Incremental VaR using Beta Co-efficient

[rahul.goyl]

Hi David,

Could you please justify on how to calculate Incremental VaR using Beta.

A $20 million portfolio consists of only two equally-weighted and uncorrelated positions in Asset A & B. Asset A ($10 million) has a volatility of 10% and Asset B (also @10 million) has a volatility of 20%. At 99% confidence, what is an approximation of incremental VaR given an additional investment of $1 million in Asset B?

A) $233,000
B) $298,000
C) 333,000
D) 416,000


Thanks in Advanced
Rahul

 

[David Harper CFA FRM]

Hi Rahul, that is a *tough* practice question

beta (b, P) = Cov(b,P)/Var(P) = [Cov(b,0.5a + 0.5b)]/Var(P) = 0.5Cov(b,b)/Var(P) = 0.5Var(b)/Var(P) = 0.5*20%^2/[0.0125]
= 1.6 beta of asset B with respect to portfolio

Marginal Var = is only a linear approxation for Incremental VaR, the writer is accurate to use "approximation" ! = potfolio VaR/Portfolio Value * beta
(see Jorion Chapter 7 or see here)

= [(portfolio volatility)(2.33 normal deviate @ 99%)]/$20 million * 1.6 beta = about 416,000

David

 

[rahul.goyl]

Hi David,

David you are very clear in explaining the Beta of 1.6, but there are few more doubts to be unleash.

Thanks for that speedy respond, however i want to understand couple on things which looks to be hidded in your explanation. Would appreciate veery much if can throw some light.

When you took 0.50 as common = [Cov(b,0.5a + 0.5b)]/Var(P)
1) You got the below, but how does 0.50a disappear ?
= 0.5Cov(b,b)/Var(P),
2) Another doubt = 0.5Var(b)/Var(P) = 0.5*20%^2/[0.0125]
How did you arrive at Var(P) = 0.0125 ?
3) = [(portfolio volatility)(2.33 normal deviate @ 99%)]/$20 million * 1.6 beta = about 416,000
In the above what is Portfolio Volatility?

David you are very clear in explaining the Beta of 1.6, but still there are few more doubts.

 

[David Harper CFA FRM]

Hi Rahul,

Sorry, beta is the hardest part of that:

1. drops out b/c correlation = 0
[Cov(b,0.5a + 0.5b)]
= 0.5Cov(b,a) + 0.5Cov(b,b), since the correlation(a,b) = 0, and Cov() = Corr()*Std()*Std(), the Cov(a,b) = 0,
so 0.5Cov(b,a) = 0.5*Corr(a,b)*StdDev(a)*StdDev(b) = 0 b/c Corr(a,b) = 0

2. This is two-asset portfolio variance, it *should* be memorized for the exam

=w^2*variance(a) + w^2*variance(b) + (2)(w)(w)Covariance(a,b)
=50%^2*10%^2 + 50%^2*20^2 + (2)(50%)(50%)(covariance = 0)
= 0.0125

3. portfolio volatiliy = SQRT[portfolio variance]
volatility% = SQRT[0.0125] = 11.18%
volatility$ = % * $20 MM = 2.23 MMM

David

 

[rahul.goyl]

Hi David,

That was the toughest one to understand, but you made it so clear & easy thru detail explanation.

There is another way which was still not clear to me.

($5.20 = Portfolio VaR)/($20 Portfolio)*(1.6 = Beta) = 0.4161. Beta part is explained by you, which is now resolved,
However there is another method which have been applied ($.40)/($2.24)*2.326 = 0.416, how this $.40 is coming inn (Clueless). $2.24 is portfolio volatility sqrt(5), which is ($10^2)(10%^2) + ($10^2)(20%^2) = $5, 2.326 is critical Z value @ pp% confidence.

How they got $.40 ?

A sincere thanks for your help david.
Rahul

 

[David Harper CFA FRM]

Hi Rahul,

Yea, I had to look that up, having forgotten the other marginal VaR formula.
Under Jorion page 168, marginal VaR also = (normal deviate)*Covariance$[B, P]/Portfolio Volatility$.
(dollar covariance as opposed to % covariance above)
Just as you have: ($.40)/($2.24)*2.326 = 0.416, so the $0.40 is dollar covariance

as in: Cov()*$20 = (0.5*20%^2)*$20 = $0.4
or in matrix form, per Jorion page 171: (sigma)X = 0*$10+20%^2*$10 = $0.40
I personally find this more difficult here in the 2-asset case, but this is following the alternative formula (Jorion 7.17):
marginal VaR = deviate*Covariance(security, portfolio)/Volatility(portfolio)

Note the difference from above, in this Covariance and volatility are in dollar ($) terms so they cancel to give the unitless marginal VaR.
The way i did above, the dollars didn't enter until the last step, when they also cancelled

Hope that helps....David

 

[rahul.goyl]

David, you are a Legend.

You explained it so well & clear my all concepts pertaining to Beta & Incremental VaR. Thanks a lot for solving that out.

Thanks
Rahul