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On 1, I suspect that is a typo and that the second formula should sum to r. On 2, that is applying well-known techniques in how to handle piece-wise linear functions in an optimizer. For instance, see page 4 of these lecture notes. It's basically doing the same thing with a few additional complications. In CVaR optimization, there are more things to sum ...

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$VaR_\alpha$ is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called $\alpha\in R$. (C.f., the program described in Theorem 2 of that paper, or the programming problem described after equation (17); alternatively, look at the structure of the choice vector $x$ on page 16 of the Yollin slides.) $VaR_\... 5 Following the comments and the edits to the question, I'll try to show how conditional Value-at-Risk (aka Expected Tail Loss) can be minimised for a portfolio. We start with the implementation suggested by Rockafellar/Uryasev: @ARTICLE{, author = {R. Tyrrell Rockafellar and Stanislav Uryasev}, title = {Optimization of Conditional Value-at-Risk}, ... 3 One: Your VaR CI relies on normal approximation and might be (very) bad depending on the number of samples and the target function (P&L). Often it is better to use the exact approach based on the empirical distribution (see here: https://stats.stackexchange.com/a/284970/8298) Two: To estimate CVaR confidence intervals you may use bootstrap confidence ... 3 If$Y=-\pi(\mu,D)$then the first formula is $$\mathrm{CVaR}_\eta(-Y)=\max_{\nu\in R}\left\{\nu+\frac1\eta E((-Y-\nu)^-)\right\}$$ where$X^-=\min (X,0)$and$X^+=\max(X,0)$. Note that$(-X)^-=-(X^+)$. If we let$1-\alpha=\eta$and$\nu=-a$this becomes (assuming$\max=\sup$, i.e. the sup is attained, and using$\sup(\mathcal A)=-\inf(-\mathcal A)$): $$\... 2 The minimum value is always attained at d=0. In this proof, I will assume that the distribution of the random variable X is absolutely continuous and monotonically increasing, and thus the CDF of X is invertible (though I believe the result holds generally). Fix \beta\in(0,1). We have that$$ \Psi(d,\alpha)\equiv\int_{\min\{d,x\}\leqslant\alpha}p(... 2 The entropic value at risk (EVaR) is a coherent risk measure, developed to tackle some computational inefficiencies of the CVaR. It is the tightest possible upper bound for traditional VaR and CVaR, obtained from the Chernoff inequality. EVaR can also be represented by using the concept of relative entropy, better known in statistics as the Kullback-Leibler (... 2 We consider the case where the distribution function$F$of$Xis strictly increasing. Then \begin{align*} VaR_{\alpha}(X) &= \inf\{x: P(X >x) \le \alpha \}\\ &=\inf\{x: F(x)\ge 1-\alpha \}\\ &=F^{-1}(1-\alpha). \end{align*} Moreover, we note that the distribution functionG$of$-Xis defined by \begin{align*} G(x) &= P(-X \le x) \\ &... 2 This sounds correct, however step 2 is a little vague, so I will try to restate the steps here for you. The assets in your portfolio must be priced with respect to a set of risk factors (e.g. interest rate curve). Each scenario consists of a value for each of your risk factors. Given the value of your risk factors you can price your portfolio. You want to ... 2 I have solved it myself. The key was to realize that forX \geq 0$and$S_X(t) = \mathbb{P}(X>t)$$$\int_0^\infty S(t) dt = \int_0^1 F_X^{-1}(u) du = \mathbb{E}\left[X \right].$$ This is elegantly explained in Characterization of$\mathbb{E}. Now this relationship can be extended for the whole real line, thus \int_0^1 F_X^{-1}(u) du = \int_0^\... 2 A slightly different take here: 2 Let F be the cumulative distribution function of X. We assume that F is continuous. Then, for x\ge 0, \begin{align*} F^{-1}(x) = \inf\{s: F(s) \ge x \}. \end{align*} Moreover, \begin{align*} \text{VaR}_{\alpha}(X) &= \inf\left\{x :1-F(x) \le \alpha\right\}\\ &=F^{-1}(1-\alpha). \end{align*} Consequently \begin{align*} E\Big(\big(X-\text{VaR}... 2 It does not even matter if it’s concave or convex wrt global optimisation, both concave and convex functions have global optimal points albeit the only difference is maximum vs minimum which is easily incorporated with just a negative sign. As for CVaR concave or convex can simply be a result of whether it’s defined on losses or gains, with positive or ... 2 CVaR is a convex function in the underlying portfolio (measured as for instance absolute value or profit). I won't get into proving anything so instead I am going to link the first result from Google search: https://pdfs.semanticscholar.org/a5df/128eed59668b525a743a4e7f3f0efe12f930.pdf In fact, one of the reasons that we in general think of CVaR as a ... 2 That is incredibly unlikely for a continuous distribution -- though possible for a distribution with a part that is not absolutely continuous, i.e. is atomic. The way to see this is to remember that the \alpha% CVaR/ES/TCE is defined as: CVaR(r,\alpha) = E(r|r\leq Var(r,\alpha)). Thus getting an \alpha-CVaR equal in magnitude to \alpha-VaR would ... 1 Using a bunch of Dirac delta functions would not be a good idea; you essentially would be assuming a distribution of point masses instead of a continuous distribution. If you work with the integral of that, the empirical CDF, you can get some answers though they may be crude or be very uncertain. You would do better using a kernel density estimate, an ... 1 Yes, conditional VaR (aka Expected Shortfall) is a coherent risk measure and thus, satisfies Monotonicity, Translation invariance, Positive homogeneity and Subadditivity. The latter means that CVaR(R_1+R_2) \leq CVaR(R_1) + CVaR(R_2) which directly extends to sums of n random variables. Sub-additivity captures the notion that diversification is ... 1 Mean-CVaR portfolio optimization is an alternative to the more widely known and simpler mean-variance model. Since there doesn't seem to be any median-variance model out there, the familiarity surrounding the traditional model stuck. The mean is an actual equation with convenient properties in statistics, whereas the median is obtained through a counting ... 1 Given the main uses of the VaR relate to risk management such as limit management, and measurement of P&L volatility, it is usually calculated under the physical/real world measure. Reason being that the risk measure are normally used to predict or explain the P&L movements from one day to another, which one can relate to their historical movements. ... 1 It really depends on the way you calculate your Var and CvaR. If your are able to get a closed form solutions for the derivative then you must use those, for faster results. Otherwise, you can use bump and reval.\frac{\partial V}{\partial d} = \frac{V(d+h) - V(d)}{h}$$1 Maybe prove that$$CVaR_\alpha (X) = \frac{1}{\alpha} \int_0^\alpha F^{-1}_X(u) du$$has the distortion function$$ g(u)= \begin{cases} \frac{u}{\alpha}, \quad \; u \leq \alpha \\ 1, \qquad u > \alpha\end{cases}\$ would be easier?

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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 ...

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