Coherence of Risk Measures - Monotonicity

[Exam_Machine]

Hello all

I simply dont understand the property of monotonicity as explained in chapter 7. Firstly, the equality signs look decidedly out of kilter. Then the explanation "If a portfolio is always worth more than another, it cannot be riskier" muddies the waters further.

Monotonicity to me means that the equality signs should be facing the same direction. This to me implies that larger losses should attract larger capital requirements, as we would expect. This is before any diversification benefit. The explanation in the notes seems to imply that larger portfolios (perhaps by virtue of being bigger and more diverse?) should be less risky than smaller ones. If this is what is meant by the notes, it makes sense, but its basically repeating the thinking behind the sub-additivity property.

I cant quite remember, but I did read books by Brehm (Guy carpenter) and one titled Actuarial Theory for Dependent Risks, and i felt much mroe comfortable with the explaanation there. their explanantion had the inequality signings facing the same way. Any thoughts? Tutors?

Or am i the slowest of all budding actuaries sitting SA3 who doest get this?

 

[Duncan Brydon]

Hi Exam_Machine

I agree with you that a monotonic function should preserve the order of the inequalities. We do point out in the ActEd Course Notes that the function shown in the Core Reading is monotonically decreasing (and hence order reversing) rather than monotonic.

When some people define coherence, they do (as you suggest) consider the random variable X to be the loss (over some time period) sustained by a portfolio. I think the names given to the coherence conditions come from this approach.

If holding portfolio A will result (with probability 1) in a greater loss than holding portfolio B, then there is greater risk associated with portfolio A than with portfolio B. That is all that the property of monotonicity is saying.

Other people define coherence in terms of a random variable X representing the future value of the portfolio. This second approach is being taken in the Core Reading and naturally results in changes in the direction of inequality signs. However, it would appear that those adopting this approach choose to retain the same names for the four coherence conditions as those thinking about coherence in terms of losses, leading to your quite logical confusion over monotonicity with reversed inequalities.

(I have also noticed that the definition of translational invariance in the Core Reading should show alpha being subtracted from the RHS of the equation rather than added, given that we are considering X to be the value of the portfolio rather than a loss random variable. We shall add this to the corrections document.)

I hope this helps.

Duncan