Ruin Probability for Dependent Risk Model with Variable Interest Rates

Transcription

1 Applied Mathematical Sciences, Vol. 5, 2011, no. 68, Ruin robability for Dependent Risk Model with Variable Interest Rates Jun-fen Li College of Mathematics and Information Science,Henan Normal University Xinxiang City, Henan rovince, , China junfen Shu-guang Zhang College of Mathematics and Information Science, Henan Normal University Jiaozuo City, Henan rovince, , China Dan-dan u College of Mathematics and Information Science, Henan Normal University Xinxiang City, Henan rovince, , China Abstract In the present paper, we establish a risk model with dependent variable interest rate in the discrete market, and study the relationship between the survival function and bankruptcy probability. Furthermore, ruin probability with the risk model are obtained by new method. Keywords: Dependence structures; Ruin probability; Variable interest rates; Mixing distributions 1 Introduction Ruin probability is an important theoretical issue in the classical risk theory, which is also a very important practical issue to the insurance companies. Therefore it is important especially to find factors to the ruin probability and to make the appropriate formula. eople study ruin probability of the company through the change in surplus changes in the time t in the classical risk model. when the surplus is negative for the first time, the insurance company is facing bankruptcy. Weng [1] (2009) studied Ruin probability in a discrete time risk medel with dependent risks of heavy tail. Albrecher [2] (1998) and Cai [3] (2002) also

2 3368 Jun-fen Li, Shu-guang Zhang and Dan-dan u studied dependent risks and ruin probabilities in insurance and ruin probability with dependent rates of interest. But the interest rate is fixed in these models. Insurance which economic behavior is long-term, the government s economic policies, so economic cycles and other factors to the insurance company will bring a degree of risk. Further the randomness of interest rate determines the insurance company Contingency reserve capacity planning and estimation of ability to pay compensation. Changes in interest rates have greater risk under certain conditions, therefore constant interest rates may have the large deviation to the actual value. To reduce this uncertainty, it is a better way to use of the variable interest rate risk model. 2 Model Suppose {X i,i =1, 2...} are random variables, Surplus process of insurance companies is U 0 = x, U i = U i 1 (1 + r i 1 ) X i,i=1, 2...,n, (1) where x( 0) represents the initial capital of insurance companies, U i represents a surplus, r i represents interest rates, X i represents a net loss of individuals in the period i. Hence,Surplus process can be expressed as U 0 = x, U i = x i 1 (1 + r k ) i i 1 X k k=1 s=k (1 + r s ),i=1, 2...,n. (2) Furthermore, we assume 1. the net losses X i to be identically distributed as a generic random variable, X i with cumulative distribution function F ( ); 2. a individual net losses to be generally dependent from the regularly varying class with index (α >0); F ( x) 3. cumulative distribution function F ( ) satisfies = 0, where the survival function =1. As can be seen from the above assumptions, the model (2) in the risk of X i are dependent risks. To facilitate the study first we give the following definition and lemma. Definition 1 For two positive functions a(x), b(x),if sup a(x) b(x) a(x) b(x) 1 and sup 1, we write a(x) b(x). Definition 2 Regular variation class R α a random variable X or its cumu- F (xy) lative distribution function on(, ) belongs to R α, if y α, for α>0,and any y>0. =

3 Ruin probability for dependent risk model 3369 Lemma 1 Non-negative random variables assumed X i with cumulative distribution function F i ( ) R α,α > 0, for any real number α > 0 and i =1, 2...,n, if i >x} = c i and i >x,x j >x} =0, i j, then { n X i >x} i=1 = n c i. (3) Lemma 2 For the random variable X i R α,α > 0 Where α>0, for any F ( x) positive constant δ, δ 1 and δ 2, = 0 and i >x,x j >x} =0, i j are equivalent to and i=1 F ( δx) = 0 (4) i >δ 1 x, X j >δ 2 x} =0, i j. (5) For model (2), the discounted value of the probability of bankruptcy following Ψ r (x, n) = { min (i 1 (1 + r k )) 1 U i < 0 U 0 = x} 0 i n = { max ( i X k ( (1 + r s )) 1 >x}. 0 i n (6) roblem (6) is the bankruptcy probability which we will study. In other words, Ψ r (x, n) bankruptcy probability in the ited time [0,n]. 3 Ruin probability for dependent risk model with variable interest rates Theorem 1For the problem (6), suppose individual net losses X i is distributed as a generic cumulative distribution function R α, satisfying conditions F ( x) = 0 and i >x,x j >x} =0, i j, then Ψ r (x, n) { max ( i X k ( (1 + r s )) 1 ) >x} 0 i n {( n X + k ( (1 + r s )) 1 ) >x} ( n ( (1 + r s )) α ). (7)

4 3370 Jun-fen Li, Shu-guang Zhang and Dan-dan u roof Since X 0 = 0, and n X k ( (1 + r s )) 1 max ( n X k ( (1 + r s )) 1 ) n 0 m n k=1 X + k ( (1 + r s )) 1, in order to obtain the desired result, it is enough to get the following expression n {( X + k ( n (1 + r s )) 1 >x} ( ( (1 + r s )) α ), (8) i {( X + k ( n (1 + r s )) 1 >x} ( ( (1 + r s )) α ). (9) According to X k R α and Lemma 2,we have {(X + k ( (1 + r s )) 1 ) >x} (1 + r s )) α, =( {( n X + k ( (1 + r s )) 1 ) >x} = n ( (1 + r s )) α {( n and X + Q k )>x( (1+r s)) 1 } ( n Q ( (1+r s)) α ) = 1. We obtain expression (8). Now we verify expression (9). For n =1,F x R α, we have 1 (1 + r 0 ) 1 >x} 1 >x(1 + r 0 )} = =1 then expression (11) is obtained. Let n 2 and v be a constant greater than 1,y = v 1 > 0, and defined A n 1 k = {X k v( (1 + r s ))x},a k = {X k

5 Ruin probability for dependent risk model 3371 y( (1 + r s ))x}. First we analyze the left side of the type (11), {( n X k ( i q+1 (1 + r s )) 1 >x} {( n X k ( (1 + r s )) 1 >x,max X k( (1 + r s )) 1 >vx} 1 k n n {( n X k ( (1 + r s )) 1 >x,x k ( (1 + r s )) 1 >vx} 1 k l n {( n X k ( (1 + r s )) 1 >x, X k ( (1 + r s )) 1 > vx, X l ( (1 + r s )) 1 >lx}. =Δ 1 Δ 2. (12) We analyze the following Δ 1 of expression (10), {( n X k ( (1 + r s )) 1 >x,x k ( (1 + r s )) 1 >vx} {( n X k ( (1 + r s )) 1 >x,x k ( (1 + r s )) 1 >vx, X q > y( (1 + r s ))x, s q n, q k} {(X k ( (1 + r s )) 1 >x,(x q ( (1 + r s )) 1 > yx} 1 [ {A k } + n k=1,q k {A q }]. (11) According to Lemma 1,for j =1, 2,when x,we have {A j } 0, a.s. (12) Hence,combining (11) and (12) yields so {( n k=1 = X k ( Q (1+r s)) 1 )>x,(x k ( Q (1+r s)) 1 )>vx} k >v(( Q (1+r s)))x} = v α ( Δ 1 = n 1 {A k } (1 + r s )) α,k =0, 2 n, (13) v α ( (1 + r s )) α. (14)

6 3372 Jun-fen Li, Shu-guang Zhang and Dan-dan u Second, we analyze Δ 2 of expression (11),According to Lemma 1,whenk l, j =1, 2 n,we have Hence, {( n k=1 X k ( Q (1+r s)) 1 >x,x k ( Q (1+r s)) 1 >vx,x l ( Q (1+r s)) 1 >lx} k ( Q (1+r s)) 1 >vx,x l ( Q (1+r s)) 1 >lx} =0. combining (10),(12) and (15) yields {( n X k ( (1 + r s )) 1 >x} k=1 Letting v 1, {( n X k ( (1 + r s )) 1 >x} k=1 Δ 2 =0. (15) n The proof of the desired result can be completed. References v α ( (1 + r s )) α. (16) n ( (1 + r s )) α. (17) [1] C.G.Weng, Ruin probability in a discrete time risk medel with dependent riskss of heavy tail, Actuarial Journal, 2009, 3: [2] H.Albrecher, Dependent risks and ruin probabilities in insureance, Interim Report, 1998, 57(5): [3] J.Cai, Ruin probability with dependent rates of interest, Journal of Applied robability: 2002, 39(2): [4] A. Juri and M. V. Wüthrich, Tail dependence from a distributional point of view, Extremes, 6 (2004), [5] D. Li, On default correlation: a copula function approach, Journal of Fixed Income, 9(2001), [6] I. H. Dinwoodie and S. L. Zabell, Large deviations for exchangeable random vectors, The Annals of robability, 3 (1992),

7 Ruin probability for dependent risk model 3373 [7] J. Galambos, The Asymptotic Theory of Extreme Order Statistics, 2nd ed, Krieger, Malabar, [8] M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks, Wiley, New York, [9] N. Whelan, Sampling from Archimedean copulas, Quantitative finance, 4 (2004), [10]. J. Schönbucher, Taken to the it: simple and not-so-simple loan loss distributions,wilmott magazine,1 (2003), Received: April, 2011