When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene...
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Transcript of When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene...
![Page 1: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/1.jpg)
When can accident years be
regarded as development years?
Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky
Speaker: Dr Glen Barnett
Senior Research Statistician, Insureware
![Page 2: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/2.jpg)
Outline of talk
• What do we mean by “the chain ladder”?
• The basic “transpose-invariance” result
• Demonstrating the result (outline of a simple proof)
• What does the result tell us? i) Structure: accident years vs development years ii) Number of parameters iii) Cross-classification structure and ordering
• What lessons are there for other ratio methods?
![Page 3: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/3.jpg)
What do we mean by “the chain ladder”?
In its standard form:
• loss development technique for cumulative & incurred arrays
• use volume-weighted average ratios (where “volume” is previous column)
• gives factor, bj = “sum of column”/“sum of previous” where sum is over observations present in both.
![Page 4: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/4.jpg)
Chain ladder for incremental arrays
You can think of the chain ladder as a way to forecast incremental arrays as well:
- take an incremental array (say incremental paid)
1 cumulate across
2 compute ratios
3 forecast
4 difference back to incrementals
① ②
③
④⊝
⃝�
![Page 5: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/5.jpg)
The basic result
Produce two tables:
1) forecast an incremental array with the chain ladder (using the 4 steps)
2) transpose the array (interchange accident and development years), then forecast that with the chain ladder and transpose back
Tables 1 and 2 are identical
![Page 6: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/6.jpg)
Another way to think about it
1) forecast an incremental array with the chain ladder 2) take an incremental array and apply new steps: 1 cumulate down
2 compute ratios running down
3 forecast down
4 difference (up) back to incrementals
Tables 1 and 2 are identical
①
② ③④ ⊝
⃝�
![Page 7: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/7.jpg)
Demonstrating the result
(i) show that an incremental forecast from the CL is the same as finding sums of incrementals in each region (A,B,C) and computing BC/A **
(ii) Note that the result is the same for a transposed array
B
Cijp̂
A
= B.C/Aijp̂
** Replace any future values in shaded regions with their CL forecasts
![Page 8: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/8.jpg)
Step (i) - (a) first, show for next diagonal,
C.L. ratio = (A+B)/A; previous cumulative = C
Forecast cumulative = (A+B)C/A
Forecast incremental = (A+B)C/A – C = B.C/A
B
Cijp̂
A
= B.C/Aijp̂
![Page 9: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/9.jpg)
= B.C/Aijp̂
B
C
A
Replace any future values in shaded regions with their CL forecasts
Step (i) - (b) for later diagonals,
Note that forecast values already “follow the ratio”, so adding them in to A and B leaves B/A unchanged.
Also, the next forecast is based on the previous cumulative forecast, so C also contains the incremental forecasts
Future forecast incremental = B.C/A
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One advantage of this fact
• Easy in Excel to forecast incrementals directly
• Single formula can be pasted to each forecast cell
• Ratios can be computed from the last forecast row
(aside)
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Step (ii)
Plainly BC/A = CB/A;
So the calculation is the same for the transposed array (B and C merely interchange their roles).
= B.C/Aijp̂
B
C
AA
B
C
= C.B/Aijp̂
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What does the result tell us?
We call this property “transpose invariance”
(more strictly, “transpose-forecast commutativity)
i) Structure: accident years vs development years
does not differentiate between accident & developmentyear directions
– chain ladder treats them identically
![Page 13: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/13.jpg)
Of course we know that development years are quite different from accident years!
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
1968 1970 1972 1974 1976 1978 1980
Acc. year
Adj Log paid
5
5.5
6
6.5
7
7.5
8
1968 1970 1972 1974 1976 1978 1980
Adj log paid
-4
-3
-2
-1
0
1
2
0 2 4 6 8
adjusted for trend in other direction
raw data
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What does this result tell us?
ii) It also tells us that there are in fact parameters in both accident and development directions;
(This has been a source of argument – e.g. Mack vs Renshaw & Verrall)
we’re aware of the parameter corresponding to the column effect (the “B” effect) – it’s the “ratio”
but we usually condition on (i.e. ignore) the row effect (the “C” effect); it’s a degree of freedom that the model has to fit the data – i.e. a parameter
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What does this result tell us?
There are parameters in both accident and development directions:
ss triangle has 2s–1 parameters for the mean
(Overparameterization)
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iii) cross classification structure (two-way-ANOVA-like) – no account of ordering
In a cross-classification structure, you can interchange the row labels (or the column labels) with no impact
- but we know that order matters!
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
Can easily tell which of these has had its labels scrambled
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iii) cross classification structure (ctd)
– in fact there’s abundant information in nearby acci yrs
e.g. – in same dev. yr: Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
If you left out a point, how would you guess what it was?
- observations at same delay very informative.
-1.5
-1
-0.5
0
0.5
1
1.5
2 3 4 5 6
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iii) cross classification structure (ctd)
– information in different dev. yrs:
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
- nearby delays also informative (smooth trends)
(could leave out whole development & still guess where it was)
-1.5
-1
-0.5
0
0.5
1
1.5
2 3 4 5 6
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iii) cross classification structure (ctd)
information in nearby accident and development yrs…
need to use more of that information!
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Are there lessons for other ratio methods?
Mack (93) and Murphy (94) are able to write a model that includes several ratio methods as wtd regression
- chain ladder uses one particular set of weights
Conditionally on previous cumulative, can write a number of other methods as weighted chain ladder:
![Page 21: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/21.jpg)
Are there lessons for other ratio methods?
Example: “average development factor”:
fix the development year (i.e. hold it constant) at j
Let yi be the cumulative in accident year i, and let xi be the previous cumulative.
b = 1/n (yi/xi) = (wi yi) / (wi xi),
where wi = 1/xi
Hence ave. devel. factor is a weighted chain ladder
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Are there lessons for other ratio methods?
…for many methods, a weighted version of this result still holds
(weighted) two-way cross-classification structure
- many of the problems carry over!
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Are there lessons for other ratio methods?
- Still parameters in both directions.
2s-1 parameters
![Page 24: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/24.jpg)
Are there lessons for other ratio methods?
- Still ignores location information in nearby accident and development years
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
Log paid
3.54
4.55
5.56
6.57
7.58
8.5
0 1 2 3 4 5 6 7 8
Dev. year
![Page 25: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/25.jpg)
ss triangle: ratio methods use 2s–1 parameters for mean
How many parameters needed to describe this:
Adj Log paid
5
5.5
6
6.5
7
7.5
8
1968 1970 1972 1974 1976 1978 1980
Adj log paid
-4
-3
-2
-1
0
1
2
0 2 4 6 8
![Page 26: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/26.jpg)
Can describe shape of curve with say 2-3 parameters
Can describe stable accident year level with 1
(many arrays are similarly simple)
for this triangle, ratio methods use 20 parameters
(and wastes those: ratios don’t predict the next increment for that array)
4
4.5
5
5.5
6
6.5
7
7.5
8
0 2 4 6 8
Incr.(1) vs Cum.(0)
Corr. = -0.205, P-value = 0.570
400 500 600 700 800 900
1,400
1,600
1,800
2,000
2,200
2,400
2,600
2,800
3,000
![Page 27: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/27.jpg)
Effects of overparameterisation
- fitting noise rather than signal
- high parameter uncertainty
- unstable forecasts
(small change in data – large change in prediction
I.e. projects and amplifies noise into the future)
![Page 28: When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,](https://reader036.fdocuments.us/reader036/viewer/2022070409/56649e965503460f94b9a14f/html5/thumbnails/28.jpg)
Conclusions
“Transpose invariance” is an important feature
– serious implications for the chain ladder;
– many of the lessons apply more generally
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Lessons
be aware of the structure of loss data
– don’t ignore what you know
need to be aware of the specific structure in a triangle
– does the model succinctly describe the main features in the data?
(diagnostics, model validation, parsimony)