Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-07-07T14:29:48.423Z Has data issue: false hasContentIssue false

Maturity Guarantees Revisited: Allowing for Extreme Stochastic Fluctuations using Stable Distributions

Published online by Cambridge University Press:  10 June 2011

G.S. Finkelstein
Affiliation:
Ernst & Young, Rolls House, 7 Rolls Buildings, Fetter Lane, London, EC4A 1NH, U.K. Tel: +44 (0)20 7951 0176; Fax: +44 (0)20 7951 8010; E-mail:gfinkelstein@cc.ernsty.co.uk

Abstract

The paper examines the suitability of the stable family of distributions with the Maturity Guarantees Working Party's stochastic investment model (Ford et al, 1980). It then examines the effect of replacing the Gaussian assumption made by the working party with a more general stable distribution. It also explains how the appropriate stable distribution can be fitted.

Type
Sessional meetings: papers and abstracts of discussions
Copyright
Copyright © Institute and Faculty of Actuaries 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Akgiray, V. & Lamoureux, C.G. (1989). Estimation of stable-law parameters: a comparative study. Journal of Business & Economic Statistics, 7, 85.Google Scholar
Bachelior, L.J.B.A. (1900). Théorie de la speculation. Paris Gauthier-Villars. Reprinted in Cootner P.H. The random character of the stockmarket. Cambridge, Mass. MIT Press, 1964.Google Scholar
Bartels, R. (1978). Generating non-Normal stable variates using limit theorem properties. Journal of Statistical Computation and Simulation, 3, 199212.CrossRefGoogle Scholar
Becker, D.N. (1991). Statistical tests of the log Normal distribution as a basis for interest rate changes. Transactions of the Society of Actuaries, XLIII, 129.Google Scholar
Bergstrom, H. (1952). On some expansions of stable disturbances. Arkiv for Matematik, II, 375378.Google Scholar
Bienayme, J. (1853). Considerations à l'appui de la découverte de Laplace sur la loi de probabilité dans la méthode des moindres carrés. Comptes Rendus Academie des Sciences de Paris, XXXVII, 309–24 (esp. 321–23).Google Scholar
Blattberg, R. & Sargent, T. (1971). Regression with non-Guassian stable disturbances: some sampling results. Econometrica, 39, 3, 501510.Google Scholar
Box, G.E.P. & Jenkins, G.M. (1970). Time series analysis, forecasting and control. Holden-Day, San Francisco.Google Scholar
Carter, J. (1991). The derivation and application of an Australian stochastic investment model. Transactions of the Institute of Actuaries of Australia, 1, 315428.Google Scholar
Chambers, J.M., Mallows, C.L. & Stuck, B.W. (1976).A method for simulating stable random variables. Journal of the American Statistical Association, 71, 340344.Google Scholar
Chatfield, C. (1975). The analysis of time series, theory and practice. Chapman Hall.CrossRefGoogle Scholar
Cline, D.B.H. (1989). Consistency of least squares regression estimators with infinite variance data. Journal of Statistical Planning and Inference, 23, 163179.Google Scholar
Cline, D.B.H. & Brockwell, P.;. (1985). Linear prediction of ARMA processes with infinite variance. Stochastic Processes and their Applications, 19, 281296.CrossRefGoogle Scholar
Cramer, H. (1962). On the approximation to the stable probability distribution. Studies in mathematical analysis and related topics: Essays in honour of George Polya. Ed. Szego, G. et al. Stanford University Press, 7076.Google Scholar
Davis, R. & Resnick, S. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Annals of Statistics, 14, 2, 533558.Google Scholar
Dumouchel, W.H. (1971). Stable distributions in statistical inference. Unpublished PhD Dissertation, Yale University, Dept of Statistics.Google Scholar
Dumouchel, W.H. (1973). Stable distributions in statistical inference 1: symmetric stable distributions compared with other long tailed distributions. Journal of the American Statistical Association, 68, 469477.Google Scholar
Dumouchel, W.H. (1975). Stable distributions in statistical inference 2: information from stably distributed samples. Journal of the American Statistical Association, 70, 386393.Google Scholar
Fama, E.F. (1963). Mandelbrot and the stable Paretian hypothesis. Journal of Business, XXXVI.Google Scholar
Fama, E.F. & Roll, R. (1968). Some properties of symmetric stable distributions. Journal of the American Statistical Association, 63, 817836.Google Scholar
Fama, E.F. & Roll, R. (1971). Parameter estimates for symmetric stable distributions. Journal of the American Statistical Association, 66, 331338.Google Scholar
Feller, W. (1971). An introduction to probability theory and its applications, Vol II. Wiley, New York, 1966, 2nd edition.Google Scholar
Flelitz, B.D. & Smith, E.W. (1972). Assymetric stable distributions of stock price changes. Journal of the American Statistical Association, 67, 813814.Google Scholar
Flnkelstein, G.S. (1997). Unpublished Master of Business Science dissertation. University of Cape Town.Google Scholar
Ford, A., Benjamin, S., Gillespie, R.G., Hager, D.P., Loades, D.H., Rowe, B.N., Ryan, J.P., Smith, P. & Wilkie, A.D. (1980). Report of the Maturity Guarantees Working Party. J.I.A. 107, 101212.Google Scholar
Geoghegan, T.J., Clarkson, R.S., Feldman, K.S., Green, S.J., Kitts, A., Lavecky, J.P., Ross, F.J.M., Smith, W.J. & Toutounchi, A. (1992). Working party report on the Wilkie stochastic investment model. J.I.A. 119, 173228.Google Scholar
Gnedenko, B.V. & Kolmogorov, A.N. (1954). Limiting distributions for sums of independent random variables. Selected English Translations with Notes by Chung., K.L.Addison-Wesley Publishing Co., Reading, Mass.Google Scholar
Hannán, E.J. & Kanter, M. (1977). Autoregressive processes with infinite variance. Journal of Applied Probability, 14, 411415.Google Scholar
Holt, D.R. & Crowe, E.L. (1973). Tables and graphs of the stable probability density functions. Journal of Reasearch, National Bureau of Standards (USA), 77B, 143198.Google Scholar
Ibragimov, I.A. & Chernin, K.E. (1959). On the unimodality of stable laws. Theory of Probability and its Applications, 4, 4, 417419.Google Scholar
Kanter, M. (1975). Stable densities under change of scale and total variation inequalities. Annals of Probability, 3, 4, 697707.Google Scholar
Kanter, M. & Steiger, W.L. (1974). Regression and autoregression with infinite variance. Advances in Applied Probability, 6, 768783.Google Scholar
Kendall, M. & Stuart, A. (1977). The advanced theory of statistics. Charles Griffen & Co, Vol 1.Google Scholar
Klein, G.E. (1993). The sensitivity of cash-flow analysis to the choice of statistical model for interest rate changes. Transactions of the Society of Actuaries, XLV, 79186.Google Scholar
Koutrouvells, I.A. (1980). Regression-type estimation of the parameters of stable laws. Journal of the American Statistical Association, 75, 918928.Google Scholar
Koutrouvelis, I.A. (1981). An iterative procedure for the estimation of the parameters of stable laws. Communications in Statistics — Simulation and Computation, 1728.Google Scholar
Leitch, R.A. & Paulson, A.S. (1975). Estimation of the stable law parameters: stock price behaviour applications. Journal of the American Statistical Association, 70, 690696.Google Scholar
Lévy, P. (1924). Théorie des erreurs. La loi Gauss et les lois exceptionelles. Bull. Soc. Math. France, 52, 4985.Google Scholar
Lévy, P. (1925). Calcul des probabilités. Gauthier-Villars, Paris.Google Scholar
Mandelbrot, B, (1963). The variation of certain speculative prices. Journal of Business, XXXVI, 395419.Google Scholar
McCulloch, J.H. (1986). Simple consistent estimators of stable distribution parameters. Communications in Statistics — Simulation, 15(4), 11091136.Google Scholar
Miamee, A.G. & Pourahmadi, M. (1988). Wold decomposition, prediction and parameterization of stationary processes with infinite variance. Probability Theory and Related Fields, 79, 145164.Google Scholar
Paulson, A.S., Holcomb, E.W. & Leitch, R.A. (1975). The estimation of the parameters of the stable laws. Biometrika, 62, 163170.Google Scholar
Praetz, P.D. (1972). The distribution of share price changes. Journal of Business, 45, 49.Google Scholar
Press, S.J. (1972). Estimation in univariate and multivariate stable distributions. Journal of the American Statistical Association, 67, 842846.Google Scholar
Ripley, B.D. (1987). Stochastic simulation. John Wiley and Sons.Google Scholar
Skorokhod, A.V. (1954). Asymptotic formulas for stable distributions. Doki. Akad. Nauk. SSSR, 98, 731735. Also in: Selected Translations in Mathematical Statistics and Probability. American Mathematical Society, 1, 1961, 157–161.Google Scholar
Skorokhod, A.V. (1954). On a theorem concerning stable distributions. Usephi. Mat. Nauk. (N.S.) 9, 189190. Also in: Selected Translations in Mathematical Statistics and Probability. American Mathematical Society, 1, 1969, 169–170.Google Scholar
Tenenbein, A. (1988). After the crash: statistical implications. The Actuary (U.S.A. Society of Actuaries), Feb 1988.Google Scholar
Thomson, R.J. (1994). A stochastic investment model for actuarial use in South Africa. Transactions of the Actuarial Society of South Africa, Annual Convention.Google Scholar
Tilley, J.A. (1990). A stochastic yield curve model for asset/liability simulations. Proceedings of the 1st AFIR Colloquium.Google Scholar
Tilley, J.A. & Mueller, M. (1991). Managing interest rate risk for long term liabilities. Proceedings of the 2nd AFIR Colloquium.Google Scholar
Walter, R.A, (1990). Lévy-stable distributions and fractal structure on the Paris market. Proceedings of the 1st AFIR Colloquium.Google Scholar
Wiener, R.A. (1975). Parameter estimation for symmetric stable Paretian distributions. Technical Repon No. 77, Ser 2. Princeton University, Dept of Statistics.Google Scholar
Wllkle, A.D. (1984). Steps towards a comprehensive stochastic investment model. Occasional Research Paper No. 36, Institute of Actuaries.Google Scholar
Wllkie, A.D. (1986). A stochastic investment model for actuarial use. T.F.A. 39, 341373.Google Scholar
Wilkie, A.D. (1995). More on a stochastic investment model for actuarial use. B.A.J. 1, 777945.Google Scholar
Yohai, J. & Maronna, R.A. (1977). Asymptotic behaviour of least-squares estimates for autoregressive processes with infinite variances. Annals of Statistics, 5(3), 554560.Google Scholar
Zolotarev, V.M. (1954). Expression of the density of a stable distribution function with exponent a greater than one, by means of a frequency with exponent less than –167.Google Scholar
Zolotarev, V.M. (1956). On analytical properties of stable laws. Vestnik Leningrad University, 11, 4952. Also in: Selected Translations in Mathematical Statistics and Probability. American Mathematical Society, 1, 1961, 207–211.Google Scholar
Zolotarev, V.M. (1966). On the representation of stable laws by integrals. Selected Translations in Mathematical Statistics and Probability, American Mathematical Society, 6, 1966, 8488.Google Scholar