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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 ...

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 $\sup(\mathcal A)=-\inf(-\mathcal A)$): $$\... 3 VaR_\alpha is a scalar choice variable in the minimization problem. In the Rockafeller-Uryasev paper, it is simply called \alpha\in R. (C.f., the program described in Theorem 2 of that paper, or the programming problem described after equation (17); alternatively, look at the structure of the choice vector x on page 16 of the Yollin slides.) VaR_\... 3 One: Your VaR CI relies on normal approximation and might be (very) bad depending on the number of samples and the target function (P&L). Often it is better to use the exact approach based on the empirical distribution (see here: https://stats.stackexchange.com/a/284970/8298) Two: To estimate CVaR confidence intervals you may use bootstrap confidence ... 2 The minimum value is always attained at d=0. In this proof, I will assume that the distribution of the random variable X is absolutely continuous and monotonically increasing, and thus the CDF of X is invertible (though I believe the result holds generally). Fix \beta\in(0,1). We have that$$ \Psi(d,\alpha)\equiv\int_{\min\{d,x\}\leqslant\alpha}p(...

2

We consider the case where the distribution function $F$ of $X$ is 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 function $G$ of $-X$ is defined by \begin{align*} G(x) &= P(-X \le x) \\ &...

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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 for $X \geq 0$ 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^\... 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 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 A slightly different take here: 1 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}... 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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