Best Estimates for Reserves
Glen Barnett
and
Ben Zehnwirth
email: GlenBarnett@insureware.com, BenZehnwirth@insureware.com,
or find us on http://www.insureware.com
Summary
I. Ratio techniques and extensions
• Ratios are regressions
• Regressions have
assumptions
assume when using ratios)
• Assumptions need to be
checked
(know what you
(When do ratios work?)
• Assumptions often don't hold
• What does this suggest?
Summary
II. Statistical modeling framework
(Probabilistic Trend Family of models)
• Model the logarithms of the incrementals
• Parameters to pick up trends in the three directions
• Probability Distribution to every cell
• Assumptions generally met
• Assessing stability of trends
(confidence about the future)
III. Reserve Figure
Ratio techniques and extensions
j-1 j
x{ }y
E(y|x) = bx
and
Var(y|x) = 2 x
(Mack 1993)
- weighted least squares with w = 1/x
- weighted average with w = x
^
b=
 xy ·1/x

x2 ·1/x
=
 y/x · x
x
=
y
x
Chain Ladder Ratio
Equivalently, y = bx + 
where E( ) = 0 and Var ( ) = 2x
y
y/x
x
trend
Ratio techniques are regression estimators
· Chain ladder is weighted regression through origin
· Derives standard errors of link ratios, forecasts for
chain ladder
· Introduces weighting (d: Var() = 2xd) parameter:
d = 2: Ave. Dev. Factor
d = 1: Chain Ladder
d = 0: Ordinary regression through origin
Regression methodology advantages
· Standard Errors of ratios and forecasts
· Testing of assumptions
Incurred losses analysed by Mack(1993)
Residuals of chain ladder ratios
Wtd. Std. Residuals
vs. Fitted Values
2
1
0
-1
0
5000
10000
15000
20000
Is E(y|x) = bx satisfied by the data?
25000
30000
1982: low incurred development 1984: high incurred development
Unadjusted Data
vs. Dev. Year
Unadjusted Data
vs. Dev. Year
2.5e4
2.5e4
2e4
2e4
1.5e4
1.5e4
1e4
1e4
5e3
5e3
0
0
0
1
2
3
4
5
6
7
8
9
1982 is underfitted
0
1
2
1
1
0
0
-1
-1
83
84
85
86
87
4
5
6
7
8
9
Wtd. Std. Residuals
vs. Acc. Year
2
82
3
1984 is overfitted
Wtd. Std. Residuals
vs. Acc. Year
81
2
88
89
81
82
83
84
85
86
87
88
89
Why is E(y|x) = bx not satisfied by the data?
y
x
The best line has an intercept
- typical with real, exposure adjusted data
y = a + bx + 
Var ( ) = 2xd
- Including an intercept will give a better fit. (Murphy, 1995)
where
- Unbiased “Development Factors”.
• Works wholly within a regression framework
• Advocates use of intercept
• Derives standard errors of forecasts for d=012
Ratios with Intercepts
ELRF Parameters
Devel.
Period
00-01
01-02
02-03
03-04
04-05
05-06
06-07
07-08
08-09
Est.
4,329.21
4,159.69
4,235.92
2,188.79
3,562.27
743.46
792.81
****
****
Intercept
S.E.
516.31
2,531.37
2,814.52
1,133.11
2,031.41
2,387.33
153.13
****
****
P-value
0.00007
0.15144
0.19266
0.12557
0.17778
0.78494
0.12147
****
****
Ratio(Slope)
Est.
Ratio-1
1.21445 0.21445
1.06962 0.06962
0.91968 -0.08032
1.03341 0.03341
0.92675 -0.07325
1.00608 0.00608
0.99109 -0.00891
1.01689 0.01689
1.00922 0.00922
S.E.
0.42131
0.35842
0.24743
0.07443
0.11023
0.12204
0.00824
0.01495
****
P-value
0.6264
0.8524
0.7586
0.67679
0.55389
0.96477
0.47483
0.46142
****
Delta (d) = 1
AIC = 760.5
In order for the test to be conducted at an overall 5% level, a parameter is
regarded as insignificant if the corresponding P-Value is greater than 0.00320.
y – x = a + (b – 1) x + 

(Venter, 1996)
Cumulative at dev. period j-1
Incremental at dev. period j
j-1 j
x{ }y
Case (i) b > 1, a = 0
Use link-ratios for projection
Case (ii) b = 1, a  0
a^ = Ave (incrementals)
Abandon Ratios - No predictive power
Plot of Development Year 1
vs Development Year 0
Cumulative (1) vs.
Incremental (1) vs.
Cumulative (0)
Cumulative (0)
12000
9000
7000
8000
5000
3000
4000
1000
0
0
0
2000
5000
2000
Corr=-0.117 P-value=0.764
5000
Weighted Standardized Residuals of Cape Cod model
y–x=a+
^a = Ave (y–x)
Wtd. Std. Residuals
vs. Dev. Year
Wtd. Std. Residuals
vs. Acc. Year
2
2
1
1
0
0
-1
-1
-2
-2
1
2
3
4
5
6
7
8
9
81
Wtd. Std. Residuals
vs. Pay. Year
82
83
84
85
86
87
88
89
Wtd. Std. Residuals
vs. Fitted Values
2
2
1
1
0
0
-1
-1
-2
-2
82
83
84
85
86
87
88
89
90
5000
10000
15000
20000
25000
Forecasts
Chain Ladder Ratios Model
Accident Mean
Standard Coeff. Of
Year
Forecast Error
Variation
1981
0
0
***
1982
155
148
0.96
1983
616
586
0.95
1984
1633
702
0.43
1985
2779
1404
0.51
1986
3671
1976
0.54
1987
5455
2190
0.4
1988
10934
5351
0.49
1989
10668
6335
0.59
1990
16360
24606
1.5
Total
52272
26883
0.51
Cape Cod Model
Accident Mean
Standard Coeff. Of
Year
Forecast Error
Variation
1981
0
0
***
1982
172
41
0.24
1983
483
464
0.96
1984
1112
498
0.45
1985
1971
1167
0.59
1986
4230
1526
0.36
1987
6908
1651
0.24
1988
10283
3231
0.31
1989
14905
3644
0.24
1990
19367
4521
0.23
Total
59430
8425
0.14
An increasing trend down the accident years will
‘induce’ a correlation between (y-x) and x.
Trend Parameter For Incrementals
1
2
w

n
j-1 j
x{ }y
y – x = a0 + a1w + (b – 1) x + 
Intercept
Acc Yr
trend
“Ratio”
where Var() = 2xd
Extended Link Ratio Family of Models
Even this extended family of models is generally inadequate:
Wtd. Std. Residuals vs. Payment Year
1
Commonly have
changing payment
year trends
0.5
0
(ABC)
-0.5
-1
-1.5
78
79
80
81
82
83
84
85
86
87
Wtd. Std. Residuals vs. Fitted Values
2
(Pan6)
Often have
non-normality
1
0
-1
0
2500000
5000000
7500000
10000000
Part II
Trends occur in three directions:
0 1
Development year
d
1
2
Payment year
t = w+d
w
Accident year
Trend properties of loss development arrays
• Trends in payment year direction project onto the
other two directions and vice versa
• Changing trends can be hard to pick up in the
presence of noise, unless main trends are
removed
first (regression as a form of adjustment)
• Modeling a changing trend as a single trend will result
in pattern in the residual plots
Underlying Trends in the Data
Projection of trends onto other directions
Changing trends hard to pick without removing main
development and payment year trend.
Distribution of data about those trends
(Data = Trends + Random Fluctuations)
Development trend for single accident year,
data on log scale:
y(0) =  + 0
y(1) =  + 1 + 1
y(2) =  + 1 + 2 + 2
2
y(d)
1


0
1
2
3
4
d
log(p(d)) = y(d) =  +  i + d
i=1
5
d
On the original (dollar) scale, each payment has a
lognormal distribution, related by the trends.
p(d)
0
1
2
3
4
5
d
All years - trends in 3 directions
d
w+d
i=1
j=1
log(p(w,d)) = y(w,d) = w+  i +  j + w,d
Different levels for
accident years
Payment year
trends
You would never use all these parameters at the same time parsimony is as important as flexibility.
A model with too many parameters will give poor forecasts.
Fitting a single trend to changing trends
Wtd. Std. Residuals vs. Payment Year
2
1
0
-1
-2
78 79 80 81 82 83 84 85 86 87 88 89 90 91
- estimates as an average trend
- changes show up in the residuals
Checking the modeling framework
• if the modeling framework “works”, it should be hard to
differentiate between real data and data simulated from an
identified model
• if you create (simulate) data, you should be able to identify the
(known) changing trends in the data; mean forecasts should
usually be within about 2 standard errors of the true mean
Smooth data can conceal changing trends
Individual Link Ratios by Delay
2.50
2.25
2.00
1.75
1.50
1.25
1.00
0-1
Very smooth data
1-2
2-3
3-4
4-5
5-6
6-7
7-8
Very smooth ratios
(real array - ABC - values in paper)
8-9
9-10
Smooth data can conceal changing trends
Wtd. Std. Residuals
vs. Dev. Year
Wtd. Std. Residuals
vs. Acc. Year
2
2
1
1
0
0
-1
-1
0
1
2
3
4
5
6
7
8
9
10
77
78
Wtd. Std. Residuals
vs. Pay. Year
2
1
1
0
0
-1
-1
78
79
80
81
82
83
84
85
80
81
82
83
84
85
86
87
Wtd. Std. Residuals
vs. Fitted Values
2
77
79
86
87
9.00
10.00
11.00
12.00
Residuals after removing all accident year and development
year trends.
Noisy data is not necessarily hard to predict
• Volatile (noisy) data can be predictable (within model
uncertainty) if trends are stable
The data in the following example is very volatile (noisy) see the paper for this real data array (Pan6).
Wtd. Std. Residuals
vs. Dev. Year
Wtd. Std. Residuals
vs. Acc. Year
1
1
0
0
-1
-1
-2
Residual plots after
removing a single
development year
and payment year
trend.
-2
0
1
2
3
4
5
86
Wtd. Std. Residuals
vs. Pay. Year
87
88
89
90
91
92
93
94
95
96
Wtd. Std. Residuals
vs. Fitted Values
1
1
0
0
-1
-1
-2
-2
86
87
88
89
90
91
92
93
94
95
96
10.00
11.00
12.00
13.00
14.00
15.00
Noisy data is not necessarily hard to predict
• change in development year trend (the trend between
0-1 is different from the later years)
• no obvious trend changes in other directions
• wider spread for first two development years
• single superimposed inflation parameter is not
significantly different from 0
 one accident year level, two development year trends,
no payment year trend, weighted regression
Noisy data is not necessarily hard to predict
• residual plots and other diagnostics for that model are good
• forecast of this model yields an outstanding mean forecast
of $20.6 million and a standard deviation of $9.3
million, so the standard deviation is high (volatile data).
• It is important to see how the forecasts compare as we
remove the most recent years (validation):
Years in
Estimation
86-96
86-95
86-94
86-93
86-92
N
44
41
35
29
25
Trend
standard
(dev period 1+)
error
0.6250
0.1432
0.6102
0.1479
0.6149
0.1681
0.5024
0.1977
0.5631
0.2143
Mean
Standard
Fcst
Error
20,352,011 9,136,870
21,410,781 9,839,127
21,037,520 10,654,173
19,755,944 11,647,274
18,567,664 11,529,359
Part III
Prediction intervals and uncertainty
• any single figure will be wrong, but we can find the probability of
lying in a range.
• include both process risk and parameter risk. Ignoring parameter risk
leads to underreserving.
• forecast distributions are accurate if assumptions about the future
remain true.
• Distribution of sum of payment year totals important for dynamic
financial analysis. Distributions for future underwriting years
important for pricing.
• For a fixed security level on all the lines combined, the risk margin
per line decreases as the number of lines increases.
Risk Based Capital
• future uncertainty in loss reserves should be based on a
probabilistic model, which might not be related to reserves
carried by the in the past.
• uncertainty for each line should be based on a probabilistic
model that describes the particular line
• experience may be unrelated to the industry as a whole.
• Security margins should be selected formally. Implicit risk
margins may be much less or much more than required.
Booking the Reserve
• extract information, in terms of trends, stability of
trends and distributions about trends, for the loss
development array. Validation analysis.
• formulate assumptions about future. If recent trends
unstable, try to identify the cause, and use any relevant
business knowledge.
• select percentile (use distribution of reserves, combined
security margin, and available risk capital). Increased
uncertainty about future trends may require a higher
security margin.
Other Benefits of the Statistical Paradigm
• Credibility - if a trend parameter estimate for an
individual company is not credible, it can be formally
shrunk towards an industry estimate.
• Segmentation and layers - often the statistical model
(parameter structure) identified for a combined array
applies to some of its segments. These ideas can also be
applied to territories etc. and to layers.