THE FAST FOURIER TRANSFORM ALGORITHM IN RUIN THEORY FOR THE CLASSICAL RISK MODEL

Transcription

1 y y THE FST FORIER TRNSFORM LGORITHM IN RIN THEORY FOR THE CLSSICL RISK MODEL Susan M. itts niversity of Cambridge bstract We focus on numerical evaluation of some quantities of interest in ruin theory, and on the practical use of the fast Fourier transform algorithm () in this context. We discuss the general application of the for stochastic models, and we illustrate this by looking again at the probability of ruin in the classical risk model and by extending this approach to evaluation of the first moment of the time to ruin in the classical model. I. RIN QNTITIES IN THE CLSSICL RISK MODEL In the classical risk model in insurance mathematics, claims arrive in a oisson process, rate, and claim sies are independent identically distributed random variables, independent of the claim arrivals process. Let denote the claim-sie distribution function, and let,, assumed finite as necessary. remiums arrive linearly in time at rate, and we write! "#%$&' (, where &, the premium loading factor, is assumed to be stricly positive. The surplus at time ) is *+",) where - CED is the initial surplus, and F",) ' is the number of claims that have arrived by time ). Then ruin is said to occur if *+",) ' ever becomes negative. Let GH"I-J'K LNMO QR)TS9*+",) ' VDW be the time of ruin. We have GX"I-J'TY[Z, with GX"I-J'\]Z if ruin does not occur. The assumptions of the above model imply that ^_"`GH"I-J'[Za'bc (see smussen (), page 6), and so the probability of (ultimate) ruin, de"i-j'9f^_"`gh"i-j'%aza', is strictly smaller than one. The ruin-time moments are dgh"i-j'i. GH"I-J' j"`gh"i-j'%fza', for kdl m on, where j",p' is the indicator function of an event p, and d ",-J'qrde"I-J'. Dalbaen (99) shows that the ruin time has finite "st$uv' st moment if and only if w/z. In this paper, we motivate and briefly describe the use of the fast Fourier transform algorithm () for some applied probability calculations, and we illustrate how this lends itself to some calculations in ruin theory for the classical risk model s examples, we consider the probability of ruin dx",-j' and the first ruin-time moment dq ",-J'. For further and more detailed discussion, see itts (5). There is a large body of literature on ruin theory, including many papers on the numerical evaluation of various ruin theoretic quantities (see, for example, Goovaerts and De Vylder (93), Ramsay (99), Ramsay and sabel (997) and Dickson and Waters ()). The use of the for the probability of ruin is discussed in Embrechts, Grübel and itts (993). Methods for reducing various types of error that occur in the application of the are developed in Grübel and Hermesmeier (999, ). In the present paper, we illustrate the naive application of the using off-the-peg routines in commonly available software packages. II. DEFECTIVE RENEWL EQTIONS Many ruin quantities can be shown to satisfy a defective renewal equation such as "I-J'T{3 } ~o ",- K J' k"i m J'/$ ƒj",- ' - Dl ()

2 y : & - : where y " ' is the unknown object of interest, is a known function on Dl Za'. Let L ", J', and assume D( a. Then is a defective probability distribution function, and ƒ is, where f is a proper probability distribution function. Then the solution to () is "I-J'T % } ~o ƒj",-k J' k"i h J' where k"` J'k?B " J' and is a compound geometric distribution (see Willmot and Lin (), Chapter 9). The probability of ruin dx",- ' satisfies the defective renewal equation x", J'k dx",-j' q$ } ~o de"i- K J' H } "I m J'X$ ", J', we have that } ", J' "` J' () } "I-J' (3) q$ & Ht",) ' m) is the claim-sie equilibrium where, writing distribution, a proper probability distribution function (Willmot and Lin ()). fter a few algebraic manipulations, the solution () gives the ollaceck Khinchine formula (smussen (), III.), with w B" q$ &m', de"i-j'b!k",-j' where?b " " ' Lin and Willmot () show that d ",- ' satisfies the defective renewal equation d_ "I-J' %$'& } ~o d_ ",- K J' } " m J'/$ ( ~ de", J' h H With ƒ ",-J' ( ~ de", J' h, and as in (4), we obtain d "I-J'b } ~ ƒ! Convolutions are involved, both for the compound geometric, and for ƒ*)+ an obvious tool for calculation of the required quantities. # } (4) ",- K J' k"i m J' (5) in (), and so the is III. THE FST FORIER TRNSFORM IN LIED ROBBILITY Let, be a discrete random variable concentrated on Q Dl.- f W for some positive integer -, with ^_"/, B', D.-, and with distribution function, say. The characteristic function of, is 34_",) '.5 5 :76 4_"I 9B' );:= and this is the discrete Fourier transform H",) ' of i "? ', where DC H",) 'b Given the values of can be found using the inversion formula?b 5 )*:E ",) ' at the - Fourier frequencies )GF {nihkjlm-, J fdl.- E, the original s DC F?B "`)/F ' 5 In applications, the non-negative random variable, may have a density (and so not be discrete). Then we approximate the distribution of, by a discrete distribution, and one way to do this is to put where } Fw Qe"RJx$ DTSm' Qeo"RJ\ DlVSm' J m C :ON } QeWKmn is a small postive discretisation parameter, which we must choose. The discretised distri- to the bution assigns to the non-negative integer J a probability mass equal to that assigned by X

3 " interval ""RJk DlVSm'3 v"rj $udlts'. The next step is to choose the truncation parameter -, so that } } } DC ' is used as the input array for the. For efficient use of the, we choose - and to reduce the discretisation error, and also the aliasing error. t the very least, this means choosing - and such that 4 "@- DlVSm' is negligible, but see also Grübel and Hermesmeier (999, ). to be a power of n. We need to choose - 3. The probability of ruin IV. SOME LICTIONS IN RIN THEORY We define the tail measure associated with a finite non-negative measure on the Borel sets of Dl Za', by ",p'] ",) Za' h). The measure has a density ",) oza'. If is the claim-sie probability measure, then the the claim-sie equilibrium distribution is given by the measure. Hence, the equilibrium distribution is continuous with density t" ', ie it is not discrete and will always need to be discretised, whether or not the original claim-sie distribution is discrete. For discretisation parameter, let }Fw vo"rj\ DlVSm'3 v"rje$ DlVSm' #} } +o" Kmn.Kmn For some claim-sie distributions (for example, the exponential distribution), an expression for the }F s can be found explicitly. When this is not possible, another approach is to evaluate the above expressions when is replaced by the distribution function, say, belonging to the discretised version of g. Then we approximate } " Je$ DlVSm' } " J/ DlTS' by }Fw F$ F C "` J' m u n where is the mass assigned to by. We introduce the notation for the discrete tail operation, so that, if is the array " DC ' and is the array " } #} DC ', then we write " '. To calculate an approximation to the compound geometric distribution, we first apply the to the array to obtain the array "!R" ', say. Then we calculate the array #%$'& "#%! J'B"#! )(* +!v" '', where ( F $ denotes multiplication, and the operations are carried out pointwise. The inverse is then applied to #,$'& to give an array # "-# # DC ', where # F is an approximation to the mass assigned by to the interval "RJbEDlVSm'3 v"rj $ DlVSm'. Then, by (4) an approximation to the ruin probability is given by d F g 4. F #, and this is an approximation to d " J $ DTSm'. The above procedure is easily F } implemented in software packages such as S-lus or R, using just a few lines of code. We illustrate this for the areto distribution with tail probabilities given by $J x", J' B"I $i J', CaD, so that the equilibrium distribution has tail probabilities given by } ", J'b B"I!$+ J', CaD. (see, Goovaerts and De Vylder (93), Ramsay (99, 3), Ramsay and sabel (997)). In this case, the discretised equilibrium distribution has an easy explicit expression, }Fw / $ F C $ $ F$ 34 Jk We apply this with and & Dl=, and the results are given in Table I, together with the values obtained in Ramsay and sabel (997) and Ramsay (3). J+rD n

4 ~ & ~ 4 Ramsay and sabel Ramsay TBLE I ROXIMTIONS TO RIN ROBBILITIES WITH RETO DISTRIBTED CLIM SIZES.! Dickson and Waters () #" #" TBLE II ROXIMTIONS TO THE FIRST CONDITIONL MOMENT %$ OF THE RIN TIME WITH RETO DISTRIBTED CLIM SIZES. B. The first moment of the time of ruin From (4), we see that dx",-j' is a density of the measure '& )(, where ( is the compound geometric probability measure with distribution function. Further, the function ƒ "I-J' is a density of )* ( &. Then, from (5), we see that dq "I-J' is a density of & WC,+ * ( )(. Starting with the array # from the calculation of d, we calculate the array -/. Lx "-#l', which is an approximation to the measure &E (. Next, we find ƒ ( "-.L`'. The convolution ƒ ( # can be carried out via, and then -.LsE WC,+ ƒ ( # is an approximation to &, so that -/.L.F43 & " J DTSm'3 v" J/$aDlVSm'. n approximation to dq " JL ' is given by d_ }F we have from () and () d " Dh'b ƒj" Dh' k"idh' % {ƒj"idh't -/.LI F, for Jk. For d_ " Dm', we note that, since k"idh', de", J' h " ", J' m f where.b" nb" % J' ' is the expectation of the compound geometric distribution. n " J' Combining approximations to dq "I-J' and de"i-j' gives an approximation to the conditional first moment & d65 "I-J'b GX"I-J'7 GH"I-J' Z. Dickson and Waters () consider d5 ",-J' for a areto distribution with tail t"` J'! 9 B" 9 $ J' ;:, EC D, with =, 9?, and with &wvdl and ]m=. Table II shows the results for - ;?@Dhn and ifdl Dhn, and for - SmnSmnBB and irdl Dhn, together with the numerical approximations in Dickson and Waters (). REFERENCES smussen, S. () Ruin robabilities. World Scientific, Singapore. Dalbaen, F. (99) remark on the moments of ruin time in classical risk theory. Insurance: Mathematics and Economics Dickson, D.C.M., and Waters, H.R. () The distribution of the time to ruin in the classical risk model. STIN Bulltin

5 Embrechts,., Grübel, R. and itts, S.M. (993) Some applications of the fast Fourier transform algorithm in insurance mathematics. Statistica Neerlandica Goovaerts, M. and De Vylder, F. (93) stable recursive algorithm for evaluation of ultimate ruin probabilities. STIN Bulletin Grübel, R. and Hermesmeier, R. (999) Computation of compound distributions I: aliasing errors and exponential tilting. STIN Bulletin Grübel, R. and Hermesmeier, R. () Computation of compound distributions II: discretiation errors and Richardson extrapolation. STIN Bulletin Lin X.S. and Willmot, G.E. () The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance: Mathematics and Economics itts, S.M. (5) The fast Fourier transform algorithm in ruin theory. In preparation. Ramsay, C.M. (99) Improving Goovaerts and De Vylder s stable recursive algorithm. STIN Bulletin Ramsay, C.M. (3) solution to the ruin problem for areto distributions. Insurance: Mathematics and Economics Ramsay, C.M. and sabel, M.. (997) Calculating ruin probabilities via product integration. STIN Bulletin Willmot, G.E. and Lin, X. S. () Lundberg pproximations for Compound Distributions with Insurance pplications. Springer, New York. 5