# Tag Info

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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 ...

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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, ...

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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 ...

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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: $\... 7$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 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 ... 4 There are 4 defining properties of coherent risk measures You can find them here as well as examples for coherent and counterexamples of those kind of risk measures. 3 I think what you are missing is simply the Vega-Gamma relation in the Black-Scholes model. Namely: $$Vega = \frac{\partial v}{\partial \sigma} = \sigma(T-t)S^2 \frac{\partial^2 v}{\partial S^2} = \sigma \tau S^2 \Gamma$$ Plugging this into your coverage error, you get its expression in terms of the Vega which is the most natural measurement of your ... 3 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. 3 Well, if you assume$X$has volatility$\sigma_X$and$Y$has volatility$\sigma_Y$, then $$\sigma_{X+Y} = \sqrt{ Var( X + Y) } = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho }$$ Then, you want to show $$\sigma_{X+Y} = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho } \leq \sigma_X + \sigma_Y$$ Squaring both sides: $$\sigma_X^2+\... 3 Coherent risk measures were created to address the problem that extant risk measures, like VaR, did not: namely that a risk measure should reward diversification. 2 I don't think that we should care if a risk measure is coherent. The reason that VaR is not coherent is because it need not be sub-additive. I'm willing to stand corrected, but I doubt that VaR is very far from sub-additive in practical situations. And I don't see a great deal of harm if it were. I have several problems with VaR but non-coherent is not ... 2 2) you only take trading days for your analysis because taking in account days on which no price changes took place would shift results in a wrong direction. For exmple, you mostly take 250 trading days p.a. 3) Your time interval up to 2007 is okay and excludes the financial crisis, which is a non-normal circumstance. Therefore, your time interval can be ... 2 We define a convex risk measure as$$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \lambda \rho( X_1 ) + (1-\lambda) \rho(X_2), $$for \lambda \in(0,1) . A coherent risk measure is subadditive and homogeneous thus for coherent \rho we get:$$ \rho( \lambda X_1 + (1-\lambda) X_2) \le \rho( \lambda X_1) + \rho( (1-\lambda) X_2) $$by subadditivity and$$ \... 1 Translation invariance of a risk measure$\rho$is defined as $$\rho(X+k) = \rho(X)-k,$$ where$X$is a random variable such that$\rho(X)$exists and$k$is a constant. The meaning is that if I add an amount$k$to my risky positions then the risk is reduced by this amount. For VaR we consider the case that$X$has a continuous distribution and that it ... 1 As always I recommend reading Rennie and Baxter for an introduction to option pricing that's not too technical and gives intuition about how it all works. 1 From Ziegel (2013) : The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called ... 1 Did the portfolio manager have the option of investing in emerging markets? If yes, use MSCI All-World. If the portfolio has holdings based in countries with "developed markets" yet has has emerging markets exposure to revenue/earnings, the convention is to use MSCI World. 1 I have never seen such an adjustment. While monthly data are irregularly sampled in time (in every way...calendar days, trading days, seconds, etc), that irregularity is likely to be a smaller effect than your choice of data frequency (monthly, weekly, daily data). That said, your question is intriguing because in other fields they do have to deal with ... 1 Normalize for trading days if possible. 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 ...

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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. ...

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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 ...

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