Cost and allocation of capital

1
Allocation of Capital in Reinsurance

• Cost and allocation of capital

Oxford Risk Workshop 19 May 2005
Matthew Eagle - Aon Re Services, Aon Re UK Based
on work carried out by Dr. Robert Verlaak, Aon Re
Belgium
2
How to evaluate the capital requirement?

• ?Risk measures
• VaR(p) most familiar measure for the banking
  sector. It represents the PML (Probable Maximum
  Loss). Based on a probability(p), the VaR
  represents the corresponding threshold value (x).

Average 23,000
VaR(95)36,100
VaR(99.9)55,500
3
How to evaluate the capital requirement?

• ?Risk measures
• TVaR(p) the expected cost in case of losses
  exceeding the VaR(p)

TVaR(95)41,800
TVaR(99.9)61,000
Average 23,000
VaR(95)36,100
VaR(99.9)55,500
4
How to evaluate the capital requirement?

• ?Risk measures
• TVaR(p) VaR(p) the expected shortfall at level
  VaR(p)

VaR(95)36,100
TVaR VaR(95) SL(36,100) / 5 41,800
5
Properties of Risk Measures

• A coherent risk measure
• Sub-additivity Risk (X Y) lt Risk(X) Risk (Y)
• Monotonicity Risk(X)ltRisk(Y) if Xlt Y with
  probability 1
• Positive homogeneity Risk(k X)k Risk(X), for
  any k fixed (gt0)
• Translation invariance Risk (k X)k Risk(X),
  for any k fixed
• !! VaR (and also Expected Shortfall) is not
  Sub-additive
• !! TVaR is a coherent risk measure
• But much more important is that TVaR is linear
  over the sub-risks suppose the partition of the
  risk XX1X2, then
• EX1X2 / X1X2gtxEX1 / X1X2gtxEX2 /
  X1X2gtx

6
Properties of Risk Measures - ordering

• Interpretation of TAIL VaR
• Stochastic dominance
• Stop-loss ordening

7
Properties of Risk Measures

• Comonotonicity
• Any risk measure that preserves Stop-Loss order
  and that is additive for comonotonic risks is
  sub-additive

8
Properties of Risk Measures

• Distortion measures
• Distortion function g
• Distortion measure

9
Properties of Risk Measures

• Any distortion measure is additive for
  comonotonic risks, positive homogeneous,
  translation invariant monotone
• Any CONCAVE distortion measure is COHERENT
• Distortion measures versus utility

10
VaR and TVaR

• VaR
• TVaR

11
Corporates vs (Re)Insurance Companies

• Remark Difference between corporate versus
  reinsurance companies
• No remuneration received for the risk (e.g.
  corporates)
• Remuneration received for the risk (e.g.
  insurance companies)

Var Capital or TVaR
?Var ?Capital or ?TVaR
?Var - ?Prem() ?Capital or ?TVaR -
?Prem()
Var Prem () Capital or TVaR- Prem()
() Prem without loading premium E(X)
12
Belgium Cat Example

• To calculate (T)VaR we need to known the
  stochastic model in detail.
• This is typical the case for Catastrophe
  Reinsurance the tools AIR, EQE, RMS, QFLAT,
• Belgium example based on the perils WINDSTORM,
  EARTHQUAKE, FLOOD
• These perils are the typical high CAPITAL
  absorbers (cumulative risk for standard policies)
• To avoid a lot of technical problems, the models
  will be simulated

13
Peril Loss Simulation

• Between 100,000 and 200,000 year simulations per
  peril and per tool. Per year we need
• The event-ID for each simulated event (to keep
  into account the geographical dependency of the
  subsidiaries)(Poisson Neg. Binomial)
• Per simulated event-ID and per subsidiary the
  parameters to simulate the severity (average
  loss, standard deviation, model of the 2dary
  uncertainty (LogNormal Beta))
• The severity per subsidiary is mutually dependent
  on each other through the event-ID (assumption of
  comonotonicity)

14
Assumed independence of perils

• Due to the dependency through the event-ID, one
  needs to analyse each peril within the same tool
  over the different subsidiaries (otherwise, one
  will loose the mutual dependency)
• The different perils can be analysed in different
  tools (assumption of peril independency)(For
  example windstorm EQE quake QFLAT flood
  RMS)
• The results of the different analyses will be
  combined in another tool (for example SAS)
  through simulation, which has the advantage that
  one can select the best tool to analyse the
  perils.

15
Example Summary Values
16
Example

• Based on a Cat Program with different layers and
  perils
• Cost of Capital versus Cost of reinsurance (1)
• Allocation of the retained risk per retention
  category (2)
• Risk lower than priority per event
• AAD
• Reinstatements
• Risk higher than the Upper Limit per event
• Allocation of the retained risks per peril (3)
• Allocation of the retained risks per subsidiary
  (4)

17
Cost of Capital vs Cost of Reinsurance
18
Cost of Capital vs Cost of Reinsurance
19
Allocation of Retained Risk through retention
20
Allocation of Retained Risk per Peril
21
Allocation of Retained Risk per Subsidary /
Business Unit
22
Cost and Allocation of Capital