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VALUE AT RISK

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**VALUE AT RISK VALUE AT RISK Definition :**

The expected maximum loss ( or worst loss ) over a target horizon within a given confidence interval Evolved through the requirements of SEC to provide a minimum capital requirement in provision for risks taken by financial institutions

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**Models to estimate VaR Historical Market Data**

assumption is that historical market data is our best estimator for future changes and that asset returns in the future will have the same distribution as they had in the past Variance – Covariance (VCV) assuming that risk factor returns are always (jointly) normally distributed and that the change in portfolio value is linearly dependent on all risk factor returns Monte Carlo Simulation future asset returns are more or less randomly simulated

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**Historical Market Data**

involves running the current portfolio across a set of historical price changes to yield a distribution of changes in portfolio value, and computing a percentile (the VaR). Benefits Simple to implement does not assume a normal distribution of asset returns Drawbacks requires a large market database computationally intensive calculation.

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**VALUE AT RISK (VAR) Market Value x Risk Variability Confidence Level =**

Worst Loss

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**DEFINITION : Market Value : Risk variability :**

Current market value of the respective transaction to be managed Mark to market at the end of time horizon Risk variability : Usually the standard deviation of the risk to be managed The higher the variability the higher is VAR

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**DEFINITION : Time Horizon : Confidence Level :**

Time period to be considered correspond to time required for corrective actions as losses starts to develop Annualized The longer the time horizon the higher is VAR Confidence Level : Confidence level of loss occurring The higher the confidence level the higher is VAR

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Example : Current market value of transaction USD 100,000 paid by 3 months usance L/C Standard deviation of Rp/USD is 10% Time horizon is 3 months Confidence level is 95% VAR = 100,000 X 0.10 X X 0.95 = USD 4, Maximum expected loss

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**WEAKNESSES OF HISTORICAL SIMULATION**

Past is not prologue, history does not always repeat it selves Trends in the data, since all data are treated equal, despite the fact that some periods might experience higher volatility For new assets or market risks no historical data available

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**MODIFICATIONS OF HISTORICAL SIMULATION**

Weighting the recent past more, assuming that the recent past is a better predictor of the immediate future than the distant past ( can be using indexes to adjust each return based on its timeline) Combining historical simulation with time series models, by fitting a time series models through the historical data Volatility updating, by comparing past volatility with recent volatility and then adjusting the past return accordingly e.g. past volatility = 0.5, recent volatility 0.75, past return 10%, recent predicted return is 0.75/0.5* 10%=15%

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**Variance – Covariance (VCV)**

Basic assumption is that the risk factors are normally distributed Hence are the returns also normally distributed

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**Variance – Covariance (VCV)**

VAR formula for VCV : σ = standard deviation of the risk Sigma for 95% confidence level is 1.645 Sigma for 99% confidence level is 2.33

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**Exchange rate (USD/Rp)**

EXAMPLE : Exchange rate (USD/Rp) Loss / Gain (p) Probability (r) d2 = r * p2 9.600 -400 0.025 4.000 9.700 -300 2.250 9.800 -200 0.050 2.000 9.900 -100 0.200 10.000 0.400 10.100 100 10.200 200 10.300 300 10.400 400 Average = S = d2 = d = With 95% confidence interval the VAR = * Rp = Rp The maximum potential loss for exchange rate risk is Rp

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**ASSESMENT OF V-CV METHOD**

Strength : simple to compute after making assumptions about the distribution of returns and inputted the means, variances and covariances or returns Weaknesses : Wrong distributional assumptions, if it turns out that the returns a re not normally distributed and the outliers are higher, computed VaR can be lower that actual VaR Input error, if data used to calculate are for example based on historical data, which is not reflecting the current situation Nonstationary variables, happens if the underlying assumed correlation does not hold anymore, e.g. interest rate is adjusted by FED

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**Monte Carlo Simulation**

Decide on N, the number of iterations to perform. For each iteration: Generate a random scenario of market moves using some market model. Revalue the portfolio under the simulated market scenario. Compute the portfolio profit or loss (PnL) under the simulated scenario. (i.e., subtract the current market value of the portfolio from the market value of the portfolio computed in the previous step). Sort the resulting PnLs to give us the simulated PnL distribution for the portfolio. VaR at a particular confidence level is calculated using the percentile function. For example, if we computed 5000 simulations, our estimate of the 95% percentile would correspond to the 250th largest loss; i.e., ( ) * 5000.

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**STRENGTH OF MONTE CARLO SIMULATION**

Does not need to rely on historical data, those historical data can still be used as benchmark and then adjust accordingly Does not need to assume normal distribution for the returns Can be used for any type of portfolio including options or option like securities

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**WEAKNESSES OF MONTE CARLO SIMULATION**

Needs to estimate the probability distribution for all the market risk variables that we want to consider Number of simulations that need to be run on the model will be substantially large

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**MODIFICATION ON MONTE CARLO SIMULATION**

Scenario simulation, only likely combination are run through the model Monte Carlo simulation with Variance-Covariance method modification, assuming normal distribution for the returns

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**Indications on method to use**

For Value at Risk for portfolios, that do not include options, over very short time periods (a day or a week) and normality can be assumed, the variance-covariance approach does a reasonably good job. If the risk source is stable and there is substantial historical data (commodity prices, for instance), historical simulations provide good estimates. In the most general case of computing VaR for nonlinear portfolios (which include options) over longer time periods, where the historical data is volatile and non-stationary and the normality assumption is questionable, Monte Carlo simulations do best.

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LIMITATIONS OF VaR : Return distributions cannot always be correctly predicted History may not be a good predictor Nonstationary correlations Only looking at the downside risk (negative side of risk) Best for calculating short term risk Difficult to use for comparing different investments

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**VaR can lead to suboptimal decision**

Overexposure to risk, managers will tend to be more bold in making risky investments while actually the rest of 5-10% probability of incurring risk might be huge Agency problem, because VaR usually uses past data, managers who knows about irregularity in the volatility can misuse them for his own advantage.

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VAR is best used for financial institutions, since they are dealing with short term assets which are related to the common market risks VAR for nonfinancial institutions should be used as a secondary measure, unless it is for short term assets.

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Chapter 21 Value at Risk Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012.

Chapter 21 Value at Risk Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012.

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