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12 votes
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How is the formula for the VEV (VaR-equivalent volatility) in the PRIIP document derived?

Let's assume T=1 and let S be a geometric gaussian process with zero drift, i.e. $\ln(S_1/S_0)$ is normally distributed with mean $-1/2\times\mathrm{VEV}^2$ and volatility VEV. Then $$\ln(\mathrm{...
Robert Schlichtner's user avatar
10 votes
Accepted

Intuitive explanation for expectiles

No reply has been given so I wanted to at least give a visualisation of the expectiles. Suppose the curvy dashed line in my picture represents a cumulative distribution function of some random ...
Raskolnikov's user avatar
  • 1,527
8 votes

non-subadditivity of VaR

Simple example where sub-additivity fails Let there be four possible outcomes $i=1,2,3,4$ that occur with equal probability $\frac{1}{4}$. Payoffs for $X$, $Y$, and $X + Y$ are given by: $$ X = \...
Matthew Gunn's user avatar
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7 votes
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Is Value At Risk additive?

The answer to your question is no. Value at Risk is not additive in the sense that $\text{VaR}(X+Y) \neq \text{VaR}(X) + \text{VaR}(Y)$. But I guess your question is more to aimed at finding a formula ...
SRKX's user avatar
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7 votes
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1 day VaR vs 10 day VaR

What are the underlying assumptions for doing this Assumption: Historical returns are lognormally distributed with no autocorrelation. can those assumptions be tested statistically Testing: $\...
amdopt's user avatar
  • 4,348
7 votes

Missing data in historical simulation VaR

This issue is incredibly important and I agree there is little practical information about it. To me, the key idea is to find the right matrix completion algorithm that best suits your needs. I work ...
Jonathan's user avatar
  • 171
7 votes
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Parametric VaR, Normality and Subadditivity

Suppose $X\sim N(\mu_X,\sigma_X^2)$ and $Y\sim N(\mu_Y,\sigma_Y^2)$ are correlated jointly normal random variables. Then, $$X+Y\sim N(\mu_X+\mu_Y,\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y).$$ ...
Kevin's user avatar
  • 16k
6 votes

Imposing Restrictions on Cointegrating Vectors, R example

I know this was asked almost two years ago, but I thought I'd answer the question. It appears that the H that you want to estimate is identical to the values you received from the Johansen test, ...
Cameron Pfiffer's user avatar
6 votes
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Expected Shortfall Formula in terms of P

Gordon's answer is spot on. Another way to see it though, would be using Bayes formula and a change of variable. \begin{align*} ES_X(p) &=E\left(X \mid X\le Q_X(1-p)\right)\\ &=\int_{-\infty}^...
Quantuple's user avatar
  • 14.7k
6 votes
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non-subadditivity of VaR

VaR is not sub-additive in general. Relying on Mark Joshi comment, there are particular cases where it can be. Such cases occur for portfolios containing elliptically distributed risk factors. Of ...
JejeBelfort's user avatar
  • 1,219
6 votes
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99.97% Percentile VaR Approximation

The 99.97% confidence is somtimes referred to as corresponding to the 1-year probability of default of 3 bps for AA-rated entities. (Here for example https://papers.ssrn.com/sol3/papers.cfm?...
Mats Lind's user avatar
  • 1,412
5 votes
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Overestimating or underestimating risk?

Yes, it is correct. Underestimation: you under-estimate the risk, so you have more VaR violations than what your model predicts. Ex: With 100 observations, and a 99% VaR, you expect 1 violation but ...
Malick's user avatar
  • 2,582
5 votes

Expected Shortfall Formula in terms of P

Note that $Q_X$ is the pseudo-inverse of the distribution function $F$, and for any uniform random variable $U$ over $[0, 1]$, the random variable $Q_X(U)$ has the same distribution as $X$. Moreover, ...
Gordon's user avatar
  • 21.2k
5 votes
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How to compute VaR of a simple equity portfolio?

There are a few different ways to calculate VaR. Historical Method For this method, you calculate the return of your portfolio each day, and get a list of daily returns over your calibration period. ...
msitt's user avatar
  • 741
5 votes
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Can portfolio Value-at-Risk be calculated analytically for multivariate t-distributed returns?

Let the $n-$dimensional vector of returns $\mathbf{r}$ have a multivariate t distribution with $\nu$ degrees of freedom. The marginal distribution of any component $r_i$ has a univariate t ...
RRL's user avatar
  • 3,700
4 votes

Extreme Value Theory in Risk Management

EVT has pluses and minuses, but (under certain conditions) provides the best estimate of extreme quantile returns in a portfolio given the data available. Probably the simplest and easiest way to do ...
RiskyScientist's user avatar
4 votes
Accepted

Value at Risk - What if an account has never suffered from a negative return

By definition, your loss cannot be positive, so you'd set the VaR to zero. But it really depends, on how you calculate your VaR. If you calculate your returns, sort them and look at the 5% quantile (...
rbm's user avatar
  • 745
4 votes

VaR estimate with Monte Carlo simlation

To answer you question "is it because X is a mixture of a continous and discrete Random Variable": the answer is no. The mean reasons are (1) the sample size (which is limited / countable) (2) the ...
rbm's user avatar
  • 745
4 votes

Ratio between Expected Shortfall and Value at Risk for $t$-distribution

Let $u=t^{-1}_v(\alpha)$ and recall that $g_v(u)=c_v(v+u^2)^{-\frac{v+1}2}$ for some constant $c_v$. By the formulas you provided, $$\begin{eqnarray*}\lim_{\alpha\to 1^-}\frac{\mathrm{ES}_\alpha(X)}{\...
Bjørn Kjos-Hanssen's user avatar
4 votes
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Risk Compensation

A linear relationship between expected returns and covariance with a risk factor is a necessary consequence of a linear asset pricing function In theory, a CAPM relationship can be derived when a ...
Matthew Gunn's user avatar
  • 6,974
4 votes

Intuitive explanation for expectiles

That picture in the other answer is pretty slick (+1), so I will just add a note on why one can interpret the colors of those areas like that: Blue: Define $Y = (X-x)_+$. This is nonnegative r.v., ...
Taylor's user avatar
  • 554
4 votes

Portfolio optimization w.r.t. value at risk: introductory or survey references

I would suggest to start with Euler capital allocation as a first step to dive into the subject, here is an example of introductory paper (Capital Allocation to Business Units and Sub-Portfolios: the ...
raptor22's user avatar
  • 608
4 votes

Portfolio VaR of a hedge portfolio (long index, short future): What total exposure to take to calculate VaR?

Firstly, your portfolio volatility of 0.74% is the variance, as the vol will be 8.6% relative your equity position. This is the Case 2 below. I will try to give you a derivation that you hopefully can ...
Pontus Hultkrantz's user avatar
4 votes
Accepted

How accurate is the square root of time rule for VaR for a portfolio containing several different types of instruments

Effectively, I sense two questions here, 1) around the validity of the $\sqrt{T}$-assumption in the scaling of the risk horizon ; and 2) the quality of the $ \Delta$-$\Gamma$-approximation in ...
Kermittfrog's user avatar
  • 6,772
4 votes
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Optimizing a portfolio whose risk is target expected shortfall

This problem can be addressed efficiently by linear programming. An (in my opinion) even better reference than the original paper by Uryasev, Rockafeller provided by noob2 is "PORTFOLIO ...
g g's user avatar
  • 2,023
4 votes

How to determine what's driving the VaR?

If you have a covariance matrix, $Q$ the VaR is a measure of the standard deviation of the portfolio, ie. $$VaR, V \propto \sqrt{S^T Q S}$$ and, $$ \frac{\partial V}{\partial S} = \frac{QS}{V} $$ ...
Attack68's user avatar
  • 10.8k
4 votes

Do the minimum VaR and minimum ES portfolios lie on the mean-variance efficient frontier?

When returns follow an elliptical distribution (e.g. the Gaussian distribution), then minimising VaR and ES is equivalent to minimising variance. See https://people.math.ethz.ch/~embrecht/ftp/pitfalls....
Enrico Schumann's user avatar
4 votes
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Correlation for Trading vs. Risk Management

Good question! I think there's some semantics to be thought about first: The word Hedging commonly implies that you want to hedge the changes in the present value of your total position ($\Pi=PV(A) +...
Kermittfrog's user avatar
  • 6,772
4 votes
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Filtered Historical Simulation VaR for swaps

Let us suppose for concreteness that the 10y swap rate is 0.5% today and was 7% a year and and 6.5% a "year minus a day" ago... reprice the swaps for each historical scenario and calculate ...
Dimitri Vulis's user avatar
4 votes

Can Heston volatility model be used to calculate VaR or CVaR?

Certainly! The Heston model is a well-known model in quantitative finance that describes the evolution of the volatility of an asset. It's a stochastic volatility model, meaning it assumes that the ...
Amit Kumar Jha's user avatar

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