Mar 21

Illustration of VaR’s failure to meet coherence

by David Harper, CFA, FRM, CIPM

FRM |

Paul Wilmott's Chapter 22 on value at risk (VaR) is new to the 2008 FRM. He lists the criteria for a coherent risk measure. The argument is that a good risk metric should meet these criteria and they include:

  • Sub-additivity: A portfolio should not be riskier than the sum of its constituents. Or, put another way, there should be at a minimum zero diversification benefits, but there should not be an anti-diversification penalty
  • Monotonicity: Higher expected value, all other things being equal, should lower risk. Note: I think the assigned text has erroneously transposed this formula. If X<Y, then rho(Y)<rho(X).
  • Positive Homogeneity: Scaling the portfolio should scale the risk. This reminds us that leverage is not a risk factor per se, rather is amplifies or magnifies risk.
  • Translation Equivalence: Adding cash (a constant) should reduce portfolio risk by the constant.

 

subadditive2

Sub-additivity

These coherence criteria may seem overly technical, but sub-additivity is arguably important. And VaR fails specifically the sub-additivity criteria (unless we insist on normal distributions, which we know isn't realistic). This is why Kevin Dowd calls VaR "merely a quintile" and not a proper risk measure!

Here is an example that I hope crystallizes the sub-additivity problem. Please note: I got the idea from Kevin Dowd's Measuring Market Risk, 2nd Edition (Chapter 16 on Model Risk is the only assigned chapter for the 2008 FRM).

Below is a simple EditGrid spreadsheet (File > Export As > Excel will open into MS Excel) but also you can click here to open a new EditGrid.

  • Assume a $100 face value bond defaults with 2% probability of default (PD). It does not default therefore with 98% (1-2%) probability. The 95% value at risk (VaR) for this single bond is $0.
  • Now combine the two bonds into a portfolio. Both are the same; i.e., 2% PD. Further, assume independence (default correlation = 0). This portfolio's 95% VaR is also $0.
  • Now combine three bonds and the 95% VaR is $100. In other words, at 95% confidence, we expect one of the bonds to default. That's because the odds of no defaults = 98%^3 = 94.1% which is less than 95%. Cumulative, with three bonds, we can no longer have 95% confidence that none will default (whereas with one bond, we could be 95% confident that it would not default).

You can probably see the problem. If a single bond has a zero VaR, then a portfolio of three bonds should not be riskier than adding the risk of each bond separately. That is, risk(bond) + risk(bond) + risk(bond) should not be less than risk (bond + bond + bond). If that is violated, VaR could penalize diversification. Yet here we have illustrated a case where 0 + 0 + 0 < $100. Here is the spreadsheet:

9/9/08 Update: Per this thread, I just added expected shortfall (ES) below the VaR to illustrate how ES satisfies sub-additivity. Note that ES(3 bonds) = 1.024 < 3* ES(1 bond) = 0.4 * 3 = 1.2.

Comments

  1. Jim Basgier 22 Mar 2008

    Shouldn’t the calculations be at the confidence level .95 or the inverse .05?  If so, the VaR of bond 1 = 5; bond 2 = 5; bond 3 = 95.  The total of 105 exceeds 100 and there is no violation of subadditive requirement.  I am sure the calculations must be more complex than in either of our examples.

  2. David Harper, CFA, FRM, CIPM 22 Mar 2008

    Hi Jim,

    95% is used, it’s the yellow input. The =CRITBINOM() functions performs the VaR quantile function. For one bond, if the PD is 2%, then the 95% VaR is 0 because it’s a discrete variable. They key here is really the behavior of discrete variables. If we move the confidence up to, say, 99.9%, then 95% VaR = 100% (or $100) and your idea applies. i agree with you that there probably are more complex approaches to estimating the quantile but i still think this satisfies as an exception…

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