12
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
$$\ln(\mathrm{...
9
votes
Accepted
Intuitive explanation for expectiles
No reply has been given so I wanted to at least give a visualisation of the expectiles.
Suppose the curvy dashed line in my picture represents a cumulative distribution function of some random ...
- 1,507
8
votes
non-subadditivity of VaR
Simple example where sub-additivity fails
Let there be four possible outcomes $i=1,2,3,4$ that occur with equal probability $\frac{1}{4}$. Payoffs for $X$, $Y$, and $X + Y$ are given by:
$$ X = \...
- 6,864
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 ...
- 11k
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, $$X+Y\sim N(\mu_X+\mu_Y,\sigma^2_X+\sigma_Y^2+2\rho\sigma_X\sigma_Y).$$
...
- 15.1k
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, ...
- 116
6
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-\...
- 13.6k
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$ ...
- 2,542
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}^...
- 14.5k
6
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 ...
- 1,219
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?...
- 1,307
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
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 ...
- 2,542
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, ...
- 21k
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
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 ...
- 3,595
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 ...
- 429
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 (...
- 735
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 ...
- 735
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 ...
- 6,864
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: $\...
- 4,728
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 ...
- 578
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 ...
- 1,256
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 ...
- 6,260
4
votes
Accepted
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 ...
- 1,973
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,
$$ \frac{\partial V}{\partial S} = \frac{QS}{V} $$
...
- 8,952
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....
- 3,176
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