FRM Fun 13: Absolute versus relative VaR versus UL

[David Harper CFA FRM]

FRM Fun 13 (this is a P2 FRM question but P1 candidates are welcome to try)

Today's question isn't really fun, but it has been asked literally dozens of times in various guise since I started the forum. A "star" goes to the most helpful answer.

Despite its weaknesses, value-at-risk (VaR) is the most prominent risk measure in the FRM curriculum. Because VaR, as a worst expected loss, associates with a time horizon (and a confidence level), the "drift" of an exposure factors into the worst expected loss. For example, if my I expect my portfolio to gain by +$2 over the month, but its worst expected loss is $10, I can either include or exclude the expected gain in a VaR number. In an equities portfolio, drift is the positive return we expect over time.

Several authors reflect this distinction--i.e., to include or exclude the drift-- by specifying an "absolute VaR" or a "relative VaR."

Two questions (both are FRM P2 although UL makes an appearance in P1) :

1. What is the relationship among relative VaR, absolute VaR and unexpected loss (UL)?
2. Under the Basel method, the three big risk buckets are market, credit, and operational risk.
   What is the presumed (i.e., default or assumed) VaR for each risk bucket; e.g., are they each absolute VaRs? are they each relative VaRs? Put another way, can we identify a conclusive set of presumed VaRs for:

• Credit risk
• Market risk
• Operational risk

 

[Aleksander Hansen]

David,

I'm just throwing this out there - or thinking out loud - but would it be an idea to, e.g. put in parenthesis whether the question pertains to P1 or P2, and if one question contains multiple sub-questions (like here), write P1 on q1 and P2 on q2?

The reason I ask is twofold:
1) if you are a candidate for part 1, you don't really need to be stressed out by scratching your head over a P2 question (unless you really want to and see it as a challenge).

2) Should a P2 candidate or FRM holder maybe give P1 candidates a "grace period" of a day or two before answering?

My reasoning here is that a P2 candidate is obviously more likely to know the answer, but importantly, by showing willingness to participate even for P1 questions the candidate might already be quite active on the forum.

Just thinking it might deter a P1 candidate to give it a shot if he/she knows that an FRM holder or P2 candidate will provide an answer first. In other words, I'm thinking a P1 candidate might give a great answer if given the opportunity, and then a day or two later P2 candidates can weigh in with additional information, add to the answer, clarify certain things and so forth.

The downside is of course that one runs the risk of having P2 candidates who are not yet comfortable with P2 material not participate as actively...

 

[ShaktiRathore]

Yeah Aleksander is right in this regard.Lets give chance to all. May be some people might come out with better answers than us. Seeing a lot of wise people visiting the forum. Nice to see such conceptual Questions.

 

[David Harper CFA FRM]

Aleks, thank you, yes excellent suggestion to distinguish between P1 and P2! Edited above and I will do so going forward, cheers,

 

[ShaktiRathore]

Absolute VaR is nothing but VaR calculated as we normally do with respect to a mean of zero as the maximum loss that can occur at a certain confidence level(CL) over a specific period of time.Relative VaR is given by at 95% CL as 1.645*volatility*Value of Portfolio. Whereas Absolute VaR will take into consideration the overall Loss including the gain from the positions that can be expected for a Given CL.For e.g. Suppose relative VaR is -12. Then as David highlighted that if there is +2 expected gain in portfolio position then the maximum loss that can occur over a period is -12+2=-10 is absolute VaR.
Unexpected Loss is the loss that can occur beyond the expected loss due to uncertain events. This losses are out of the blue losses that can jeopardize the finance of the institution and banks always set aside some capital for facing these losses.
As far as I know all the three VaRs are absolute VaRs. The credit VaR is calculated at 99.9% CL using Internal Ratings Based Approach wherein ratings are internally generated for Bonds and debt using complex models and then risk weights are assigned according to rating and VaR is finally arrived through summation of products of Risk weights and corresponding outstanding debt,
Market VaR at 99% CL for 1 year (250 days) using Internal Models Approach wherein all the portfolio market positions are combined into one portfolio and the total risk is given by portfolio volatility(takes into account diversification of asset classes)
and finally operational VaR at 99.9% CL over a year using Advanced measurement Approach uses a combination of LDA approaches.Here separate distributions of frequency of losses(poisson distribution) and lognormal distribution of severity of losses. The combined distribution of above will give us the operational VaR. All the three VaRs are assumed over a period and have conclusive VaRs that are drawn. Backtesting can be employed to test the models that are employed in VaRs calculations.

 

[nilay14]

Hi david,
I have gone through above question and I am agree with the explanation given by ShaktiRathore. but
when I visited http://forum.bionicturtle.com/threads/relative-var-tracking-error.8/ link it created confusion for absolute and relative VaR concept
please go through below excerpt from your thread...

Relative versus Absolute VaR
For our purposes (the exam), relative VaR refers to VaR relative to the expected value of the portfolio. Absolute VaR refers to VaR relative to zero. So, if you start today with portfolio value of $100, expected annual return of 10% and (annualized) standard deviation (of returns) of 10%, the one-year 95% RELATIVE VAR = ($100)(10%)(-1.645) = -$16.45. The one-year 95% ABSOLUTE VAR = ($100)[(-1.645)(10%)+10%] = -$6.45. See the difference? At the end of the year, we expect the portfolio to grow to $110. But $16.45 is "at risk." Relative VaR is the full $16.45 and Absolute VaR, giving credit to the gains that theoretically relate to the risk, speaks to loss versus initial value. Another way to view it: either way, our 1-year 95% VaR (note how i am careful to qualify the VaR with both a time horizon and a confidence level, because there is a different VaR for different combinations of time & confidence) says the risk is a final value of $93.55. The difference is whether we count the loss relative to where we started ($100) or where we expect to end ($110).

Which one is correct?

 

[ShaktiRathore]

Hi Nilay,

Yes I wrongly in hurry interchanged Absolute and Relative Vars definitions was really hurried up to finish this long answer. Sorry for the mistake. But i have made the changes above.
thanks

 

[nilay14]

Hi Shakti,

No need to sorry... that mistake cleared my doubt. I was having misconception regarding Absolute and Relative VaR. Thanks for the correct ans.

 

[David Harper CFA FRM]

Just wanted to close the loop on question 2 (since I'm linking to this thread):

Under Basel II advanced approaches (IRB for credit risk, IMA for market risk and AMA for operational risk), the defaults are:

• Credit risk capital charge is relative CVaR since it is a charge for UL which excludes EL (in credit, EL is an ongoing expense that Basel expects to be expensed by provisions. If, however, there is a gap, capital must first address the gap)
• Market risk capital charge can be argued as either, IMO, because the drift is uniquely positive: maybe better is Absolute VaR where the drift is assumed to be zero; or, if you like, relative VaR where the drift is omitted.
• Operational risk capital is absolute OpVaR because, unlike credit, otherwise coverage of EL is not presumed (i.e., it is not ordinary practice to expense OpRisk EL):

Basel II Operational Risk, AMA Quantitative Standards 669(b): "Supervisors will require the bank to calculate its regulatory capital requirement as the sum of expected loss (EL) and unexpected loss (UL), unless the bank can demonstrate that it is adequately capturing EL in its internal business practices. That is, to base the minimum regulatory capital requirement on UL alone, the bank must be able to demonstrate to the satisfaction of its national supervisor that it has measured and accounted for its EL exposure"

 

[bpdulog]

Hi Nilay,

Yes I wrongly in hurry interchanged Absolute and Relative Vars definitions was really hurried up to finish this long answer. Sorry for the mistake. But i have made the changes above.
thanks

Based on the formula I have derived from the BT practice questions, absolute VaR doesn't seem to take into account the gain:



Or is it a difference between actual return vs. expected return? Does absolute var use E(R) as the mean and relative var assume a mean of 0? Or am I mixing up the two? Thanks in advance

 

[David Harper CFA FRM]

Hi @bpdulog Your equation is exactly correct! Your questions are also instructive. Your statement "absolute VaR doesn't seem to take into account the gain" could be interpreted either way, in my opinion (it's true, but depends I think on interpretation).

I prefer to think of the distinction in terms of time, today versus the end of the horizon (e.g., in a one-day VaR, the end of the horizon is tomorrow; in a one-year VaR, the end of the horizon is next year). Ignoring the time scaling, say µ is 1.0% and σ=10%

• Absolute VaR is the worst potential loss relative to our position today, so the 95% absolute VaR of 15.45% = -1%+10%*1.645 is reduced by 1.0% because, as of today, we expect some positive return on the position. The potential loss is said to be "relative to the current position." Mathematically, drift (ie, expected return, mean) is included in the formula, so it does "account" for the gain. In this way, also, absolute VaR is superior as the return-adjusted VaR measure: if the position has greater risk, arguably we'd want to "include" the offsetting impact of greater risk?
• Relative VaR is the worst potential loss relative to our position at the end of the horizon, so the 95% relative VaR of 16.45% = 10%*1.645 is greater by exactly the drift term of 1.0%. The potential loss is said to be "relative to the expected future position." However, whereas absolute VaR is a return-adjusted risk measure, relative VaR is just a risk measure (in my opinion). Further, if the expected return is zero, then relative VaR = absolute VaR, which is the common assumption if the horizon is daily.

In this way, you can see how this VaR is really a future loss. Some practitioners discount the VaR (e.g., at the risk free rate) to translate into a present value (as we often do in finance). When they do this, if the drift is risk-free rate, the absolute VaR can also reduce to (be approximated by) the relative VaR. But the FRM, to my knowledge, does not discuss this nuance.

Re: "Or is it a difference between actual return vs. expected return?" This is a interesting question. It's always an expected return; in contrast to risk-neutral assumptions associated with derivatives valuation, VaR (as risk measurement) is essentially preoccupied with a future distribution.

I hope that clarifies!

 

[bpdulog]

Thanks @David Harper CFA FRM for the very detailed explanation. Some of my angst has to do with the way GARP is wording the question. They ask us to calculate VaR, but there are a few different types! They will signal when to use the delta normal method, but I haven't seen any distinction between absolute and relative.

 

[David Harper CFA FRM]

Sure @bpdulog I agree, I noticed the 2016 p2 practice exam refers to either absolute and relatives as just "VaR." I will definitely include this feedback in our next update to GARP. As a practical matter, when you see a one-day VaR, it's okay to assume relative VaR (and in most of these questions, the one-day VaR omits a return, so you have no drift to assume anyways!). Thanks,

 

[saurabh.alkari]

Hi @David Harper CFA FRM ,
Thank you for clearing relative VaR & Absolute VaR concepts.
I still have one doubt. What effect will Profit/Loss Distribution and Loss/Profit Distribution have on Absolute VaR and Relative VaR?
I am assuming above answer is assuming a Profit/Loss Distribution.

Thanks.

 

[David Harper CFA FRM]

Hi @saurabh.alkari sure thing! You make a good point and you are correct that the above presumes profit/loss (P/L). The P/L convention assumes the inputs mathematically retain their meaning: profits are +$P and losses are -$L. Hence our typical absolute VaR implicitly does assume P/L and is given by, if i momentarily disregard time scaling: aVaR = -µ + σ*Z.

If instead the input format is L/P such that losses are +$L and profits are negative -$P, then the aVaR = µ + σ*Z. This is reviewed in Dowd Chapter 3 and presumes that our VaR is a worst expected loss that is expressed as a positive. In this way, the default approach (by which I mean: explicit specification always takes priority, but if no specification is indicated) is:

• Inputs are P/L, which I think of as the "mathematically true" way; ie, losses are negatives
• Output VaRs are typically a worst expected loss (which is mathematically negative) but expressed as a positive.

Simple example: P/L assumptions µ = +3.0%, σ = 2.0% imply that equivalent L/P assumptions are µ = -3.0%, σ = 2.0%, and:

• P/L 99.0% aVaR = -µ + σ*Z = -(3.0%) + 2.0%*2.33 = 1.66%
• L/P 99.0% aVaR = µ + σ*Z = (-3.0%) + 2.0%*2.33 = 1.66%. I hope that's helpful!

 

[ivan.jarmanovic]

Hi all,
In P2.T9 Jorion's Ch 7 & 17 "Value at Risk: the new benchmark for managing financial risk":

- I guess there we are talking about absolute VaR when dealing with individual VaR, etc?; and
- the drift (i.e. mean) is assumed to be zero throughout the full chapter?

Thanks
-Ivan

 

[evelyn.peng]

Hi all,
In P2.T9 Jorion's Ch 7 & 17 "Value at Risk: the new benchmark for managing financial risk":

- I guess there we are talking about absolute VaR when dealing with individual VaR, etc?; and
- the drift (i.e. mean) is assumed to be zero throughout the full chapter?

Thanks
-Ivan

If drift is not included, it indicates relative VaR, not absolute VaR. I don't think Jorion necessarily indicated all the VaR he is referring to are necessarily relative VaR (i.e., without drift).
Try this thread for further explanation on absolute vs. relative VaR: https://forum.bionicturtle.com/thre...ersus-relative-var-versus-ul.6020/#post-18763

Here is my summary including all the VaR defs for each of the categories: