Monotonicity property of coherent risk measures.

Hello friends, A basic query here, thanks in advance to all of you. Can any of you please check this out and let me know the rationale behind Monotonicity property ? I mean if a portfolio is greater than other, how does that imply any effect on the correlations? Or is it something else that I am overlooking and misunderstanding? Am I understanding the property correctly? Thanks!


higher return = higher risk, that is probably the underpinning.

think of Z as loss, if in any scenario portfolio 2 results in less loss than portfolio 1, it makes sense portfolio 2 is less risky.

Thanks CSK, Z = size of the portfolio (Or is it the return?) Now I agree that the risk is more if the return is more under no-arbitrage situation. But, since correlations are not the only factor affecting the risk of the portfolio (the individual risks also matter) and here we do NOT know the background (which portfolios etc.), is it logical to justify the theory. Could you make an effort to maybe put it in other words. Thanks.

Size of the portfolio is not a random variable. Return is.

Z is a random variable, usually denotes loss from a portfolio

actuaryalfred Wrote: ------------------------------------------------------- > Z is a random variable, usually denotes loss from > a portfolio i didnt know that :(, had to reason something out, Z for me means all integers :slight_smile:

if you skim through the original famous coherent risk paper, Z is not a random variable for return, it’s the net-worth of the portfolio. But nowadays people sometimes use the loss instead of the net-worth, which is why Monotonicity in the wiki is not the same as the original paper

How can it be a net-worth of a portfolio if it said to be a random function?

it’s the net-worth 1 period later, so your portfolio value in say 10 days is going to be a random variable right?

@ Actuaryalfred, Perfect. Z is the loss. But, is risk = correlations? Higher correlations mean more risk, ceterus perebus. But, are we assuming that everything else is the same except the correlations? Because that is a very unrealistic assumption. Lets assume there is a portfolio of small cap penny stocks, comprising companies from totally different sectors. The correlation is low, but risk is high. Compare this to a portfolio of Blue chip stocks which are from the same sector. Risk is lower, but correlation is high. If loss is more, then risk is high, but can the same be said about correlations?

actuaryalfred Wrote: ------------------------------------------------------- > it’s the net-worth 1 period later, so your > portfolio value in say 10 days is going to be a > random variable right? yeah, i just didnt exactly understand what do you mean by net-worth, now it is clear

i would say risk =/= correlation in general, but it seems to be true that high correlation implies high risk if you can keep everything else unchanged (think of normal assumption and two stocks)

Okay, on second thought, I think it does make sense because here we are talking of a specific class called “coherent risk measures” whose property is monotonicity. This is not true in general. Now it makes perfect sense.

Does it? That paper is available on the internet, and the first thing is that the definition of a risk measure in that paper excludes correlation because risk measures have the same units as portfolio net worth. For example, VaR is a risk measure (though not coherent without some added assumptions) but average correlation is not. That paper is turgid, btw, and proposing a set of nice looking axioms to describe risk measures makes no sense to me at all. In fact, I think even calling those things axioms is BS. They aren’t axioms; they are “desirable properties” or some s%^t and the original paper makes it pretty obvious that they aren’t so desirable.