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Coherent risk measure


New Member

Apologies if I am just not grasping something obvious but I have a question in relation to an apparent inconsistency between "Monotonicity" and "Positive Homogeneity". I may be just misunderstanding what the different symbols represent. I will use "p" for "rho" below and "L" for "lambda". L is just a scalar whereas p is the risk measure (and can be thought of as the cash needed to be added to make the portfolio acceptable). I use "<" below but it means less-than-or-equal-to.

Positive homogeneity: for L>0, p(L*X) = L*p(X) . If we double the portfolio, we double the amount of risk.
Monotonicity: If X<Y for all possible future states then p(Y) < p(X). So we have a lower risk for the portfolio that will have the greater value regardless of the future state of the system.

OK, so my question is, what if we have a two portfolios X and Y, which X is exactly double the Y portfolio. I.e X=2*Y . Then by monotonicity Y <X and hence p(X)<p(Y). But by positive homogeneity, p(X) = p(2Y) = 2p(Y).

So we have 2p(Y) < p(Y) ???

I am obviously just confused with something.

Thanks in advance

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Morrin,

Great observation, I hadn't noticed the discrepancy. Before Dowd was assigned (years ago), Wilmott was assigned for VaR and he gave monotonicity as:
if X < Y, then rho (X) < rho (Y).

I happen to be currently reviewing "Loss Models" (http://www.amazon.com/Loss-Models-Decisions-Proability-Statistics/dp/0470487437; to assist in our joint course with pristine on OpRisk) and the authors here, like Wilmott did, give monotonicity as if X < Y, rho (X) < rho (Y). This version, of course, satisfies your test. (Mathematically, I cannot find a way for you to be wrong)

I think the difference is the definition of X and Y. The "Loss Models" text says "it is useful to think of the random variables X and Y as the loss random variables for two divisions and X + Y as the loss random variable for the entity created by combining the two key divisions" ... and importantly "monotonicity means that if one risk always has greater losses than another risk, the risk measure should always be greater." (Klugman et al)

... while Dowd (FRM assigned; he gives your expression above, that I copied of course) says "Let X and Y represent any two portfolios’ P/L (or future values, or more loosely, the two portfolios themselves)" and, in the footnote, "Monotonicity means that a random cash flow or future value Y that is always greater than X should have a lower risk: this makes sense, because it means that less has to be added to Y than to X to make it acceptable, and the amount to be added is the risk measure." (FRM assigned Dowd)

It seems to me Dowd has inverted the interpretation of 'X' and 'Y' to render the inconsistency. . What i mean is, Dowd seems to mean something like: if X < Y (as noted), then E[FV(X)] < E[FV(Y)] --> absolute loss (X) > absolute loss (Y) --> X > Y (as positive loss variables).
... It's a little off topic, but Dowd uses absolute VaR = -drift + volatility*sigma. So, it's almost like his (X < Y) is when the drift of Y is higher than X.

The best i can figure, then, is either (i) Dowd (page 33) just has the formula incorrect or (ii) not unrelated, the meaning of random variable (X) changes its definition, strictly speaking, from monotonicity to positive homogeneity in such a way that under Dowd's monotonicity it isn't really a loss variable per se but an input into (FV) the loss variable. Sorry i don't have a cleaner path....



Hi David,

This may be a very simple question but at the risk of displaying my ignorance...

Dowd defines VaR as P[1-e^(u - standard deviation * Z value)] I just want to make sure I understand this correctly. It's the multiplication of portfolio value with cumulative probability of default. I get the sense that this is similar to what's given in the chapter on Poisson where PD was given as (1-e^lambda*time), although I don't know how the portions in bold relate to one another. Question 1, am I generally correct in interpreting this formula, and how do the bold parts relate to one another?

Question 2: Dowd then assumes mu to be 0. Why's that?

Many thanks

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @southeuro

Question (1): I'm not aware of such a link between VaR and Poisson-based "waiting time" except they share the elegance mathematical properties. Dowd's P[1-e^(u - standard deviation * Z value)] is the analog to our more familiar P*[-u + standard deviation*Z] but, rather than assuming simple (arithmetic) returns, the returns are assumed to be (much like Black Scholes) continuously compounded such that future P(t) has a lognormal distribution. So, this alternative Dowd VaR is a really a consequence of using returns = ln(r[t]/r[t-1]), and assuming they are normal, rather than r[t]/r[t-1] - 1.

Question (2): there is a lot of forum discussion on this (search "drift" and "absolute vs relative var") but, in general, if returns are daily, it's common to round the drift (mu) down to zero, esp in market risk. Mu is analagous to EL in credit risk, but in market risk, drift is expected to be positive, so zero mu makes the absolute VaR more conservative. In daily returns, it's typical to assume mu = 0. Even scaled over ten days. The issue becomes longer time frame and hence the distinction between absolute VaR (i.e., non zero mu) and relative VaR (zero mu). I hope that helps!


Well-Known Member
Try to read Prt2 and Prt3, It seems that Dowd was right after all, he was referring to Artzner I think.

Well, I was confounded by two not unrelated queries here,
If Positive Homogeneity states that p(n*X)=n*p(X), then there was a problem with Subadditivity as well p(X+X)<=p(X)+p(X), that was resolved by the below link. Positive homogeneity refers to identical portfolios I think while Subadditivity and Monotonicity refers to different portfolios, which I think resolves the question of RM1.

If Monotonicity states that if Y<X, then p(X)<p(Y), that I thought to be somewhat contradictory as higher the value of the portfolio, higher returns are generated and the risk should be more, extrapolating from David's argument and the below site, it seems that David's conjecture is correct and Dowd is referring to X and Y as loss instances with Y having a greater frequency of losses and thus having higher risk and capital/collateral requirements. The site also gives a new meaning to it, saying "We interpret this to mean that if the portfolio (which is our eventual goal) is decreased from X to Y, then our Risk won't increase." My brain ran out after that, can someone explain the previous statement?

I thought that this site might be of interest: http://www.financialwisdomforum.org/gummy-stuff/CoherentRisk2.htm

I am sorry if its a basic question

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi @QuantMan2318 I cannot interpret that finally sentence, which appears by itself: "We interpret this to mean that if the portfolio (which is our eventual goal) is decreased from X to Y, then our Risk won't increase." Sorry :(

Dowd actually "X and Y represent any two portfolios' P/L (or future values, or more loosely, the two portfolios themselves)". I am not certain, but one way I have attempted to conceptualize this (implied above) is to rely on Dowd's (absolute) VaR as the risk measure. Consider two assets, X and Y, each with price of $100 and volatility of 10%. Let the expected return of X = 5%, but the expected return of Y = 6%. In this way, E(Y) > E(X); which, as somebody else has pointed out elsewhere, is flawed, because the condition is Y > X not E(Y) > E(Y). But it satisfies me as a concept. Now consider the absolute VaR of each:
  • 95% aVaR(X) = (-5% + 10%*1.65 )*100 = $11.45
  • 95% aVaR(Y) = (-6% + 10%*1.65 )*100 = $10.45
This is just to illustrate a possible interpretation of monotonicity: the expected value of Y is greater than X, such that it's (absolute VaR) risk is lower; put another way, for a given risk threshold, we need to add less cash to the Y asset, than the X asset, for a given risk threshold. Thanks,

Pedro Cazorla

New Member
I think I have the answer.
2p(Y) < p(Y)
This inequality holds as long as p(Y) is negative. So, the implication is that when Y is greater than zero (the only way that x=2Y is going to be greater than Y is when Y is greater than zero), p(Y) es negative. In fact, in Acerbi (Spectral Measures of risk: A coherent representation of subjective risk aversion, 2002), the author says that monotonocity can be replace by the positive axiom: If X is greater than zero, then p(X) is less than zero.