Understanding Loss Reserve Risk
Seminar on Reinsurance
May 19, 2008
Spencer M. Gluck, FCAS
www.guycarp.com
Historical Index
Workers Compensation
Net 5 Year Index
Base = 2
30%
Gain (Loss)
20%
10%
0%
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
-10%
-20%
-30%
-40%
Index
Index vs. Predicted Index
Accident Year
Gain on the Index represents downward reserve development.
Gain on Index vs. Predicted Index represents more favorable development than predicted.
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Guy Carpenter
1
Maybe the chain ladder’s not so bad
 Industry carried reserves appear to be heavily influenced by the cycle.
 Simple chain-ladder projections dramatically out-perform Industry
carried reserves.
 So let’s assume that it’s the risk in the chain-ladder estimates that we
need to be concerned with.
 So on the following slides, we examine the historical performance of
chain-ladder estimates by tracking their progression over time.
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Guy Carpenter
2
One Year Reserve Movement in 3-D
Note the pattern following the calendar year of development.
6
x 10
1.5
1
0.5
0
-0.5
-1
0
-1.5
10
-2
0
2
20
4
6
8
10
30
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Guy Carpenter
3
Same data – movement as a percentage of reserves
0.6
0.4
0.2
0
-0.2
-0.4
0
-0.6
10
-0.8
0
20
2
4
6
8
10
30
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Guy Carpenter
4
Now the reserve movement divided by the square root of opening
reserves
400
300
200
100
0
-100
-200
0
-300
-400
10
-500
0
2
20
4
6
8
10
30
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Guy Carpenter
5
Individual accident year movements still have some kind of cycle in
them.
Paid Basis
Incurred Basis
One Accident Year Estimate Development
Age 1, 1 Year Later
15%
Adverse Reserve
Development
10%
5%
0%
-5%
-10%
-15%
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Statement Year
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Guy Carpenter
6
Multiple accident years in the same calendar year are remarkably
similar to the individual accident year
Paid Basis
Incurred Basis
Calendar Year Estimate Development
AY's Age 1-9, 1 Year Later
15%
Adverse Reserve
Development
10%
5%
0%
-5%
-10%
-15%
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005
Statement Year
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Guy Carpenter
7
Tracked over 5 years, the cyclical pattern becomes even clearer
(but what cycle is this?)
Paid Basis
Incurred Basis
Calendar Year Estimate Development
AY's Age 1-9, 5 Years Later
25%
Adverse Reserve
Development
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Statement Year
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Guy Carpenter
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How much does general inflation explain?
(A bit for paid, nothing for incurred).
Paid Basis
Incurred Basis
Calendar Year Estimate Development, Inflation - Adjusted
AY's Age 1-9, 5 Years Later
25%
Adverse Reserve
Development
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Statement Year
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Guy Carpenter
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AY Age 1
Adverse Reserve Development
AY's Age 2+
All AY's
Calendar Year Estimate Development - Paid Basis
AY Comparison, 5 Years Later
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Statement Year
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Guy Carpenter
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AY Age 1
AY's Age 2+
Calendar Year Estimate Development - Incurred Basis
AY Comparison, 5 Years Later
Adverse Reserve Development
All AY's
30%
25%
20%
15%
10%
5%
0%
-5%
-10%
-15%
-20%
1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Statement Year
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Guy Carpenter
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Can we measure the cycle without looking at losses?
(to some degree but it’s not perfect)
US Industry Net Earned Premium (log scale)
Deflated and Adjusted for Long Term Residual Trend
0.8
0.6
0.4
0.2
0
1976
1978
1980
1982
1984
1986
1988
1990
1992
1994
-0.2
1996
1998
2000
2002
2004
2006
WC
PPA
CA
CMP
MM
GLGT
-0.4
-0.6
Accident Year
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Guy Carpenter
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Starting point for the model: almost the same as Zehnwirth’s trend
parameter structure1
 Cij are incremental paid losses,
 Ei are exposures of some kind,
 And yij = Cij/Eij
j
i j
k 1
l 2
yij  AYi   DYk   CYl  eij
 AYi are accident year scale parameters2
 DYk and CYl are trend parameters2
1 – The difference is that the errors are additive and “over-dispersed Poisson”
rather than multiplicative and log-normal.
2 – It’s a modeling “framework”. The subscripted parameters would never be all
different. You choose when to make them different.
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Guy Carpenter
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The calendar year parameters are the key
 Changing the calendar year parameters (when necessary) allows you
to model crucial features of the data not possible in many other
methods.
 We can try to differentiate between the calendar year effects (loss
trend cycle) and the accident year effects (pricing/underwriting cycle).
 But if calendar year trends sometimes change in our data, might they
not change (unpredictably) in the future?
Possible changes in future calendar years are both
unpredictable and non-diversifying.
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Guy Carpenter
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Additional model features
 The ODP version allows zeros as legitimate values for paid data, and
variance proportional to the mean fits many data sets without
additional adjustment.
 We model outstanding losses as well, with inter-dependent paid and
outstanding. The outstanding losses influence the projection to the
extent of their demonstrated predictive power.
 The accident year parameters can be smoothed with a “Kalman filter”
(an assumed random walk in the underlying parameters).
 The calendar year trends can be smoothed assuming an underlying
first-order autoregressive (AR-1) time series.
– This provides parameter smoothing, a predicted future
trend, and a probabilistic structure to reflect the risk of
future trend changes
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Guy Carpenter
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WC Residuals – Paid Model
0
0
2 4 6 8 10 12 14 16 18 20 22 24 26 28
Accident Year Index
2
4
6
8
Development Year Index
10
0
2
4
6
8 10 12 14 16 18 20 22 24 26 28
Calendar Year Index
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Guy Carpenter
16
WC Residuals – Paid/Outstanding Model
0
0
2 4 6 8 10 12 14 16 18 20 22 24 26 28
Accident Year Index
2
4
6
8
Development Year Index
10
0
2
4 6
8 10 12 14 16 18 20 22 24 26 28
Calendar Year Index
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Guy Carpenter
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Some results -- Fitted Paid Loss Trends
25%
20%
15%
GL
10%
WC
5%
0%
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
-5%
 Correlation is 84%
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Guy Carpenter
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Some results -- Fitted Underwriting Cycle
GL
WC
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
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Guy Carpenter
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