Aggregating VAR across portfolio and firm

[sleepybird]

Hi David,
If you're given 1-year VARs of a firm for market risk, operational risk, and credit risk (they are uncorrelated to each other and all calculated at same significance level), can you simply sum them to get the overall 1-year firm VAR?

For a portfolio VAR (referring to Q#11 on page 285 of FRM curriculum Valuation and Risk Management), we use (Portfolio VAR)^2 = (Stock VAR)^2 + (Bond VAR)^2, assuming no correlation between bond and stock. Should we apply same formula to calculate the firm VAR described above?

With regard to the formula for portfolio VAR, if there is correlation between bond and stock, say 0.3, how can we incorporate this into the formula?

Thanks
Sleepybird

 

[David Harper CFA FRM]

Hi sleepybird,

(I don't understand your reference to page 285? you do not appear to reference the FRM handbook ...)

1. No! Summing the VaRs implicitly assumes they are PERFECTLY correlated. If you are explicitly told the risks are uncorrelated (ZERO correlation) then you can use VAR^2 = VaR^2 + CVaR^2 + OpRiskVaR^2, but this is less than summing. But the question must tell you "zero correlation" or "uncorrelated". For example, Basel implicitly assumes perfect correlation by adding the three charges.

2. The general form is SumVaR^2 = VaR1^2 + VaR^2 + 2*VaR1*VaR2*correlation, such that:
if ZERO correlation: SumVaR^2 = VaR1^2 + VaR^2, or
if PERFECT correlation: SumVaR^2 = VaR1^2 + VaR^2 + 2*VaR1*VaR2 = (VaR1+VaR2) ^2 --> SumVaR = VaR1 + VaR2
(... but this applies only in the limiting case of mean-variance where VaRs are unrealistically normal)

If correlation = 0.3, then a good question should remind this is unrealistically where VaRs are normal, but then:
SumVaR^2 = VaR1^2 + VaR^2 + 2*VaR1*VaR2*0.3

Thanks,

 

[sleepybird]

David, thanks.
I was referring to Q#11 of the sample exam questions at the end of Valuation and Risk Models book.

 

[sleepybird]

Hi David,
This might be a stupid question but why do we ignore the weights of the stocks and bonds when aggregating portfolio VAR in the formula above? Thanks.

 

[David Harper CFA FRM]

Hi sleepybird,

We don't need to, but position ($) = Portfolio Value($P) * Weight (%), so the weights are "embedded:"

From http://forum.bionicturtle.com/threads/portfolio-var.4846/

VaR(P$) = W(P$)*deviate*SQRT[w(a%)^2*sigma(a%)^2 + w(b%)^2*sigma(b%)^2 + 2*w(a%)*w(b%)*COV(a,b)],
VaR(P)^2 = W(P$)^2*deviate^2*[w(a%)^2*sigma(a)^2 + w(b%)^2*sigma(b)^2 + 2*w(a%)*w(b%)*COV(a,b),
VaR(P)^2 = [W(P$)^2*deviate^2*w(a%)^2*sigma(a)^2)] + (W(P$)^2*deviate^2*w(b%)^2*sigma(b)^2) + W(P$)^2*deviate^2*2*w(a%)*w(b%)*COV(a,b),

as W($P)^2*deviate^2*w(a%)^2*sigma(a)^2 = [W(P)*deviate*w(a%)*sigma(a)]^2, and W($P)*w(a%) = w($a):
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + W(P$)^2*deviate^2*2*w(a%)*w(b%)*COV(a,b),
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + [W(P$)*w(a%)*deviate] * [W(P$)*w(b%)*deviate]*2*COV(a,b); as COV = sigma(a)*sigma(b)*correlation(a,b):
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + [W(P$)*w(a%)*deviate*sigma(a)] * [W(P$)*w(b%)*deviate*sigma(b)]*2*correlation(a,b),
VaR(P)^2 = VaR(a$)^2 + VaR(b$)^2 + 2*VaR(a$) * VaR(B$) * correlation(a,b),
VaR(P$) = SQRT[VaR(a$)^2 + VaR(b$)^2 + 2*VaR(a$) * VaR(B$) * correlation(a,b)]

Thanks, David

 

[sleepybird]

Hi David,
Thanks. That makes sense, but I think only if we are aggregating the VAR in $ terms?

There's a question in the 2010 GARP practice exam where we were given the volatilitities of 5% and 12% for Bond A ($25M) and Bond B($75M), respectively, with correlation of 0.25. We were then asked to calculate the gain from diversification for a VAR estimated at the 95% level for the next 10 days.

The answer key calculates the following:
Undiversified VAR:
1.645*5%*SQRT(10/250) + 1.645*12%*SQRT(10/250)=3.723%

Here's we're not multiplying the portfolio value. Shouldn't we apply the weights here?

The answer key goes on to calculate below
Diversified VAR:
SQRT[(0.25)^2*(5%)^2 + (0.75)^2*(12%)^2+2(0.25)(0.75)(5%)(12%)]=0.09308

Difference is 0.283 or $283,000.

Thanks.
Sleepybird

 

[David Harper CFA FRM]

Hi sleepybird,

(Why aren't yield volatilities multiplied by duration? Bond VaR wants yield volatility * modified duration, otherwise this is just a VaR of the yield volatilities not the bond values? in any case ... we can assume these are price volatilities [sic])

as above, gets to the same place, i think?
Individual bond A VaR = 5%*sqrt(10/250) * $25 * 1.645 = $0.41125,
Individual bond B VaR = 12%*sqrt(10/250) * $75 * 1.645 = $2.96, then:

SQRT[$0.41125^2 + $2.96^2 + 2* $0.41125* $2.96 * 0.25] = 3.089; i.e., 3.372 - 3.089 = ~$283K

Thanks,

 

[sleepybird]

Hi David,
Sorry this is still not very clear to me.

Diversified SumVaR^2 = VaR1^2 + VaR^2 + 2*VaR1*VaR2*0.25 = 0.41125^2 + 2.96^2 + 2*0.41125*2.96*0.25 --> we get DIVERSIFIED VAR of $3.089 as you calculated above.

Then shouldn't undiversified SumVaR^2 = VaR1^2 + VaR^2, i.e., dropping out the last term. In this case undiversified VAR^2 = 0.41125^2 + 2.96^2 --> we get UNDIVERSIFIED VAR of $2.989.

Then the difference is $0.1.

Why did you subtract $3.089 from $3.372 rather than $2.989? i.e., why the UNDIVERSIFIED VAR is calculated 1.645*5%*SQRT(10/250) + 1.645*12%*SQRT(10/250)=3.3723%*$100=3.372 rather than SQRT(0.41125^2 + 2.96^2)=2.989?


And sorry, few corrections to my original post above:
1. Yes, those are yield volatilties
2. 1.645*5%*SQRT(10/250) + 1.645*12%*SQRT(10/250)=3.3723% not 3.723%
3. SQRT[(0.25)^2*(5%)^2 + (0.75)^2*(12%)^2+2(0.25)(0.75)(5%)(12%)(0.25)]=0.093908 <--missing the correlation 0.25 and final answer 0.093908 rather than 0.09308.

Thanks.
Sleepybird

 

[David Harper CFA FRM]

Hi Sleepbird,

I am using millions, I think we agree that, if correlation = 0.25, diversified VaR = $3.089 million.

But undiversified VaR assumes correlation = 1.0. Your $2.989 implicitly assumes ZERO correlation and is therefore a diversified VaR under (an unstated) assumption of zero correlation:

• General form: VaR(P)^2 = VaR(A)^2 + VaR(B)^2 + 2*VaR(A)*VaR(B)*correlation.
• If correlation = zero, VaR(P)^2 = VaR(A)^2 + VaR(B)^2 + 2*VaR(A)*VaR(B)*0 = VaR(A)^2 + VaR(B)^2 --> VaR(P) = SQRT[ VaR(A)^2 + VaR(B)^2]. In this example, under zero correlation, diversified VaR = 2.989
• If correlation = 1.0 (i.e., "undiversified VaR"): VaR(P)^2 = VaR(A)^2 + VaR(B)^2 + 2*VaR(A)*VaR(B)*1.0 = [VaR(A)^2 + VaR(B)^2]^2 --> VaR(P) = VaR(A) + VaR(B).
  In this example, undiversified VaR = $3.372 regardless of the correlation, b/c undiversified VaR does not give credit for imperfect correlation.