VALUE-AT-RISK (VAR) (PART 6)

Value-at-Risk (VaR)

by Simon Benninga and Zvi Wiener

Monte Carlo Approach
This method is based on the assumption that we have some information about the joint distribution of market changes. Then using this distribution we can draw randomly a large number of scenarios and price the portfolio for each scenario. A rich set of scenarios will give a good approximation for the distribution of final value of the portfolio. The lowest q-quantile of this distribution can be used as an approximation to VaR. Moreover this method allows a dynamic improvement. One can run a small set of simulations, get a preliminary result and then improve it by running additional simulations if necessary. In the example below we choose the simplest form of the joint distribution - all market parameters are distributed jointly normal with the same mean and covariance as we have measured above. In a general case one can provide any reasonable distribution and use the same method.
nor[mu_,sig_]:=
Random[ NormalDistribution[mu,sig] ];
MCvarAdd[portf_, mrkt_, n_:10]:=
Module[
{tbl, simulatedParam, values},
tbl = Table[ sqrtAddCovar.Table[ nor[0,1],
{Length[AddCovMx]}] + AddMeansVec, {n} ];
simulatedParam = Table[mrkt,{n}] + tbl;
values =
Map[valueP[ portf,#]&, simulatedParam ];
{Mean[values], StandardDeviation[values],
StandardErrorOfSampleMean[values],
values, simulatedParam }
];
To generate a vector of correlated normally distributed random variables we generate first a vector of independent random variables (normally distributed) and then multiply it by the square root of the covariance matrix. It is left as an exercise to verify that this leads to the required correlation. The tbl table is the resulting table of market changes.
The following function incorporates all the necessary steps for Monte Carlo approach to VaR. The parameters of this function are: portf portfolio, mrkt current market 6 MathematicainEducationandResearch Vol. 7 No. 4 1998
VALUE-AT-RISK (VAR)
data, quant quantile, n number of generated scenarios.
This function returns the VaR estimate and the list of all changes in the value of the portfolio that were observed in this simulation.
MCApproachAdd[ portf_, mrkt_, quant_, n_:10 ]:=
Module[{currentValue, changes},
currentValue = valueP[ portf,mrkt];
changes = MCvarAdd[portf, mrkt, n][[4]] -
currentValue;
{Quantile[ changes, quant], changes}
];
To get consistent results this function can be combined with the random seed generator. For 1000 simulations (about 1 minute of computational time) we get:
SeedRandom[1];
MCApproachAdd[ portfolio, mrkt9Feb97, 0.2,
1000][[1]]
-3.71068
Running this simulation again we get a similar result:
SeedRandom[7];
MCApproachAdd[ portfolio, mrkt9Feb97,
0.2, 1000][[1]]
-3.76359
Discussion
The historical simulation method is useful when the amount of data is not very large and we do not have enough information about the profit and loss distribution. It is usually very time consuming, but its main advantage is that it catches all recent market crashes. This feature is very important for risk measurement.
The variance covariance method is the fastest. However it relies heavily on several assumptions about the
distribution of market data and linear approximation of the portfolio. It is probably the best method for quick estimates of VaR. However one should be very careful when using this method for a non-linear portfolio, especially in the case of high convexity in options or bonds. The Monte Carlo simulation method is very slow, but it is probably the most powerful method. It is flexible enough to incorporate private information together with historical observations. There are many methods of speeding calculations, so-called variance reduction techniques. The results of all three methods are similar and our goal was to demonstrate a very basic approach to risk measurement techniques using Mathematica.
REFERENCES
BEDER, TANYA (1996), VAR: Seductive But Dangerous, Financial Analyst Journal, September-October, pp. 12–24.
GRUNDY, BRUCE D. and ZVI WIENER. 1996. The Analysis of VAR, Deltas and State Prices: A New Approach. mimeo The Wharton School, University of Pennsylvania.
HULL, JOHN C. Options Futures, and Other Derivatives, Third Edition. Prentice-Hall, 1997.
JORION, PHILIPPE, Value at Risk, the New Benchmark for Controlling Market Risk, McGraw-Hill, 1997.
LINSMEIER, THOMAS and NEIL PEARSON. Risk Measurement: An Introduction to Value at Risk. mimeo, University of Illinois, 1996.
Several Web sites offer information on value-at-risk:
http://pw2.netcom.com/ bschacht/varbiblio.html—an excellent source of VaR related materials.
http://www.riskmetrics.reuters.com/—Reuters materials on riskmetrics.
http://www.jpmorgan.com/RiskManagement/RiskMetrics/RiskMetrics.html
ABOUT THE AUTHORS
The authors acknowledge grants from Wolfram Research, the Krueger and Eshkol Centers at the Hebrew University, and the Israeli Academy of Science. Wiener’s research has benefited from a grant from the Israel Foundations Trustees, the Alon Fellowship and the Eshkol grant.
Simon Benninga is professor of finance at Tel-Aviv University (Israel) and the Wharton School of the University of Pennsylvania. He is the author of Financial Modeling (MIT Press, 1997) and of Corporate Finance: A Valuation Approach (with Oded Sarig, McGraw-Hill, 1997); he is also the editor of the European Finance Review.
Simon Benninga
Faculty of Management
Tel-Aviv University, Tel-Aviv, Israel
benninga@post.tau.ac.il
http://finance.wharton.upenn.edu/ benninga Zvi Wiener is assistant professor of finance at the business school of the Hebrew University of Jerusalem. His finance research concentrates on the pricing of derivative securities, Value-at-Risk, computational finance and stochastic dominance. He wrote this article while visiting at the Olin School of Business at Washington University in St. Louis.
Zvi Wiener
Finance Department, Business School
Hebrew University, Jerusalem, Israel
mswiener@mscc.huji.ac.il
http://pluto.mscc.huji.ac.il/ mswiener/zvi.html
ELECTRONIC SUBSCRIPTIONS
Included in the distribution for each electronic subscription is the file varisk.nb, containing Mathematica code for the material described in this article.
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8 MathematicainEducationandResearch Vol. 7 No. 4 1998