An Introduction to Stochastic
Reserve Analysis
Gerald Kirschner, FCAS, MAAA
Deloitte Consulting
Casualty Loss Reserve Seminar
September 2004
Presentation Structure



Background
Chain-ladder simulation methodology
Bootstrapping simulation methodology
Arguments against simulation


Stochastic models do not work very well
when data is sparse or highly erratic.
Stochastic models overlook trends and
patterns in the data that an actuary
using traditional methods would be able
to pick up and incorporate into the
analysis.
Why use simulation in reserve
analysis?



Provide more information than
traditional point-estimate methods
More rigorous way to develop ranges
around a best estimate
Allows the use of simulation-only
methods such as bootstrapping
Simulating reserves stochastically
using a chain-ladder method



Begin with a
traditional loss
triangle
Calculate link
ratios
Calculate mean
and standard
deviation of the
link ratios
Acc.
Year
1
2
3
4
Development Age
12
24
36
48
1,000
1,500
1,750
2,000
1,200
2,000
2,300
1,800
2,500
2,100
Acc.
Link Ratios
Year
12 - 24 24 - 36 36 - 48
1
1.500
1.167
1.143
2
1.667
1.150
3
1.389
Mean
1.500
1.157
1.143
Std. Deviation 0.1179 0.0082
0
Simulating reserves stochastically
using a chain-ladder method


Think of the observed link ratios for
each development period as coming
from an underlying distribution with
mean and standard deviation as
calculated on the previous slide
Make an assumption about the shape of
the underlying distribution – easiest
assumptions are Lognormal or Normal
Simulating reserves stochastically
using a chain-ladder method

For each link ratio that is needed to square
the original triangle, pull a value at random
from the distribution described by
1.
2.
3.
Shape assumption (i.e. Lognormal or Normal)
Mean
Standard deviation
Simulating reserves stochastically
using a chain-ladder method
Lognormal Distribution, mean 1.5, standard deviation 0.1179
% of Total Observations
30.0%
Acc.
Year
1
2
3
Mean
Std. Deviation
Link Ratios
12 - 24 24 - 36 36 - 48
1.500
1.167
1.143
1.667
1.150
1.389
1.500
1.157
1.143
0.1179 0.0082
0
Acc.
Year
1
Link Ratios
12 - 24 24 - 36 36 - 48
1.500
1.167
1.143
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
1.168 1.249 1.330 1.411 1.492 1.573 1.654 1.735 1.816 1.897
Random draw
2
1.667
1.150
1.143
3
4
1.389
1.419
1.163
1.145
1.143
1.143
Simulated values are shown in red
Simulating reserves stochastically
using a chain-ladder method



Square the triangle using the simulated
link ratios to project one possible set of
ultimate accident year values. Sum the
accident year results to get a total
reserve indication.
Repeat 1,000 or 5,000 or 10,000 times.
Result is a range of outcomes.
Enhancements to this
methodology

Options for enhancing this basic
approach


Logarithmic transformation of link ratios
before fitting, as described in Feldblum et
al 1999 paper
Inclusion of a parameter risk adjustment as
described in Feldblum, based on Rodney
Kreps 1997 paper “Parameter Uncertainty
in (Log)Normal distributions”
Simulating reserves stochastically
via bootstrapping


Bootstrapping is a different way of
arriving at the same place
Bootstrapping does not care about the
underlying distribution – instead
bootstrapping assumes that the
historical observations contain sufficient
variability in their own right to help us
predict the future
Simulating reserves stochastically
via bootstrapping
Actual Cumulative Historical Data
Acc.
Year
1
2
3
4
Ave Link Ratio
Development Age
12
24
36
48
1,000
1,500
1,750
2,000
1,200
2,000
2,300
1,800
2,500
2,100
1.500
1.157
1.143
1.
2.
Keep current
diagonal intact
Apply average link
ratios to “backcast” a series of
fitted historical
payments
Recast Cumulative Historical Data
Acc.
Year
1
2
3
4
Development Age
12
24
36
48
1,008
1,512
1,750
2,000
1,325
1,988
2,300
1,667
2,500
2,100
Ex: 1,988 =
2,300
Simulating reserves stochastically
via bootstrapping
3.
Actual Incremental Historical Data
Acc.
Year
1
2
3
4
Development Age
12
24
36
1,000
500
250
1,200
800
300
1,800
700
2,100
48
250
Recast Cumulative Historical Data
Acc.
Year
1
2
3
4
Development Age
12
24
36
1,008
504
238
1,325
663
312
1,667
833
2,100
4.
Convert both actual and
fitted triangles to
incrementals
Look at difference
between fitted and actual
payments to develop a set
of Residuals
Residuals
48
250
Acc.
Year
1
2
3
4
Development Age
12
24
36
(0.259) (0.183) 0.801
(3.437) 5.340 (0.699)
3.266 (4.619)
0.000
48
0.000
Simulating reserves stochastically
via bootstrapping
Residuals adjusted for # degrees of freedom
5.
= Residual * [n / (n-p) ]^0.5
n = # data points
p = # Parameters to be estimated = (2 * number of AY) - 1
Acc.
Development Age
Year
12
24
36
48
1
(0.473)
(0.335) 1.462
0.000
2
(6.275)
9.749 (1.275)
3
5.963
(8.433)
6.
4
0.000
n
p
DF
10
7
1.82574
Adjust the residuals to
include the effect of
the number of degrees
of freedom.
DF adjustment =
n n  p 
where n = # data
points and p = #
parameters to be
estimated
Simulating reserves stochastically
via bootstrapping
Random Draw from Residuals
Acc.
Year
1
2
3
4
Development Age
12
24
36
1.462
(0.335) 5.963
9.749
(8.433) (0.473)
(1.275)
(6.275)
9.749
7.
48
1.462
False History
= [residual * (fitted incremental ^ 0.5)] + fitted incremental
Acc.
Development Age
Year
12
24
36
48
1
1,055
497
330
273
2
1,680
445
304
3
1,615
652
4
2,547
Create a “false history”
by making random
draws, with
replacement, from
the triangle of
adjusted residuals.
Combine the random
draws with the recast
historical data to come
up with the “false
history”.
Simulating reserves stochastically
via bootstrapping
Cumulated False History
Acc.
Year
1
2
3
4
Ave Link Ratio
8.
Development Age
12
24
36
1,055
1,551
1,881
1,680
2,125
2,429
1,615
2,267
2,547
1.367
1.172
1.145
48
2,154
9.
Squaring of the Cumulated False History
Acc.
Year
1
2
3
4
12
1,055
1,680
1,615
2,547
Development Age
24
36
1,551
1,881
2,125
2,429
1,615
1,893
3,480
4,080
48
2,154
2,782
2,168
4,673
Calculate link ratios
from the data in
the cumulated false
history triangle
Use the link ratios
to square the false
history data
triangle
Simulating reserves stochastically
via bootstrapping



Could stop here – this would give N different
possible reserve indications.
Could then calculate the standard deviation of
these observations to see how variable they
are – BUT this would only reflect estimation
variance, not process variance.
Need a few more steps to finish incorporating
process variance into the analysis.
Simulating reserves stochastically
via bootstrapping
10.
Calculate the scale
parameter Φ.
Incorporate Process Variance in the model
Calculate scale parameter Φ = Pearson chi-squared statistic / # degrees of freedom
Pearson χ2 = sum of the squares of the unscaled Pearson residuals
DF = # data points / # parameters to be estimated
Acc.
Year
1
2
3
4
Development Age
12
24
36
0.067
0.034
0.641
11.811
28.514
0.488
10.667
21.333
0.000
Φ=
4
0
.
2
8
7
9
48
0.000
Simulating reserves stochastically
via bootstrapping
Calculate Incremental Future Payments
Acc.
Development Age
Year
12
24
36
1
2
3
278
4
934
600
11.
48
353
275
592
Pull random draws from a series of Gamma distributions
mean = incremental future payment from the previous step
variance = Φ * mean
Acc.
Development Age
Year
12
24
36
48
1
2
313
3
213
280
4
1,047
597
501 12.
RESERVE = sum of random draws =
2,951
Draw a random
observation from
the underlying
process
distribution,
conditional on the
bootstrapped
values that were
just calculated.
Reserve = sum of
the random draws
Pros / Cons of each method
Chain-ladder Pros
 More flexible - not
limited by observed
data
Chain-ladder Cons
 More assumptions
 Potential problems
with negative values
Bootstrap Pros
 Do not need to
make assumptions
about underlying
distribution
Bootstrap Cons
 Variability limited to
that which is in the
historical data
Selected References for
Additional Reading




England, P.D. & Verrall, R.J. (1999). Analytic and bootstrap
estimates of prediction errors in claims reserving. Insurance:
Mathematics and Economics, 25, pp. 281-293.
England, P.D. (2001). Addendum to ‘Analytic and bootstrap
estimates of prediction errors in claims reserving’. Actuarial
Research Paper # 138, Department of Actuarial Science and
Statistics, City University, London EC1V 0HB.
Feldblum, S., Hodes, D.M., & Blumsohn, G. (1999). Workers’
compensation reserve uncertainty. Proceedings of the
Casualty Actuarial Society, Volume LXXXVI, pp. 263-392.
Renshaw, A.E. & Verrall, R.J. (1998). A stochastic model
underlying the chain-ladder technique. B.A.J., 4, pp. 903923.