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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

It sounds like the P&L's you are given are not really the historical P&L's. Rather, you have some portfolio and market data currently; you have 260 days of historical market data changes; and you calculate what the P&L of the present portfolio would have been if the market moved as it did on that historical date from the current market data. You'...


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One way to look at answering this question is VAR Contribution. Evaluate VAR of the Portfolio, and then evaluate VAR of the Portfolio without the asset. The largest difference of VAR with the asset - VAR of the portfolio without the asset would be the asset which is contributing the most to VAR. You may want to correct the size of the portfolio for each ...


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Here is the excel formula with steps: =NORMSDIST((NORMSINV(0.02)+NORMSINV(0.999)×SQRT(0.1))/SQRT(1−0.1)) =NORMSDIST((−2.054+3.09×SQRT(0.1))/SQRT(1−0.1)) =NORMSDIST(-1.135) =12.8% They keep changing the names of the function - e.g., NORMSDIST is NORM.S.DIST(-1.135,TRUE)in the recent versions, and same for NORMSINV = NORM.S.INV


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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 find an intuition for. Your portfolio consists of two assets A basket/collection of equity with a market price $S$ USD per unit of equity. Assume you hold $...


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I would think that you would treat this as computing the VAR of a two asset case. In your case these assets would be 1/ your GBP bond and 2/ an FX position in EURGBP. You already have a vol measure for asset 1. Once you have a vol measure for the FX, you should be able to obtain the standard deviation of the combined asset via $$\sigma_{X+Y}=\sqrt{\sigma_X^2+...


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Since this question does not seem to be a duplicate, I will make up a simple (but not entirely unrealistic) numeric example. Suppose some asset is now trading at some observable price, and suppose further that you have written two options: a put and a call that are slightly out of the money, i.e. whose strikes are, for concreteness, within 1 historical ...


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That is incredibly unlikely for a continuous distribution -- though possible for a distribution with a part that is not absolutely continuous, i.e. is atomic. The way to see this is to remember that the $\alpha$% CVaR/ES/TCE is defined as: $CVaR(r,\alpha) = E(r|r\leq Var(r,\alpha))$. Thus getting an $\alpha$-CVaR equal in magnitude to $\alpha$-VaR would ...


1

VaR is a loss function calculated from what's available in step 1, whose value is a magnitude, and whose sign indicates whether there is a portfolio loss, or a negative loss (which is actually a gain, given that VaR, as a loss, is ordinarily reported as a negative number). So to ask which asset, whose returns are available in step 2, is driving this loss ...


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Here are two examples: Büyükşahin, B., Harris, J.H., 2011. Do speculators drive crude oil futures prices?. Energy J. 32 (2), 167–202. Fujihara, R.A., Mougou, M., 1997. An examination of linear and nonlinear causal relationships between price variability and volume in petroleum futures markets. J. Futures Mark. 17 (4), 385–416.


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I think first understanding what the mean is and what a quantile is would be helpful. The 0.05 quantile is the value for which for a given distribution only 5% of the values are expected to be lower, for example in the standard normal distribution this quantile is roughly -1.65. Now to understand a relationship between those two you have to look at the ...


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In a Full reval scenario, you would 1) identify your risk factors (ATM point? Skew? Surface? SABR?) Say you want to simulate all surface points. Then, after you have applied your surface returns you need to first make sure that your new IV surface is free of arbitrage. That is an art in itself, though. Then you do the valuation as is usual.


1

As I see it, in both, a (MC) simulation or a historical simulation, risk estimators (VaR, iVaR, mVaR) suffer from the instability of the quantile. If we had a sufficiently “dense” set of observations around the $(1-\alpha)$ quantile strip, we could compute a weighted average around that quantile and find risk factor and instrument contributions. Yet, in ...


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You might want to have a look at the conjugate priors to the normal distribution (with known mean) Your setup will result in a $t$-distributed return with updated dispersion parameter.


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The historical volatilities of the market factors is not the same as the implied volatilty used to price the options. The "implied volatility" is just one of the model inputs. It does not need to be similar to the historical volatility of the underlying. The mark to market of an option is the premium that one would have to pay in the market for this option. ...


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Without any additional information, and disregarding potential problems of VaR subadditivity, adding VaR figures is usually deemed conservative. For a meaningful risk measure, we usually require $$ Risk(X+Y) \leq Risk(X) + Risk(Y) $$ In your example: Simply adding the VaR figures per currency is sufficiently conservative, if no correlation is assumed. ...


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Multiple packages are available in R for extreme value analysis.If you are looking at extreme value theory in regards to stock prices there is full implementation of libraries in the rMetrics team's fExtremes library in the R statistical script language. This and others are covered in the book on R for finance by Gilleland, Ribatet, and Stephenson (2013). In ...


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The paper "Calculating CVaR and bPOE for Common Probability Distributions With Application to Portfolio Optimization and Density Estimation" by Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2018) gives a large number of CVAR analytical formula with full proof. Most of them can also be found on the Expected shortfall (aka CVAR) Wikipedia page.


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