Approximations to Ruin Probability

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Approximations to Ruin Probability

• By Flora Chan

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Outline

• Research objectives and motivations
• Background knowledge
• Different methods of approximation
• Applications on two claims distributions
• Comparison and analysis of the various methods

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Research Objectives

• Closed form solutions to ruin probabilities are
  not always possible
• Investigate 4 methods of approximating ruin
  probabilities
• Matching Moments
• Modified De Vylders Approximation
• Tijms Approximation
• Original De Vylders Approximation

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Classical Risk Model (1)

• Surplus process U(t)t0
• u is the initial surplus
• c is the rate of premium income per time unit
• N(t) is the number of claims up to time t
• Xi is the amount of the i-th claim

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Classical Risk Model (2)

• Total number of claims up to time t, N(t)t0 is
  a Poisson process with parameter ?
• Aggregate claims process, S(t)t0,
• is a compound Poisson process with Poisson
  parameter ?

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Classical Risk Model (3)

• Premium loading
• c (1?) ? p1 (pk E(Xki))
• Probability of ruin, ?(u)

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Maximum Aggregate Loss (1)

• Aggregate loss process, L(t)t0
• Maximum aggregate loss, L
• Relationship between ?(u) and L
• i.e. ?(u) gives the tail probability of L

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Maximum Aggregate Loss (2)

• L is the sum of individual new record highs of
  L(t)
• Process has stationary and independent
  increments, hence probability of each record high
  is the same
• L follows a compound geometric distribution with
  q ?(0) 1/(1?)

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Maximum Aggregate Loss (3)
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Approximations to ?(u)

• Propose a two-term exponential function as ?(u)
• Commonly used one-term exponential approximation
• Lundbergs inequality
• Cramers asymptotic formula

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Matching Moments (1)

• Derive the first three moments of the max. agg.
  loss r.v. L in 2 ways
• Direct computation of n-th moment by
• where Pr(Lgtx) is given by ?(u)
• E(L), E(L2) and E(L3) in terms of ?1, ?2, R1 and
  R2.

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Matching Moments (2)

• View L as sum of record highs
• n-th moment of Li depends on moments of
  individual claims distribution pn1
• Apply Panjers recursion formula since L is
  compound geometric
• E(L), E(L2) and E(L3) in terms of p1, p2, p3, p4
  and ?

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Matching Moments (3)

• Note that
• 4 equations, 4 unknowns ? solve simultaneously
  for ?1, ?2, R1 and R2 for particular values of ?

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Matching Moments (4)

• Does not involve complicated computations
• Easily solved using spreadsheets
• Only first 3 moments are required
• Obtain from past claims data
• Involves solving quadratic equations
• Constraints on parameters chosen for p(x)
• Applies to most claims distributions

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Original De Vylders Method

• Define a new classical risk process with Poisson
  parameter , and individual claims
• Approximate ?(u) by
• Parameters can be determined by characteristic
  functions of the two surplus processes

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Modified De Vylder (1)

• Similar to the original method, but new classical
  risk process has two-term mixed exponential
  individual claims distribution
• where ? 1 ?2 1, ?1, ?2 gt 0.
• Equate characteristic functions of two surplus
  processes

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Modified De Vylder (2)

• Approximation adopts new Poisson parameter ?
  and loading factor
• ? 6 unknowns
• Note that ?1 ?2 1, need first 5 terms in
  characteristic functions
• ? 6 equations, 6 unknowns
• Solve simultaneously

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Modified De Vylder (3)

• Does not involve complicated computations
• Easily solved using spreadsheets
• One more moment is required
• Different surplus process in approximation
• Involves solving quadratic equations
• Constraints on parameters chosen for p(x)
• Applies to most claims distributions

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Tijms Approximation

• Approximate ?(u) by
• where
• C is calculated as in Cramers asymptotic formula
  
• R is the adjustment coefficient
• R1 is chosen by matching the first moment of L

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Numerical Illustration (1)

• Two individual claims distribution
• 3-term mixed exponential function
• P(x) 1 - 0.0039793e-0.014631x
• - 0.1078392e-0.190206x
• - 0.8881815e-5.514588x
• 2. Gamma distribution
• p(x) ?? x?-1e-?x/?(?), ?, ? 0.1

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Numerical Illustration (2)
Properties of the distributions
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Numerical Illustration (3)

• Five methods of computing ?(u)
• Exact calculation
• Method of Matching Moments
• Modified De Vylders Approximation
• Tijms Approximation
• Original De Vylders Approximation
• Example ? 0.5, ? 1

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Distribution 1 (1)
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Distribution 1 (2)

• ?(u) for different methods

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Distribution 1 (3)

• General conclusions
• Quality of approximation improves as initial
  surplus relative to the first moment increases
• Changing loading factor does not affect
  performance
• Method (3) gives the best approximation of ?(u)
• Method (2) and (4) gives similar results
• Method (5) has the worst performance

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Distribution 2 (1)
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Distribution 2 (2)

• ?(u) for different methods

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Distribution 2 (3)

• General conclusions
• Distribution has low variance, ?(u) quickly goes
  to 0
• All four methods perform equally well
• As initial surpluses increase, approximations
  quickly approach the exact values

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Summary (1)

• The method of matching moments and modified De
  Vylders approximation can be applied to many
  kinds of distributions
• Tijms approximation only exists when m.g.f.
  exists
• e.g. not for Lognormal, Pareto

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Summary (2)

• Quality of approximation
• Modified De Vylders method performs relatively
  better than the others
• Method of matching moments and Tijms
  approximation give similar results
• Approximations improve as initial surplus
  increases
• Poor approximation for heavy-tailed distributions
  (for those without m.g.f.)