Measuring Loss Reserve Uncertainty
William H. Panning
EVP, Willis Re
Casualty Actuarial Society
Annual Meeting, November 2006
2006 Hachemeister Award Presentation
Agenda
1. What is Loss Reserve Uncertainty (LRU) and why is it
important?
2. How does this new method for measuring LRU differ
from existing methods?
3. How does this new method work and how do we know it
is accurate?
4. What are the practical advantages and limitations of this
method?
2
1. What is Loss Reserve Uncertainty (LRU)
and why is it important?
3
1.1 What is LRU and why is it important?
• LRU is a measure of potential loss reserve development -the degree to which actual future loss payments may
ultimately deviate – favorably or unfavorably – from the
currently forecast amounts that constitute loss reserves
• LRU differs from pricing uncertainty -- the potential
deviation between forecast loss (when a policy is written)
and paid losses plus estimated reserve (at the end of an
accident year).
4
1.2 Why does measuring LRU matter?
• ERM: LRU is a key component of Enterprise Risk
Management, which requires measuring and managing all
of a firm’s significant risks
• Capital needs: knowing LRU assists a firm in determining
the appropriate amount of capital to hold and whether some
risk reduction action may be appropriate
• Interpreting calendar year deviations: LRU can tell us
whether their magnitude is significant and worthy of
attention
• Comparisons with other firms: are we better or worse?
5
1.3 To whom should LRU matter?
Issues
Audiences
• Estimating surplus adequacy
• Management
• Rating agencies and regulators
• Analysts and investors
• Capital allocation and pricing
• Management: actuary, CFO, CEO
• Managerial feedback: is this
deviation significant?
• Management
• Rating agencies
• Reinsurance
• Management
6
2. How does this new method for measuring LRU
differ from existing methods?
7
2.1 How does this new method differ from existing ones?
Existing Methods
New Method
• Most are ad hoc (algorithmic
democracy), lack criteria for fit
• Based on a standard, minimum
squared error linear regression
• Others require costly, opaque
software, specialized expertise
• Simple, transparent, implemented
in Excel spreadsheet
• None have been validated
• Validated using simulated data
• Chain ladder focus creates bias
• Use of regression minimizes bias
• Susceptible to statistical pitfalls
• Avoids these pitfalls
8
2.2 Additional objectives & features of the new method
• It should be based on widely-available public data
• Schedule P Part 3 Paid Loss triangles
• The results should enable comparisons
• Across different lines of business within a firm
• For the same line of business across different firms
• Between forecast and actual calendar year payments
• The results should be scalable
• Unaffected by irrelevant differences in the size of reserves
• Applicable to reserves estimated by other methods
9
3. How does this new method work and
how do we know it is accurate?
10
3.1 The starting point: Paid Loss Triangles
Year
Losses
Were
Incurred
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
0
1
2
Development Year
3
4
5
6
7
8
9
10+
Note that development years start with DY0
624
695
668
696
770
690
544
563
593
621
1,595
1,503
1,477
1,540
1,670
1,515
1,321
1,355
1,416
?
2,066
1,975
1,968
2,055
2,225
2,051
1,859
1,852
?
?
2,366
2,295
2,263
2,357
2,583
2,436
2,191
?
?
?
2,559
2,496
2,447
2,551
2,822
2,666
?
?
?
?
2,685
2,631
2,562
2,699
2,985
?
?
?
?
?
2,765 2,818 2,860 2,895
2,727 2,784 2,831 ?
2,645 2,707 ?
?
2,806 ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
The cumulative numbers in each row converge to ultimate values.
Reserve = sum of ultimates minus boxed diagonal values
11
?
?
?
?
?
?
?
?
?
?
3.2 Essential steps in estimating LRU
Step 1: Use the numbers already available to find a common
underlying pattern for estimating future loss payments
•
The chain ladder method does this by calculating link ratios: the
average ratio of numbers in a DY to the corresponding numbers
in the preceding DY
Step 2: Measure the variability of the available numbers around
this underlying pattern
Step 3: Use this measure to estimate the variability and
correlation of forecast future payments and total reserve
NOTE: Steps 2 and 3 depend crucially on doing Step 1 correctly
12
3.3 Three statistical pitfalls in Step 1, and their solutions
1. The chain ladder has no objective criterion for measuring and
maximizing goodness of fit to existing data
•
Solution: use linear regression, to minimize total squared error
2. The use of cumulative data creates serial correlation
•
Solution: use incremental data
Development year
0
Acc Yr:
(A+ea)
1
2
. . .
(A+ea)+(B+eb) (A+ea)+(B+eb)+(C+ec) . . .
3. Heteroskedasticity (non-constant SD): σ(ea) ≠ σ(eb) ≠ σ(ec)
•
13
Solution: analyze each development year separately
3.4 Estimating reserves from incremental paid loss data
Year
0
1
2
3
4
5
6
7
8
9
SD
0
624
695
668
696
770
690
544
563
593
621
70
1
971
808
809
844
900
825
777
792
823
61
Development Year
2
3
4
5
6
7
8
9
471
300 193 126
80
53
42
35
473
319 201 135
96
57
47
491
295 184 115
83 Use
63 these numbers (X)
515
302 194 148 107
555
358 239 162
To fit these numbers (Y)
536
384 231
537
332
Then use these numbers
497
?
To forecast these numbers
?
31
33
22
19
12
5 <--Note Heteroskedasticity
• Use linear regression to fit paid losses in future DY’s (up to DY7), with
DY0 as the independent variable in all cases
• Use estimated regression coefficients to forecast future loss payments
14
3.5 Steps in calculating the standard deviation of reserves
Table 3: Accident Year x Development Year
Incremental Paid Losses
Accident
Year
0
1
2
3
4
5
6
7
8
9
0
624
695
668
696
770
690
544
563
593
621
1
971
808
809
844
900
825
777
792
823
2
471
473
491
515
555
536
537
497
?
?
Development Year
3
4
5
300 193 126
319 201 135
295 184 115
302 194 148
358 239 162
384 231
332
6
80
96
83
107
7
53
57
63
8
42
47
9
35
• Use the Salkever (textbook) method to calculate the standard deviation
of each forecast future paid loss
• Use the linear regression results to calculate the variance-covariance
matrix of forecast errors for each future DY
• Finally, aggregate these results to obtain LRU by Development Year,
Calendar Year, and Total Reserve
15
Calculating the SD of Forecast Paid Losses for DY2
Step 1: Assemble the Input Data: X, X0, s, I
DY0
X= 624
695
668
696
770
690
544
563
X0=593
621
The standard error of
the estimate, seest,
shown in Tables 4 and 5.
s= 59
The identity matrix
(For DYn it is n x n)
I=
1
0
0
1
Step 2: Calculate the Variance-Covariance Matrix VCV
VCV= s2[I + X0(X'X)-1X0']
= 3,838 369
369 3,872
Step 3: Calculate the square root of Σ VCV
(Σ VCV)1/2 = 8,4471/2 = 92
16
3.6 How do we know that the method is accurate?
• We created 10,000 simulated paid loss triangles where we
knew the true underlying values (ultimate paid losses as well
as the actual paid losses at any given point in the process)
• We used the new method to estimate ultimate paid losses
• We compared the known true values to the estimates
obtained from the simulated triangles
• These estimates were, on average, identical to the true
values underlying the simulation
• For estimated reserves
• For loss reserve uncertainty, which includes parameter risk
• NOTE: No other method for estimating LRU has been
validated (to the best of my knowledge)
17
3.7 Validation statistics
DY
2
3
4
5
6
7
Total
CY
44
43
-1
-2.3%
1598
1622
24
1.5%
800
796
-4
-0.5%
Reserve: Sum of Future Loss Payments
True Value
Estimated Value
Difference
Difference %
400
300
200
125
397
298
198
124
-3
-2
-2
-1
-0.8% -0.7% -1.0% -0.8%
75
75
0
0.0%
LRU: Standard Deviation of Sum of Future Loss Payments
True Value
Estimated Value
Difference
Difference %
18
89
97
8
9.0%
77
82
5
6.5%
60
62
2
3.3%
43
45
2
4.7%
30
32
2
6.7%
21
19
-2
-9.5%
189
180
-9
-4.8%
127
118
-9
-7.1%
4. What are the practical advantages and
limitations of this method?
19
4.1 Avantage: The results of this method are scalable
• This method measures LRU in dollars. A better measure is the
coefficient of variation, or CV, which is LRU as a % of the
estimated reserve
• CV is unaffected by reserve size, and so can be compared
• across different business lines for the same firm
• or across the same line for different firms
• I believe that the CV can be legitimately applied to reserve
estimates obtained in other ways (e.g., using claims data)
20
4.2 Advantage: Comparisons across lines of business
ABC Insurance
Homeowners
Private Passenger Auto
Commercial Auto
Workers Compensation
Comm'l Multiple Peril
Σ Coefs E(Res) SD/E(Res) SD/E(CY)
0.45
2.26
3.95
3.14
1.48
50
134
238
211
240
9.0%
6.5%
7.1%
5.4%
23.6%
17.8%
10.1%
9.2%
5.6%
7.9%
Σ Coefs: Ratio of
remaining payments to
dollars paid in initial
development year
(measures length of
payout)
E(Res): Estimated
Reserve
SD/E(Res): Coefficient of
Variation, or standard
deviation of reserve
divided by estimated
reserve
SD/E(CY): Coefficient of
Variation of forecast
payments in the next
Calendar Year
NOTE: All results are
based on 2003 Schedule
P Part 3
21
4.3a Advantage: Comparisons across firms: PPA
Private Passenter Auto
Industry Aggregate
A
B
C
D
E
F
G
H
J
K
L
M
N
P
Q
R
S
22
Σ Coefs E(Res) SD/E(Res) SD/E(CY)
1.52 72,008
1.07 2,853
1.31 14,254
1.68
344
1.80
503
1.61 9,497
1.68 1,283
2.16 1,981
1.97
600
1.41
181
1.58
166
3.33
200
2.26
134
1.47
255
1.82
55
1.48 2,305
1.30 2,842
1.51
67
1.5%
1.8%
2.0%
2.4%
2.4%
2.4%
3.1%
3.8%
4.2%
4.6%
6.2%
6.2%
6.5%
6.6%
6.7%
11.9%
14.7%
27.8%
2.3%
2.7%
3.4%
3.6%
3.0%
2.7%
4.0%
4.2%
5.7%
5.4%
7.9%
8.2%
10.1%
10.4%
8.5%
8.4%
13.7%
41.3%
Σ Coefs: Ratio of
remaining payments to
dollars paid in initial
development year
(measures length of
payout)
E(Res): Estimated
Reserve
SD/E(Res): Coefficient of
Variation, or standard
deviation of reserve
divided by estimated
reserve
SD/E(CY): Coefficient of
Variation of forecast
payments in the next
Calendar Year
NOTE: All results are
based on 2003 Schedule
P Part 3
4.3c Advantage: Comparisons across firms: WC
Workers Compensation
A
Industry Aggregate
B
C
D
E
F
G
H
J
K
L
M
N
P
Q
R
23
Σ Coefs E(Res) SD/E(Res) SD/E(CY)
2.52
312
3.05 35,332
3.32
313
2.95
222
3.43 1,306
3.64 1,833
2.64
168
2.88 3,628
2.75 3,981
3.21
114
3.14
211
2.55
97
2.59
49
1.70
70
3.36 2,001
3.11
81
3.57
2
1.2%
2.1%
2.6%
2.7%
2.9%
4.0%
4.0%
4.6%
4.7%
5.1%
5.4%
5.5%
5.5%
6.0%
8.2%
12.6%
18.8%
1.6%
4.3%
4.3%
4.8%
6.0%
8.8%
7.1%
8.5%
8.6%
8.2%
5.6%
6.7%
9.1%
6.8%
15.0%
23.4%
26.4%
Σ Coefs: Ratio of
remaining payments to
dollars paid in initial
development year
(measures length of
payout)
E(Res): Estimated
Reserve
SD/E(Res): Coefficient of
Variation, or standard
deviation of reserve
divided by estimated
reserve
SD/E(CY): Coefficient of
Variation of forecast
payments in the next
Calendar Year
NOTE: All results are
based on 2003 Schedule
P Part 3
4.4a Comparisons of forecast versus actual paids: WC
Acc
Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
t
0
1
2
3
4
5
6
7
8
9
0
18
25
24
24
28
26
30
25
24
20
ACTUAL 2004
FORECAST 2004
Difference
1
26
32
28
30
34
34
37
32
30
0
30
31
0
2
14
15
14
16
21
21
23
19
0
0
19
18
1
Development Year
3
4
5
8
4
2
8
5
4
7
5
4
12
8
7
14
7
5
15
7
0
15
0
0
0
0
0
0
0
0
0
0
0
15
13
2
7
7
1
5
5
0
6
2
2
3
3
0
0
0
0
0
0
3
2
1
Forecast based on 2003 Schedule P Part 3
24
7
1
2
2
0
0
0
0
0
0
0
2
2
1
8
1
2
0
0
0
0
0
0
0
0
9
1
0
0
0
0
0
0
0
0
0
2
1
1
Sum
1
85
0
78
1
7
9%
4.4b Comparisons of forecast versus actual paids: CMP
Acc
Year
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
t
0
29
54
47
60
65
55
50
44
63
56
ACTUAL 2004
FORECAST 2004
Difference
SE: Standard Error
Difference/SE
1
18
26
24
27
33
29
25
26
37
0
37
32
5
2.5
1.8
2
8
9
11
11
15
13
16
19
0
0
19
10
9
2.5
3.6
Development Year
3
4
5
10
9
5
15
12
6
10
11
5
11
7
6
15
12
9
19
17
0
25
0
0
0
0
0
0
0
0
0
0
0
25
13
13
3.4
3.8
17
11
7
3.4
1.9
9
8
1
Forecast based on 2003 Schedule P Part 3
25
6
2
2
3
4
0
0
0
0
0
0
4
4
0
7
2
2
4
0
0
0
0
0
0
0
4
2
2
8
1
3
0
0
0
0
0
0
0
0
3
1
2
9
1
0
0
0
0
0
0
0
0
0
Sum
1
120
1
81
1
39
6.4
6.0
48%
4.5a Limitations of this method: data peculiarities
Top Ten Insurer
Acc
Yr
t
0
1
2
3
4
5
6
7
8
9
26
0
22
29
-10
56
49
57
68
70
54
52
1
26
-33
94
81
71
98
105
112
96
0
2
-25
74
81
75
79
106
102
97
0
0
Development Year
3
4
5
6
38
23
13
6
45
31
17
5
51
34
17
12
53
35
18
7
49
32
15
0
73
1
0
0
59
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
0
6
4
0
0
0
0
0
0
0
8
2
2
0
0
0
0
0
0
0
0
9
1
0
0
0
0
0
0
0
0
0
4.5b Limitations of this method: data peculiarities
Top Twenty Insurer
Acc
Yr
t
0
1
2
3
4
5
6
7
8
9
27
0
898
1,111
1,081
931
1,105
1,279
1,553
1,790
1,354
1,161
1
2
276 56
357 57
393 63
319 65
401 78
360 64
602 128
975 204
478
0
0
0
Development Year
3
4
5
6
7
28 -78 12
6
6
-27 -349
7
5
4
32
13
6 10 -394
33
19
8 -343
0
51
22 19
0
0
44
23
0
0
0
58
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
2
2
0
0
0
0
0
0
0
0
9
15
0
0
0
0
0
0
0
0
0
4.5c Limitations of this method: data peculiarities
Top Ten Insurer
Acc
Yr
t
0
0
456
1
386
2
773
3
413
4
579
5
821
6
821
7
881
8
879
9 1,070
28
1
95
119
134
113
40
199
261
247
246
0
2
14
12
-55
-38
32
37
42
41
0
0
Development Year
3
4
5
6
9 -22 -46
-1
1 -42
-2
3
-51
4
5
2
10
9
6
3
11
10
8
0
17
8
0
0
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7
1
1
1
0
0
0
0
0
0
0
8
1
1
0
0
0
0
0
0
0
0
9
1
0
0
0
0
0
0
0
0
0
4.6 What are the limitations of this method?
• Some versions of this method permit a gradual speedup or
slowdown in payment patterns over time, but otherwise it
assumes a stable past and future environment with regard to
•
•
•
•
Underwriting criteria
Exposure types
Reinsurance parameters
Legal and Regulatory environments
• Estimating the tail is difficult with this method as with others
• It necessarily relies on public data, and so does not reflect
claims-level data available only to the firm
29
5. Conclusions
30
5.1 Conclusions
•
•
•
•
•
•
•
•
•
•
This method produces validated results
It is the only method that has been validated
It is reasonably simple
It avoids serious statistical pitfalls
It is based on standard textbook methods
It can be implemented in a spreadsheet
It can be explained to colleagues and superiors
It enables intra-firm comparisons of different lines of business
It enables comparisons of a line of business across firms
It enables detection of emerging problems that need attention
31
5.2 How can I learn more about this method?
Send me an email at
Bill.Panning@Willis.com
32