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# Absolute vs Relative VaR

#### Jiew Kwang

##### Member
Hi guys, I wanna ask a simple question regarding VaR.

Relative VaR$= Portfolio value * (volatility * normal deviate) Absolute VaR$ = Portfolio value * (-E(R) + volatility * normal deviate)

From what i think i understood, the relative VaR is loss expected to final wealth. Does this imply that the portfolio value already holds the expected return?

On the other hand, for the absolute VaR, it is the loss to initial wealth. That -E(R) <-- Again, does that imply that the portfolio already holds the expected return since you want to deduct it off?

Can someone help me crystallize this? Thanks!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Jiew,

I don't know if I can crystallize (it does not seem simple, many arguments about this….) but my short answer is, of course your relative/absolute VaR formulas are correct, it maybe helps to keep in mind they are deltas (difference) between wealth levels not wealth levels themselves. Confusion seems to arise when compared to Jorion's formulas which concern levels:

Following Jorion's variables, with examples and simple returns
Let w0 = initial value/wealth = \$100
Let r = expected return = +10%
Let r* = threshold VaR return = -22.90%
Let sigma = volatility = 20%
Confidence = 95% --> normal deviate = 1.645

Expected future value w1 = w0*(1+r) = 100*(1+10%) = 110
Worsted expected future value (VaR threshold) = 100*(1 - 22.9%) = 77.1
Relative VaR = Loss relative to final wealth (as you say) = Expected future value - worst expected future value = 110 - 77.1 = 32.9
This is the same as w0*sigma*deviate (100*20%*1.645 = 32.9) and, IMO, "reconciliation" between these two is the mental key

Same as:
Expected future value - worst expected future value
w0*(1+r) - w0*(1+r*) = w0*[(1+r) - (1+r)] = w0*(r-r*) = 100*(10% - -22.9%) = 32.9; i.e., demonstration that w0 does not include the expected return
(Jorion shows the equivalent 5.2: = -w0*(r* - r))

And absolute VaR is
Initial (today's value) - worst expected future value;
w0 - w0*(1+r*) = w0(1-1-r) = -w0*r*

Hope that helps, David

#### Jiew Kwang

##### Member
Hi David,

I have a few quick questions on VaR.

I referred to the notes:

Type: Ex ante
Ease of calculation: No
Ease of explanation: Yes
Aggregation: Yes

My question is what is meant by aggregation in this context and why is it 'yes'. Also, what is the reason for 'difficult to calculate'.

Thanks!

regards,
jk

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi jk,

That is Jorion's Table 1-3, where his no/yes choices are informed mostly by the (somewhat) hierarchical presentation of risk metrics (or risk limits). His order then, from primitive to modern (my terms, his concept):
Stop loss >> Notional >> Exposure (Sensitivities; e.g., duration) >> VaR

Aggregation typically refers to (horizontally) the combinations of assets or positions into a portfolio (Jorion's example: equities + bonds) OR (vertically) position >> portfolio >> group >> business unit >> conglomerate (less Jorion's point in this case). So, his point is: you can't easily add (aggregate) exposure limits (eg, duration, delta), especially if they do not incorporate correlations. However, VaR naturally handles.
… in a way, I think, this is really almost "by definition." VaR is just a distributional quantile, so it EXPECTS the distribution to do the aggregating; for example, I think the OpRisk LDA case study is a good example of "aggregating" all sorts of risk types and business units, rolling them up into a single (piece-wise) OpRisk VaR distribution.

Re: calculate. Well…. He's got stop loss and notional, as "easy to calculate" risk limits. Then he's got VaR and exposures as hard (not easy) to calculate. I guess that a relative statement (harder than the others?). I am not sure I am keen to defend it, maybe I don’t understand or agree. I personally think VaR has two aspects: 1. the potentially *difficult* methodologies of getting to the distribution (MCS, HS and variants, and analytical), then 2. once you have the distribution, getting the VaR (quantile) is easy. As I think the historical simulation variations are "not easy," I do halfway agree with him on ease of calculation…

Hope that helps, good luck Saturday!!

David

#### Jiew Kwang

##### Member
Hi David,

Thanks for your explanation once again.

fyi, I have not registered with GARP yet as I'm planning to sit for next May's exams. I'm really looking forward to next yr's materials and also, learning more from you through the forums.

jk