Question:

Assume a portfolio of two equally-weighted assets that happen to also have equivalent VaRs (value of risk) of $100 each; i.e., VaR(A) = $100 and VaR(B) = $100.

(i) If VaR(A + B) >  VaR(A) + VaR(B), what risk measure criteria is violated?
(ii) If we assume normality, and the correlation between A & B is 1.0, what is VaR (A+B)? (an unfair question at this junction, not yet covered)
(iii) If we assume normality, and the correlation between A & B is zero, what is VaR (A+B)? (also unfair)

Answer:
(i) In this case, the VaR is not sub-additive, which renders the VaR not coherent.
(ii) If correlation = 1.0, then portfolio VaR = $100 + $100 = $200
(iii) if correlation = zero (i.e., independence), portfolio VaR = SQRT[100^2 + 100^2] = $141 (note diversification benefits)

Hi David,

How can VAR(A+B) > VAR A + VAR B because even if the corr is 1 it is max to VAR A + VAR B ? Request you to kindly explain.

Rgrds,
OM

Hi OM,

VaR (A+B) <= VaR(A) + VaR(B) is necessarily true only under Delta Normal VaR; specifically, when the returns are elliptically distributed which includes normally distributed (see Jorion Chapter 7, p 164).

Otherwise, VaR is not necessarily sub-additive and it’s possible that VaR(A+B) > VaR(A) + VaR(B)

Here is a practice question that exploits the lack of normality: option payoffs are non-normal so their VaR is not sub-additive.

David

Hi David,

Just wondering, what would VaR (A+B) be if correlation between A & B is -1.

Thanks!

Hi FRM Wannabe,

Althought Jorion doesn’t give that per se, this would *almost* be a fair question given our toolset, good thought:

portfolio VaR = SQRT[VaR1^2 + VaR2^2 - 2*VaR1*VaR2]
portfolio VaR = SQRT[(VaR1 - VaR2)^2]
portfolio VaR = VaR1 - VaR2, if correlation is -1

keep in mind:
variance (a+b) = var(a) + var (b) + cov(a,b)
variance (a-b) = var(a) + var (b) - cov(a,b)

David

hi david,

i am not very clear with the formulas for var aggregation on a portfolio. That is if there are 2 assets in a portfolio with var1 and var2 how do we aggregate var up for the portfolio ....

Hi Surajm,

Can I refer you to this learning spreadsheet:
http://www.bionicturtle.com/premium/spreadsheet/4.a.1_two_asset_var_relative_vs_absolute/

Note we use the variance formula: variance (a+b) = var(a) + var (b) + cov(a,b)
this gives the 2-asset variance, so we take the square root for the 2-asset volatilty. Then we can mulitiply by the normal deviate (e.g., 1.65 @ 95% confidence)

Thanks, David

Hi david,

I understand the VAR(A+B) formula but what instead of individual variances we are given individual Value at Risk (VaR) for the assets. Is there a forumla for VaR(A+B)  ?

Yes, following from above:

SQRT[VAR1^2 + VAR2^2 + 2*VAR1*VAR2*correlation]
VAR(A+B) = SQRT[VAR(A)^2 + VAR(B)^2 + 2 *VAR(A)*VAR(B)*correlation]

David