optimal insurance and reinsurance portfolios, implied ...u u ul ug ucap c cvar capital ucvar rate...

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Optimal Insurance and Reinsurance Portfolios, Implied Pricing, Allocating

Retrocessional Cost and Capital Allocation

Presenter: Morton Lane Ph.D.Co-author: Jerome Kreuser Ph. D.

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SUMMARY

A Basic Model- Objectives/ Risk Preferences/Opportunity Set

B Important Features of Optimal Solution- Implied Probabilities/Threshold Prices/ Marginal Value Of Capital

C Multi- Zone- Allocating the Cost of Retrocession usingRisk-Adjusted Probabilities

D Large Scale Applications- Questions Posed and Answers Derived

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Max Expected Ending Funds

Subject to;

Definitions of Funds at End Period…….all ScenariosPremiums – Retro Costs – Losses + Recoveries

Risk Preference ConstraintsConditional Value at Risk Levels……….all Levels

Opportunity, LimitsDeals, Retrocession, Ratings, Balance Sheets

Non Negativities, etc. over all scenarios

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Model Inputs Name Identification bda Bid/ask spread.

kc Confidence level expressed as a decimal for risk level k

capital Starting capital

kcvar Percent limit on loss of capital for risk level k

,i jloss Unit loss of deal j in scenario i

jprice Price of deal j as a percent

iρ Probability of scenario i

rate Rate of return on investments

tm Percentage of capital as limit on total ceded premiums

trs Transaction costs as a percent

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Decision Variables Determined by Model Name Identification

kα Alpha value for risk k, which turns out to be VaR for active constraints

jdeal Amount of premium of deal j written

0funds Beginning period funds net of capital 1ifunds End-of-period funds net of capital in scenario i

igains Gains from recoveries in scenario i

ilosses Losses in scenario i

jretro Amount of premium of deal j ceded

,k iz Excess loss over VaR of funds in scenario i for risk level k

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0

, ,

min, , , , (1 ) , , ,

i i i k k k ik K i I

k k i j j k i

j jj J j J

u u ul ug ucap c cvar capital ucvar rate capital uucvar uz ubd ubr uzl

tm capital ucap ubd ubr

∈ ∈

∈ ∈

− × × + ×

+ × × + +

∑ ∑

∑ ∑

0funds 00 (1 ) ii I

u rate u∈

= − + ∑1ifunds

,i i k ik K

u uzρ∈

= −∑

jdeal ,00 (1 )(1 ) i jj i ji

p trs bda u loss ul ubd= − − − − +∑

jretro ,00 (1 )(1 ) i jj i ji

p trs bda u loss ug ucap ubr= + + − + +∑

ilosses igains00

i i

i i

u ulu ug

= += − +

kα ,0 (1 )k i ki I

uz c ucvar∈

= − + −∑

,k iz,0 k i i kuz ucvarρ≤ − +

,

0

j

k i

k

uoneucapuz

ucvar

≤

The dual objective is:

Corresponding to the

Corresponding to the

Corresponding to the

Corresponding to the

Corresponding to the and

Corresponding to the

Corresponding to the

We have the following constraints on the dual variables:

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(1) 1 (1 )

i i k k k ii I k K i I

j jj J j J

funds c cvar capital ucvar rate capital u

tm capital ucap ubd ubr

ρ∈ ∈ ∈

∈ ∈

= − × × + ×

+ × × + +

∑ ∑ ∑

∑ ∑

EQUIVALENCE OF PRIMAL-DUAL OBJECTIVES

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Multiple Zones

FloridaGeorgia – MaineRest of USNationwide Wind

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OpportunitySet

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Maximum written Limits Maximum retrocession Limits

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Risk Preferences

The risk preference constraints used are as follows:

Probability 30%, CVaR limit of 10% loss of risk capitalProbability 20%, CVaR limit of 20% loss of risk capitalProbability 0.1%, CVaR limit of 100% loss of risk capital

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Optimal Solution

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Threshold Prices

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Threshold Prices

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Threshold Prices

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Threshold Prices

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Optimal Solution

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How should Retro Premium Be Allocated ?

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How should Retro Premium Be Allocated ?

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How should Retro Premium Be Allocated ?

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Allocating Capital

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Allocating Capital

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Allocating Capital

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Allocating Capital

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END

MLANE@LANEFINANCIALLLC.COM

KRESEUR@RISKONTROLLER.COM