ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY DEALING IN A BS-MARKET

Transcription

1 Teor Imov rtamatemstatist Theor Probability and Math Statist Vip 74, 6 No 74, 7, Pages 11 3 S 94-97)693-X Article electronically published on June 5, 7 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY DEALING IN A BS-MARKET UDC 5191 A V BAEV AND B V BONDAREV Abstract We study a mathematical model of an insurance company that shares its capital by investing it in both stocks and bonds The basic tool to describe the evolution of the stock price is the Ornstein Uhlenbeck process We construct an estimate for the ruin probability of an insurance company as a function of the initial capital The distribution of the capital between stocks and bonds is found for which this estimate is minimal 1 Introduction We study a mathematical model of an insurance company that invests a part of its capital in stocks and the rest in bonds The basic tool to describe the evolution of the stock price is the Ornstein Uhlenbeck process it is shown in 1] that this is an arbitragefree model) We assume that a constant fraction of the capital is invested in stocks; this constant is chosen to minimize the probability of ruin of the insurance company over a certain time interval It is well known that if an insurance company does not deal in a BS-market, then the Lundberg estimate holds for the probability of ruin of the company over an infinite time interval; namely, P inf ξ t e Rξ t<+ where R> is a nonzero solution of the equation λ e xy df y) =cx+λ Some questions related to the probability of ruin of an insurance company dealing in a BS-market are studied by A Melnikov for the cases where the evolution of a risky asset is described by classical models Melnikov 1] studies the probability of ruin of an insurance company for two cases, namely if either 1) the company invests the capital to bonds only, or ) its investment is a mixture The evolution of the stock price is described by the Wiener process in 1]; that is, the Samuelson model is treated in 1] The following results are obtained in the present paper We construct an estimate for the ruin probability of an insurance company as a function of its initial capital and find a constant distribution of the capital between investments in stocks and in bonds for which the estimate is minimal Some remarks concerning the model An investor purchasing both risky and nonrisky assets is said to deal in a BS-market An insurance company is treated as an investor in this paper In what follows we assume Mathematics Subject Classification Primary 6E15, 6H5, 6H3; Secondary 6P5 Key words and phrases Poisson measure, stochastic integral, investor s portfolio, ruin probability 11 c 7 American Mathematical Society

2 1 A V BAEV AND B V BONDAREV that the law of the evolution of a bank account process B t is given by 1) db t = rb t dt where B is the initial capital invested in bonds We consider the model ) S t = S exp µt σ t + ση t to describe the evolution of the stock price see 1]) where η t, t, is the Ornstein Uhlenbeck process; that is, 3) dη t = γη t dt + σdw t, η = The Samuelson model 4) S t = S exp µt σ t + W t is widely used, 3] for this purpose Nevertheless we propose a different model that eliminates some of disadvantages of the Samuelson model Below we mention a few of them First, the price S t of a risky asset in the Samuelson model is represented as a solution of the stochastic differential equation 5) ds t = S t µdt+ σdw t ], and thus the local return at the moment t, t ; T ], is such that S t+ t S t 6) lim = µdt+ σdw t t S t We see that the local return contains two terms being of different nature, namely µ t is nonrandom, while σw t+ t W t ] are independent random variables The assumption that the changes in the price are independent is not realistic from the point of view of practice Our idea is to substitute the Ornstein Uhlenbeck process instead of the Wiener process W t in the model 4) Second, it follows from 5) that the total return of a stock over an interval ; T ]is T ds t 7) = µt + σw T, S t whence T ds T t ds t 8) E = µt, Var = σ T S t S t This means that the variance of the return coefficient of a stock over an interval ; t] increases with t On the other hand, this phenomena is not detected by statistical analysis of several types of stocks Moreover one can observe an irregular behavior of the square deviation of the return Using ) and 3) we get t 9) S t = S exp Let Then or ds e χ t = S e χ t dχ t = ) µ γη s σ ds + σ t ) µ γη t σ dt + σdw t dw s ) µ γη t σ dt + σ S e χ t dt + σs e ξ t dw t ds t = S t µ γη t ) dt + σdw t ],

3 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY 13 whence the return coefficient is given by T ds T T T = µdt+ γη t dt + σdw t ]=µt + η T, S T while its mean value is T ds t E = µt S t Thus the variance of the return is equal to Var T ds t S t =Varη T = σ 1 e γt ] ; γ that is, the variance of the return approaches σ /γ and does not increase with t It is proved in 1] that ) is an arbitrage-free model and thus it can be used to describe the evolution of the price of a risky asset 3 Balance equations The setting of the problem The following problem is considered in this paper An insurance company, as an investor, divides the capital in a certain proportion A fraction u 1 of the capital is used to purchase stocks, while the rest 1 u 1 is used to purchase bonds The corresponding amounts of the capital are uξ t and 1 u)ξ t, respectively If the price of one share of the stock at the moment t is S t, then the insurance company purchases uξ t /S t shares for uξ t dollars, while the rest of the capital 1 u)ξ t is used to purchase bonds At the moment t + t, a share of the stock will cost 1) S t+ t S t 1 + µ t γη t t + σ W t ) according to model ), while any bond will cost 11) 1 u)ξ t 1 + r t) The wealth of the company at the moment t + t consists of two parts, namely of uξ t 1) S t+ t = uξ t S t 1 + µ t + σ W t )=uξ t 1 + µ t γη t t + σ W t ) S t S t due to the investment in stocks and of 13) 1 u)ξ t 1 + r t) due to the investment in bonds We assume in what follows that the premium rate of the insurance company is constant and equals c> The accumulated claim process in the interval t, t + t) is modelled by a compound Poisson process ς t and is given by 14) ζ t = Z t+ t k=z t +1 where Z t is a standard Poisson process with parameter λ>, and the claim sizes ς k are nonnegative identically distributed random variables that do not depend on Z t It is well known see, for example, 4, 5]) that the random variable Z t k=1 ς k k=1 ς k =) can be represented as the stochastic integral with respect to the Poisson measure More precisely, Z t t 15) ς k = yνdy, ds) k=1 ς k

4 14 A V BAEV AND B V BONDAREV where νa, t) is a Poisson measure such that λt A 16) P νa, t) =i = df x)] i ] exp λt df x) i! A Here Pς k <x = F x) ande νa, t) =λt df x) In what follows we assume that the A measure νa, t) and Wiener process W t are independent and either c aλ or c>aλ We have 17) ξ t+ t = uξt 1 + µ t γη t t + σ W t )+1 u)ξ t 1 + r t)+c t yνdy, t) in view of equalities 1) 15) where νa, t) =νa, t + t) νa, t) Using 17) we obtain the balance equation for the model ): ) 18) dξ t = ξ t uµ +1 u)r uγηt dt + uξt σdw t + cdt yνdy, dt) We seek a number u 1 such that the estimate of the ruin probability for an insurance company over the interval, + ) is minimal The estimate should approach zero as ξ + the zero load c aλ is also possible) 4 Solution of the balance equation Consider the process see 6]) ξt =exp uµ +1 u)r] t + u σ t uη t 19) t =exp uµ +1 u)r] t + γu η s ds + u σ t uσw t whose stochastic differential is given by ) dξt = ξt uµ +1 u)r uγηt ] dt + u σ ) dt uσ dw t If then θ t = ξ t ξ t, dθ t = ξt ξt uµ +1 u)r uγη t ) ) dt + uξ t σdw t + cdt yνdy, dt) + ξ t ξt uµ +1 u)r uγηt ] dt + u σ ) dt uσ dw t u σ ξt ξ t dt ) = ξt cdt yνdy, dt), whence 1) ξ t = ξt ] 1 ξ + t ξsc aλ) ds ] y νdy, ds) where νa, t) =νa, t) E νa, t), a = ydfy) =E ς k,andc aλ

5 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY 15 5 Moment inequalities for the stochastic integral over a Poisson measure The following result is well known 7]: if ν is the jump measure of a process, π is its compensator, µ = ν π, and ϕ C<+, then ) E ξt p p/ cp E ξ + C p ξ ) for the process 3) ξt = and all p 3where 4) t t ϕu, s) µdu, ds) t ξ t = ϕ u, s) πdu, ds) The condition ϕ C<+ does not hold in a number of cases; however, the moments of order m of characteristic 4) are bounded for some integer m> Let t ξ t = αξs νdα, ds) where νa, t) =νa, t) λt df x) A Here νa, t) is a Poisson measure with mean λt A df x)andξ t is defined by relation 19) Theorem 1 Let µ>r> If for some integer m>, then 1 5) sup E ξτ m b m t<+ E ς k ] m < + λc m d ) m 1+ t d λc m ] mm 1) where b =E ςi m ] 1/m) and r + µ r) σ d = 1+m), µ r] σ 1+m) < 1, r + σ 1+m) µ r], σ 1+m) 1 Proof Applying the generalized Itô formula6]weget E t ξt m = E ξs + αξ ) m ) m ) ] m 1 s ξs m ξs α ξ s λfdα) ds t m 6) = Cm k E ξs ) m k ) αξ k s λfdα) ds k= t m λ Cm k E ξ ) m m k)/m) s α k E ξs] m ) k/m) F dα) ds Since k= Var η t = σ 1 e γt ] γ 1 Editorial Note In what follows, the authors use the C-notation for binomial coefficients; eg, Cm k = m) k

6 16 A V BAEV AND B V BONDAREV and the random variable η t η = σ γ 1 e γt ] has the standard Gaussian distribution, it follows that E ξ ] m t = E exp muµ +1 u)r]t +m u σ t muη t =exp muµ +1 u)r]t +m u σ 7) t E exp muη t =exp muµ +1 u)r]t +m u σ t +m u σ γ 1 e γt] exp muµ +1 u)r]t + mu σ t1 + m) Relation 6) implies that 8) E t ξt m m λ Cm k E ξs m ) m k)/m) ] E ς m k/m) i E ξs] m ) k/m) ds in view of k= ] k/m) α k F dα) α m F dα) = E ς m ] k/m) Now we find < ū 1 minimizing the right hand side of 7) It is easy to see that µ r]+u σ 1 + m) = Thus the right hand side of 7) attains its minimum at the point u µ r] = σ 1 + m) We take ū = u if u < 1 In this case, the minimum of the right hand side of 7) is 9) E ] ξt ] m µ r) exp m r + σ t 1 + m) If u 1, then µ r σ 1 + m) andweget µ σ 1 + m) r>, µ σ 1 + m) r + σ 1 + m) Thus the right hand side of 7) attains the minimum at ū = 1, which corresponds to the investment of the whole capital to risky assets Note that the minimum does not exceed the bound ] exp m r + σ 1 + m) t Therefore E ] ξt ] m exp m r + σ 1 + m) t The latter relation together with 9) implies the bound where E ξ t ] m exp mdt r + µ r) σ d = 1+m), r + σ 1+m), µ r] σ 1+m) < 1, µ r] σ 1+m) 1 i

7 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY 17 It follows from 6) that 3) Let sup E ξτ m τt λ t m Cm k sup τs k= m λcm k sup τt k= m λcm k sup τt k= Then relation 3) implies 31) x m Since C k m m k= C m Ck m E ξ m τ E ξ m τ ) m k)/m) ] E ς m k/m) exp kds ds ) m k)/m) E ς m i i ] k/m) 1 exp dkt dk E ξ ) m m k)/m) ] τ E ς m k/m) 1 i dk x = sup E 1/m) ξτ m) τt λcmx k m k b k 1 dk b λcm 1 d = we derive from 31) that 3) x m b λcm 1 d m!!m )! k )! m k)! k!m k)! m!m )! = b λcm 1 d = b λcm 1 m k= m i= d xm C k m xm k ) b k m Cm k xm k ) b k k= C k m C m Ck m =! 1, k, k 1) Cm x i m i b i = b λcm 1 b + x)m d 1+ x) b m If x>, we deduce from 3) that x b λcm 1 1+ b m, d x) whence λc 33) x b m 1+ b m 1 d x) λcm Consider the graphs of the functions y = x and y = b d 1 + b x )m 1 see Figure 1) Let R> be a solution of the equation λc x = b m 1+ b m 1 ; d x) that is, λc 34) R = b m 1+ b ) m 1 d R

8 18 A V BAEV AND B V BONDAREV λcm b d y = x λcm y = b d 1 + b x )m 1 R x Figure 1 Then x R is a solution of inequality 33) It is easy to see that λc R>b m d The function λc y = b m 1+ b ) m 1 d x is increasing; thus ) m 1 λc 35) R<b m d 1+ d 36) sup E ξτ m b m t<+ λc m d λc m as x This implies the following bound: ) m 1+ d λc m ] mm 1) Theorem 1 is proved Remark It is easy to see that ] m 1) d 1+ ] m 1) d = 1+ λmm 1) that is, λc m b m λc m d as m + ) m 1+ ] d = 1+ λmm 1) d λc m ] mm 1) λmm 1) 4m 1) d d λmm 1) dλ m + e 1 ; ) m λmm 1) b e dλ 1 d 6 Inequalities for the ruin probability of an insurance company over the time interval, + ) We distinguish between the three cases: ū =,ū = u,andū = 1 that correspond to different investment strategies

9 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY 19 We follow the idea of 9] Let ū = This means that the entire capital is invested in bonds Assume that c aλ > We have 37) ξ t =exp rt 1 Let 38) ψt, z) =λ t ] e zαξ s 1 zαξ s F dα) ds where z>andξ s > We also consider the function 39) ϕβ) =λ e βα 1 βα ] F dα) βc aλ) for β It is easy to see that ϕ) = Since ϕ β) =λ α e βα 1 ] F dα) c aλ), t zξ sc aλ) ds we have ϕ ) = c aλ) < This means that the function ϕβ) starts from zero and decreases in particular, ϕ is negative for some period of time) It is also obvious that the derivative ϕ β) =λ α e βα 1 ] F dα) c aλ) is positive after some β> Thus the function ϕβ) hasarootr different of ; that is, Then λ e Rα 1 Rα ] F dα) Rc aλ) = 4) ϕβ) =λ e βα 1 βα ] F dα) βc aλ) for <β R Inequalities 37) and 4) imply that ψt, z) forz R Let z = R Considering 1) we get P inf ξ t tt ] t )] = P inf ξ 1 t ξ + c aλ) ds y νdy, ds) 41) = P = P tt inf tt sup tt ξ + R P sup exp R tt e Rξ t ξs t ξ s c aλ) ds )] y νdy, ds) ξsy νdy, ds) t ] λ e Ryξ s 1 Ryξ s ξsy νdy, ds) t ] λ e Ryξ s 1 Ryξ s t ] F dy) ds Rξ ] F dy) ds e Rξ

10 A V BAEV AND B V BONDAREV The latter inequality follows from the Kolmogorov inequality, since the stochastic process t t ] ] ρt) =exp R ξsy νdy, ds) λ e Ryξs 1 Ryξ s F dy) ds is a martingale If ū = u or ū =1inthecaseofc aλ, then P inf ξ ] t t = P inf ξ 1 t ξ + ξs c a) ds tt tt = P inf = P E ξ tt sup tt ξ + t ξs c a) ds t ] ξsy νdy, ds) T E ν ξ s y νdy, ds) m ξ + T ξ sc aλ) ds )] y νdy, ds) )] y νdy, ds) T ξ + ξsc aλ) ds T E ξ E ν ] m ξ s y νdy, ds) m ξ ] m, since the Wiener process W t and Poisson measure νa, t) are independent Taking 5) into account we obtain P inf ξ t tt T E ξ E ν ξ s y νdy, ds) m ξ ] m 4) ) b m λc m ] ) mm 1) m d 1+ d λcm ξ ] m r + µ r) σ where d = 1+m), µ r] σ 1+m) < 1, r + σ 1+m) µ r], σ 1+m) 1 Passing to the limit as T + we derive from 41) that P inf ξ t = lim P inf ξ t e Rξ t<+ T + tt see 8]), while 4) implies that P inf ξ t t<+ = lim P T + b m λc m d Thus we have proved the following result inf ξ t tt ) m 1+ d λcm ] ) mm 1) ξ ] m Theorem Let ξ be the initial capital of an insurance company dealing in a BSmarket Let the evolution of a risky asset be described by relation ) Let <r<µ where r is the bank interest Let the accumulated claim process be described by the compound Poisson process 14) where λ> is the intensity of a standard Poisson process Z t modelling the number of claims over the interval,t] Put F x) =Pς k <x where <ς k are independent identically distributed random variables that do not depend on the Poisson process Z t and have finite exponential moments Put E ς k = a Then

11 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY 1 1) if c>aλand the entire capital is invested in bonds, that is ū =, then the ruin probability admits the following estimate: 43) P inf ξ t e Rξ t<+ where R> satisfies the equation 44) λ e Rα 1 Rα ] F dα) Rc aλ) =; ) if E ςi m < +, c aλ, andµ r]/σ 1 + m)) 1, that is, the entire capital is invested in stocks and ū =1, then the ruin probability admits the following estimate: 45) P inf ξ t t<+ where 46) d = r + σ 1 + m) ) b m λc m ] ) mm 1) m d 1+ d λcm ξ ] m, b = E ςi m ] 1/m) ; 3) if E ςi m < +, c aλ, andµ r]/σ 1 + m)) = u < 1, that is, ū = u and the fraction ū 1% of the initial capital is invested in the risky asset and the rest of the capital is used to purchase bonds, then the ruin probability can be estimated as follows: 47) P inf ξ t t<+ where 48) d = r + ) b m λc m ] ) mm 1) m d 1+ d λcm µ r) σ 1 + m), ξ ] m b = E ςi m ] 1/m) 7 Concluding remarks It is well known that if an insurance company invests all the capital in bonds, then the Lundberg bound holds for the ruin probability over an infinite interval: 49) P inf ξ t e Rξ t<+ where R> is a nonzero solution of the equation λ e xy df y) =cx + λ Some similar problems are considered in 1] for an insurance company dealing in a BS-market for the case where the evolution of the price of a risky asset is described by the classical models The Wiener process is the basic tool used in 1] to describe the dynamics of the price of a risky asset; that is, the Samuelson model is studied in 1] The probability of ruin of an insurance company is considered in the paper 1] for the case where either the entire capital is invested in bonds or the capital is divided for mixing investments It is shown in 9] that if the capital is invested in bonds, then the Lundberg bound 49) holds In the general case as well as in the case of the Samuelson model, inequality 49) is not yet proved The case considered in this paper is of a more complicated nature, since the underlying stochastic process describing the evolution of the capital of a company is more complicated

12 A V BAEV AND B V BONDAREV One of the important problems for the above model is the estimation of the probability of ruin of an insurance company for which the constant c>isequalto aλ = λ E ς k, which corresponds to the zero load If µ r ū = σ < 1, m =, 1,, 1 + m) µ r that is, the fraction σ 1+m) 1% of the capital is invested in the risky asset and the rest is invested in bonds, then E ξt ] 1 = E exp ūµ +1 ū)r] t ū σ t +ūη t =exp ūµ +1 ū)r] t ū σ σ t +ū 1 e γt] γ µ r) m exp rt + σ 1 + m) t > 1 If t>s,then E ξt ] 1 ξ s = E exp ūµ +1 ū)r]t s) ū σ t s)+ūη t η s ) =exp ūµ +1 ū)r]t s) ū σ s)+ū t Varη t η s ) ) µ r) m exp r + σ 1 + m) t s) > 1, whence 5) E ξ t = t E ξ t ] 1 ξ s c aλ) ds + ξ > c aλ)t + ξ for t> If µ r σ 1, m =, 1,, 1 + m) that is, if the entire capital is invested in the risky asset obviously, µ σ 1 + m) in this case), then E ξt ] 1 = E exp µt σ t + η t =exp µt σ t + σ 1 e γt] > 1 γ for t> If t>s,then E ξt ] 1 ξ s = E exp µt s) σ t s)+η t η s exp µt s) σ t s) > 1 and 51) E ξ t = E ξt ] t 1 c aλ) ds + ξ > c aλ)t + ξ Relations 5) and 51) imply that the mean capital of an insurance company dealing in a BS-market is greater than that of a company investing the capital only in bonds This result does not hold with probability 1 in which case we would obtain bound 49)) Inequalities 45) and 47) of Theorem provide bounds for the ruin probability of an

13 ON THE RUIN PROBABILITY OF AN INSURANCE COMPANY 3 insurance company dealing in a BS-market The case of a zero load is also possible; that is, c>mayequalaλ = λ E ς k Bibliography 1 A V Baev and B V Bondarev, Ornstein Uhlenbeck process and its applications in problems of actuarial mathematics, Applied statistics Actuarial and Financial Mathematics ), no 1, 3 8 P A Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review ), A N Shiryaev, Essentials of Stochastic Finance Facts, Models, Theory, Fazis, Moscow, 1998; English transl, World Scientific Publishing Co, Inc, River Edge, NJ, 1999 MR e:9185) 4 A V Skorokhod, Lectures on the Theory of Stochastic Processes, Lybid, Kyiv, 199; English transl, VSP/TViMS, Utrecht, Netherlands/Kiev, Ukraine, 1996 MR d:61) 5 B V Bondarev, Mathematical Models in Insurance, Apeks, Donetsk, Russian) 6 I I Gikhman and A V Skorokhod, Stochastic Differential Equations and their Applications, Naukova dumka, Kiev, 198 Russian) MR j:63) 7 I I Gikhman and A V Skorokhod, Stochastic Differential Equations, Naukova Dumka, Kiev, 1968; English transl, Springer-Verlag, New York Heidelberg, 197 MR :7777); MR :1165) 8 I I Gikhman, A V Skorokhod, and M I Yadrenko, Probability Theory and Mathematical Statistics, Vyshcha Shkola, Kiev, 1988 Russian) 9 A V Baev and B V Bondarev, An insurance company dealing in a B, S) market, Applied statistics Actuarial and Financial Mathematics 3), no 1, AVMelnikov,Risk Analysis in Finance and Insurance, Ankil, Moscow, 1; English transl, Chapman & Hall/CRC, Boca Raton, FL, 4 MR1335 4i:914) Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics, Donetsk State University, Universitets ka Street 4, 8355 Donetsk, Ukraine address: tv@matfakdongudonetskua Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics, Donetsk State University, Universitets ka Street 4, 8355 Donetsk, Ukraine address: bvbondarev@cablenetluxorg Received 11/JAN/5 Translated by V V SEMENOV