Incremental VaR - Jorian Chapter 7 - AIM 54.4

[Hardy Noman]

Hi Shakti ,

Can you please elaborately explain the below question on Incremental VaR (from schweser pg. 30 Book 4 - investment)

Que) A portfolio consist of Asset A & B. these assets are the risk factors in the portfolio.
Volatility of A = 6% , B = 14%.
There are $4 million invested in A and $2 million invested in B. Assuming no correlation, Compute the VaR of the portfolio using a confidence parameter of Z = 1.65

Ans) using matrix notation to derive the $ Variance of the Port.
Variance(portfolio).V^2 = [$4 $2] [0.06^2 0 ] [$4] = 0.0576 + 0.0784 = 0.136
....................................................[ 0 0.14^2] [$2]

(Where does ''V^2" come from...what does it mean?)

This value is in ($ Millions)^2. VaR is then the square root of the portfolio variance times 1.65
VaR = (1.65)($386,782) = $608,490

My working) According to me we could have just calculated the standard deviation of the Portfolio and then calculated portfolio VaR from there.
Weight of Asset A = 0.67 (4/6) , Asset B = .33 (2/6)

Variance (Portfolio) = (.67)^2 . (0.06)^2 + (.33)^2 . (0.14)^2 + 2(.67)(.33)(.06)(.14)0
= 0.00161604 + 0.00213444 = 0.00375048

So portfolio Std Dev = Sqr rt (0.00375048) = 0.0612 or 6.12%

Using this we can find portfolio VaR as follows

1.65 * 6.12% * $6 million = $64,341.

Doubt)
1) Why method is correct mine or the authors?
2) What method does the answer carry out, it seems the text book answer dosent consider weights

Pls help!!!

Thank you, Hardy.

 

[David Harper CFA FRM]

Hi Hardy, Your approach looks good to me (I also get a portfolio standard deviation of ~ 6.12% (actually 6.1464%) but I think your final multiplication is just off by an order of magnitude: 1.65 * 6.12% * $6 million = $605,80. Otherwise, your method is correct.
( ... with exact numbers, including a precise normal deviate, I get $606,592, fwiw). Thanks,

 

[Hardy Noman]

Hi David.

Thanks for that!.

Can you please explain me the approach taken by schweser (for the above question)?
because they seem to carry these type of calculations at 4-5 different questions in the notes.

Even Their calculation of covariance also seems bizarre (which is also seen quite a few times in this topic)
for eg. [Cov (Ra, Rp)] = [0.06^2 0] [$4] = [0.0144]
............[Cov (Rb, Rp)]....[0 0.14^2]. [$2]... [0.0392].
** Please read the above as a matrix

they use these calculations to calculate MvaR and other calculations too
Here the covariance with the portfolio is show as if its 0.06^2 * 4 + 2*0 = 0.0144 , 4*0 + 0.14^2*3 = 0.0392 !!
but it should be -

*

* stolen from investopedia
Also please explain :-
1) what does V^2 stand for in the earlier schweser question "Variance(portfolio).V^2"

Thanks for your time , Hardy

 

[David Harper CFA FRM]

Hi Hardy,

Sure no problem. First, the investopedia formula is correct but it is a sample covariance; i.e., we need it to return an estimate (estimator) when we don't know the population distributions. But this problem is not "based in data:" it presumes normality and gives us the assumptions which implicitly assume we have the population knowledge as represented by a function. So the covariance that ultimately applies--one of the most important formulas because it generalizes the variance--is: COV(x,y) = E(xy) - E(x)E(y).

Schweser is using matrix notation, is all. [0.0144] is what I would call "dollar covariance"

I don't know why "Ra" notation is employed since returns are not involved that i see, but nevertheless:

COV(a, P) = COV[A, w(a)A + w(b)B] where Portfolio (P) = w(a)A + w(b)B; the difficult idea is that this is a covariance between an asset and the portfolio that also contains the asset itself
COV[A, w(a)A + w(b)B] = COV[A,w(a)A] + COV[A, w(b)B] = w(a)COV(A,A) + w(b)COV[A,B] = w(a)VAR(A) + w(b)COV[A,B];
i.e., covariance(A,P) = 67% weight(A)*6.0%^2 + 33% weight B*0 = 0.00240
dollar covariance = 0.00240 * $6.0 million = $0.01440, with V^2 I assume referring to fact that variance/covariance are in squared-units. I hope that helps, thanks,

 

[Hardy Noman]

couldn't have got a better explanation! you just made it so simple....Thanks a million!

For Marginal VaR...could you explain how the final formula evolved?
i mean MVaR(i) = Change in VaRp / Change in ($ investment i) = Z. change in Std Dev (p) / change in w(i) = Z. Cov (Ri,Rp)/ Std Dev (p)
I got how the Red formula came up, but failed to understand how the Blue formula evolved out of the Red one.
i mean how is change in Std Dev (p) / change in w(i) = Cov (Ri,Rp)/ Std Dev (p)

Thank you.
Hardy

 

[ShaktiRathore]

Std Dev (p)=sqrt(w1^2*stdDev(R1^2)+...wi^2*stdDev(Ri^2)+..)
d(Std Dev (p))/dwi=(1/2Std Dev (p))*2*wi*stdDev(Ri)^2=(1/Std Dev (p))*wi*stdDev(Ri)^2...differentiating both sides w.r.t wi
=>d(Std Dev (p))=(1/Std Dev (p))*wi*stdDev(Ri)^2*dwi=change in Std Dev (p)...(1)
substituting value of change in Std Dev (p) from (1) in Z. change in Std Dev (p) / change in w(i)
Z. change in Std Dev (p) / change in w(i) =Z*(1/Std Dev (p))*wi*stdDev(Ri)^2*dwi/change in w(i)
dwi is change in weight of assets i so cancel equalt things
Z. change in Std Dev (p) / change in w(i) =Z*(1/Std Dev (p))*wi*stdDev(Ri)^2...(2)
Rp=w1R1+...wiRi..... return of portfolio is weighted mean of assets returns taking covariance on both sides.
=>Cov(Rp,Ri)=wi*Cov(Ri,Ri)=wi(stdDev(Ri)^2) =>wi*stdDev(Ri)^2)=Cov(Rp,Ri)...(3) assuming the asset i has o correlation with others etc..I have to take this assumption otherwise it seems difficult to derive the formula which seems unrealistic on one side and on the other side seeing that portfolio should be properly diversified than it seems possible that with some assets the asset i has positive correlation while with others it has negative correlation so overall the correlation of asset i approaches 0 with all other assets which are diversified assets in the portfolio it seems logical enough and nonetheless it works to arrive at the final result
sunbstituting value of wi*stdDev(Ri)^2) in 2 from 3
Z. change in Std Dev (p) / change in w(i) =Z*(1/Std Dev (p))*Cov(Ri,Rp)
thanks