Var for two asset portfolio using volatility

[sudeepdoon]

Hi David,
I was going through a series of questions and came across one where we were to calculate the VAR for a portfolio of 2 asset having some volatility each...

My solution was to calc the volatility of the portfolio using 2 asset model and then using calc the VAR as Z * volatility..

but the soln was different and tbe answers.
It had calc VAR in %age terms for each of the asset and then used these in the 2 asset formula to give a value which he called as VAR of the portfolio...

My questions are:
1. Is my solution correct
2. Is it correct to use the VAR like vol. in the formula. what he did was
var(a+b)= wa^2*var(a) + wb^2 * var(b) + 2*var(a)*var(b)*wa*wb*cov(a,b)

-- Sudeep

 

[David Harper CFA FRM]

Hi Sudeep,

I don't *think* that will work, but it should be easy to confirm.
Note semi-related question here: http://forum.bionicturtle.com/viewthread/1800/

where I showed you can just use $VaRs as an alternative to *your* method. So, there is your method (which, of course is correct), then you can also operate on the dollar VaRs of each position, in which case this applies:
VaR = SQRT( VaR_asset1^2 + VaR_asset2^2 + 2*VaR_asset1*VaR_asset2*correlation)

and this XLS demonstrates: http://public.sheet.zoho.com/public/btzoho/sep8-portfoliovar
i.e., see column E
E29 is your (standard) method; E31 uses formula above; i.e., both give $6.15 for 50%/50% portfolio with 0.5 correlation

...now i don't *think* this third method will work, but you might just test it to see if that method gets $6.15 also...thanks, David

 

[sucheta_isi]

David,

I have one question. What should be the answer to a question like this:


If portfolio A has a VaR of 100 and portfolio B has a VaR of 200, then the VaR of the portfolio C (= A + B) ?

I know that VaR is not subadditive. But in case we assume normal distribution of loss then it will be subadditive.

SO what will be the exact answer for the above question.
Under non normal condition?
Under normal condition?

Thanks,
sray

 

[David Harper CFA FRM]

sray,

under normal, you can use: "diversified" VaR = SQRT( VaR_asset1^2 + VaR_asset2^2 + 2*VaR_asset1*VaR_asset2*correlation)
e.g., if correlation = 1.0, then VaR C = 300; if correlation = 0, then VaR C = SQRT(100^2 + 200^2); etc
"undiversified VaR" is by definition rho = 1, so undiversified VaR = 300

under non-normal, need more info, or may need to simulate

David

 

[sucheta_isi]

David,

Thanks for such a super fast reply.

So this means that the VaR of the combined portfolio will be less than or atleast equal to VaR(A)+ VaR(B)

Am I right?

 

[sucheta_isi]

Also I have another question:

If nothing is specified in a question should we assume that the loss distribution follow NORMAL distribution and so that we can say that VaR will be subadditive?

 

[David Harper CFA FRM]

sray,

sure. on 1) the VaR of the combined portfolio will be less than or at least equal to VaR(A)+ VaR(B): this true, but as you've said, *only if* the distribution is elliptical (normal), as that's the precondition for subadditivity

2) great question. the 2009 should specify (right, given all the attention on non-normality, it should never assume that) ... if it asks for calculations, as you can see in the samples, many of the questions implicitly assume normality, so for calc purposes, no harm in assuming that (e.g., any parametric VaR is highly likely to use 95% 1.645 and 99% 2.33 deviates, which are normal) ... beyond calcs, though, I would *not* make that assumption (e.g., credit and ops are non-normal) and i would instead be ready for "violations" of normal.

David

 

[sucheta_isi]

So, in reference to my first question..if nothing is specified then we can NOT (rather SHOULD NOT) conclude anything about the VaR(C). It can be gt/lt/equal to VaR(A)+ VaR(B). Right?


Another question:


We know that if we increasing the confidence level the VaR increases. Can we draw any conclusion about the rate of this increase?

Thanks a ton .

 

[David Harper CFA FRM]

sray,

strictly speaking, given the formal attention in AIMs to coherence and to VaR weaknesses, agreed: should not make gt/lt/= assumption ... i would be ready to invoke lack of subaddivity (which btw, as Dowd shows, does not apply to ES: ES is subadditive!) ... but in regard to calculations, at the same time, i would be ready to continue in the tradition of the practice questions, where normality is used just for tractability. When a question asks for a calculation, I would not get bogged down in distributional assumptions unless the question insists; there are only a few non-normal distributions we can quantitatively engage in the time span of an exam question; e.g., Poisson, binomial, uniform

or put another way, in learning calcs we are typically using normal, but qualitatively we know it's wrong and qualitatively we should be able to say why it's wrong (e.g., emprically, asset returns are heavy-tailed)

rate of increase: that's basically the first derivative of the CDF, so (i) it depends on distribution, but (ii) except for a uniform distribution, it's nonlinear

David

 

[sucheta_isi]

David,

I am sorry for asking question on the same topic again and agian. But , this is the last.

Suppose in the exam there are two questions:

Ques 1:

There are two asstes A and B. C is the combined ortflio of these two assets. Then which one is the correct

a. VaR(C) < VaR(A) + VaR(B)
b. VaR(C) < VaR(A) + VaR(B)
c. VaR(C) = VaR(A) + VaR(B)
d. VaR(C) = VaR(A) + VaR(B)
e. Can not say

Ques 2:

VaR(A)=100 and VaR(B)= 200. C = A+B .Then which one is correct

1. VaR(C) is less than or atleast equal to 300
2. VaR can be greater than 300

What will be the correct answers to score on these questions.

Thanks,
sray

 

[David Harper CFA FRM]

Hi sray,

1. without distributional caveat/qualifier (e.g., normal, delta normal), per lack of subadditvity: cannot say
2. without distributional caveat/qualifier: (2) because it allows for the rare case where diversification is penalized
...but neither are good practice questions, IMO, not representative of good FRM questions

David

 

[sucheta_isi]

David,

I am back again with VaR. If we are given 2 day VaR then can we calculate 10 day VaR just by multiplying it with Root over 5?

 

[David Harper CFA FRM]

Hi sray,

Yes, you can scale a 2-day VaR to 10-days by multiplying by SQRT(10/2) = SQRT (new days/n days) ... this because scaled variance = n-day variance * new days/n
... and (eg) Basel II IMA permits this to get the 10-day market risk charge VaR
... but the square root rule (SRR) assumes i.i.d. returns so you just want to be aware of the assumptional violations: are returns independent? probably not really ... is volatility constant ("identical")? probably not

David

 

[JayC]

sray,

under normal, you can use: "diversified" VaR = SQRT( VaR_asset1^2 + VaR_asset2^2 + 2*VaR_asset1*VaR_asset2*correlation)
e.g., if correlation = 1.0, then VaR C = 300; if correlation = 0, then VaR C = SQRT(100^2 + 200^2); etc
"undiversified VaR" is by definition rho = 1, so undiversified VaR = 300

under non-normal, need more info, or may need to simulate

David

Hello David,
Could you please help me on this calculation : I am trying to calculate a portfolio VaR (5 or 6 assets).
I have already done it for 2 assets using the following formula : VaR (asset 1 + asset 2) = sqrt (VaR asset1 ^2 + VaR asset2 ^2+ 2 * VaR asset1 * VaR asset 2 * correlation12).
What similar formula shoud I use for 5, 6 or n assets ? Thks, Jay​

 

[ShaktiRathore]

Hi jay use the Var-Covar matrix for this with which i think you should be aware of.
VARp= (Var matrix)*(VaR-CovarMatrix)*(Var matrix transpose) use formula of this sort and try out yourself...
VaR-CovarMatrix for 2 asset=
var1 covar1,2 ;
covar 2,1 var 2
for 3 asset:
var1 covar1,2 covar1,3
covar 2,1 var 2 covar2,3
covar3,1 covar2,3 var3

try out thanks....