Reinsurance and the Distribution of Term Insurance Claims

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Reinsurance and the Distribution of Term
Insurance Claims
NZ Society of Actuaries Conference Queenstown -
November 2006

• Richard Bruynel FIAA, FNZSA
• BNZ Life Insurance Ltd

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Warning!!!

• Work in Progress

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Consider a set of term insurance policies
Eg Rk 40 Sk (Quota Share)Eg Rk
Min(Sk, R) (Surplus Risk)
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Why use Reinsurance?

• Reduce risk of blow-out in claims total claims
  significantly above expected claims
• We are concerned with both the magnitude and
  probability of a blow-out in claims
• We need to understand the distribution of claims
  to measure and control this risk.

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Downside of Reinsurance

• The expected cost of reinsurance is the
  Reinsurers Margin the amount built into the
  reinsurers premium rates to allow for their
  expenses and profit.
• Reinsurers premium rate

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Optimal Reinsurance Strategy

• An Optimal Reinsurance Strategy is that set of
  retention choices (Rks) that minimises the risk
  of excess net claims for a given level of
  expected cost.
• The Optimal Reinsurance Strategy depends on the
  risk measure chosen.

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Optimal Reinsurance Strategy

• If variance of net claims is the risk measure
  then
• 
  for a constant c
• Claim probability qk is usually small so this is
  close to
• If the Reinsurers Margin is constant then
  Surplus Risk reinsurance is near optimal.

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Monte Carlo Simulation of Claims Distribution

Distributions scaled to have the same mean

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Effect on Capital of Retention levels

• To examine the effect on total capital
  requirements of changing retention levels, we
  need a measure of marginal capital
• Define Capital for Claims Fluctuations (CFCF) as
• CFCF 10 of expected net claims, plus
• (99.5-ile less the mean) of the net claims
  distribution
• Net claims means net of reinsurance, so if
  Retention 0 then
• CFCF 0

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Effect on Profit of Retention levels

• To examine the effect on expected profit of
  changing retention levels, we need a measure of
  marginal profit
• If retention increases expected profit arises
  from two sources
• Reinsurers Margin that is not paid away, and
• Interest on additional capital held (CFCF)
• Assumptions
• Reinsurers Margin is 15 of expected claims
• 5 p.a. interest is earned on CFCF
• Start from a zero base if Retention 0 then
• Expected Profit 0

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Optimal Retention Level
In theory Optimal Retention is where marginal
return on capital equals the companys target
return rate. In practice can only determine
incremental return on capital Example Portfolio
of 5,000 policies Incremental RoC (moving
from 300k to 500k retention) (0.668
0.451) / (3.73 2.47) 17.2

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Modelling a Pandemic

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Effects of modelling pandemics

• Claims distributions become very skewed,
  especially for larger portfolios.
• Magnitude and probability of extreme events is
  completely dependent on parameters chosen for
  pandemics.
• Need extremely high reserves to handle 1 in 200
  years claims.
• Insurers are unlikely to make adequate return on
  this level of capital.
• Challenging for insurers or reinsurers to achieve
  genuine diversification of their books in the
  face of possible pandemics.