GARP.2010.PQ.P1 Diversified/undiversified VaR (garp10-p1-16)

[Kavita.bhangdia]

Hi David,
What I understand is

Incremental VaR is diversified (and hence sum does not equal to total VaR)
Component VaR is undiversified (and hence sum equals total VaR).

Am I correct?
Please clarify

Thanks,
Kavita

 

[QuantMan2318]

Hi @Kavita.bhangdia, there is a small difference in the statement that you made above.
Undiversified VaR does sum to Total VaR but is not the same as Component VaR.

Take the example of a Portfolio VaR, it is alpha*portfolio Standard Deviation where the formula for the portfolio S.D is as follows:
sqrt(w1*sigma1^2+w2*sigma2^2+2*w1*w2*sigma1*sigma2*rho)
So if the coefficient of correlation is one, i.e., the positions are perfectly correlated (which I think cannot exist but can come close) we get Undiversified VaR that is the sum of the individual position VaR. (Then why should a Risk Manager encourage traders or the treasury to have a perfectly correlated position?, because we get no benefit here)

Component VaR on the other hand is a different animal altogether, it has been created to capture the additive property that is lost when diversification benefits appear, therefore it is unrelated to the Undiversified VaR. When there is diversification, i.e., the rho above is less than 1, we see that the Portfolio VaR is a complex number that appears within a square root, so how can you make it additive? by using Differential Calculus of course, the Component VaR is the Marginal VaR times the dollar weight where we use Marginal VaR to decompose/find the derivative of the SD of the Portfolio wrt to change in the component weight, this gives Component VaR that adds to Total VaR

Incremental VaR is the change in the Total/Portfolio VaR when a individual position is added to the portfolio, it is neither Marginal nor Component VaR, it required recalculating the Portfolio VaR when an additional position is added

 

[Kavita.bhangdia]

Thanks Quantman.. I need some more time to digest this though..

Regards,
Kavita

 

[David Harper CFA FRM]

I think @QuantMan2318 gives an excellent response, I would just like to add a framing perspective (as these terms are all in Jorion Chapter 7 which has been in the FRM syllabus for several years).

• Diversified VaR and Undiversified VaR are Portfolio VaR concepts, but
• Component VaR and Incremental VaR are Position (within the portfolio) VaR concepts.

So we would not typically refer to either the diversified/undiversified VaR of a position nor the component/incremental VaR of a portfolio (because it is not sufficiently specific).

As QuantMan implies diversified VaR (in the traditional mean-variance framework approach) includes correlations (i.e., imperfect correlations capture the benefit of diversification) and we can think of undiversified VaR as the special case of diversified VaR when correlations spike to 1.0 (so, Jorion assumes this to stress the portfolio)

Compent and Incremental VaRs do tend to assume the diversified portfolio, so they are more consistent, as position concepts, with diversified VaR. Component VaRs sum to diversified portfolio VaR; incremental VaRs are not additive. But i hope my comment is additive, get it?

 

[QuantMan2318]

Component VaRs sum to diversified portfolio VaR; incremental VaRs are not additive. But i hope my comment is additive, get it?

Sure it is more than that, it is multiplicative as in Geometric Progession

 

[WhizzKidd]

Hi David,

I need your help.

I found this thread on diversified and undiversified VaR. How does one calculate the diversified VaR for the above question? I have the undiversified components, which are added but need to subtracted from the Var_diversified?

This is a 2010 GARP practice question (answer: b)

 

[Arnaudc]

Good morning,
thanks for your question because it made me practice and realize my matrix calculation at first was really rusty...
How I would do that:
Undiversified VaR is = VaR Bond A + VaR Bond B = 1.645 x 5% x 25 x sqrt(10/250) + 1.645 x 12% x sqrt(10/250) x 75= 0.41125 + 2.961 = 3.372250

To find the diversified VaR, you need the covariance between A and B which is given by Correlation x StDev A x StDev B = 0.25 x 5% x 12% = 0.0015

Now we need some matrix notation:
(0.05² 0.0015) (25) = (0.05² x 25 + 0.0015 x 75) = (0.175)
(0.0015 0.12²) (75) (0.0015 x 25 + 0.12² x 75) (1.1175)

The finally multiply this matrix obtained by your position : ( 25 75) (0.175) = (25 x 0.175 + 75 x 1.1175) = 88.1875. This is the variance. Get the StDev = 9.390820 (1.1175)

Diversified VaR = 1.645 x 9.39082 x sqrt (10/250) = 3.08958
EDIT:
The diversified VaR (in the case of 2 assets can be conveniently found as: Sqrt ( VaR² A + VaR² B + 2 x Correlation x VaR A x VaR B) = Sqrt (0.41125²+2.961² + 2 x 0.25 x 0.41125 x 2.961) = 3.0896 (much faster solution here)
end of EDIT

The diversification gain = 3.372250 - 3.08958 = 0.282670 in Millions = 282.670 rounded to 283.00

I hope it helps...

Kind regards,

 

[Nicole Seaman]

View attachment 936

Hi David,

I need your help.

I found this thread on diversified and undiversified VaR. How does one calculate the diversified VaR for the above question? I have the undiversified components, which are added but need to subtracted from the Var_diversified?

This is a 2010 GARP practice question (answer: b)

In addition to @Arnaudc's helpful response above, this question is also discussed in detail here: https://forum.bionicturtle.com/thre...-a-var-valuation-garp10-p1-16.2108/#post-7397 (paid only).

Thank you,

Nicole

 

[David Harper CFA FRM]

Per the link shared by @Nicole Seaman there is a known error in this old question. @Arnaudc shows a solid method but with one key difference which I typed below in red. The problem is that GARP's answer uses the yield volatility directly, but we should multiply yield volatility by (modfied) duration to get price volatility. Thanks!

How I would do that:
Undiversified VaR is = VaR Bond A + VaR Bond B = 1.645 x 5% x (2/1.05) * 25 x sqrt(10/250) + 1.645 x 12% x (13/1.03) x sqrt(10/250) x 75= 0.789 + 37.369 = 38.152

To find the diversified VaR, you need the covariance between A and B which is given by Correlation x StDev A x StDev B = 0.25 x 5% x 12% = 0.0015

Now we need some matrix notation:
(0.05² 0.0015) (25) = (0.05² x 25 + 0.0015 x 75) = (0.175)
(0.0015 0.12²) (75) (0.0015 x 25 + 0.12² x 75) (1.1175)

The finally multiply this matrix obtained by your position : ( 25 75) (0.175) = (25 x 0.175 + 75 x 1.1175) = 88.1875. This is the variance. Get the StDev = 9.390820 (1.1175)

Diversified VaR = 1.645 x 9.39082 x sqrt (10/250) = 3.08958 <<- I didn't bother to edit the good-looking matrix math; I believe if you replace yield volatility with price volatility, you'll get a diversified VaR of ~ $37.57, per Nicole's link the ΔVaR should be $594,000 (semi-annual discounting) or about $580,000 (annual discounting as I just did)
EDIT:
The diversified VaR (in the case of 2 assets can be conveniently found as: Sqrt ( VaR² A + VaR² B + 2 x Correlation x VaR A x VaR B) = Sqrt (0.41125²+2.961² + 2 x 0.25 x 0.41125 x 2.961) = 3.0896 (much faster solution here)
end of EDIT

The diversification gain = 3.372250 - 3.08958 = 0.282670 in Millions = 282.670 rounded to 283.00