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15

You have the correct approach. (1) The simulation generates sampled portfolio values, $P_1,P_2, \dots, P_n$ at time $t=T$. VaR is specified as a left-tail percentile. Order the sample as $$P_{(1)} \leq P_{(2)} \leq \dots \leq P_{(n)}.$$ If you are considering $VaR_\alpha$ at the $100(1-\alpha) \% $ confidence level , then choose the smallest integer $k$ ...


9

Values of VaR are just the inverses of the cumulative distributions. CVaR is not a very commonly used term, its more frequently used synonym is Expected Shortfall. See http://www.maths.manchester.ac.uk/~saralees/chap17.pdf for the list of Expected Shortfall values for more than 20 distributions.


7

It's very common to work in spreads rather than price for this calculation. The simplest approach would be to get an implied spread for each bond, and then allow the spreads to vary in simulation according to an equity-style factor model. Each spread simulation can then be mapped back to bond prices by reversing the formula. A few points: If you can, you ...


7

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 the fundamental assumptions (together with exogeneity). Then estimators are unbiased, optimal (exhibit the minimum variance within the class of unbiased ...


7

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 for your investments than to look at the property itself. I think the only way to get a nice formula for this is to assume that both assets are: Normally ...


7

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 mostly with equity time series and there are substantial missing values issues due to, e.g., as you cite, IPOs with limited history. Recently I have had good ...


5

It doesn't make sense to use option price series data for computing option risk anyway. Since they are derivatives (i.e. their value is derived from other securities) it is more basic and reasonable to handle the underlying risks. As hinted by John, the risks to an option portfolio are generally considered in the context of inputs to a pricing model (which ...


5

The given matrix can not represent a covariance matrix since it would imply that asset 1 is negatively correlated to asset 2 and asset 3. But asset 2 is negatively correlated to asset 3 which contradicts the first statement. In general a covariance matrix has to be positive semi-definite and symmetric, and conversely every positive semi-definite symmetric ...


5

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 sophisticated but the underlying idea is the same. VaR is used for reporting/ceremonial business decisions as much as (or perhaps even more than) it is for trading ...


5

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, with the exception of rows 1:4 and columns 2:4. You only need to set those values to zeroes and ones, which is fairly easy considering that the diagonal is (very ...


5

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. Once you have this, then you find the 5th percentile to give the 95% VaR. The advantage of this method is that it is the most straightforward to compute, and ...


5

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: $$ X = \begin{bmatrix}-1\\0\\1\\2 \end{bmatrix} \quad Y = \begin{bmatrix}0\\-1\\1\\2 \end{bmatrix} \quad X + Y = \begin{bmatrix}-1\\-1\\2\\4 \end{bmatrix}$$ What's the ...


4

Both approaches have drawbacks, so if one must choose among the two then one shall compare those drawbacks in the specific case. Or another way would be devising a hybrid of the two (e.g. adding statistics of historical deviations of the fund portfolio from the (1) view etc...). Among the drawbacks of (1): trading costs, rebalancings, management fees etc ...


4

Do $N$ MC simulations of $M$ samples, calculating your estimate of VaR for each one $\{\widehat{VaR}_i\}_{i=1}^N$ and you now have an IID sample! Take the sample (or unbiased) standard deviation for your estimate of VaR (this is probably what you mean by error) $SD(\widehat{VaR})=\sqrt{\frac{1}{N-1} \sum_{i=1}^N (\widehat{VaR}_i - \overline{VaR})^2}$ and of ...


4

No specific history. I'm not aware who introduced this measure initially. Most probably it came up as an example in the research papers on coherent risk measure. All names make sense to some extent: Expected shortfall - as it's an expectation of losses Conditional Value at Risk - as it can be written as $E[X |X >VaR_α(X)]$, i.e. conditional expectation ...


4

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.


4

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 $t-2$ observation by $(1-\lambda)*\lambda^{1}$ and so on so forth. For $\lambda= 0.94 $ : The first observation is weighted by = $(1-0.94) * 0.94^0 =0.06%$ The second ...


4

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 this is to use the peak over threshold method and fit the Generalized Pareto Distribution (GPD). The GPD is very convenient for calculating VaR and ES. A good ...


4

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 (which, as you say, may be positive), then you'd simply set your VaR to zero. But if you treat your returns as realizations of some (unknown) random variable, ...


4

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 fact that you're trying to get the tail value and (3) the shape of the KDE (distribution).The theoretical value of 3.088 will be emprirically calculated if and ...


4

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 course the normal distribution is among the elliptical distributions family. The latter can be helpful for analytical VaR modelling as an elliptical model is ...


4

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)}{\mathrm{VaR}_\alpha(X)}&=&\lim_{\alpha\to 1^-} \frac{g_v(t^{-1}_v(\alpha))}{(1-\alpha)(v-1)\left(\frac{t^{-1}(\alpha)}{v+(t^{-1}(\alpha))^2}\right)}\\ =\...


4

What are the underlying assumptions for doing this Assumption: Historical returns are lognormally distributed with no autocorrelation. can those assumptions be tested statistically Testing: $\sqrt{xy} = \sqrt{x} \sqrt{y}$ Substitute time $t$ and variance $\sigma^2$ for $x$ and $y$ respectively $\sqrt{t\sigma^2} = \sqrt{t} \sqrt{\sigma^2} = \sigma\...


4

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, $$ \frac{\partial V}{\partial S} = \frac{QS}{V} $$ Suppose you had 3 assets, with large positions in the first two assets, and small position in the third, AND that the first two were perfectly negatively correlated,...


3

Given that by delta means that if the price goes up by 0.01% i.e. one basis point, you gain 15 and vice versa if the price goes down by one basis point. You know that the daily standard deviation is 2.2%, than again you know that $ 220*15 = 3300$ is the standard deviation of your portfolio. So, since we are using a normal distribution you can look at a table ...


3

In general I would answer your question in the following way: Alternatives to VaR which share most of its helpful properties but not its shortcomings are the so called coherent risk measures. They have the following properties: monotonicity sub-additivity homogeneity and translational invariance One example would be the conditional value-at-risk. You ...


3

VaR is not a good measure of risk taking, in my opinion. It suffers from inherent faulty assumptions (check out VaR Wiki to start) and it omits many other important aspects of risk measurement. When I evaluate an asset's risk and return I like to start looking at the following: Historical risk and returns of an asset. This leads to the Sharpe Ratio, ...


3

First of all you need a model to generate future returns, I assume you already have this. Since its just a model, there will be an unexplained component in the predictions made for every period $t$ and for every asset $i$. Let $\varepsilon_{t, i}$ denote this random innovation and $\mathrm{E}[r_{t, i}] = f(\varepsilon_{t, i})$ the expected asset return as ...


3

You need to isolate the risk factors that impact your forward contract, which is your spot fx rate, and the two rates of each currency that underlies the forward contract. You therefore need to estimate the VaRs of each of those risk factors. You also need the correlations between the underlying risk factors. For example, a forward to buy USD in exchange ...


3

No matter how you calculate the VaR (historical simulation, covariance approach, MC) I assume that you work on historical data or data derived from the history of assets, risk factors and theresuch. If this assumption is correct then I would use approach (1). If you know the exact positions today of the (sub-)funds, then (except from some technicalities) ...


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