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Demonstrate that VaR violates the requirements of a coherent risk measure.

FRM® Part I · Valuation and Risk Models · Measures of Financial Risk · FRM1.VR.1.f

The idea

VaR is not coherent because it is not guaranteed to be subadditive: in some cases the VaR of a combined portfolio exceeds the sum of the stand-alone VaRs. The standard demonstration uses two independent bonds, each losing USD 10 million with probability 4 percent and nothing otherwise. At 95 percent, each bond has a VaR of zero, since a 4 percent chance is inside the allowed 5 percent tail. The combined portfolio has a 1 − 0.96² = 7.84 percent chance of some loss, so its 95 percent VaR is USD 10 million, which exceeds 0 + 0. VaR is subadditive for elliptical distributions such as the normal. With zero means, the combined VaR is as follows. Combined = √(VaRA² + VaRB² + 2 × ρ × VaRA × VaRB). Here ρ is the correlation, which cannot exceed 1. In the failing cases a firm could cut reported risk by splitting into units. Exam point: the failure arises with skewed, discrete or fat-tailed losses.

Formulas

VaR of two positions under normality

VaR(A+B) = √(VaRA² + VaRB² + 2 × ρ × VaRA × VaRB)

VaRA, VaRB
the stand-alone VaRs of the two positions
ρ
the correlation between their returns

Holds for elliptical distributions such as the joint normal, where VaR is subadditive and can never exceed VaRA + VaRB. With skewed, discrete or fat-tailed losses there is no such guarantee.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which property of a coherent risk measure does VaR fail to satisfy in general?

  1. MonotonicityWhy not: VaR satisfies monotonicity: if one portfolio always loses at least as much as another, its loss quantile is at least as large.
  2. Subadditivity - correct
  3. Positive homogeneityWhy not: VaR satisfies positive homogeneity: scaling a position by k scales every loss, and so the quantile, by k.
  4. Translation invarianceWhy not: VaR satisfies translation invariance: adding k of riskless cash shifts the loss quantile down by k.

VaR meets monotonicity, positive homogeneity and translation invariance, but the VaR of a combined portfolio can exceed the sum of the stand-alone VaRs, violating subadditivity.

A harder one

Two independent bonds each lose USD 10 million with probability 4% and lose nothing otherwise. The 95% VaR of each bond alone is zero. The 95% VaR of a portfolio holding both bonds is:

  1. USD 0Why not: Assumes the portfolio VaR equals the sum of the stand-alone VaRs (0 + 0), which is exactly what subadditivity-violating cases break.
  2. USD 0.8 millionWhy not: Reports the expected loss of the portfolio, 2 x 0.04 x 10, rather than a quantile.
  3. USD 10 million - correct
  4. USD 20 millionWhy not: Takes the worst case in which both bonds default, although that has only 0.16% probability.

P(no loss) = 0.96² = 92.16%, so P(some loss) = 7.84%, which exceeds 5%. The 95% VaR is therefore the smallest loss with at most 5% exceedance, USD 10 million (P(loss = 20) is only 0.16%). 10 > 0 + 0, so VaR is not subadditive.

The hardest tier

A portfolio holds ten independent bonds. Each loses USD 1 million with probability 2% and loses nothing otherwise. The 95% VaR of each bond alone is zero. Which statement about the 95% VaR of the ten-bond portfolio is correct?

  1. It is USD 0, equal to the sum of the stand-alone VaRs, so subadditivity holds.Why not: Assumes the portfolio VaR is zero; P(at least one default) is 1 - 0.98^10 = 18.3%, far above 5%.
  2. It is USD 1 million, above the USD 0 sum of stand-alone VaRs, which violates subadditivity. - correct
  3. It is USD 2 million, because two defaults are the most likely loss event.Why not: Reads two defaults as typical; P(at most one default) is already 98.4%, which covers the 95% level.
  4. It is USD 0.2 million, the expected loss, which is below the sum of stand-alone VaRs.Why not: Uses the expected loss of 10 x 0.02 x 1 and confuses it with a quantile; a loss of 0.2 million cannot occur.

P(0 defaults) = 0.98^10 = 81.7%, P(at most 1) = 98.4%. The 95% level is first reached at one default, so VaR = USD 1 million. The stand-alone VaRs sum to 10 x 0 = 0, so 1 > 0 and VaR is not subadditive.

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