Compound Sums and their Applications in Finance
Compound sums arise frequently in insurance (total claim size in a portfolio) and in accountancy (total error amount in audit populations). As the normal approximation for compound sums usually performs very badly, one may look for better methods for approximating the distribution of a compound sum, e.g. the bootstrap or empirical Edgeworth / saddlepoint approimations. We sketch some recent developments and indicate their relevance in finance. Second, we propos and investigate a simple estimator of the probability of ruin in the Poisson risk model, for the special case where the claim sizes are assumed to be exponentially distributed.
S. Asmussen (2000) Ruin Probabilities. World Scientific.
K. Croux & N. Veraverbeke ( 990) Nonparametric estimators for the probability of ruin. Insurance: Mathematics and Economics 9, 127-130.
P. Embrechts, C. Kluppelberg & T. Mikosch (1997) Mdelling External Events for Finance and Insurance. Springer.
B. V. Gnedenko & V. Y. Korolev (1996) Random Summation. CRC Press.
A. Gut (1983) Complete convergence and convergence rates for randomly indexed partial sums with an application to some first passage times. Acta Mathematica Hunganco 42,225-232.
R. Helmers (2000). Inference on rare errors using asymptotic expansions and bootstrap calibration. Biometrika 87, 689-694.
R. Helmers, B.Y. Jing, & W. Zhou (2002). Saddlepoint approximation for studentized Poisson compound sum with no moment conditions (in preparation).
R. Helmers, & B. Tarigan (2002). A symptotic expansions for negative moments of zero truncated Poisson random variables, with applications (in preparation).
R. Helmers, & B. Tarigan (2002). Statistical estimation in the Poisson risk model, part I (in preparation).
R. Helmers, & B. Tarigan (2002). On the Edgeworth expansion and the bootstrap approximation for a Studentized Poisson compound sum (in preparation).
C. Hipp (1989). Estimation and bootstrap confidence interval for ruin probabilities. Astin Bulletin 19, 57-70.
J.L. Jensen (1995) Saddlepoint Approximations. Oxford University Press, New York.
Reiss,R. -D. (1993) A Course on Point Processes. Springer Series in Statistics.
T. Rolski, H. Schmidli, V. Schmidt & J. Teugels (1999) Stochastic Processes for Insurance and Finance, Wiley, Chichester.