When can accident years be regarded as development years? Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky Speaker: Dr Glen Barnett Senior Research Statistician,

When can accident years be

regarded as development years?

Glen Barnett, Ben Zehnwirth, Eugene Dubossarsky

Speaker: Dr Glen Barnett

Senior Research Statistician, Insureware

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?

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.

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⊝

⃝�

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

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 ⊝

⃝�

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

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

= 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

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)

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

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

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

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

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)

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

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

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

iii) cross classification structure (ctd)

information in nearby accident and development yrs…

need to use more of that information!

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:

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

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!

Are there lessons for other ratio methods?

- Still parameters in both directions.

2s-1 parameters

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

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

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

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)

Conclusions

“Transpose invariance” is an important feature

– serious implications for the chain ladder;

– many of the lessons apply more generally

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)