What coherent risk measures are, why some people think coherence is important, and why I don’t.

## The rules

A risk measure is considered to be coherent if it satisfies some mathematical properties. They are formulated in various ways — here is one set:

- (
**monotonicity**) If the value of portfolio X is always bigger than the value of portfolio Y, then the risk of X is less than or equal to the risk of Y.**Update**: this should be rephrased: If the return of portfolio X is always bigger than the return of portfolio Y, then the risk of X is less than or equal to the risk of Y; I think there needs to be a condition that the two portfolios have equal value as well. - (
**cash invariance**) The risk of the portfolio that is X without some amount of cash is equal to the risk of X plus the amount of cash. [ risk(X - m) = risk(X) + m ] - (
**homogeneity**) The risk of some multiple of X is the multiple times the risk of X. [ risk(c * X) = c * risk(X) ] - (
**sub-additivity**) The risk of the portfolio of X and Y together is less than or equal to the risk of X plus the risk of Y. [ risk(X+Y) <= risk(X) + risk(Y) ]

## Why make rules?

I see three main reasons for creating something like “coherent risk”:

- it makes the task clearer
- mathematics is fun
- it stokes the academic paper machine

Clarity is, of course, a good thing — I have no problems with that.

I’m all for having fun too.

The main impact I see, though, is that it provides an opportunity for lots more academic papers to be written. It’s not that I’m against papers, but I’m dubious that the increment in papers due to risk coherence has been a net positive (counting opportunity cost).

## Quality of the rules

#### monotonicity

Fair enough.

#### cash invariance

The key effect of the cash invariance axiom is to make sure that the risk measure is in terms of monetary units.

This is a restriction on what is meant by “risk measure”. In some cases the restriction is undoubtedly useful.

#### homogeneity

The third — homogeneity — is **wrong**. I call this the Amaranth axiom. It turns out that being all of one side of a whole market is really risky.

You might counter that this is — for most entities — a hypothetical issue. I don’t think so. Fund managers habitually restrict the size of equity positions based on the number of days of average volume. The market impact of trading is known to be worse than linear.

An implication of this axiom is that the risk of having nothing is zero.

This looks like it violates the monotonicity axiom: Let X be a long-only portfolio and let Y be the empty portfolio. X will always have positive value, bigger than the value of Y. The Expected Shortfall of X is positive which is **not** less than the Expected Shortfall of Y. Doesn’t this violate the monotonicity axiom?

**Update**: The argument here is wrong in that it is in terms of value and should be in terms of returns. The empty portfolio always has zero return, the long-only portfolio will sometimes have negative returns.

The homogeneity axiom exists — I think — because that is what our risk measures do. It seems hard to make a general purpose risk measure that violates this axiom — but perhaps it is worth a try.

#### sub-additivity

The idea of this axiom is that there is a diversification effect.

This is the axiom where all the action has been. The one instance of a real risk measure that is not coherent is Value at Risk, and it is not coherent because of this axiom.

Apparently VaR does follow the axiom for elliptical distributions. This means VaR is likely to obey the rule for portfolios that don’t include options (but there are examples of non-sub-additivity with bond portfolios).

## Other voices

A reasonably accessible discussion of coherence with a positive spin is by Glenn Meyers.

## Summary

There are reasons to not be keen on Value at Risk. However, its failure at being coherent is not a very compelling one in my opinion.

I don’t see much practical importance of coherence.

## Questions

Where have I gone wrong?

Are there other real-life risk measures that are cash-invariant but not coherent?

## Epilogue

*Add a touch of mystique where the writing gets weak*

*Break up coherence with a cut-cut-cut up technique*

from “Folklore” by James

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