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Professor Jorion, chapter 7

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Hi everybody,

I am confused about a concept in chapter 7 of Jorion. In study notes (topic 52, 2012 edition), there is a sentence "both a correlation of zero and one will place a lower and upper bound on the portfolio total VaR". I can understand one will act as upper bound. How about -1, which I think will be lower bound instead of zero? Thanks!


Well-Known Member
-2*VaR1*VaR2<0 given that VaR1,VaR2>0
so its true that VaR1-VaR2<sqrt(VaR1^2+VaR2^2)
or that VaRp(rho=-1)<VaRp(rho=0).
the sentence should not be a general one but the author wants to highlight a case that correlation of zero provides lower limit and corr. of 1 will provide a upper limit to portfolio VaR.


David Harper CFA FRM

David Harper CFA FRM
Staff member
Interesting. I do not agree with (and am not aware where Jorion says) that zero correlation is a lower bound (in the two-asset mean variance VaR).

The general form is VaR(P) = SQRT[VaR(1)^2 + VaR(2)^2 + 2*VaR(1)*VaR(2)*correlation]
  • VaR(1)^2 and VaR(2)^2 are always positive (increasing portfolio VaR),
  • The directional impact on Portfolio VaR therefore depends on the third term: +2*VaR(1)*VaR(2)*correlation
    • Where the product of weights is positive (e.g., long + long, short + short), lower correlation lowers portfolio VaR with -1.0 as lower bound
    • Where the product of weights is negative (e.g., long + short, higher correlation decreases portfolio VaR with +1.0 as lower bound on portfolio VaR
  • In this way
    • In a (typical) long + long portfolio, bounds are correlation -1.0 (lower) and +1.0 (upper)
    • In a hedged portfolio, long + short, bounds are correlation +1.0 (lower) and -1.0 (upper); i.e., i don't see where 0 rho is ever a bound.

      In the formula, this is because the short position has negative VaR(2) owing to its negative weight.
Jorion's interesting point (p 165, Chap 7) is, i think, that the hedged portfolio is counter-intuitive. Mathematically, again, it's because the short is represented by a negative weight.

I tested this in our XLS, see https://www.dropbox.com/s/2islu84rn567e69/0827_correlations.xlsx

And here is the chart. This plots 99% 2-asset (mean-variance) portfolio VaR when:
  • 200% weight in Asset (A) with vol = 10%,
  • -100% weight in Asset (B) with vol 20%; i.e., long/short 200/100.
  • Note: perfect hedge occurs at correlation = +1.0
Pg 18 of the study note refers to example on Jorion's 2 currency portfolio. Tha variance of the portilio is mentioned as 0.00271, however i am getting as 0.00269. Can you confirm the accuracy of the calculation? Also Pg. 19 shows the Dollar Variance as $24,400, i want to know how is this derived?

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Ruchir,

Your 0.00269 appears uses rounded weights of 0.67 and 0.33, which are more precisely 2/3 and 1/3 (e.g., 0.66667). The exact variance is given by:
(2000/3000)^2*5%^2 + (1000/3000)^2*12%^2 + (2000/3000)*(1000/3000)*5%*12%*0 = 0.0027111...

Dollar variance = 0.0027111 * 3,000^2, or maybe more intuitively = 2,000^2*5%^2 + 1,000^2*12%^2 + 0 = $24,400; i.e., dollar weights rather than percentages
Dollar variance can be used to compute beta (Jorion's beta after 7.19, page 167):
Beta = W * Dollar Covariance (i,P) / Dollar Variance(P)

Here is the underlying XLS, which includes the variants of beta/marginal VaR that depend on "dollar variance" and "dollar covariance:"

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @stephenjohn Yes, I apologize: it is a mistake in the calculation, see previous comment here at https://www.bionicturtle.com/forum/...ormula-for-2-asset-question.10481/#post-50339 (snippet below)
... further, my fresh calculation is exactly the same as yours (=0.0432×3+0.0867×3=0.3897), see XLS at https://www.dropbox.com/s/osftut2k9vdozbf/0517-t8-jorion-matrix.xlsx?dl=0
@kevolution I apologize but our matrix math is incorrect here. Of course you are correct the two methods should produce the same result: after all, your formula is the reduced version of the matrix approach for the special case of only two assets. I do agree with your result, the portfolio variance (in returns^2) should be 0.01083. I'm not sure how we mistakenly got 0.04 in the matrix math. Our mistake in σ^2(p) = x'*cov()*x is that you have to post-multiply then pre-multiply. I entered into Excel super quickly here at https://www.dropbox.com/s/osftut2k9vdozbf/0517-t8-jorion-matrix.xlsx?dl=0 See below, the first step (1. post multiply) returns the column vector in purple [0.0432, 0.0867]; then (step 2) the pre-multiply returns dollar variance of 0.3897, which matches return (%) StdDev of 10.40% such that 10.404%^2 = 0.0108. Thank you. cc @Nicole Seaman
Please note that Deepa is currently revising this Jorion note (all four P2 Jorion notes, in fact) with much improvement, and this will be fixed in a better P2.R77 Study Note to be published ASAP. Thanks!