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NUMERICAL EXPERIMENT FOR RUIN PROBABILITY IN RISK MODEL WITH DEPENDENT FINANCIAL AND INSURANCE RISKS
G. Tsitsiashvili, M. Osipova
IAM, FEB RAS, Vladivostok, Russia •
e-mails: guram@iam.dvo.ru, mao1975@list.ru
ABSTRACT
For discrete time risk model with dependent financial and insurance risks numerical experiment with recurrent procedure of ruin probability calculation is made. It shows that suggested recurrent procedure is much faster than application of usual Monte-Carlo method.
INTRODUCTION
In (Tsitsiashvili, 2010) for discrete time risk model with dependent financial and insurance risks recurrent algorithm of ruin probability calculation is constructed on a base of hyperexponential approximation of insurance losses distributions. Special methods of symbol forms transformation and economical procedure of enumeration of vectors with integer components and fixed sum are developed
In this paper constructed recurrent algorithm is tested in numerical experiment. Comparison analysis showed that suggested algorithm is significantly faster then application of usual Monte-Carlo method.
1. PRELIMINARIES
In (Tsitsiashvili 2010) recurrent discrete time risk model with initial capital x, x > 0, nonnegative lossesZn, n = 1,2,..., and inflation factor Y"1 > 1 from n -1 to n year:
S0 = X Sn = Y- Sn-1 + 1 -Zn, n = l2-^
is considered. Here Xn = Zn -1 is insurance risk and Yn is financial risk. Suppose that the sequence {Yn, n > 0} is stationary reversible Markov chain with state set {rq_1,q e Q}, Q = {1,...,m}, with transition matrix LqJ| , and with initial distribution
II qq llq,q'eQ
P (Yn = r- )= Pq > 0, I Pq = 1.
qeQ
Introduce dependence between financial and insurance risks by a conditional distribution of random variable Zn
n
Fq (t)=P ÍZn >/Y = r-11. qe ß. t > 0.
V / n q y
In this model finite horizon ruin probability
vn (x) = P (inf {n = 1,2,...: Sn < 0 SQ = x}< n). Following recurrent algorithm of its calculation is constructed. Denote (5j is Kronecker symbol)
1q = (,...,5mq), K = (ki,...,km), k e {0,1,...} , i = 1,...,m, rK = n rk , Kz=z kq.
qeQ qeQ
Theorem. If
and
rK Xi * X i, i > 1, j < l, Ks > 1,
_ i i Fq (t) = Z aqi exp ("XitX -œ < aqi < ^ Z ^ = 1, q G Q Î > 0, i=1 i=1
when for n > 1
Wn (t) =
andfor q, q' g Q, i = 1,...,l
Z Z Z B}K,i,q, t > 0,
qGQ 1<KE<n i=1
1 + Z Z Z B£,q (exp (-rKXit) -1), t < 0,
qGQ 1<KE<n i=1
(1)
BÎIq = Pqaqi eXP (-Xi ) , BÎ,Î q = 0, 9 * tf',
B1q = z
"n +1,J, q ~
q GQ
^q'qaqieXP (-X,0
( i BK, ' ^
Xi Z Z -KX^Y+p
1<kE<« j=1 r X ; - X
j '
, q = q ', Bi+1,i,q =0, q * q',
BK =- Z q gQ
Kq'qI ( >0)exp (-rK-1qXi ) z -K-
l ¿/OVXj
, 1 < Ky< n + 1.
'j=1 r q Xi -X;
Remark 1. In (Tsitsiashvili 2010) there is some inaccuracy in recurrent formulas for coefficients B^ iq, BK+1,j,q. Here odd multiplier pq' is cancelled from these formulas.
In (Tsitsiashvili 2010) recurrent algorithm of matrix \nq>q\ ,gq .generation is replaced by a
random choice of elements of symmetric matrix l-Ay^j=1 for fixed probabilities {Pj,..., pm} from formulas
max
j-1
i-1 m
0, Pi-Z Ak-Z Ps+ZZ Ask
s=j+1 s=1 k=j+1
k=1
< A.. < min ij
i-1
j-1
k=1
k=1
pj-Z A, Pi-Z A k
Am = pi -Z 4k ,1 < i < m,
i < j < m -1,
m-1
k=1
then TTij = — ,1 < i, j < m.
Pt
In (Tsitsiashvili 2010) a problem of an enumeration of all vectors of the set k = {K = (k1,..., km ): ki = 0,1,..., i = 1,..., m, 1 < Ks < n} is solved via recurrent calculation of the sets of vectors
Ki = {K = (k1,...,kj): ki = 0,1,..., i = 1,..., j, Ks= i}, 0 < i < n, 1 < j < m,
k0 = {0}, k = {i}, 1 < i < n, k{+1 = U {(K, t): K e k{_t}, 0 < i < n,1 < j < m -1.
t=0
Then
n
k = u k m
i=1
and calculation complexity of this algorithm is not larger than
(n + 1)m+1.
2. NUMERICAL EXPERIMENT
Suppose that m = 2, Q = {1,2}, p1 = 0.25, p2 = 0.75, r1=1.03, r2=1.08,
w11 = 5/9, rc12 = 4/9, ^21 = 4/27, ^22 = 23/27 and consider Pareto distributions of insurance losses
F1 (t) = (1 + 5 x)-12, F2 (t) = (1 + 0.83 x)-22, t > 0. We approximate Pareto distributions by hyperexponential (Anja Feldman & Ward Whitt 1998)
_ 27 _ 27
F1 (t) « S a1i exp (-^it), F2 (t) « S a2i exp (-^it), i=1 i=1
with parameters
i a1i a2i
1 0.089437 0 23.304
2 0.533823 0 6.516
3 0.307218 0 1.546
4 0.059768 0 0.306
5 0.008462 0 0.057
6 0.001122 0 0.01
7 0.000147 0 0.002
8 0.0000192 0 0.00035
9 2.5x10-6 0 0.000065
10 3.27x10-7 0 0.000012
11 4.27x10-9 0 2.2x 10-6
12 5.56x10-10 0 3.9x 10-7
13 7.18x10-10 0 6.8x 10-8
14 8.37x10-11 0 8.3x10-9
15 0 0.193963 4.491
16 0 0.651199 1.422
17 0 0.147814 0.371
18 0 0.006832 0.076
19 0 0.000188 0.014
20 0 4.61x 10-6 0.003
21 0 1.11x10-7 0.0005
22 0 2.65x10-9 0.000088
i aii a2i
23 0 6.35X10"11 0.000016
24 0 1.52x10"12 2.9x10"6
25 0 3.36x10"14 5.4x10"7
26 0 8.51x10"16 9.7x10"8
27 0 1.72x10"17 1.58x10"8
Results of numerical experiments for ruin probability (x) with step h =0.5 by x using
Monte-Carlo method with N=10000000 realizations and by the formula (1) are represented on the table
X Monte-Carlo method the formula (1)
0.5 0.421138 0.422476
1 0.347786 0.348598
1.5 0.291625 0.292944
2 0.247464 0.249402
2.5 0.212054 0.214347
3 0.183264 0.185541
3.5 0.159391 0.161535
4 0.139627 0.141338
4.5 0.123078 0.124237
5 0.109101 0.109692
Time of calculation by Monte-Carlo method is approximately 2 hours and 15 minutes and time of calculation by the formula (1) is 15 seconds.
Results of numerical experiments for ruin probability y50 (x) with step h =0.5 by x using
Monte-Carlo method with N=10000000 realizations and by the formula (1) are represented on the table
X Monte-Carlo method the formula (1)
0.5 0.442933 0.444793
1 0.38219 0.38395
1.5 0.335 0.336951
2 0.296502 0.298993
2.5 0.264418 0.267366
3 0.237423 0.240461
3.5 0.214501 0.217261
4 0.19457 0.197082
4.5 0.177139 0.179427
5 0.162125 0.163917
Time of calculation by Monte-Carlo method is approximately 10 hours and 30 minutes and time of calculation the formula (1) is 26 minutes.
REFERENCES
1. Tsitsiashvili, Gurami 2010. Algorithmic problems in discrete time risk model. Reliability: Theory and Applications (3) 1: 29-37.
2. Feldman, Anja & Whitt, Ward 1998. Fitting mixtures of exponentials to long-tail distributions to analyze network performance models. Performance evaluation (3) 1: pp. 245-279.
