optimal insurance and reinsurance portfolios, implied ...u u ul ug ucap c cvar capital ucvar rate...
TRANSCRIPT
1
Optimal Insurance and Reinsurance Portfolios, Implied Pricing, Allocating
Retrocessional Cost and Capital Allocation
Presenter: Morton Lane Ph.D.Co-author: Jerome Kreuser Ph. D.
2
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
3
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
4
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
5
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
6
7
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:
8
(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
9
Multiple Zones
FloridaGeorgia – MaineRest of USNationwide Wind
10
11
OpportunitySet
12
Maximum written Limits Maximum retrocession Limits
13
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
14
15
Optimal Solution
16
Threshold Prices
17
Threshold Prices
18
Threshold Prices
19
Threshold Prices
20
21
22
23
24
25
26
27
28
Optimal Solution
29
30
31
32
33
34
How should Retro Premium Be Allocated ?
35
How should Retro Premium Be Allocated ?
36
How should Retro Premium Be Allocated ?
37
Allocating Capital
38
Allocating Capital
39
Allocating Capital
40
Allocating Capital
