Transcript Ruin probabilities : classical versus credibility

Ruin Probabilities :
Classical Versus Credibility
Cary Chi-Liang Tsai
Department of Finance, National Taiwan University
Gary Parker
Department of Statistics and Actuarial Science
Simon Fraser University, Canada
Classical Discrete Time Risk Model
U n  u  n  c  Sn ,
• u = U0 : the initial surplus
• c : the amount of premiums received each period
• Sn : the total claims of the first n periods
S n  W1  W2    Wn ,
• assume Wi is the aggregate claims in period i, and
W1, W2, …, Wn are non-negative i.i.d. random variables
• c= (1+η)×E[W], η is the relative security loading
• the time of ruin T = min{n: Un < 0}
• the probability of ruin Ψ(u) = Pr{T < ∞ | U0 = u}
Experience Rating
• Constant premiums: U n1  U n  c  Wn1 ,
• Dynamic premiums: U n1  U n  cn1  Wn1 ,
• Renewal premiums charged in casualty insurance are
usually based on past experience.
• cn+1= (1+η)× E[Wn+1| W1=w1, W2=w2, …, Wn=wn].
• The determination of cn+1 is based on credibility theory
• Question: Can the dynamic credibility premium scheme
significantly affect the probability of ruin ?
Buhlmann’s Credibility Premium
• For j = 1, 2,…, n+1, the distribution of each W j depends
on a parameter Θ
• Given Θ, the random variables W1, W2, …, Wn+1 are
conditionally independent and identically distributed.
• Denote µ(Θ) = E[W j |Θ] and ν(Θ) = Var[W j |Θ].
• µ = E[µ(Θ)] = E[W j], a = Var[µ(Θ)] and ν = E[ν(Θ)]
• 
 n 1  E[Wn 1 | W1 ,W2 ,,Wn ]  Z n  W n  (1  Z n )  
1 n
• Buhlmann’s credibility factor Zn = n/(n+ ν /a), w n   wi
n i 1
• cn+1= (1+η)×  n 1
One More Dynamic Premium Approach
• since Zn= n/(n+ ν /a) is increase to 1 as n goes to ∞,
 n1  w n (the sample mean) as n approaches to ∞.
• If n is large, there is very little change from  n to  n 1
when one more aggregate claim wn is observed, which
implies the renewal premium cn+1 is very stable and the
credibility impact disappear for large n.
• idea : consider the most recent k periods of claim
experiences to renew the premium for the next period
•
n1,m  E[Wn1 | Wh  wh ,,Wn  wn ]  Z n,m  w n,m  (1  Z n,m )  
• m = min (n, k), h = max (n-k, 0) + 1 = n – m + 1,
Zn,m = m/(m+ ν/a),
w n,m
1 n
  wi
m i h
Distribution Assumptions
n
if X1, X2, …, Xn are i.i.d. Exp(β),  X i~ Gamma(n,β)
i 1
assume Λ follows Poisson(λ), and
given Λ, W1, W2, …, Wn+1 are independent Gamma(Λ,β).
µ(Λ) = E[W,j |Λ] = Λβ, ν(Λ) = Var[W j |Λ] = Λβ2,
µ = E[µ(Λ)] = E[W j] = λβ , ν = E[ν(Λ)] = λβ2, and
a = Var[µ(Λ)] = λβ2.
• Zn = n/(n+1), Zn,m = m/(m+1)
•  n1  E[Wn1 | W1  w1 ,W2  w2 ,,Wn  wn ]  n  w n  1   .
•
•
•
•
•
 n1,m
n 1
n 1
m
1
 E[Wn1 | Wh  wh ,,Wn  wn ] 
 w n ,m 
  .
m 1
m 1
Parameter Assumptions
• assume three sets of parameters for Gamma distribution
(λ, β) = (100, 1) (high frequency and low severity),
(λ, β) = (10, 10) (mid frequency and mid severity) and
(λ, β) = (1, 100) (low frequency and high severity)
(λ, β)
(100,1)
(10,10)
(1,100)
E(Λ)
100
10
1
Var(Λ)
100
10
1
E(Wi)
100
100
100
• Sample histograms of Λ and Wi
Var(Wi)
200
2000
20000
Simulation Results
• perform Monte-Carlo simulation 1000 times
• the study period is 100 (that is, if Un > 0 for all n = 1, 2,
…,100, then ruin is assumed not occur for this simulation).
• ruin probability = number of ruins/1000
• the relative security loading η =0.1
• Simulation results (tables and figures)
Conclusions and Future Research
• “The credibility scheme can reduce ruin probability” is
not always true, and only holds under some condition
(large enough the initial surplus).
• Different lines of business of the insurer can yield
significantly different ruin probabilities even though the
claim amount of these business comes from the same
distribution with different parameters but the same
expectations.
• The low frequency and high severity case is far worse
than the high frequency and low severity one
• Several lines of business portfolio, each line has its own
claim distribution and parameters
==> apply the Buhlmann-Straub credibility theory to
obtain the renewal premiums.