# CAS Ratemaking Seminar RCM-1 Logic, Fallacies, and Paradoxes in Risk/Profit

```CAS Ratemaking Seminar
RCM-1
Logic, Fallacies, and Paradoxes in Risk/Profit
Introductory Remarks by Glenn Meyers
ISO Innovative Analytics
March 17, 2008
Insurer Risk and Capital
Management Perspective
• Risk based capital
– The insurer's risk, as measured by its
stochastic distribution of outcomes, provides a
meaningful yardstick that can be used to set
capital requirements.
• Insurer risk management
– The insurer manages its business to get the
best return on its cost of capital.
Steps for Insurer Capital
Management in Ratemaking
•
•
•
•
Determine total capital for insurer
Determine rate of return on that capital
holding that allocated capital
– My remarks today address this part of the
problem.
Insurer Risk Management
• Reserve Risk contributes to the need for
capital and hence it contributes to the (\$)
cost of capital.
• How long you need to hold capital is a
consideration in determining an
acceptable price.
The Cash Flow for
Underwriting Insurance
• Investors provide capital
– Through the insurer they:
• Pay losses and other expenses
– Invested at interest rate i%
• Release capital as liabilities become
certain.
Net out the loss
and expense payments
• Investors provide capital
– Through the insurer they:
• Receive investment income from capital as it
is being held.
• Release capital as liabilities become certain.
• We want the present value of the income to
be equal to the capital invested at the rate
of return for equivalent risk
Define Terms
• Allocated Capital
invested in year t
• Provision for Profit
• Insurer’s return on
invested assets
• Insurer’s target return on
capital
Ct
P
i
r
Calculating the Profit Provision
Time
0
Capital Allocated
at time t
C0
Amount Released
at time t
0
1
C1
Rel1 = C0 (1+i) – C1
---
---
---
t
Ct
Relt = Ct-1 (1+i) – Ct
---
---
--
Then P  C0  
t 1
Rel t
1  r 
t
Calculating the Profit Provision
Another Formula

P  C0  
Ct 1 1  i   Ct
1  r 
C0 1  r  1  i  C1 1  r  1  i  C2 1  r  1  i 



 ...
2
3
1 r
1  r 
1  r 
t
t 1

 r  i  
t 0
Ct
1  r 
t 1
Sample Calculation
t
0
1
2
3
4
5
6
7
8
Disc
Ct
TVaR
LDisc
t
t
60,502 72,015 11,512
36,519 46,990 10,471
19,540 27,430 7,890
9,138 14,756 5,618
3,616 7,379 3,762
1,141 3,502 2,361
256 1,632 1,376
32
869
838
1
99
98
Profit = 1,348 = 2.2% of Initial Liability
Prediction
This how actuaries will include the cost
of capital in future insurance costing.
Main Obstacles to Overcome
• Fuzzy relationship between risk and capital
– Insurers are starting to build internal capital models
– Examples EU Solvency II, British FSA, S&amp;P
• Quantification of all risks
–
–
–
–
Underwriting risk and reserving risk
Asset risk - Several commercial models
Operational risk
Other
• Consensus
– Will not come until above issues are substantially settled.
```