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7

The von Neumann-Morgenstern utility axioms are normative criteria for rational choice. In contrast, he Artzner/Uryasev axioms are normative criteria that some argue must hold for any measure that aims to measure portfolio risk. What they have in common is simply that they are normative criteria. The substance of the axioms are different, however, since they ...


7

I found this paper: Conditional value-at-risk for general loss distributions by Rockafellar and Uraysev http://dx.doi.org/10.1016/S0378-4266(02)00271-6 which says CVaR is coherent for general loss distributions, including discrete distributions. I think that I was confused by other authors who were also confused with the definitions of CVaR. In particular, ...


6

$VaR^\alpha$ is not a coherent risk measure because it fails sub-additivity (a coherent risk measure is monotonic, sub-additive, positive homogenous, and translation invariant). The expectation operator $E[\cdot]$ is linear, so it meets sub-additivity, as well as the other three properties, so $CVaR$ is a coherent risk measure.


5

I'm just providing a global answer to the question, as I think it can be interesting for some beginners in quant finance. The properties given by TheBridge: Normalize $\rho (\emptyset)=0$ This means you have no risk in taking no position. Sub-addiitivity $\rho(A_1+A_2) \leq \rho(A_1)+\rho(A_2)$ Having a position in two different can only decrease the ...


4

Conditional VaR (CVaR), which is also called Expected Shortfall, is a coherent risk measure (although being derived from a non-coherent one, namely VaR). See this paper: Expected Shortfall: a natural coherent alternative to Value at Risk from Carlo Acerbi and Dirk Tasche http://www.bis.org/bcbs/ca/acertasc.pdf EDIT: I just saw that you emphasized ...


4

It is common. The smaller the tail area you are considering the harder it is to be right because the effect of the assumption on the distribution becomes more important. Think about it in the other direction: if your level is 50%, then pretty much any distributional assumption will do. The other issue is the length of the time horizon. As the horizon ...


3

As you inferred, this is related to the concept of diversification as a risk-mitigation tool. In short, think of $\rho$ as representing some risk measure, and $\rho(x)$ as the risk of asset $x$ under that measure. If subadditivity holds, then the risk of holding assets 1 and 2 simultaneously must be less than or equal to the sum of their individual risks: ...


2

Rewriting the condition as $$\rho\left({X_1+X_2 \over 2}\right) \leq {\rho(X_1) + \rho(X_2) \over 2}$$ You can interpret it as a portfolio containing the average holdings of two other portfolios has at most the risk of the average risk of the two other portfolios. There is no need to have any concept of anyone actually holding any of the portfolios.


1

Compounding the monthly excess returns won't provide the annual excess return. You need to compute the difference between the annual return of the portfolio and the annual return of the benchmark. To illustrate this let's look at an example. Consider the following two situations: The benchmark performs well with a $2\%$ return each month; The benchmark ...


1

2) you only take trading days for your analysis because taking in account days on which no price changes took place would shift results in a wrong direction. For exmple, you mostly take 250 trading days p.a. 3) Your time interval up to 2007 is okay and excludes the financial crisis, which is a non-normal circumstance. Therefore, your time interval can be ...


1

See the paper "On the conditional value at risk Probability dependent utility function" by Alexandre Street, on Theory and Decision, 2010. It shows that the well know CVAR fails in the independence axiom but it also provides good insights for that. The CVAR (redefined for revenues and not for losses - see the above paper) is convex in the probability set. ...


1

If your problem is an equity portfolio without options, then I would vote for variance. This uses all the information rather than just looking at the tail. In 1999 the semi-variance became popular because it showed very small risk for telecom, media and tech stocks. They were just going up -- how is that risky? Equities are pretty close to having ...


1

I don't think that we should care if a risk measure is coherent. The reason that VaR is not coherent is because it need not be sub-additive. I'm willing to stand corrected, but I doubt that VaR is very far from sub-additive in practical situations. And I don't see a great deal of harm if it were. I have several problems with VaR but non-coherent is not ...



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