11 votes
Accepted

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{...
  • 161
9 votes

Where can I find a list of VaR and CVaR formulas for continuous distributions?

Values of VaR are just the inverses of the cumulative distributions. CVaR is not a very commonly used term, its more frequently ...
  • 346
9 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 ...
  • 1,487
7 votes

Ran multivariate linear regression, checked normal probability plot, residuals are not normal. What can I do?

Regression analysis, as a minimization of the sum of squared errors, does not require normality of the error term. The requirements are that errors are homoscedastic and uncorrelated. And these are ...
  • 4,247
7 votes
Accepted

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 ...
  • 10.9k
7 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 = \...
  • 6,394
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 ...
  • 171
7 votes
Accepted

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, $$X+Y\sim N(\mu_X+\mu_Y,\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y).$$ ...
  • 14k
6 votes

Do people actually use VaR in professional settings?

As discussed, banks do use VaR for risk management. They will have something modified for the specific use (i.e. probably not your VaR from a fitted normal distribution), it's likely more ...
  • 5,081
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, ...
6 votes
Accepted

RiskMetrics VaR Volatility Sample Size

Depending of $\lambda$, pasts observations will be weighted differently, if you compute the volatility at time $t$ , the $t-1$ observation will be weighted by $(1-\lambda)*\lambda^{0}$, the $t-2$ ...
  • 2,534
6 votes
Accepted

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}^...
  • 14.1k
6 votes
Accepted

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?...
  • 1,277
5 votes
Accepted

Portfolio VaR with Copula?

You don't really have a multivariate case: we can only define VaR (in its usual sense) for a one-dimensional output. Recall that $$ \operatorname{VaR}_\alpha(X) = \inf\{v:F_X(v)\geq \alpha\} $$ and ...
  • 1,474
5 votes

Stressed Value at Risk vs Value at Risk

The most important difference is that the calculations are based on a "stressed" historical period in the markets as opposed to the most recent X number of years.
  • 51
5 votes
Accepted

Calculate VaR for a liabilty taking a exponential distribution?

The VaR of level $\alpha$ a loss random variable (the bigger the worse) is the quantity $q$ such that the loss is bigger with probability $1-\alpha$. Thus we need a $q$ such that $$ P[L>q] = 1-\...
  • 13.3k
5 votes
Accepted

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 ...
  • 2,534
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, ...
  • 20.5k
5 votes
Accepted

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. ...
  • 741
5 votes
Accepted

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 ...
  • 1,209
5 votes
Accepted

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 ...
  • 3,470
4 votes
Accepted

Historic Value at Risk - Ratios vs. Differences

As a short summary and adaption of the question: You better redefine $\hat{r}_i= \frac{S_{i-1}}{S_1}-1$ and $\hat{S}_i = (1+\hat{r}_i)S_0$. The above definition of $\hat{S}_i$ yields a sample of ...
  • 13.3k
4 votes
Accepted

How to extrapolate VaR?

It depends on the method by which you calculate VaR. Some models (t-distributuion, normal) lead to a form of VaR such that it is just scaled volatility: $$ VaR = c \sigma $$ with some proper $c$ (e.g. ...
  • 13.3k
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 ...
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 (...
  • 725
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 ...
  • 725
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)}{\...
4 votes
Accepted

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 ...
  • 6,394
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., ...
  • 534
4 votes
Accepted

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: $\...
  • 3,880

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