# Tag Info

12

There are many advantages and flaws to each quoted method by Shane(presuming that I understand them properly), the first one has the big main advantage that it doesn't need any evaluation of probability law, it is just some kind of evolved scenario re-playing "as of" today using the history of (usually) one day market evolutions over one or two year. So ...

9

I am implementing a method in Java to calculate the variance, covariance, and value at risk for a portfolio, which should be flexible for use with any number of assets in a portfolio. I am struggling with how to calculate the covariance of the assets as I can only find formulae to do so for two or three sets of values. Are you sure you ...

8

Perhaps you may want to consider article by D. Levine - Modeling Tail Behavior with Extreme Value Theory who gives practicale example on how EVT can be used to calculate probabilities on returns in tails with use of the Pickands-Balkema-de Haan Theorem and generalized Pareto distribution. It also contains some criterias and points on other methods that can ...

8

One approach is Conditional Value at Risk (CVaR) a.k.a. Expected Shortfall (ES). It does, as you suggest, take into account the whole set of returns. However, instead of traditional VaR which asks "what is the worst 1% or 5% loss I can expect" in a given time frame, conditional VaR asks "assuming I sustain losses of at least 95% or 99% (and perhaps am ...

7

Well all that you have cited seems quite all you can do with scenario maybe I can add another one which is portfolio dependent. Instead of looking to arbitrary scenarios you first decompose the factor to which you portfolio is the most sensitive to, and then look for scenarios that are specifically impacting this combination of risk factors. ...

5

There are several methods to compute VAR: i) historical, ii) variance-covariance, and iii) monte carlo. iv) copula techniques. I assume you are asking about approach (ii). If the data are not multivariate normal and i.i.d. then the variance-covariance approach will not reflect true risk. For example, if there is serial correlation then risk is understated. ...

5

In general you don't need copulas to calculate VaR on portfolio. You can use historical method if you have time series of returns for the assets in your portfolio. If you have sufficiently enough data this will allow you to take into account correlation risk, non-normality of returns. Example of code in R for equally weighted portfolio without assuming any ...

5

Standard (read: regulators will accept it) could be a one day, 99% VaR calculated with two years of historical data. A minimum of one year of history is needed although this is not the norm. Typically the one-day VaR is transformed into a 10-day VaR by scaling the calculation by sqrt(10). However, the new market risk rule governs that one justify their ...

5

In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have ...

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 ...

4

The VaR constraint is convex and quadratic and can be handled with any solver supports quadratic constraints, like Guribi, cplex (from IBM) or xpress (from FICO). The CVaR can be formulated as a linear program if you are able to perform monte-carlo simulations on the returns. Briefly, the LP model is \begin{eqnarray*} c &\ge& \alpha + {1 \over ...

4

The Historical Method, which I would call Historical Simulation requires that you have a reasonably clean and accurate time series of data for the underlying asset. Essentially, you are using the past performance of the asset to model its likely behaviour over a time frame of typically 1 to 10 days. Choosing and updating your time series data set needs to ...

4

Using the delta-gamma approximation is still significantly faster even if you incorporate all of the nonzero cross-gammas. This speedup comes from the fact that you're using just the delta/gamma/cross-gamma parameters to calculate your VaR instead of 10,000+ Monte Carlo simulations. The calculations of each cross-gamma to the overall VaR, furthermore, can be ...

4

A probabilistic view on your full scale simulation. In the steps 1-3 you calculate the 0.99 quantile of the lognormal distribution with parameters $\ln N(\ln S_0 +(\mu - \frac{\sigma^2}{2})t,\sigma^2 t^2)$. The cdf of lognormal distribution is $\Phi(\frac{\ln x-\mu}{\sigma})$ Thus, you can calculate $V_p$ through $V_p=e^{\ln S_0 +(\mu - ... 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 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 ... 3 Usually when it is for (market) risk management purposes it is quite standard to have 1 day horizon with (allegedly ;-) ) 99% confidence level. As far as I know when it is for regulatory or economic capital requirement and/or Asset Liability Management then horizons might be much longer up to one year and confidence levels are usually 99% and 95%. Regards 3 It depends on the assets which copula is best and other methods may still be better and comparable in complexity. If you want to use copula's for equities you can take a look at Clayton copula. While the Gaussian copula is symmetric the Clayton copula has asymmetric tail dependency. This makes modeling the increase in correlation during a crisis possible. 3 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, ... 3 I'm guessing you're simulating rate curves etc. inside your system, and you want to reprice your instruments over the simulated curves using QuantLib. In this case, most of the logic is in your system already, and you have to plug pricing functionality in. If so, I don't think there's many steps involved besides, well, pricing the instrument on the ... 3 If$z_\alpha$is the so-called standard normal$z$-score of the significance level$\alpha$such that $$\frac 1 {\sqrt{2\pi}}\int_{-\infty}^{z_\alpha} e^{-\xi^2/2}d\xi=\alpha$$ and we assume normality, (ignoring skewness and kurtosis,) then we can estimate the$\alpha$quantile of a distribution with cdf$\Phi$as$$\Phi^{-1}(\alpha)=\mu + \sigma ... 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 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 ...

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) ...

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 ...

2

You can find a good example on CVaR optimization in the book "Portfolio Optimization with R/Rmetrics" By Diethelm Wuertz, Yohan Chalabi, William Chen, Andrew Ellis. #load library fPortfolio library(fPortfolio) #use indicies LPP2005, see http://www.pictet.com/en/home/lpp_indices.html lppData <- 100*LPP2005.RET[,1:6] #create portfolio specification ...

2

You could define your optimization problem as a typical linear risk/return optimization problem(1), then use some predicted return as the return component and VaR or cVaR as the risk component. This will not be linear, however you can use an evolutionary algorithm, or some other exotic search algorithm, to maximize for return given a limit on VaR or cVar. ...

2

I think what you're looking for is a type of solver called a second-order cone program (SOCP) solver. This is just like a quadratic program (QP) solver, except the constraints can be quadratic as well as the objective function. There is an open-source implementation in python via the CVXOPT module.

2

The time scaling of higher moments for ordinary (discrete) returns as per the Wingender paper is illustrated in Excel and VBA in the following spreadsheet demonstration files: Terminal-Wealth-Time-Horizon-Calcs-Normal-and-Modified-VBA and; Liqudity-VaR-With-Correct-Time-Scaling-of-Higher-Moments Available here For more on the weaknesses of the Cornish ...

2

Skewness decays with time, but the rate of that skewness decay will vary based on the instruments and how they are traded, so a simple estimator such as the square root of time rule is not appropriate. I typically recommend that to scale VaR or ES it makes more sense to lower your confidence level (raise the alpha parameter) to one that makes sense for your ...

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