR = UL
Default assumption: drift is negative, but unlike credit, banks often do not provision for operational EL. Therefore, following Basel, default is EC = UL + EL
A brief note about a classic confusion:
There is a confusion here, often and yesterday in my comments section, because (e.g) Crouhy shows relative VaR (and Jorion has a similar variation) as Expected profit - Worst case loss @ 99%. So, this confuses because the "absolute" include expected profit. But please note: e.g., if initial portfolio value (w0) = $100, and expected return = +5% with return volatility = 10%, then
relative VaR = 10%* 1.645 @ 95% * 100 = $16.45
and this is the same as Crouhy's relative VaR = +$5 - (- 11.45)... because the loss cutoff is 100 - 11.45 = 88.55, which i find personally confusing
absolute VaR = -5% + 10%*1.645 = 11.45% = $11.45
in market risk VaR, where the drift is always positive (expected return > 0), the absolute VaR must be less than (<) the relative VaR
If the relative/absolute distinction is still confusing, please maybe keep in mind we can have two loss perspectives, starting with today (T0)
* absolute: what is the worst expected loss from my current (today's initial) position?
* relative: what is the worst expected loss from my expected future position; i.e., which includes the positive drift in the case of market risk and negative drift (EL) in the case of OpRisk and Credit risk?
Finally,
* Although CVaR can include/exclude EL, UL is straightforward. This is why I would prefer to move to Crouhy's unambiguous: UL = WCL(confidence) - EL
* If you are not sure, your best guess is EC = UL (b/c that will be the case except maybe for OpRisk)
* However, the treatment of EL should be given!
sorry for length, i hope collecting the summarized view is helpful...David
gizmo - no, incremental VaR and RAROC are part of L2 - David
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