Incorporating Reinsurance and Risk Loads
in Primary Insurance Pricing
Presented By:
David Appel, Ph.D.
CAS Ratemaking Seminar
March 11, 2004
Objectives
1.
2.
3.
Briefly describe problems with direct
ratemaking
Outline methodology for incorporating
reinsurance costs into rates
Discuss allocation of profit based on
residual risk
Milliman USA
Perspective is Bureau Ratemaking
(but applicable to any direct ratemaking)

Direct premiums, losses and expenses statewide
(entire industry viewed as single aggregate company)

Cost of Capital (i.e., fair return) = “r” – based on
capital market data

Degree of capitalization (i.e., leverage) = “P/S” ratio based on actual industry data for primary companies
Milliman USA
Ratemaking in Simplest Terms
Direct Premium = Direct Losses + Expenses + r * (Dir Prem/(P/S))
or,
Direct Premium =
Direct Losses + Expenses
[1-r/(P/S)]
Given r and P/S, rates depend solely on direct losses and expenses
Implicitly assumes all layers of loss distribution supported by same
amount of capital
Milliman USA
Problems with Implicit Assumption




Amount of capital based on observed P/S for primary
companies
Cost of capital based on observation in capital market
(principally for primary companies)
Primary company cost of capital and amount of capital
depend on reinsurance availability
Reinsurers (especially Cat reinsurers) have more
supporting capital than primary insurers
The cost of this additional capital must be reflected in rates
Milliman USA
Simple Example to Illustrate Problem
Assume, without loss of generality:
1.
2.
3.
4.
No expenses or investment income – premium depends
solely on E[loss] and required profit
Required return determined in capital markets
Amount of capital (surplus) based on E[loss], and
E[loss]/surplus ratio based on observed data
E[loss] = $1.25 million, split 80%/20%, primary/reinsurer
Milliman USA
Illustration of Problem: Failure to Reflect Total
Capital Supporting Insurance Transaction
Current
Procedure based
on Direct Loss
Expected Loss
E[Loss]/Surplus
Surplus
Return on Equity
Required Profit
Premium
Alternative Split into Primary and
Reinsurance Layers
$1,250,000
2
625,000
10%
Primary
$1,000,000
2
500,000
10%
Reinsurance
$250,000
0.5
500,000
10%
62,500
50,000
50,000
1,312,500
1,050,000
300,000
Milliman USA
Setting Bureau Rates to Include Cost of
Reinsurance
Required Premium
= Direct loss + expense +
net cost of reinsurance
= Direct loss + expense +
reinsurer expense & profit
However, because this is a bureau rate, there is no actual
reinsurance contract; don’t know ceded premium and loss
Therefore, to get reinsurer expense and profit, must estimate
competitive market price for reinsurance
Milliman USA
Estimating a Competitive Reinsurance Premium

Assume statewide reinsurance program
Attachment point = 2x annual average loss
Limit = 1 in 100 year event (99th %ile of loss distribution
10% quota share retained by primary co. within this layer


Estimate ceded losses, given cat loss distribution
and assumed program
Add reinsurer expenses and profit to ceded loss
Milliman USA
Cat Reinsurance Premium
Calculation of Reinsurance Cost
Statewide Total
Total
(1) Hurricane Losses
(2) Loss Adjustment Expense Factor
(3) Hurricane Losses and Loss Expenses
(1) x (2)
(4) Percent Reinsured
(5) Reinsured Losses and Loss Expenses
(3) x (4)
(6) Reinsurance Expense Factor
(7) Reinsurance Loss+Expenses
(5) / (6)
(8) Reinsurance Expense Cost
(7)-(5)
(9) Reinsurance Premium to Surplus Ratio
(10) Reinsurer Underwriting Return Percent of Surplus
(11) Reinsurer Underwriting Return Percent of Premium
(10) / (9)
(12) Reinsurance Premium
(7) / (1.000-(11))
(13) Reinsurance Profit Cost
(12) - (7)
(14) Direct Losses
(15) Direct Losses and LAE
(14) x (2)
(16) Direct Variable Expense (Excl Reinsurance)
(17) Direct Premium Including Reinsurance Cost
((15) + (12) + (8)) / (1.000-(16))
(18) Reinsurance Expense Cost as % of Direct Premium
(8) / (17)
(19) Reinsurance Profit Cost as % of Direct Premium
(13) / (17)
(20) Reinsurance Premium as % of Direct Premium
(11) / (17)
31,909,123
1.154
36,823,128
0.511
18,823,846
0.90
20,915,385
2,091,538
0.37
15.4%
41.5%
35,752,015
14,836,630
48,447,610
55,908,541
32.00%
107,112,809
1.95%
13.85%
33.38%
Milliman USA
Statewide Premium Incl. Reinsurance Costs
Statewide Total
Total
(1) Expected Value of Net Losses
(2) Expected Value of Ceded Losses
(3) Expected Value of All Losses
(1)+(2)
(4) Commission and Brokerage
(5) Other Acquisition
(6) General
(7) Taxes Licenses and Fees
(8) Deviations
(9) Reinsurance Expense Cost
(10) Reinsurance Profit Cost
(11) Net Profit and Contingencies
(12) Loss Adjustment Expense Factor
(13) Total Indicated Premium
((3) x (12)) / (1-Sum[(4) to (11)])
(14) Total Indicated Underwriting Profit
(11) x (13)
32,135,784
16,311,825
48,447,610
10.00%
5.00%
5.00%
2.00%
5.00%
1.95%
13.85%
5.00%
1.154
107,112,809
5,355,640
Milliman USA
Allocating Reinsurance Costs and
Profit Proportional to Risk
Reinsurance costs vary by region, but…
even after reinsurance, residual risk varies by region.
Therefore, must allocate both reinsurance cost and profit
relative to regional contribution to total risk.
 Identify risk metric
 Compute risk measure for each region
 Allocate costs proportional to risk
Milliman USA
Possible Risk Measures

Variance of losses
 Standard deviation of losses
 Probability of ruin
 Expected policyholder deficit
 Any others you might like!!
Milliman USA
Allocation Using Standard Deviation
Zone 1
Allocation of Primary Company Amounts
(1) Standard Deviation of Net Losses
(2) Allocation Percent [(1) / Sum(1)]
(3) Expected Profit to Allocate
(4) Expected Losses
(5) Loss Adjustment Expense Factor
(6) Expected Losses and Loss Expenses [(4) x (5)]
(7) Expected Investment Income on Policy Reserves Percent
(8) Underwriting Profit and Contingencies
(3) - (6) x (7)
(9) Variable Expense Percent
Allocation of Reinsurer Amounts
(10) Standard Deviation of Ceded Losses
(11) Allocation Percent [(1) / Sum(1)]
(12) Expected Profit to Allocate
(13) Expected Ceded Losses
(14) Loss Adjustment Expense Factor
(15) Expected Losses and Loss Expenses [(4) x (5)]
(16) Expected Investment Income on Policy Reserves Percent
(17) Reinsurer Profit
(12) - (15) x (16)
(18) Reinsurer Expenses
(Total (18) allocated with (13))
Summary of Expense Provisions
(19) Indicated Premium [((6) + (8) + (17) + (18)) / (1.000 - (9))]
(20) Underwriting Profit and Contingencies (Percent)
(8) / (19)
(21) Reinsurer Proift (Percent)
(17) / (19)
(22) Reinsurer Expenses (Percent)
(18) / (19)
Zone 2
Zone 3
Sum
59,635,323
85.2%
6,226,869
14,879,388
1.154
17,170,814
5.3%
5,322,087
6,998,894
10.0%
730,795
7,941,806
1.154
9,164,844
5.3%
247,872
3,371,907
4.8%
352,080
9,314,590
1.154
10,749,037
5.3%
(214,319)
70,006,124
100.0%
7,309,744
32,135,784
1.154
37,084,695
5.3%
5,355,640
27.00%
27.00%
27.00%
27.00%
63,536,239
82.9%
13,127,070
14,283,493
1.154
16,483,151
5.3%
12,258,523
9,376,989
12.2%
1,937,357
1,648,536
1.154
1,902,411
5.3%
1,837,113
3,698,257
4.8%
764,088
379,796
1.154
438,284
5.3%
740,994
76,611,486
100.0%
15,828,515
16,311,825
1.154
18,823,846
5.3%
14,836,630
1,831,461
211,379
48,698
2,091,538
72,693,200
7.32%
18,306,328
1.35%
16,113,280
-1.33%
107,112,809
5.00%
16.86%
10.04%
4.60%
13.85%
2.52%
1.15%
0.30%
1.95%
Milliman USA
Summary of Profit Factors Under
Alternative Allocations
Summary - Reinsurance Cost and Profit by Zone
Standard
Deviation
Method
Variance
Method
Underwriting Profit and Contingencies
Reinsurer Proift (Percent)
Reinsurer Expenses (Percent)
Zone 1
Zone 2
7.32%
1.35%
16.86% 10.04%
2.52%
1.15%
T otal Profit plus Reinsurance Cost
26.70%
12.54%
3.57%
20.80%
8.14%
18.88%
2.37%
-2.52%
1.55%
1.39%
-3.70%
0.20%
0.33%
5.00%
13.85%
1.95%
29.39%
0.42%
-3.17%
20.80%
6.11%
17.64%
2.54%
3.61%
8.89%
1.13%
1.70%
2.85%
0.30%
5.00%
13.85%
1.95%
T otal Profit plus Reinsurance Cost
26.30%
13.63%
4.84%
20.80%
Average Across Methods
27.46%
8.86%
1.75%
20.80%
Underwriting Profit and Contingencies
Reinsurer Proift (Percent)
Reinsurer Expenses (Percent)
T otal Profit plus Reinsurance Cost
Probability
of Ruin
Method
Underwriting Profit and Contingencies
Reinsurer Proift (Percent)
Reinsurer Expenses (Percent)
Zone 3
-1.33%
4.60%
0.30%
Sum
5.00%
13.85%
1.95%
Milliman USA