July 29, 2003
CS-12 IAA Progress on RBC
Life Case Study
Les Rehbeli
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
1
Introduction

Purpose of case study
– To demonstrate approaches to determine solvency provisions
for various risks
– To illustrate concepts for advanced internal modeling
– To highlight issues a factor-based approach must address
2
Internal Modeling

Develop models to quantify various risks being considered
– Analyze each risk separately
 Generate scenarios in which liabilities vary only on the risk
being measured
– Aggregate into total company solvency requirement

Focus on total solvency provisions
– Sum of reserves and capital
3
Internal Modeling

Model cash flows over time horizon appropriate to risk being modeled
– Systematic (non-diversifiable) risks over entire term of liability
– Non-systematic (diversifiable) risks over shorter horizon

Liabilities defined as present value of future liability cash flows
discounted at risk-free rate

Solvency provision defined as difference between average liabilities of
worst 1% of scenarios and best estimate liabilities
– CTE(99) minus CTE(0)
 approximately equivalent to 99.5th percentile
4
Risks Analyzed in the Case Study

Mortality (systematic risks)
– Mortality level risk
– Mortality trend risk

Lapse (systematic risks)
– Lapse level risk

Non-systematic insurance risks
– Mortality volatility risk
– Mortality catastrophe risk
– Lapse volatility risk

Market risks
– Credit risk
– Mismatch risk
5
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
6
The Insurance Company

Medium-sized insurance company
– term, whole life and immediate annuity non-participating
products

Assets managed at the segment level
– segments for insurance products, annuity products and surplus
– liabilities supported by high grade fixed income securities
– surplus also invested in stocks

Various reinsurance arrangements in place
7
The Insurance Company
Company Segmentation
Product Code
Type of Product
Number of
Lives
Sum Assured or
Monthly Payment
ALC 1001
Term to 100 Insurance
56,971
3.6 billion
ALC 1002
Non-Par Whole Life
5,000
0.9 billion
ALC 1003
Term to 100 Insurance
94,560
9.0 billion
ALC 1004
1 Year Renewable Term
7,463
1.4 billion
ALC 1005
5 Year Renewable Term
3,450
0.5 billion
ALC 1006
Payout Annuities
250
1.5 million / month
8
Total Solvency Provisions
($ millions)
Systematic Insurance Risks
Non-Systematic Insurance
Risks
Market Risks
Product
Segment
Mortality
Level
Mortality
Trend
Lapse
Level
Mortality
Volatility
Mortality
Catastr.
Lapse
Volatility
Mismatch
Default
Total
T100 – 1
43.1
50.1
28.9
3.4
6.2
3.5
-
-
73.7
Whole Life
43.8
17.4
7.1
3.3
3.8
3.2
-
-
49.2
T100 – 2
105.7
163.6
103.3
9.5
35.1
10.9
-
-
227.5
1 yr YRT
53.1
37.6
39.9
21.5
3.5
12.8
-
-
86.3
5 yr YRT
8.6
5.8
3.9
3.9
4.4
2.1
-
-
14.8
Total Ins.
-
-
-
-
-
-
335.7
3.8
335.7
Annuities
16.8
8.7
-
0.2
(0.1)
-
15.7
1.4
24.7
Surplus
-
-
-
-
-
-
-
26.7
26.7
Total
178.8
265.8
152.8
29.7
53.0
26.1
351.4
30.5
512.4
9
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
10
Mortality Risks

Level risk
– misestimation of the mean

Trend risk
– deterioration of the mean

Volatility risk
– statistical fluctuations

Catastrophe risk
– spike in mortality experience
11
Mortality Level Risk

Misestimation of the mean

Mortality assumptions based on mortality studies and industry data
– but mortality studies are based on observations that are volatile

In a mortality study, we may presume that historical observations
represent the best estimate level of mortality
– but it is possible that the observations are in the tail of the true
mortality distribution
12
Mortality Level Risk
Setting of Best Estimate Mortality Assumption
20%
30%
40%
50%
60%
70%
80%
90%
100%
110%
120%
% of Industry Table
13
Mortality Level Risk

The smaller the portfolio, the larger the range of possible outcomes for
future mortality
– might also partially rely on industry data

To evaluate mortality level risk, assume that observations were actually
at, say, 99th percentile of the true distribution
– by using inverse Normal Power approximation
– or by simulating claims experience and using 99th percentile

For case study, revalue liabilities with mortality assumption distribution to
calculate CTE(99)
– or simply revalue liabilities at 99.5th percentile of assumptions
14
Mortality Level Risk
Liabilities ($ millions)
Mortality
Assumption
Percentile
T100 – 1
Non-Par
Whole Life
T100 – 2
1 year YRT
5 year YRT
Payout
Annuities
5.0
124.4
31.2
736.3
(267.1)
(27.8)
271.9
25.0
144.2
46.8
787.0
(241.6)
(24.0)
267.9
50.0
157.2
57.7
824.2
(225.8)
(21.4)
263.8
75.0
170.0
68.9
860.6
(211.1)
(19.0)
255.6
95.0
185.2
84.9
900.8
(191.5)
(15.8)
252.5
99.0
195.4
95.7
921.4
(179.2)
(13.7)
251.0
99.5
198.7
99.8
926.8
(174.9)
(13.2)
248.0
99.9
204.2
110.5
934.8
(167.1)
(12.1)
243.0
CTE(99) –
CTE(0)
43.1
43.8
105.7
53.1
8.6
16.8
15
Mortality Trend Risk



Deterioration of the mean
– misestimation of the trend
We can estimate a “best estimate trend” based on past observations and
expert opinions
– uncertain due to volatility in past observations
– also due to systematic changes in the trend
Quantify trend uncertainty by revaluing liabilities under other trend
assumptions
16
Mortality Trend Risk


For case study, assume annual rate of
mortality improvement is normally distributed
– mean and standard deviation of 0.50%
improvement per year
– limit improvement to 40 years
– limit range to -3.0% and 3.0%
Apply to all products simultaneously
– determine which direction will increase
liabilities on a company basis
– consider reinsurance
Percentile
Annual
Mortality
Improvement
0.5
1.77%
1.0
1.66%
5.0
1.32%
10.0
1.14%
30.0
0.76%
50.0
0.50%
70.0
0.24%
90.0
-0.14%
95.0
-0.32%
99.0
-0.66%
99.5
-0.76%
17
Mortality Trend Risk
Liabilities ($ millions)
Mortality
Trend
Percentile
T100 – 1
Non-Par
Whole Life
T100 – 2
1 year
YRT
5 year
YRT
Payout
Annuities
Total
5.0
123.4
44.9
715.2
(249.4)
(25.2)
257.3
867.2
25.0
142.8
52.5
779.2
(235.6)
(23.1)
254.1
972.9
50.0
156.6
57.4
826.1
(225.9)
(21.6)
251.9
1,046.0
75.0
170.3
62.2
870.5
(216.5)
(20.0)
249.6
1,116.9
95.0
189.1
68.7
928.9
(202.7)
(17.9)
246.4
1,212.9
99.0
201.2
72.7
966.3
(193.0)
(16.5)
243.8
1,274.1
99.5
204.7
74.2
982.2
(189.9)
(16.0)
242.9
1,296.1
99.9
214.0
76.8
1,014.5
(182.2)
(15.0)
241.4
1,339.0
CTE(99) –
CTE(0)
50.1
17.4
163.6
37.6
5.8
8.7
262.5
18
Mortality Volatility Risk

Statistical fluctuations around the expected assumptions
– assume that the best estimate assumption is correct

Time horizon
– level and trend risks were measured over the entire term of the liability
– volatility risk can be diversified by management action
 project out for a two year time horizon

Simulation approach taken for case study
– analytic methods are also feasible to quantify volatility risk
19
Mortality Volatility Risk
Claims over two year horizon ($ millions)
Mortality
Volatility
Percentile
T100 – 1
Non-Par
Whole
Life
5.0
10.5
25.0
T100 – 2
1 year
YRT
5 year
YRT
Payout
Annuities
Total
Correlated
Total
Independent
4.9
60.1
15.9
3.5
44.6
139.5
144.6
11.2
5.5
62.4
17.3
3.9
44.7
144.8
147.8
50.0
11.8
6.0
64.2
18.6
4.3
44.7
149.6
150.4
75.0
12.5
6.7
66.2
20.4
4.8
44.8
155.5
153.4
95.0
13.7
7.9
69.7
25.1
5.9
44.9
166.4
159.1
99.0
14.7
9.0
72.5
32.1
7.2
44.9
176.7
165.5
99.5
15.1
9.3
73.6
37.0
7.9
45.0
180.7
170.0
99.9
16.1
10.1
75.6
54.1
9.9
45.0
190.3
182.7
CTE(99)
– CTE(0)
3.4
3.3
9.5
21.5
3.9
0.2
31.7
22.7
20
Mortality Volatility Risk
($ millions)
Product
Capital Based on Two
Years Claims
Capital Based on All
Liability Cash Flows
T100 – 1
3.4
6.2
Whole Life
3.3
5.4
T100 – 2
9.5
16.8
1 Year YRT
21.5
23.9
5 Year YRT
3.9
12.9
Annuities
0.2
7.6
21
Mortality Catastrophe Risk

One-time spike in mortality experience
– for example, Spanish Flu

Highly subjective

Deterministic approach taken for case study
– doubling of mortality for one year

Interaction between catastrophe risk and volatility risk
– capital for catastrophe risk is difference between CTE(99) at higher
mortality and CTE(99) at normal mortality
22
Mortality Catastrophe Risk
Claims over two year horizon ($ millions)
Risk
Measure
Expected
Mortality
Basis
CTE(99)
CTE(0)
T100 – 1
Non-Par
Whole Life
T100 – 2
100%
15.3
9.5
100%
11.9
Capital for
volatility
1 year YRT
5 year
YRT
Payout
Annuities
74.0
40.8
8.3
45.0
6.2
64.5
19.4
4.4
44.7
3.4
3.3
9.5
21.5
3.9
0.2
CTE(99)
200%
21.5
13.3
109.0
44.3
12.8
44.9
CTE(99)
100%
15.3
9.5
74.0
40.8
8.3
45.0
Capital for
catastrophe
6.2
3.8
35.1
3.5
4.4
(0.1)
Total
9.6
7.2
44.6
24.9
8.3
0.1
23
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
24
Lapse Risks

Can be analyzed in similar fashion to mortality risks

But several other factors to consider:
– lapse rates may be correlated with economic assumptions for some
portfolios
 very difficult to model
– lapse assumption highly dependent on product and how it is sold
– impact to company can vary for different policy durations and products

Case study analyzes inaccuracies due to statistical error
25
Lapse Risks

Level risk
– Misestimation of the best estimate

Volatility risk
– Statistical fluctuations
26
Lapse Level Risk

Misestimation of the best estimate

From lapse studies, we can determine best estimate lapse rates and their
standard deviations
– we can assume a distribution for the lapse rates and solve for lapse
rates at alternate percentiles
 e.g. assume lapses are normally distributed and grade from 10% to
1% over 12 years
– 90th percentile lapse assumption may be 12.4% grading to 1.2%
– 10th percentile lapse assumption may be 8.7% grading to 0.8%

Need to account for policyholder behavior / economic environment

Statistical error may not always be one-sided
27
Lapse Level Risk
Liabilities ($ millions)
Lapse
Level
Percentile
T100 – 1
Non-Par
Whole
Life
Lapse
Rates
5.0
Higher
138.1
25.0
Higher
50.0
T100 – 2
1 year
YRT
5 year
YRT
Total
Correlated
Total
Independent
49.2
742.5
(178.4)
(17.1)
965.3
951.0
148.7
52.3
787.6
(187.9)
(17.7)
1,006.1
999.7
Exp.
155.9
54.5
818.1
(196.8)
(18.6)
1,033.7
1,032.2
75.0
Lower
163.2
56.5
847.0
(216.2)
(20.5)
1,061.8
1,064.6
95.0
Lower
173.9
59.1
884.7
(224.2)
(21.3)
1,097.5
1,105.6
99.0
Lower
181.3
60.7
910.3
(228.1)
(21.7)
1,119.7
1,133.8
99.5
Lower
183.8
61.3
917.0
(236.1)
(22.6)
1,126.7
1,143.1
99.9
Lower
188.9
62.4
933.4
(250.4)
(24.2)
1,147.4
1,160.7
CTE(99) – CTE(0)
28.9
7.1
103.3
39.9
3.9
97.2
115.2
28
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
29
Market Risks

Mismatch risk
– ALM risk

Asset default risk
– credit risk
30
Mismatch Risk

ALM risk
– the risk that best estimate asset cash flows do not match best estimate
liability cash flows
– reinvestment and disinvestment risk
– the risk that the market price of assets changes unfavorably at a time
when those assets need to be liquidated

Case study projects best estimate asset and liability liabilities under many
future reinvestment rate scenarios
31
Mismatch Risk
Assets Required to Back Liabilities ($ millions)
Percentile
Insurance
Annuities
5.0
294.6
221.0
25.0
406.0
226.3
50.0
489.2
230.4
75.0
577.0
236.5
95.0
807.9
243.6
99.0
841.9
246.1
99.5
842.7
246.6
99.9
843.3
247.0
CTE(99) – CTE(0)
335.7
15.7
32
Asset Default Risk

Credit risk

Case study uses factors derived from existing regulatory regime

Since other provisions for risk use the risk-free discount rate, the
provision for credit risk on assets backing liabilities is not necessary
 included all assets in case study for demonstration purposes
33
Asset Default Risk
Capital Requirements ($ millions)
Capital for Asset Default
Asset Type
Book Value
of Assets
Credit Risk
Factors
Insurance
Annuity
Surplus
Total
Bank Notes
77.5
0.25%
0.2
0.0
0.0
0.2
Corp. Bonds AAA
134.5
0.25%
0.2
0.1
0.0
0.3
Corp. Bonds AA
263.7
0.50%
0.9
0.4
0.0
1.3
Corp. Bonds A
286.4
1.00%
1.2
0.5
1.1
2.9
Corp. Bonds BBB
99.5
2.00%
0.9
0.4
0.6
2.0
Mortgage Residential
4.0
2.00%
0.1
0.0
0.0
0.1
Mortgage Commercial
8.7
4.00%
0.3
0.0
0.0
0.3
Common Stocks
145.8
15.00%
0.0
0.0
21.8
21.8
Preferred Stocks
63.5
2.00%
0.0
0.0
1.3
1.3
Real Estate
15.8
4.00%
0.0
0.0
0.6
0.6
Other
12.5
8.00%
0.0
0.0
1.0
1.0
Total
1,576.8
3.8
1.4
26.7
31.9
34
Contents
1. Introduction
2. The Insurance Company
3. Mortality Risk
4. Lapse Risk
5. Market Risk
6. Effects of Reinsurance
35
Effects of Reinsurance

Factor-based systems cannot fully capture the characteristics of the risks
a company faces
– especially when reinsurance is used

Case study analyzes six reinsurance arrangements:
– YRT 45% coinsurance at neutral reinsurance rates
– YRT excess reinsurance at neutral insurance rates
– YRT 90% coinsurance at neutral reinsurance rates
– YRT 45% coinsurance at low reinsurance rates
– YRT excess reinsurance at low insurance rates
– Quota share
36
Effects of Reinsurance
Capital for Mortality Risks ($ millions)
Reinsurance
Type
Ceded
Reinsurance
Premiums
Gross Basis
Level
Trend
Volatility
Catastrophe
43.1
50.1
3.4
6.2
Coins.
45%
70% Table
20.9
20.3
1.8
3.4
Excess
Retention
> $50K
70% Table
22.3
21.7
0.9
3.5
Coins.
90%
70% Table
2.2
9.2
0.3
0.6
Coins.
45%
45% Table
23.3
23.4
1.9
3.5
Excess
Retention
> $50K
45% Table
23.6
25.2
0.9
3.6
Quota
Share
45%
N/A
24.3
27.2
1.9
3.4
37
Conclusions

Advanced models can be developed to better understand the net risks
faced by an insurance company

These models can be used to develop a standardized approach for risks
that are well understood and for which there is ample historical data
– difficult to accurately capture the impact of reinsurance

Must exercise care for risks not modeled in the case study:
– impact of policyholder behavior
– complex options in policies
– complex interactions between risks
38