CIA Annual Meeting
Ottawa June 2006
Peter Douglas
University of Regina douglas@math.uregina.ca

Stochastic Models
Application to LTD

When is a stochastic model appropriate?

Why stochastic LTD?

How?

Stochastic LTD

Ideas presented are very Blue Sky

My goal to provoke thought

Not that hard to do

Excel model

When Are Stochastic Models
Appropriate?

When the loss model has
 a long or heavy right tail

A “ cliff ” or trigger point

Dependencies of Risk

Looking At LTD Models

Traditionally viewed as a life annuity a x : 65
 x |

65
 t

 x
0 v t t p x

We can also view these as a random variable with probability distribution

0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.09
0.08
0.07
0.06
How Often Do We Look at LTD
This Way?
Duration on Claim (CIA Termination Rates M42)
24 48 72 96 120 144 168
Months of Payment
192 216 240 264

0.05
0.04
0.03
0.02
0.01
0
0
0.1
0.09
0.08
0.07
0.06
How Often Do We Look at LTD
This Way?
Duration on Claim (CIA Termination Rates M42)
24
Trigger
Point
48 72
Heavy tail
96 120 144 168
Months of Payment
192 216 240 264

1.20
1.00
0.80
0.60
0.40
0.20
0.00
42
Why Use the Same Pricing
Formula for These Loss Models?
Survival Functions For M42
47 62 52
Age
CIA LTD GAM
57

Dependencies of Risks

LTD Experience is influenced by


Economic conditions
Geographic location



CPP policy
Court decisions
Legislation

All of these lead to a dependency between risks (i.e. claims)

Sample Group 100 Males
(all duration 0)
Age
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
Number Ave Benefit
3 $ 1,324.91
7
11
$
$
2,152.00
2,042.81
13
13
14
14
14
11
$
$
$
$
$
$
2,064.21
2,709.21
2,942.74
3,279.58
3,551.44
3,605.74

Simple Stochastic Model



1000 Trials
Each trial represents one possible outcome for the portfolio
For each trial



Simulate the time on claim for each life
Use CIA LTD table to determine distribution of time on claim
Sum the PVs for all lives

Sample Group Results
Percentile
50%
67%
80%
90%
95%
99%
PV
$ 8,255,576
$ 8,727,606
$ 9,231,784
$ 9,713,406
$ 10,115,280
$ 11,191,097
Ratio to Mean
100%
106%
112%
118%
123%
136%
Mean $ 8,241,102

Sample Group Frequency
Distribution
20%
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
5-5.5
6-6.5
7-7.5
8-8.5
9-9.5 10-10.5 11-11.5 12-12.5
PV ($millions)

Sample Group With 3 Yr
Duration Pooling
Percentile
50%
67%
80%
90%
95%
99%
PV
$ 4,190,965
$ 4,559,952
$ 4,947,502
$ 5,351,332
$ 5,700,433
$ 6,395,653
Ratio to Mean
100%
109%
118%
127%
136%
152%
Mean $ 4,198,533

3 Yr Pooling – Frequency
Distribution
24%
22%
20%
18%
16%
14%
12%
10%
8%
6%
4%
2%
0%
2-2.5
3-3.5
4-4.5
5-5.5
PV ($ millions)
6-6.5
7-7.5
8-8.5

Ideas for Incorporating
Dependencies of Risks


Add a random variable

 allow for good or poor years affects all lives equally

Key impact in early years
Modification to termination probability for each year

Use same modification for each life

Recalculate distributions for time on claim with each trial

Uses For Stochastic LTD Models

Supplement not replace deterministic models

Better understanding of risks

Pooling charges