Paul Margus, FSA, MAAA

Case Study: Reinsurance Retention Limits

Choosing Reinsurance Retention Limits is often more art than science. A careful study of the risks can help us make better business decisions.  Such decisions make a trade-off between

  • lower profits with greater certainty, and
  • higher but more volatile profits.

In this discussion:

  • The term “Claims” means direct claims ignoring reinsurance.
  • “Claim Cost” means direct claims, minus reinsurance reimbursements, plus reinsurance costs.

The methods discussed below work well for all lines of insurance.   However, Life Insurance claims are relatively large and rare.   This can mean more severe fluctuations in claims.   So, consider a block of 10,000 newly-issued Life Insurance policies with a total Face Amount of about $5 billion, priced for a 10% pre-tax profit margin.

Table 1 — 10,000 newly-issued Life Insurance policies

Breakdown

One year’s Issues

10-year

Expected Claims

Policy Count

Face Amount

Average Size

Issue Age

0 – 30

1,661

$861 m

16.6%

$518 K

$3,301 K

31 – 40

3,472

$1,798 m

34.7%

$518 K

$10,436 K

41 – 50

3,376

$1,731 m

33.4%

$513 K

$23,841 K

51 – 60

1,287

$681 m

13.1%

$529 K

$20,762 K

61+

204

$115 m

2.2%

$561 K

$8,210 K

Total

10,000

$5,186 m

100.0%

$519 K

$66,551 K

Face Amount

0 – 100 K

491

$40 m

0.8%

$81 K

$461 K

100 – 200 K

1,259

$207 m

4.0%

$165 K

$2,542 K

200 – 500 K

4,092

$1,459 m

28.1%

$357 K

$18,019 K

500 – 750 K

2,152

$1,344 m

25.9%

$625 K

$16,977 K

750 – 1,000 K

1,094

$960 m

18.5%

$878 K

$12,689 K

1,000 K +

912

$1,176 m

22.7%

$1,289 K

$15,863 K

Total

10,000

$5,186 m

100.0%

$519 K

$66,551 K

Sex

F

4,447

$2,287 m

44.1%

$514 K

$23,757 K

M

5,553

$2,899 m

55.9%

$522 K

$42,793 K

Total

10,000

$5,186 m

100.0%

$519 K

$66,551 K

 

Because this is a small block, we may expect a great deal of uncertainty.    Even over 10 years, that implies significant deviations from our expected $66.5 million of claims.

Reinsurance can reduce this uncertainty.   Imagine a reinsurance treaty that imposes a 10% load on expected claims, covering the reinsurer’s profit and expenses.   That is, if we reinsure the entire risk, the reinsurance cost is exactly 110% of expected claims.   Instead of $66.5 million of expected claims (± a random deviation) we incur $73.2 million of reinsurance cost (with no deviation).    For less than full reinsurance, the actual claim cost is some combination of actual claims and fixed reinsurance cost.

The 10% load sounds modest[1].   But if the expected loss ratio is 60%, then full reinsurance increases total costs to 66%.   Our expected 10% pre-tax profit margin drops to 4%.   The trade-off is clear enough: do we prefer an uncertain 10% profit margin or a guaranteed 4% margin?      Or do we prefer something in between?

To see what really happens, I did 5,000 random trials, assuming a range of retention limits.   The results appear in Table 2, Figure 1, and Table 3.

Table 2 and Figure 1 show the trials individually.   Adding reinsurance

  • decreases the standard deviation,
  • but increases the expected cost.

In the “full reinsurance” case, all 5,000 trials weigh in at a uniform $72.5 million.   Because the entire risk is transferred to the reinsurer, the claim cost is just the cost of reinsurance.   Figure 1 shows this case as a vertical yellow line.

Table 3 summarizes our findings.

  • Without reinsurance,
    • There’s an 0.6% chance that claims exceed 130% of expected, resulting in a pre-tax profit margin of -8.0% or worse.
    • There’s a 9.1% chance that claims will exceed 115% of expected.   Here, the profit margin is 1.0% or less.
    • And 54.0% (100.0% minus 46.0%) of the time, the profit margin exceeds 10%.
  • Full Reinsurance costs 10% of expected claims.   But without reinsurance, actual claims exceed the reinsurance cost only 17.4% of the time.   In other words, over a 10-year period, there’s an 82.6% probability of a “loss” from reinsurance.

In this example, we have analyzed a volatile block of Life Insurance.   The 10-year study period is long enough to mitigate this somewhat.   Most companies have a much larger policy count, further stabilizing the results.

Often, the reinsurance decision is based on a reluctance to be caught in a “worst case” situation (however improbable it may be), in which claims greatly exceed expectations.     But if we have a good handle on the claim rates, we have shown that the risks can be quantified.

Most lines of insurance are much less volatile and may not need much reinsurance.

  • Compared to our Life insurance Example, Health Insurance claims are much more frequent and generally much smaller.
  • Similar considerations apply to Group LTD and Individual Disability Income, but perhaps to a lesser extent.
  • Long Term Care is a hybrid.    In its early policy years, it resembles Life Insurance.   Claims are rare but large.    Subsequently, claims become much more frequent.

Some factors are not covered in the above analysis.

  • When the risk is not fully understood, we may well benefit from the reinsurer’s underwriting expertise.
  • Epidemics and major disasters are not included in the above analysis.  But in extreme situations, a deluge of claims may render the reinsurers unable to pay.

 

Table 2 — 5,000 random trials on 10,000 policies

Dollar Amount

Retention Limit

no reins

$1,000 K

$600 K

$200 K

full reins

Expected Claim Cost

$66.6m

$66.9m

$67.9m

$70.8m

$73.2m

Standard Deviation

$7.0m

$6.4m

$5.1m

$2.2m

$0.0m

Claim Cost in a Single Random Trial

Pre-tax Profit Margin

Number of Trials

Amount

% of Expected Claims

$43m

-

$44m

65%

+31.0%

1





$44m

-

$45m

67%

+29.8%






$45m

-

$46m

68%

+29.2%

1





$46m

-

$47m

70%

+28.0%

2

1




$47m

-

$48m

71%

+27.4%

5

1




$48m

-

$49m

73%

+26.2%

9

2




$49m

-

$50m

74%

+25.6%

10

6




$50m

-

$51m

76%

+24.4%

12

9




$51m

-

$52m

77%

+23.8%

34

15




$52m

-

$53m

79%

+22.6%

37

20

1



$53m

-

$54m

80%

+22.0%

43

29

2



$54m

-

$55m

82%

+20.8%

62

44

17



$55m

-

$56m

83%

+20.2%

90

65

13



$56m

-

$57m

85%

+19.0%

90

80

32



$57m

-

$58m

86%

+18.4%

136

105

39



$58m

-

$59m

88%

+17.2%

171

135

73



$59m

-

$60m

89%

+16.6%

184

166

107



$60m

-

$61m

91%

+15.4%

193

189

135



$61m

-

$62m

92%

+14.8%

235

225

164



$62m

-

$63m

94%

+13.6%

267

268

248



$63m

-

$64m

96%

+12.4%

270

298

261



$64m

-

$65m

97%

+11.8%

287

307

316

8


$65m

-

$66m

99%

+10.6%

288

310

378

38


$66m

-

$67m

100%

+10%

273

339

399

121


$67m

-

$68m

102%

+8.8%

275

289

424

282


$68m

-

$69m

103%

+8.2%

258

298

381

516


$69m

-

$70m

105%

+7.0%

234

253

349

815


$70m

-

$71m

106%

+6.4%

245

270

322

938


$71m

-

$72m

108%

+5.2%

215

215

292

885


$72m

-

$73m

109%

+4.6%

202

214

257

647


$73m

-

$74m

111%

+3.4%

162

180

212

415

5,000

$74m

-

$75m

112%

+2.8%

141

158

172

209


$75m

-

$76m

114%

+1.6%

112

127

140

82


$76m

-

$77m

115%

+1.0%

98

104

88

31


$77m

-

$78m

117%

-0.2%

98

64

60

9


$78m

-

$79m

118%

-0.8%

56

48

28

4


$79m

-

$80m

120%

-2.0%

49

52

34



$80m

-

$81m

121%

-2.6%

37

27

21



$81m

-

$82m

123%

-3.8%

28

21

13



$82m

-

$83m

124%

-4.4%

21

19

7



$83m

-

$84m

126%

-5.6%

18

14

8



$84m

-

$85m

127%

-6.2%

12

8

2



$85m

-

$86m

129%

-7.4%

11

11

4



$86m

-

$87m

130%

-8.0%

11

7

1



$87m

-

$88m

132%

-9.2%

4

3




$88m

-

$89m

133%

-9.8%

7

1




$89m

-

$90m

135%

-11.0%

2

1




$90m

-

$91m

136%

-11.6%


1




$91m

-

$92m

138%

-12.8%

1





$92m

-

$93m

139%

-13.4%

2





$93m

-

$94m

141%

-14.6%






$94m

-

$95m

142%

-15.2%


1




$95m

-

$96m

144%

-16.4%






$96m

-

$97m

146%

-17.6%






$97m

-

$98m

147%

-18.2%

1





Total Trials


 

5,000

5,000

5,000

5,000

5,000

















 

Figure 1 — 5,000 random trials on 10,000 policies (data from Table 1)

New Bitmap Image

 

Table 3 — 5,000 random trials on 10,000 policies (Summary)

Dollar Amount

Retention Limit

no reins

$1,000 K

$600 K

$200 K

full reins

Expected Claim Cost

$66.6m

$66.9m

$67.9m

$70.8m

$73.2m

Standard Deviation

$7.0m

$6.4m

$5.1m

$2.2m

$0.0m

 

Percent of Aggregate Risk Reinsured

0.0%

5.4%

20.6%

64.1%

100.0%

Percent of Direct Premium

Expected Loss Ratio, based on Claim Cost

60%

60.3%

61.2%

63.8%

66.0%

Expected Profit Margin

10%

9.7%

8.8%

6.2%

4.0%

Standard Deviation

6.3%

5.8%

4.6%

2.0%

0.0%

Actual Claim Cost

Pre-tax Profit Margin

Probability that Actual Claim Cost exceeds Indicated Amount.

Amount

% of Expected Claims

$66.6m

or more

100%

or more

+10%

or less

46.0%

47.7%

56.3%

96.7%

100.0%

$69.9m

or more

105%

or more

+7.0%

or less

30.7%

30.9%

33.2%

64.4%

100.0%

$73.2m

or more

110%

or more

+4.0%

or less

17.4%

16.9%

15.8%

15.0%

100.0%

$76.5m

or more

115%

or more

+1.0%

or less

9.1%

7.6%

5.3%

0.9%

--

$79.9m

or more

120%

or more

-2.0%

or less

3.1%

2.3%

1.1%

--

--

$83.2m

or more

125%

or more

-5.0%

or less

1.4%

0.9%

0.3%

--

--

$86.5m

or more

130%

or more

-8.0%

or less

0.6%

0.3%

0.0%

--

--










 



[1] Many real reinsurance treaties impose considerably higher costs.    And we should also factor in our own reinsurance administrative costs.

  © 2009 Paul Margus
 
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