Standard Normal Table

References

Abouarghoub, Wessam (2013). Implementing the new science of risk management to tanker freight markets, doctoral thesis, University of the West of England.

Alexander, Carol O. (2001). Market Models, Chichester: John Wiley & Sons.

Alexander, Carol O. and A. M. Chibumba (1997). Multivariate orthogonal factor GARCH, working paper.

Allen, M. (1994). Building a role model, Risk, 7 (9), 73-80.

Bai, Jushan, and Shuzhong Shi (2011). Estimating high dimensional covariance matrices and its applications, Annals of Economics & Finance, 12 (2), 199-215.

Baxter, Martin and Andrew Rennie (1996). Financial Calculus: An Introduction to Derivative Pricing, Cambridge: Cambridge University Press.

Basel Committee on Banking Supervision (1995). An Internal Model-Based Approach to Market Risk Capital Requirements.

Basel Committee on Banking Supervision (1996a). Amendment to the Capital Accord to Incorporate Market Risks.

Basel Committee on Banking Supervision (1996b). Supervisory Framework for the Use of “Backtesting” in Conjunction With the Internal Models Approach to Market Risk Capital Requirements.

Beck, Kent (2002). Test Driven Development: By Example, Reading: Addison-Wesley.

Beck, Kent and Cynthia Andres (2004). Extreme Programming Explained: Embrace Change, 2^nd Edition, Reading: Addison-Wesley.

Berkowitz, Jeremy (2001). Testing density forecasts, with applications to risk management, Journal of Business & Economic Statistics, 19 (4), 465-474.

Berkowitz, Jeremy, Peter Christoffersen and Denis Pelletier (2011). Evaluating value-at-risk models with desk- level data, Management Science 57 (12), 2213–2227.

Berkowitz, Jeremy and James O’Brien (2002). How accurate are value‐at‐risk models at commercial banks? Journal of Finance, 57 (3), 1093-1111.

Bernstein, Peter L. (1992). Capital Ideas: The Improbable Origins of Modern Wall Street, New York: Free Press.

Black, Fischer and Myron S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81 (3), 637-654.

Black, Fischer (1976). The Pricing of Commodity Contracts, Journal of Financial Economics, 3, 167-179.

Bollerslev, Tim (1986). Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307-328.

Bollerslev, Tim (1990). Modelling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model, Review of Economics and Statistics, 72, 498-505.

Box, George E. P. and Norman R. Draper (1987). Empirical Model-Building and Response Surfaces, New York: Wiley.

Britten-Jones, Mark and Stephen M. Schaefer (1999). Nonlinear value-at-risk, European Finance Review, 2 (2), 161-187.

Brockwell, Peter J. and Richard A. Davis (2010). Introduction to Time Series and Forecasting, 2^nd ed., New York: Springer.

Burden, Richard L. and J. Douglas Faires (2010). Numerical Analysis, 9^th ed., Boston: PWS Publishing Company.

Burghardt, Galen and Bill Hoskins (1995). A question of bias, Risk, 8 (3), 63-70.

Campbell, Rachel, Kees Koedijk and Paul Kofman (2002). Increased correlation in bear markets, Financial Analysts Journal, 58 (1), 87-94.

Campbell, Sean D. (2005). Finance and Economics Discussion Series, Washington: Federal Reserve Board.

Cárdenas, Juan, Emmanuel Fruchard, Etienne Koehler, Christophe Michel, and Isabelle Thomazeau (1997).value-at-risk: One Step Beyond, Risk, 10 (10), 72-75.

Cárdenas, Juan, Emmanuel Fruchard, Jean-François Picron, Cecilia Reyes, Kristen Walters, and Weiming Yang (1999). Monte Carlo within a day, Risk, 12 (2), 55-59.

Chew, Lillian (1993). Made to measure, Risk, 6 (9), 78-79.

Christoffersen, Peter (1998). Evaluating interval forecasts. International Economic Review, 39 (4), 841-862.

Christoffersen, Peter and Denis Pelletier (2004). Backtesting value-at-risk: a duration-based approach, Journal of Financial Econometrics, 2 (1), 84-108.

Cockburn, Alistair (2000). Writing Effective Use Cases, Reading: Addison-Wesley.

Cornell, Bradford (1981). A note on taxes and the pricing of Treasury bill futures contracts, Journal of Finance, 36 (12), 1169-1176.

Cornell, Bradford and Marc R. Reinganum (1981). Forward and futures prices: Evidence from the foreign exchange markets, Journal of Finance, 36 (12), 1035-1045.

Cornish, E. A. and Ronald A. Fisher (1937). Moments and cumulants in the specification of distributions, Review of the International Statistical Institute, 5, 307-320.

Corrigan, Gerald (1992). Remarks before the 64^th annual mid-Winter meeting of the New York State Bankers Association, January 30, Waldorf-Astoria, New York City: Federal Reserve Bank of New York.

Cotter, John and François Longin (2007). Implied Correlations fromvalue-at-risk. Working paper, University College Dublin.

Coveyou, R. R. and R. D. MacPherson (1967). Fourier analysis of uniform random number generators, Journal of the Association for Computing Machinery, 14, 100-119.

Cox, John C., Jonathan E. Ingersoll, Jr., and Stephen A. Ross (1981). The relation between forward prices and futures prices, Journal of Financial Economics, 9, 321-346.

Crnkovic, Cedomir and Jordan Drachman (1996). Quality control, Risk, 9 (9), 138-143.

Culp, Christopher (2001). The Risk Management Process: Business Strategy and Tactics, New York: John Wiley & Sons.

da Silva, Alan Cosme Rodrigues, Claudio Henrique da Silveira Barbedo, Gustavo Silva Araújo and Myrian Beatriz Eiras das Neves (2006). Internal Model validation in Brazil: analysis ofvalue-at-risk backtesting methodologies, Revista Brasileira de Finanças, 4 (1), 363-384.

Dahlgren, Robert, Chen-Ching Liu and Jacques Lawarrée (2003). Risk assessment in energy trading. IEEE Transactions on Power Systems, 18 (2), 503-511.

Dale, Richard (1996). Risk and Regulation in Global Securities Markets, Chichester: John Wiley & Sons.

Davidson, Clive (1996). The data game, Firmwide Risk Management, special supplement to Risk, 9 (7), 39-44.

Davies, Robert B. (1973). Numerical inversion of a characteristic function, Biometrika, 60, 415-417.

Dennis, J. E. and Robert B. Schnabel (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Englewood Cliffs: Prentice-Hall.

Ding, Z. (1994). Time series analysis of speculative returns, PhD thesis, San Diego: University of California.

Doherty, Neil A., (2000). Integrated Risk Management: Techniques and Strategies for Managing Corporate Risk, New York: McGraw-Hill.

Dowd, Kevin (2005). Measuring Market Risk, 2nd ed., Chichester: John Wiley & Sons.

Dusak, Katherine (1973). Futures trading and investor returns: an investigation of commodity market risk premiums, Journal of Political Economy, 81, 1387-1406.

Eckhardt, Roger (1987). Stan Ulam, John von Neumann, and the Monte Carlo method, Los Alamos Science, Special Issue (15), 131-137.

Eichenauer, J. and J. Lehn (1986). A non-linear congruential pseudo random number generator, Statistical Papers, 27, 315-326.

Eichenauer-Herrmann, J. (1993). Statistical independence of a new class of inversive congruential pseudorandom numbers, Mathematics of Computation, 60, 375-384.

Engle, Robert F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, 987-1008.

Engle, Robert F. (2000). Dynamic conditional correlation—A simple class of multivariate GARCH models, working paper.

Engle, Robert F. and K. F. Kroner (1995). Multivariate simultaneous generalized ARCH, Econometric Theory, 11, 122-150.

Engle, Robert F., Simone Manganelli (2004). CAViaR: conditional autoregressive value-at-risk by regression quantiles. Journal of Business and Economic Statistics, 22 (4), 367–381.

Engle, Robert F. and Kevin Sheppard (2001). Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, working paper.

Evans, Michael and Tim Swartz (2000). Approximating Integrals Via Monte Carlo and Deterministic Methods, Oxford: Oxford University Press.

Fallon, William (1996). Calculating value-at-risk, working paper.

Filliben, James J. (1975). Probability plot correlation coefficient test for normality, Technometrics, 17 (1), 111–117.

Fincke, U. and M. Pohst (1985). Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Mathematics of Computation, 44 (170), 463-471.

Finger, Christopher (1997). A methodology for stress correlation, Risk Metrics Monitor (Fourth Quarter), 3-11.

Finger, Christopher (2006). How historical simulation made me lazy, RiskMetrics Group Research Monthly.

Fishman, George S. (1996). Monte Carlo: Concepts, Algorithms, and Applications, New York: Springer-Verlag.

Forbes, Catherine, Merran Evans, Nicholas Hastings, and Brian Peacock (2010). Statistical Distributions, 4^nd ed., New York: John Wiley & Sons.

Francis, Stephen C. (1985). Correspondence appearing in: United States House of Representatives (1985). Capital Adequacy Guidelines for Government Securities Dealers Proposed by the Federal Reserve Bank of New York: Hearings Before the Subcommittee on Domestic Monetary Policy of the Committee on Banking, Finance and Urban Affairs, Washington: US Government Printing Office, 251-252.

Franses, Philip Hans (1998). Time Series Models for Business and Economic Forecasting, Cambridge: Cambridge University Press.

Franses, Philip Hans and Dick van Dijk (2000). Non-Linear Time Series Models in Empirical Finance, Cambridge: Cambridge University Press.

French, Kenneth R. (1980). Stock returns and the weekend effect, Journal of Financial Economics, 8, 55-69.

French, Kenneth R. (1983). A comparison of futures and forward prices, Journal of Financial Economics, 12, 311-342.

French, Kenneth R. and Richard Roll (1986). Stock return variance: the arrival of information and the reaction of traders, Journal of Financial Economics, 17, 5-26.

Fuglsbjerg, Brian (2000). Variance reduction techniques for Monte Carlo estimates of value-at-risk, working paper.

Garbade, Kenneth D. (1986). Assessing risk and capital adequacy for Treasury securities, Topics in Money and Securities Markets, 22, New York: Bankers Trust.

Garman, Mark B. and Steven W. Kohlhagen (1983). Foreign currency option values, Journal of International Money and Finance, 2, 231-237.

Gärtner, von Bernd (1999). Ein reinfall mit computer-zufallszahlen, Mitteilungen der Deutschen Mathematiker-Vereinigung, 2, 55-60.

Geiss, Charles G. (1995). Distortion-free futures price series, Journal of Futures Markets, 15 (7), 805-831.

Gentle, James E. (1998). Numerical Linear Algebra for Applications in Statistics, New York: Springer-Verlag.

Gibbons, Michael R. and Patrick Hess (1981). Day of the week effect and asset returns, Journal of Business, 54, 579 – 596.

Glasserman, Paul (2003). Monte Carlo Methods in Financial Engineering, Springer: New York.

Glasserman, Paul, Philip Heidelberger, and Perwez Shahabuddin (2000). Variance reduction techniques for estimating value-at-risk, Management Science, 46 (10), 1349 – 136.

Goldfeld, Stephen M and Richard E. Quandt (1973). A Markov model for switching regressions, Journal of Econometrics, 1, 3-16.

Goldman Sachs and SBC Warburg Dillon Read (1998). The Practice of Risk Management, London: Euromoney Books.

Golub, Gene H. and Charles F. Van Loan (1996). Matrix Computations, 3^rd ed., Baltimore: Johns Hopkins University Press.

Gridgeman, N. T. (1960). Geometric probability and the number p, Scripta Mathematica, 25, 183-195.

Group of 30 (1993). Derivatives: Practices and Principles, Washington: Group of 30.

Group of 30 (1994). Derivatives: Practices and Principles, Appendix III: Survey of Industry Practice, Washington: Group of 30.

Guldimann, Till M. (1995). Risk measurement framework, RiskMetrics—Technical Document, 3^rd ed., New York: Morgan Guaranty, 6-45.

Guldimann, Till M. (2000). The story of RiskMetrics, Risk, 13 (1), 56-58.

Gupta, Anurag and Marti G. Subrahmanyam (2000). An empirical examination of the convexity bias in the pricing of interest rate swaps, Journal of Financial Economics, 55 (2), 239-279.

Haas, Marcus (2001). New methods in backtesting. Mimeo, Financial Engineering Research Center Caesar, Friedensplatz, Bonn.

Haas, M., 2005. Improved duration-based backtesting of value-at-risk. Journal of Risk, 8 (2), 17–38.

Han, Chulwoo, Frank C. Park, and Jangkoo Kang (2007). Efficient value-at-risk estimation for mortgage-backed securities, Journal of Risk 9 (3), 37-61.

Hall, Asaph (1873). On an experimental determination of p, Messenger of Mathematics, 2, 113-114.

Hamilton, James D. (1993). Estimation, inference and forecasting of time-series subject to changes in regime, in G. S. Maddala, C. R. Rao and H. D. Vinod (editors), Handbook of Statistics, vol. 11: Econometrics, New York: North-Holland.

Hamilton, James D. (1994). Time Series Analysis, Princeton: Princeton University Press.

Hammersley, J. M. and D. C. Handscomb (1964). Monte Carlo Methods, New York: John Wiley & Sons.

Hartman, Joel, and Jan Sedlak (2013). Forecasting conditional correlation for exchange rates using multivariate GARCH models with historical value-at-risk application, working paper.

Haug, Espen G. (1997). The Complete Guide to Option Pricing Formulas, 2^nd ed., New York: McGraw-Hill.

Hellekalek, P. (1998). Good random number generators are (not so) easy to find, Mathematics and Computers in Simulation, 46, 485-505.

Hendricks, Darryll (1996). Evaluation of value-at-risk models using historical data, Federal Reserve Bank of New York Economic Policy Review, April.

Higham, Nicholas J. (2002). Computing the nearest correlation matrix—a problem from finance, IMA Journal of Numerical Analysis 22(3), 329-343.

Heron, Dan and Richard Irving (1996). Banks graspvalue-at-risk nettle, A Risk Special Supplement, Risk, June, pp. 16–21.

Holton, Glyn A. (1998). Simulating value-at-risk, Risk, 11 (5), 60-63.

Holton, Glyn A. (2004). Defining risk, Financial Analysts Journal, 60 (6), 19–25.

Hughston, Lane (1996). Vasicek And Beyond: Approaches to Building and Applying Interest Rate Models. London: Risk Publications.

Hughston, Lane (1999). Options: Classic Approaches to Pricing and Modelling. London: Risk Books.

Hull, John C. (2011). Options, Futures, and Other Derivatives, 8^th ed., Englewood Cliffs: Prentice Hall.

Hull, John and Alan White (1998). Incorporating volatility updating into the historical simulation method forvalue-at-risk, Journal of Risk, 1 (1), 5-19.

Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables, Biometrika, 48, 419-426.

James, Jessica and Nick Webber (2000). Interest Rate Modelling, Chichester: John Wiley & Sons.

Jamshidian, Farshid and Yu Zhu (1997). Scenario simulation: Theory and methodology, Finance and Stochastics, 1 (1), 43-67.

Jarrow, Robert A. (editor) (1998). Volatility: New Estimation Techniques for Pricing Derivatives, London: Risk Books.

Jarrow, Robert A. and George S. Oldfield (1981). Forward contracts and futures contracts, Journal of Financial Economics, 9, 373-382.

Jaschke, Stefan R. (2001). The Cornish-Fisher-expansion in the context of delta-gamma-normal approximations, Journal of Risk, 4(4), 33-52.

Jaschke, Stefan R. and Peter Mathé (2004). Stratified sampling for risk management, unpublished manuscript.

Johnson, Dallas E. (1998). Applied Multivariate Methods for Data Analysts, Pacific Grove: Duxbury Press.

Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation, Biometrika, 36, 149-176.

Judge, George G., R. Carter Hill, William E. Griffiths, Helmut Lütkepohl, and Tsoung-Chao Lee (1988). The Theory and Practice of Econometrics . 2^nd ed., New York: John Wiley & Sons.

Kercheval, Alec N. (2008). Optimal covariances in risk model aggregation, Proceedings of the Third IASTED International Conference on Financial Engineering and Applications, ACTA Press, Calgary, 30-35.

Klaassen, Franc (2000). Have exchange rates become more closely tied? Evidence from a new multivariate GARCH model, working paper.

Knuth, Donald E. (1997). Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3^rd ed., Vol. 2. Reading: Addison-Wesley.

Kolb, Robert W. (2006). Understanding Futures Markets, 6^th ed., Malden: Blackwell.

Korn, Ralf and Mykhailo Pupashenko (2015). A new variance reduction technique for estimating value-at-risk, Applied Mathematical Finance, 22(1), 83-98.

Kupiec, Paul H. (1995). Techniques for verifying the accuracy of risk measurement models, Journal of Derivatives, 3 (2), 73–84.

Larman, Craig (2003). Agile and Iterative Development: A Manager’s Guide, Reading: Addison-Wesley.

Lad, Frank (1996). Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction, New York: John Wiley & Sons.

Laplace, Pierre Simon Marquis de (1878-1912). Oeuvres Complètes de Laplace, Paris: Gauthier-Villars.

Leavens, Dickson H. (1945). Diversification of investments, Trusts and Estates, 80 (5), 469-473.

L’Ecuyer, Pierre. (1998). Random number generation, in Jerry Banks (editor), Handbook of Simulation: Principles, Methodology, Advances, Applications, and Practice, 1998, New York: John Wiley & Sons.

L’Ecuyer, Pierre. (1999). Good parameter sets for combined multiple recursive random number generators, Operations Research, 47 (1), 159-164.

L’Ecuyer, P., F Blouin and R. Couture (1993). A search for good multiple recursive random number generators, ACM Transactions on Modeling and Computer Simulation, 3 (2), 87-98.

Leffingwell, Dean and Don Widrig (1999). Managing Software Requirements: A Unified Approach, Reading: Addison-Wesley.

Lehmann, E. L. and Joseph P. Romano (2005). Testing Statistical Hypotheses, 3^rd ed., New York: Springer.

Lehmer, D. H. (1951). Mathematical methods in large-scale computing units, Proceedings of a Second Symposium on Large-Scale Digital Calculating Machinery. Cambridge: Harvard University Press, 141-146.

Leipnik, R. B. (1991). Lognormal random variables, Journal of the Australian Mathematical Society, Series B, 32, 327-347.

Leong, Kenneth S. (1996). The right approach, Value-at-Risk, A Risk Special Supplement, Risk Magazine, June, 9–14.

Levine, Robert S. (2007). Implementing Systems Solutions for Financial Risk Management, London: Risk Books.

Lewis, P. A. W., A. S. Goodman and J. M. Miller (1969). A pseudo-random number generator for the System/360, IBM Systems Journal, 8 (2), 136-145.

Li, Qingna, Donghui Li and Houduo Qi (2010). Newton’s method for computing the nearest correlation matrix with a simple upper bound, Journal of Optimization Theory and Applications, 147 (3), 546-568.

Lietaer, Bernard A. (1971). Financial Management of Foreign Exchange: An Operational Technique to Reduce Risk, Cambridge: MIT Press.

Linsmeier, Thomas J. and Neil D. Pearson (1996). Risk Measurement: An Introduction to Value at Risk, unpublished manucript.

Lintner, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics, 47: 13-37.

Ljung, G. M. and G. E. P. Box (1978). On a measure of lack of fit in time series models, Biometrika, 65 (2), 297-303.

Longerstaey, Jacques (1995). Mapping to describe positions, RiskMetrics—Technical Document, 3^rd ed., New York: Morgan Guaranty, 107-156.

Lopez, Jose A. (1999). Methods for evaluating value-at-risk models, Federal Reserve Bank of San Francisco Economic Review, 2, 3-17.

Lyons, Richard K. (1995). Tests of microstructure hypotheses in the foreign exchange market, Journal of Financial Economics, 39, 321-351.

Macaulay, Frederick R. (1938). The Movements of Interest Rates. Bond Yields and Stock Prices in the United States since 1856, New York: National Bureau of Economic Research.

Ma, Christopher K., Jeffrey M. Mercer and Matthew A. Walker (1992). Rolling over futures contracts: A note, Journal of Futures Markets, 12 (2), 203-217.

Malz, Alan M. (2011). Financial Risk Management: Models, History, and Institutions, Chichester: John Wiley & Sons.

Mark, Robert (1991). Units of management. Balance Sheet (distributed in Risk, 4 (6)), 3-7.

Markowitz, Harry M. (1952). Portfolio Selection, Journal of Finance, 7 (1), 77-91.

Markowitz, Harry M. (1959). Portfolio Selection: Efficient Diversification of Investments, New York: John Wiley & Sons.

Markowitz, Harry M. (1999). The early history of portfolio theory: 1600-1960, Financial Analysts Journal, 55 (4), 5-16.

Marsaglia, G. (1968). Random numbers fall mainly in the planes, Proceedings of the National Academy of Sciences USA, 61, 25-28.

Marshall, Chris and Michael Siegel (1997). Value at risk: implementing a risk measurement standard, Journal of Derivatives, 4 (3), 91-110.

Mathai, A. M. and Serge B. Provost (1992). Quadratic Forms in Random Variables, New York: Marcel Dekker.

McLeod, A. I. and W. K. Li (1983). Diagnostic checking ARMA time series models using squared residual autocorrelations, Journal of Time Series Analysis, 4 (4), 269–273.

Metropolis, Nicholas (1987). The beginning of the Monte Carlo method, Los Alamos Science, Special Issue (15), 125-130.

Metropolis, Nicholas and Stanislaw Ulam (1949). The Monte Carlo method, Journal of the American Statistical Association, 44 (247), 335-341.

Mills, Terence C. (1999). The Econometric Modelling of Financial Time Series, 2^nd ed., Cambridge: Cambridge University Press.

Mina, Jorge and Andrew Ulmer (1999). Delta-Gamma four ways. Technical report, RiskMetrics Group.

Mittnik, Stefan (2014).value-at-risk-implied tail-correlation matrices. Economics Letters, 122 (1), 69-73.

Molinari, Steven L. and Nelson S. Kibler (1983). Broker-dealers’ financial responsibility under the Uniform Net Capital Rule—a case for liquidity, Georgetown Law Journal, 72 (1), 1-37.

Morgan, Byron J. T. (1984). Elements of Simulation. London: Chapman & Hall.

Morgan Guaranty (1996). RiskMetrics—Technical Document, 4^th ed., New York: Morgan Guaranty.

Mossin, Jan (1966). Equilibrium in a capital asset market, Econometrica, 34, 768-783.

Niederreiter, Harald (1992). Random Number Generation and Quasi-Monte Carlo Methods. Philadelphia: Society for Industrial and Applied Mathematics.

Office of the Comptroller of the Currency (2000). OCC Bulletin 2000–16: Model Validation, Washington: Office of the Comptroller of the Currency.

O’Neil, Catherine (2010). Measuring CDS value-at-risk. Risk Metrics Working Papers.

Opschoor, Anne, Dick van Dijk and Michel van der Wel (2013). Predicting covariance matrices with financial conditions indexes (No. TI 13-113/III, pp. 1-43). Tinbergen Institute Discussion Paper Series.

Pan, Jun and Darrell Duffie (1997). An Overview of value-at-risk, Journal of Derivatives, 4 (3), 7-49.

Park, Hun Y. and Andrew H. Chen (1985). Differences between futures and forward prices: A further investigation of the mark-to-market effects, Journal of Futures Markets, 5 (1), 77-88.

Park, Stephen K. and Keith W. Miller (1988). Random number generators: good ones are hard to find, Communications of the ACM, 31 (10), 1192-1201.

Patel, Jagdish K. (1996). Handbook of the Normal Distribution, 2^nd ed., New York: Marcel Dekker.

Pérignon, Christophe, Zi Yin Deng and Zhi Jun Wang (2008). Do banks overstate their value-at-risk? Journal of Banking & Finance, 32 (5), 783-794.

Pérignon, Christophe and Daniel R. Smith (2010). The level and quality of Value-at-Risk disclosure by commercial banks, Journal of Banking & Finance, 34 (2), 362-377.

Pichler, Stefan and Karl Selitsch (2000). A comparison of analyticvalue-at-risk methodologies for portfolios that include options, Model Risk, Concepts, Calibration and Pricing, Rajna Gibson (editor), London: Risk Books.

Press, W., S. Teukolsky, W. Vetterling and B. Flannery (1995). Numerical Recipes in C: The Art of Scientific Computing, 2nd ed., Cambridge: Cambridge University Press.

Pritsker, Matthew (2006). The hidden dangers of historical simulation, Journal of banking & finance, 30 (2), 561-582.

Pupashenko, Mykhailo (2014). Variance reduction technique for estimating value-at-risk based on cross-entropy, Journal of Mathematics and System Science, 4(1), 37-48.

Qi, Houduo and Defeng Sun (2010). Correlation stress testing for value-at-risk: an unconstrained convex optimization approach, Computational Optimization and Applications, 45 (2), 427-462.

Questa, Giorgio S. (1999). Fixed Income Analysis for the Global Financial Market: Money Market, Foreign Exchange, Securities, and Derivatives, New York: John Wiley & Sons.

Rebonato, Riccardo and Peter Jäckel (1999). The most general methodology to create a valid correlation matrix for risk management and option pricing purposes, Journal of Risk, 2(2), 17-27.

Reuters (2000). An Introduction to The Commodities, Energy & Transport Markets, Singapore: John Wiley & Sons.

Rota, Gian-Carlo (1987). The lost cafe, Los Alamos Science, Special Issue (15), 23-32.

Rouvinez, Christophe (1997). Going Greek withvalue-at-risk, Risk, 10 (2), 57-65.

Roy, Arthur D. (1952). Safety first and the holding of assets, Econometrica, 20 (3), 431-449.

Røynstrand, Torgeir, Nils Petter Nordbø, Vidar Kristoffer Strat (2012). Evaluating power of value-at-risk backtests, masters thesis, Norwegian University of Science and Technology.

Rubinstein, Reuven Y. (2007). Simulation and the Monte Carlo Method, 2^nd ed. New York: John Wiley & Sons.

Saff, E. B. and A. B. J. Kuijlaars (1997). Distributing many points on a sphere, Mathematical Intelligencer, 19 (1), 5-11.

Schneider Geri and Jason P. Winters (2001). Applying Use Cases: A Practical Guide, 2^nd ed. Reading: Addison-Wesley.

Schrock, Nichols W. (1971). The theory of asset choice: simultaneous holding of short and long positions in the futures market, Journal of Political Economics, 79, 270-293.

Schwaber, Ken and Mike Beedle (2001). Agile Software Development with Scrum, Upper Saddle River: Prentice Hall.

Sharpe, William F. (1963). A simplified model for portfolio analysis, Management Science, 9, 277-293.

Sharpe, William F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (3), 425-442.

Shirreff, David (1992). Swap and think, Risk, 5 (3), 29 – 35.

Singh, Manoj K. (1997). Value-at-risk using principal components analysis, Journal of Portfolio Management, 24 (1), 101-112.

Sironi, Andrea and Andrea Resti (2007). Risk Management and Shareholders’ Value in Banking, Chichester: John Wiley & Sons.

Solomon, H. and M.A. Stephens (1977). Distribution of a sum of weighted chi-square variables, Journal of the American Statistical Association, 72, 881-885.

Spanos, Aris (1999). Probability Theory and Statistical Inference: Econometric Modeling with Observational Data, Cambridge: Cambridge University Press.

Steinberg, Richard M. (2011). Governance, Risk Management, and Compliance, Chichester: John Wiley & Sons.

Stoyanov, Jordan (1997). Counterexamples in Probability, 2^nd ed., Chichester: John Wiley & Sons.

Strang, Gilbert (2005). Linear Algebra and Its Applications, 4^rd ed., Brooks Cole.

Stuart, Alan and J. Keith Ord (1994). Kendall’s Advanced Theory of Statistics, Volume 1: Distribution Theory, London: Arnold.

Student (1908a). The probable error of a mean. Biometrika, 6, 1-25.

Student (1908b). Probable error of a correlation coefficient. Biometrika, 6, 302-310.

Tobin, James (1958). Liquidity preference as behavior towards risk, The Review of Economic Studies, 25, 65-86.

Todhunter, Isaac. (1865). History of the mathematical theory of probability from the time of Pascal to that of Laplace. Cambridge: Cambridge University Press. Reprinted (1949), New York: Chelsea.

Treynor, Jack (1961). Towards a theory of market value of risky assets, unpublished manuscript.

Tsay, Ruey S (2013). Multivariate Time Series Analysis: With R and Financial Applications, John Wiley & Sons.

Vasey, G. M. and Andrew Bruce (2010). Trends in Energy Trading, Transaction and Risk Management Software 2009 – 2010, CreateSpace Independent Publishing Platform.

Viswanath, P. V. (1989). Taxes and the futures-forward price difference in the 91-day T-bill market, Journal of Money, Credit and Banking, 21 (2), 190-205.

Walmsley, Julian (2000). The Foreign Exchange and Money Markets Guide, 2^nd ed., New York: John Wiley & Sons.

Wilmott, Paul, Jeff Dewynne, and Sam Howison (1993). Option Pricing: Mathematical Models and Computation, Oxford: Oxford Financial Press.

Wilson, Thomas (1992). Raroc remodeled, Risk, 5 (8), 112-119.

Wilson, Thomas (1993). Infinite wisdom, Risk, 6 (6), 37-45.

Wilson, Thomas (1994a). Debunking the myths, Risk, 7 (4), 67-73.

Wilson, Thomas (1994b). Common methods of calculating capital at risk, Risk, 7 (10), 78-80.

Wilson, William W., William E. Nganje and Cullen R. Hawes (2007). Value-at-risk in bakery procurement, Review of Agricultural Economics, 29 (3), 581-595.

Zangari, Peter (1994). Estimating volatilities and correlations, RiskMetrics—Technical Document, 2^nd ed., New York: Morgan Guaranty, 43-66.

Zangari, Peter (1996a). Data and Related Statistical Issues, RiskMetrics—Technical Document, 4^th ed., New York: Morgan Guaranty, 163-196.

Zangari, Peter (1996c). Market risk methodology, RiskMetrics—Technical Document, 4^th ed., New York: Morgan Guaranty, 105-148.

Ziggel, Daniel, Tobias Berens, Gregor Weiss and Dominik Wied (2014). A new set of improved value-at-risk backtests, Journal of Banking and Finance, 48:29-41.

Endnotes

Chapter 1

1. Dowd (2005) discusses ETL metrics.
2. Recall that standard deviation is the square root of variance.
3. Gradient approximations are discussed in Section 2.3.
4. As obtained with a Monte Carlo transformation.
5. Some value-at-risk measures make simplifying assumptions that render the value of ⁰p unnecessary—it drops out of the calculations. Others either accept ⁰p as an input or calculate it based on current values of key factors.
6. See Dale (1996) pp. 60 – 61 and Molinari and Kibler (1983) footnote 41.
7. See Dale (1996), p. 78.
8. The Basel Committee on Banking Supervision is a standing committee comprising representatives from central banks and regulatory authorities. Over time, the focus of the committee has evolved, embracing initiatives designed to define roles of regulators in cross-jurisdictional situations; ensure that international banks or bank holding companies do not escape comprehensive supervision by some “home” regulatory authority; and promote uniform capital requirements so banks from different countries may compete with one another on a “level playing field.” Although the Basel Committee’s recommendations do not themselves have force of law, G-10 countries have often implemented those recommendations as statutes or regulations.
9. Personal correspondence with the author.
10. These value-at-risk measures are described by Chew (1993).
11. Founded in 1978, the Group of 30 is a nonprofit organization of senior executives, regulators, and academics. Through meetings and publications, it seeks to deepen understanding of international economic and financial issues. Results of the Price Waterhouse study are reported in Group of 30 (1994).
12. This incident is documented in Shirreff (1992). See Corrigan (1992) for a full text of the speech.
13. The name “dollars-at-risk” appears as early as Mark (1991) and “capital-at-risk” as early as Wilson (1992).
14. The above discussion of RiskMetrics is based upon Guldimann (2000), the author’s own recollections, and private correspondence with Till Guldimann.

Chapter 2

1. It should be apparent from context when parentheses ( ) are being used to indicate an interval as opposed to an ordered pair.
2. It should be clear from context whether a prime indicates differentiation of a function as opposed to transposition of a vector or matrix.
3. Our name reflects the method’s similarity to the method of ordinary least squares. See Exercise 2.16.
4. In [2.61] if b = 0, set b = |b| = 0.
5. Consider the equation z² + 4z + 5 = 0. Factoring the left side, we obtain (z + 2 + i)(z + 2 − i) = 0, indicating the two solutions z = −2 − i and z = −2 + i. Now consider the equation z² − 4z + 10 = 6. Subtracting 6 from both sides and factoring, we obtain (z − 2)(z − 2) = 0. This has two solutions, but they coincide. We say that the equation has the repeated solution z = 2.
6. If the vertical-bar notation of [2.131] is unfamiliar to you, it is read as “evaluated at”, so the left side of the approximation indicates a derivative evaluated at a specific point x^[0].
7. See Dennis and Schnabel (1983) for a more sophisticated solution.

Chapter 3

1. For technical reasons, we should qualify [3.2] and say that it may fail to hold on a set of values for X of probability 0.
2. Technically, f must be measurable for f (X) to be a random variable.
3. The use of subscripts in the notation η₁ and η₂ for skewness and kurtosis is unfortunate because it can lead to confusion if subscripts are also employed to distinguish between different random variables. We use the notation because it is well established.
4. We could force uniqueness by defining the q-quantile as the supremum of all values satisfying the definition provided in the text.
5. An alternative would be to derive a mean vector based upon interest rate parity.
6. A set of vectors is orthonormal if they are orthogonal and normalized (e.g. of length 1).
7. Treatment of the noncentrality parameter is not standardized in the literature. Some authors define the parameter as in [3.114] but denote it simply δ. Others define the parameter differently, for example, taking a square root in [3.114] or dividing the sum by 2.
8. The gamma function is defined for any y > 0. It is related to the factorial function by Γ(y) = (y – 1)! for y ∈ .
9. See Stoyanov (1997) for more counterexamples relating to the joint-normal distribution.
10. We discuss random variate generation in Chapter 5.
11. We lose no generality by assuming Σ is positive-definite. If Σ were singular, we could perform dimensional reduction as described in Section 3.6.1 to obtain a positive-definite joint-normal random vector.
12. This and the analysis of Exhibit 3.28 were performed with the Monte Carlo method, which we describe in Chapter 5.
13. See Spanos (1999) for a detailed discussion including historical notes.
14. I am indebted to Arcady Novosyolov for this simplification.

Chapter 4

1. See Holton (2004) for an in-depth discussion of subjective vs. objective probabilities in the context of risk.
2. Usage of the term “sample variance” is inconsistent. Some authors define estimator [4.27] as the sample variance.
3. We consider only discrete processes. With continuous processes, t takes on real values.
4. Usage of the term “white noise” is not uniform. Some authors use the term to mean Gaussian white noise.
5. Such a realization can be constructed using techniques of random variate generation described in Chapter 5.

Chapter 5

1. Ulam and Teller were fierce rivals during their tenure at Los Alamos. Rota (1987) indicates that Ulam’s significant contribution to the design of the hydrogen bomb resulted coincidentally from his efforts to prove Teller’s design infeasible.
2. This incident is described in Eckhardt (1987).
3. W. S. Gossett, who published under the pen name “Student,” randomly sampled from height and middle-finger measurements of 3000 criminals to simulate two correlated normal distributions. He discusses this methodology in both Student (1908a) and Student (1908b).
4. Laplace had previously described the potential for statistical sampling to approximate solutions to nonrandom problems, including the valuation of definite integrals. See Chapter V of his Théorie Analytique des Probabilités and a 1781 memoir, both available in his published between 1878 and 1912.
5. See Eckhardt (1987) and Metropolis (1987) for historical accounts of this early work.
6. Metropolis and Ulam (1949).
7. Buffon communicated this problem to the Academy in 1733. See Todhunter (1865) for an historical account of Buffon’s work.
8. See Chapter V of his Théorie Analytique des Probabilités, available in his collected works published between 1878 and 1912.
9. Fox’s experiment is reported by Hall (1873).
10. The approximation may be too good. Gridgeman (1960) documents a number of historical implementations of the needle dropping experiment. He includes a statistical analysis of the plausibility of Fox’s reported results.
11. In constructing a realization of a sample of size m, we have m degrees of freedom. These allow us to simultaneously satisfy m independent conditions. However, the existence of infinitely many possible tests means there are infinitely many conditions to satisfy. With a continuous distribution, infinitely many of the tests will be independent.
12. The particular generator used in this analysis was the so-called DRAND48 linear congruential generator, which has parameters η = 2⁴⁸, a = 25,214,903,917 and c = 0. We discuss linear congruential generators next.
13. Lehmer (1951) considers the case c = 0. Obviously, if c = 0 and z^[k–1] = 0, then all subsequent values z^[k], z^[k+1], z^[k+2], … will equal 0. This can’t happen as long as 0 is not used as a seed and η is not divisible by a.
14. This is the name given to the generator in IBM’s System/360 Scientific Subroutine Package, Version III, Programmer’s Manual, 1968.
15. See, for example, Park and Miller (1988)
16. More generally, all that is required is that b not be divisible by η.
17. Knuth (1997, p. 103) indicates that the number of calculations for dimension n is on the order of 3ⁿ. Fincke and Pohst (1985) provide a more detailed complexity analysis.
18. See J. M. Hammersley and D. C. Handscomb (1964). Monte Carlo Methods, New York: John Wiley & Sons., p. 50.

Chapter 6

1. Time differences reflect periods when daylight savings time is nowhere in effect.
2. The two exchanges have an offset arrangement that allows an open contract on one exchange to be closed on the other, so the two exchanges’ contracts are truly fungible.
3. Open outcry trading for both contracts ends at 2:00 PM each day.
4. Trading closes in Singapore at 7:00 PM local time.
5. Log returns are used. Data from days when any of the exchanges were closed is omitted from the calculation. Results are inferred using the method of uniformly weighted moving averages (UWMA) discussed in Chapter 7.
6. The CSCE is a subsidiary of the New York Board of Trade (NYBOT).
7. A normalized delta is an option’s delta divided by the option’s notional amount. For vanilla options, normalized deltas are between –1 and 1.

Chapter 7

1. The effect is partially offset if short-term interest rates are more volatile than long-term interest rates.
2. Exponentially weighted moving average estimation had been used in time series analysis for some time. Zangari’s contribution is to propose its use in value-at-risk analyses.
3. A histogram of a time series can be treated as a realization of a sample from the unconditional distribution of the underlying stochastic process if the process is strictly stationary.

Chapter 8

1. Classic papers include French (1980), Gibbons and Hess (1981) and French and Roll (1986).
2. If there are no intervening nontrading days, a overnight loan is a loan that commences today and matures tomorrow. A tom-next (tomorrow-next) loan commences tomorrow and matures the next day. A spot-next loan commences in 2 days (spot) and matures the next day. Such loans are convenient for extending an existing loan by a day.

Chapter 9

1. For simplicity, we assume the portfolio is due to receive no fixed cash flows from caplets whose rate-determination dates have already passed.
2. Notation ⁰E( ) indicates an expected value conditional on information available at time 0. See Section 0.4. The vertical bar to the right of each partial derivative is read “evaluated at”, so both partial derivatives are “evaluated at ⁰E(¹R)”. See Section 2.2.4.
3. Any such portfolio would also have exposures to interest rates and implied volatilities. For this example, we treat these as constant.
4. Formula [9.30] defines an ellipsoid as long as ^1|0Σ (and hence ^1|0Σ^–1) is positive definite.
5. A trivial solution is to space the points at equal intervals about the equator of the sphere. This works in all cases and is perfectly symmetrical, but it is uninteresting for our purpose.
6. It is not uniquely defined. Two stable arrangements of 16 electrons are possible. However, one of these has a lower potential energy as defined by [9.35].
7. The algorithm is not intended to reproduce the exact motion of l – 1 electrons. To the precision dictated by our stopping condition, the result will be a locally minimum-energy configuration.
8. Corresponding put deltas are –.75, –.50, and –.25.
9. These may have negative or imaginary values due to roundoff error.
10. I am indebted to Craig Dibble, formerly of Bankers Trust, for bringing Garbade’s paper to my attention.
11. If the basis point covariances seem large, remember that they are based on data from the 1980s.
12. Because all eigenvectors have length 1, it is meaningful to directly compare variances of corresponding principal components.
13. In practice, we might not apply a principal-component remapping to eliminate just two dimensions. We apply the remapping here for practice.
14. For expositional convenience, we change our units of measure from the earlier example.

Chapter 10

1. In Chapter 3, we adopted the inverse CDF notation Φ^-1(q) for quantiles. This was because, if a random variable has unique quantiles, they equal corresponding values of the inverse CDF. A random variable has unique quantiles for q ∈ (0,1) if it is continuous with a PDF that is nonzero on some interval (which can be unbounded or all of ) and zero elsewhere. In essentially all value-at-risk applications, random variables ¹L and ¹P have conditional distributions that satisfy this criterion. Contrived exceptions are possible; consider, a portfolio composed entirely of expiring digital options.
2. Value-at-risk measures that employ these have sometimes been called delta-gamma value-at-risk measures, reflecting an assumption that the transformation procedure would be proceeded by a quadratic remapping based on delta-gamma approximations. The name is unfortunate because, as explained in Sections 9.3.6 – 9.3.7, quadratic ramappings should be based on less localized approximations, such as those obtained by interpolation or the method of least squares.
3. To clarify notation, in Section 1.8.1 we mathematically defined a portfolio as an ordered pair (⁰p, ¹P). Accordingly, notation (53600, ¹P) tells us that ⁰p = 53600.
4. This is worth emphasizing. For a given sample size and value-at-risk metric, standard error depends entirely upon the PDF of ¹P. The Monte Carlo transformation procedure works by constructing a realization of a sample for ¹P. The actual mechanics of how that realization is constructed are unimportant for standard error. Factors such as the composition of the portfolio, the number of key factors upon which it depends, or the portfolio mapping affect standard error only to the extent that they shape the PDF of ¹P. If we know the PDF of ¹P, we don’t need to consider these other factors. Understand this, and you will understand why the Monte Carlo method does not suffer from the curse of dimensionality.
5. Each Monte Carlo analysis was performed with sample size m = 1000. Standard errors for a sample size of m = 20000 were estimated by taking the sample standard deviation of the results and dividing by the square root of 20.

Chapter 11

1. Allen (1994) and Wilson (1994b) had already described variants of the approach. Also, as part of the public rollout of RiskMetrics, J.P. Morgan distributed a document entitled RiskMetrics – Directory of Products. This listed third-party vendor products that were compatible with RiskMetrics. One of the vendors, Sailfish, was indicated as offering historical simulation in addition to a value-at-risk measure styled on RiskMetrics.
2. Wilson (1994b).
3. Heron and Irving (1996).

Chapter 12

1. Such a crude value-at-risk measure would probably treat implied volatilities as constant. It could capture vega effects by modeling implied volatilities as key factors. In my work, I have come across a number of linear value-at-risk measures inappropriately applied to non-linear portfolios. All lacked the sophistication to model implied volatilities as key factors.

Chapter 14

1. Source: 2008 phone interview with Till Guldimann.
2. Values of n greater than 1 generally don’t come into play.
3. Since a continuous distribution is being used to approximate a discrete one, a case could be made that rounding the lower solution up and the higher one down would be more consistent with [14.8], but we present the test as Kupiec specified it.
4. They refer readers to Press et al. (1992) for a description of Kuiper’s statistic.
5. See Berkowitz and O’Brien (2002) and Pérignon, Deng and Wang (2008).

About the Author

Glyn A. Holton is an author and consultant specializing in financial risk management. He is known for his groundbreaking paper Defining Risk. He wrote the definitive book on value-at-risk and distributes the second edition of that book freely online. He blogs at GlynHolton.com.

Holton, Glyn A. (2014). Value-at-Risk: Theory and Practice, second edition, e-book published by the author at www.value-at-risk.net.

Holton, Glyn A. (2003). Value-at-Risk: Theory and Practice, San Diego: Academic Press.