Working with Triangles

Working with Triangles¶

Getting Started¶

Welcome! We drafted these tutorials to help you get familiar with some of the common functionalities that most actuaries can use in their day-to-day responsibilities. The package also comes with pre-installed datasets that you can play with, which are also used in the tutorials here.

The tutorials assume that you have the basic understanding of commonly used actuarial terms, and can independently perform an actuarial analysis in another tool, such as Microsoft Excel or another actuarial software. Furthermore, it is assumed that you already have some familiarity with Python, and that you have the basic knowledge and experience in using some common packages that are popular in the Python community, such as pandas and numpy.

This tutorial is linted using black via nb_black. This step is optional.

%load_ext lab_black

All tutorials and exercises rely on chainladder v0.8.12 and later. It is highly recomended that you keep your packages up-to-date.

For more info on how to update your pakages, visit Keeping Packages Updated.

import pandas as pd
import numpy as np
import chainladder as cl

print("pandas: " + pd.__version__)
print("numpy: " + np.__version__)
print("chainladder: " + cl.__version__)

pandas: 1.5.3
numpy: 1.24.3
chainladder: 0.8.16

Since we will be plotting for quite a bit, here’s a magic function in IPython, which sets the backend of matplotlib to the ‘inline’ backend. With this backend, the output of plotting commands is displayed inline within frontends like the Jupyter notebook, directly below the code cell that produced it. The resulting plots will then also be stored in the notebook document.

%matplotlib inline

Disclaimer¶

Note that a lot of the examples shown might not be applicable in a real world scenario, and is only meant to demonstrate some of the functionalities included in the package. The user should always follow all applicable laws, the Code of Professional Conduct, applicable Actuarial Standards of Practice, and exercise their best actuarial judgement.

Working with a Triangle¶

Let’s begin by looking at an unprocessed triangle data and load it into a pandas.DataFrame. We’ll use the data raa, which is available from the repository.

raa_df = pd.read_csv(
    "https://raw.githubusercontent.com/casact/chainladder-python/master/chainladder/utils/data/raa.csv"
)
raa_df.head(20)

	development	origin	values
0	1981	1981	5012.0
1	1982	1982	106.0
2	1983	1983	3410.0
3	1984	1984	5655.0
4	1985	1985	1092.0
5	1986	1986	1513.0
6	1987	1987	557.0
7	1988	1988	1351.0
8	1989	1989	3133.0
9	1990	1990	2063.0
10	1982	1981	8269.0
11	1983	1982	4285.0
12	1984	1983	8992.0
13	1985	1984	11555.0
14	1986	1985	9565.0
15	1987	1986	6445.0
16	1988	1987	4020.0
17	1989	1988	6947.0
18	1990	1989	5395.0
19	1983	1981	10907.0

The data has three columns:

development: or valuation time, in this case, valuation year
origin: or accident date, in this case, accident year
values: the values recorded for the specific accident date at the specific valuation time (such as incurred losses, paid losses, or claim counts)

A table of loss experience showing total losses for a certain period (origin) at various, regular valuation dates (development), reflect the change in amounts as claims mature and emerge. Older periods in the table will have one more entry than the next youngest period, leading to the triangle shape of the data in the table or any other measure that matures over time from an origin date. Loss triangles can be used to determine loss development for a given risk.

Let’s put our data into the triangle format.

raa = cl.Triangle(
    raa_df,
    origin="origin",
    development="development",
    columns="values",
    cumulative=True,
)
raa

	12	24	36	48	60	72	84	96	108	120
1981	5,012	8,269	10,907	11,805	13,539	16,181	18,009	18,608	18,662	18,834
1982	106	4,285	5,396	10,666	13,782	15,599	15,496	16,169	16,704
1983	3,410	8,992	13,873	16,141	18,735	22,214	22,863	23,466
1984	5,655	11,555	15,766	21,266	23,425	26,083	27,067
1985	1,092	9,565	15,836	22,169	25,955	26,180
1986	1,513	6,445	11,702	12,935	15,852
1987	557	4,020	10,946	12,314
1988	1,351	6,947	13,112
1989	3,133	5,395
1990	2,063

You can also load the example data directly, using load_sample:

raa = cl.load_sample("raa")
raa

	12	24	36	48	60	72	84	96	108	120
1981	5,012	8,269	10,907	11,805	13,539	16,181	18,009	18,608	18,662	18,834
1982	106	4,285	5,396	10,666	13,782	15,599	15,496	16,169	16,704
1983	3,410	8,992	13,873	16,141	18,735	22,214	22,863	23,466
1984	5,655	11,555	15,766	21,266	23,425	26,083	27,067
1985	1,092	9,565	15,836	22,169	25,955	26,180
1986	1,513	6,445	11,702	12,935	15,852
1987	557	4,020	10,946	12,314
1988	1,351	6,947	13,112
1989	3,133	5,395
1990	2,063

A triangle has more properties than just what is displayed. For example we can see the underlying link_ratios, which represent the multiplicative change in amounts from one development period to the next.

raa.link_ratio

	12-24	24-36	36-48	48-60	60-72	72-84	84-96	96-108	108-120
1981	1.6498	1.3190	1.0823	1.1469	1.1951	1.1130	1.0333	1.0029	1.0092
1982	40.4245	1.2593	1.9766	1.2921	1.1318	0.9934	1.0434	1.0331
1983	2.6370	1.5428	1.1635	1.1607	1.1857	1.0292	1.0264
1984	2.0433	1.3644	1.3489	1.1015	1.1135	1.0377
1985	8.7592	1.6556	1.3999	1.1708	1.0087
1986	4.2597	1.8157	1.1054	1.2255
1987	7.2172	2.7229	1.1250
1988	5.1421	1.8874
1989	1.7220

We can also view (and manipulate) the latest_diagonal of the triangle.

raa.latest_diagonal

	1990
1981	18,834
1982	16,704
1983	23,466
1984	27,067
1985	26,180
1986	15,852
1987	12,314
1988	13,112
1989	5,395
1990	2,063

raa.latest_diagonal / 1000

	1990
1981	18.83
1982	16.70
1983	23.47
1984	27.07
1985	26.18
1986	15.85
1987	12.31
1988	13.11
1989	5.39
1990	2.06

The latest diagonal also corresponds to a valuation_date. Note that ‘valuation_date’ is a datetime that is at the terminal timestamp of the period (i.e. the last split second of the year).

raa.valuation_date

Timestamp('1990-12-31 23:59:59.999999999')

We can also tell whether our triangle:

is_cumulative: returns True if the data across the development periods is cumulative, or False if it is incremental.
is_ultimate: returns True if the ultimate values are contained in the triangle.
is_val_tri: returns True if the development period is stated as a valuation data as opposed to an age, i.e. Schedule P style triangle (True) or the more commonly used triangle by development age (False).
is_full: returns True if the triangle has been “squared”.

print("Is triangle cumulative?", raa.is_cumulative)
print("Does triangle contain ultimate projections?", raa.is_ultimate)
print("Is this a valuation triangle?", raa.is_val_tri)
print('Has the triangle been "squared"?', raa.is_full)

Is triangle cumulative? True
Does triangle contain ultimate projections? False
Is this a valuation triangle? False
Has the triangle been "squared"? False

We can also inspect the triangle to understand its data granularity with origin_grain and development_grain.

print("Origin grain:", raa.origin_grain)
print("Development grain:", raa.development_grain)

Origin grain: Y
Development grain: Y

The package supports monthly (“M”), quarterly (“Q”), semester (or semi-annually), (“S”) and yearly (“Y”) grains for both origin_grain and development_grain.

The Triangle Structure¶

The triangle described so far is a two-dimensional (accident date by valuation date) structure that spans multiple cells of data. This is a useful structure for exploring individual triangles, but becomes more problematic when working with sets of triangles. Pandas does not have a triangle dtype, but if it did, working with sets of triangles would be much more convenient. To facilitate working with more than one triangle at a time the chainladder.Triangle acts like a pandas dataframe (with an index and columns) where each cell (row x col) is an individual triangle. This structure manifests itself as a four-dimensional space. Let’s take a look at another dataset clrd.

clrd_df = pd.read_csv(
    "https://raw.githubusercontent.com/casact/chainladder-python/master/chainladder/utils/data/clrd.csv"
)
clrd_df

	GRCODE	GRNAME	AccidentYear	DevelopmentYear	DevelopmentLag	IncurLoss	CumPaidLoss	BulkLoss	EarnedPremDIR	EarnedPremCeded	EarnedPremNet	Single	PostedReserve97	LOB
0	86	Allstate Ins Co Grp	1988	1988	1	367404	70571	127737	400699	5957	394742	0	281872	wkcomp
1	86	Allstate Ins Co Grp	1988	1989	2	362988	155905	60173	400699	5957	394742	0	281872	wkcomp
2	86	Allstate Ins Co Grp	1988	1990	3	347288	220744	27763	400699	5957	394742	0	281872	wkcomp
3	86	Allstate Ins Co Grp	1988	1991	4	330648	251595	15280	400699	5957	394742	0	281872	wkcomp
4	86	Allstate Ins Co Grp	1988	1992	5	354690	274156	27689	400699	5957	394742	0	281872	wkcomp
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
42840	44598	College Liability Ins Co Ltd RRG	1995	1996	2	343	249	82	397	0	397	1	630	othliab
42841	44598	College Liability Ins Co Ltd RRG	1995	1997	3	839	575	190	397	0	397	1	630	othliab
42842	44598	College Liability Ins Co Ltd RRG	1996	1996	1	125	6	98	257	0	257	1	630	othliab
42843	44598	College Liability Ins Co Ltd RRG	1996	1997	2	95	17	28	257	0	257	1	630	othliab
42844	44598	College Liability Ins Co Ltd RRG	1997	1997	1	83	4	77	256	0	256	1	630	othliab

42845 rows × 14 columns

Let’s load the data into the sets of triangles.

clrd = cl.Triangle(
    clrd_df,
    origin="AccidentYear",
    development="DevelopmentYear",
    columns=[
        "IncurLoss",
        "CumPaidLoss",
        "BulkLoss",
        "EarnedPremDIR",
        "EarnedPremCeded",
        "EarnedPremNet",
    ],
    index=["GRNAME", "LOB"],
    cumulative=True,
)
clrd

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(775, 6, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet]

Since 4D strucures do not fit nicely on 2D screens, we see a summary view instead that describes the structure rather than the underlying data itself.

We see 5 rows of information:

Valuation: the valuation date.
Grain: the granularity of the data, O stands for origin, and D stands for development, OYDY represents triangles with accident year by development year.
Shape: contains 4 numbers, represents the 4-D structure. This sample triangle represents a collection of 775x6 or 4,650 triangles that are themselves 10 accident years by 10 development periods.
- 775: the number of segments, which is the combination of index, that represents the data segments. In this case, it is each of the GRNAME and LOB unique combination.
- 6: the number of triangles for each segment, which is also the columns [IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet]. They could be paid amounts, incurred amounts, reported counts, loss ratios, closure rates, excess losses, premium, etc.
- 10: the number of accident periods.
- 10: the number of valuation periods.
Index: the segmentation level of the triangles.
Columns: the value types recorded in the triangles.

To summarize the set of triangles:

We have a total of 775 segments, which are at the GRNAME and LOB level.
Each segment contains 6 triangles, which are IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet.
Each triangle is 10 accident years x 10 development periods.

Using index.head() allows us to see the first 5 segments in the set of triangles. Note that as the data is loaded, the triangle are sorted by index, in this case, GRNAME first, then by LOB.

clrd.index.head()

	GRNAME	LOB
0	Adriatic Ins Co	othliab
1	Adriatic Ins Co	ppauto
2	Aegis Grp	comauto
3	Aegis Grp	othliab
4	Aegis Grp	ppauto

Under the hood, the data structure is a numpy.ndarray with the equivalent shape. Like pandas, you can directly access the underlying numpy structure with the values property. By exposing the underlying ndarray you are free to manipulate the underlying data directly with numpy should that be an easier route to solving a problem.

print("Data structure of clrd:", type(clrd.values))
print("Sum of all data values:", np.nansum(clrd.values))

Data structure of clrd: <class 'numpy.ndarray'>
Sum of all data values: 3661713596.0

Keep in mind though, the chainladder.Triangle has several methods and properties beyond the raw numpy representation and these are kept in sync by using the chainladder.Triangle directly.

Pandas-Style Slicing¶

As mentioned, the 4D structure is intended to behave like a pandas DataFrame. Like pandas, we can subset a dataframe by referencing individual columns by name.

clrd[["CumPaidLoss", "IncurLoss", "BulkLoss"]]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(775, 3, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[CumPaidLoss, IncurLoss, BulkLoss]

We can also boolean-index the rows of the Triangle.

clrd[clrd["LOB"] == "wkcomp"]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(132, 6, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet]

We can even use the typical loc, iloc functionality similar to pandas to access subsets of data. These features can be chained together as much as you want.

clrd.loc["Allstate Ins Co Grp"].iloc[-1]["CumPaidLoss"]

	12	24	36	48	60	72	84	96	108	120
1988	70,571	155,905	220,744	251,595	274,156	287,676	298,499	304,873	321,808	325,322
1989	66,547	136,447	179,142	211,343	231,430	244,750	254,557	270,059	273,873
1990	52,233	133,370	178,444	204,442	222,193	232,940	253,337	256,788
1991	59,315	128,051	169,793	196,685	213,165	234,676	239,195
1992	39,991	89,873	114,117	133,003	154,362	159,496
1993	19,744	47,229	61,909	85,099	87,215
1994	20,379	46,773	88,636	91,077
1995	18,756	84,712	87,311
1996	42,609	44,916
1997	691

Pandas-Style Arithmetic¶

With complete flexibility in the ability to slice subsets of triangles, we can use basic arithmetic to derive new triangles, which is commonly used as diagnostics to explore trends.

clrd["CaseIncurLoss"] = clrd["IncurLoss"] - clrd["BulkLoss"]
clrd["PaidToInc"] = clrd["CumPaidLoss"] / clrd["CaseIncurLoss"]
clrd[["CaseIncurLoss", "PaidToInc"]]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(775, 2, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[CaseIncurLoss, PaidToInc]

We can also aggregating the values across all triangles into one triangle.

clrd["CumPaidLoss"].sum()

	12	24	36	48	60	72	84	96	108	120
1988	3,577,780	7,059,966	8,826,151	9,862,687	10,474,698	10,814,576	10,994,014	11,091,363	11,171,590	11,203,949
1989	4,090,680	7,964,702	9,937,520	11,098,588	11,766,488	12,118,790	12,311,629	12,434,826	12,492,899
1990	4,578,442	8,808,486	10,985,347	12,229,001	12,878,545	13,238,667	13,452,993	13,559,557
1991	4,648,756	8,961,755	11,154,244	12,409,592	13,092,037	13,447,481	13,642,414
1992	5,139,142	9,757,699	12,027,983	13,289,485	13,992,821	14,347,271
1993	5,653,379	10,599,423	12,953,812	14,292,516	15,005,138
1994	6,246,447	11,394,960	13,845,764	15,249,326
1995	6,473,843	11,612,151	14,010,098
1996	6,591,599	11,473,912
1997	6,451,896

We can also construct a paid loss ratio triangle against EarnedPremNet.

clrd["CumPaidLoss"].sum() / clrd["EarnedPremNet"].sum()

	12	24	36	48	60	72	84	96	108	120
1988	0.2581	0.5093	0.6368	0.7115	0.7557	0.7802	0.7932	0.8002	0.8060	0.8083
1989	0.2686	0.5230	0.6526	0.7288	0.7727	0.7958	0.8085	0.8166	0.8204
1990	0.2728	0.5249	0.6546	0.7287	0.7674	0.7888	0.8016	0.8080
1991	0.2540	0.4896	0.6093	0.6779	0.7152	0.7346	0.7453
1992	0.2595	0.4927	0.6074	0.6711	0.7066	0.7245
1993	0.2645	0.4959	0.6061	0.6687	0.7021
1994	0.2709	0.4942	0.6005	0.6614
1995	0.2651	0.4755	0.5737
1996	0.2635	0.4586
1997	0.2552

Aggregating all segments together is interesting, but it is often more useful to aggregate across segments using groupby. For example, we may want to group the triangles by line of business and get a sum across all companies for each industry.

clrd.groupby("LOB").sum()

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(6, 8, 10, 10)
Index:	[LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet, CaseIncurLoss, PaidToInc]

The shape is (6, 8, 10, 10) because now we have 6 LOBs with 8 triangles for each LOB. We can also note that index is now on [LOB] only.

np.unique(clrd["LOB"])

array(['comauto', 'medmal', 'othliab', 'ppauto', 'prodliab', 'wkcomp'],
      dtype=object)

The aggregation functions, e.g. sum, mean, std, min, max, etc. don’t have to just apply to the index axis. You can apply them to any of the four axes in the triangle object, which are segments (axis 0, or the index axis), columns (axis 1, the various financial fields), origin (axis 2, across triangle rows), and development period (axis 3, across triangle columns). You can also use either the axis name or number. Let’s try to sum all of the segments ([GRNAME, LOB]) and columns ([IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet]) using axis name and number, respectively.

clrd.sum(axis="index").sum(axis="segments")

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(1, 8, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet, CaseIncurLoss, PaidToInc]

clrd.sum(axis=0).sum(axis=1)

	12	24	36	48	60	72	84	96	108	120
1988	56,387,734	59,928,520	61,653,458	62,644,631	63,182,710	63,438,080	63,440,904	63,452,111	63,522,981	63,518,183
1989	62,840,448	66,705,570	68,666,928	69,702,283	70,188,013	70,355,679	70,389,857	70,453,158	70,483,269
1990	70,064,967	74,084,388	75,879,276	76,812,990	77,179,572	77,240,321	77,283,937	77,345,588
1991	74,611,611	78,513,121	80,235,908	80,967,061	81,178,876	81,185,484	81,278,636
1992	81,213,791	85,089,370	86,443,882	86,783,107	86,908,610	87,086,637
1993	87,896,227	91,685,456	93,018,485	93,167,810	93,473,079
1994	94,593,702	98,130,718	99,071,781	99,809,122
1995	97,722,814	101,192,332	102,056,680
1996	98,497,932	100,917,690
1997	96,832,221

Accessor Methods¶

Pandas has special “accessor” methods for str and dt. These allow for the manipulation of data within each cell of data:

# splits lastname from first name by a comma-delimiter
df['Last_First'].str.split(',')

# pulls the year out of each date in a dataframe column
df['Accident Date'].dt.year 

chainladder also has special “accessor” methods to help us manipulate the origin, development and valuation vectors of a triangle.

We may want to extract only the latest accident period for every triangle.

clrd[clrd.origin == clrd.origin.max()]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(775, 8, 1, 10)
Index:	[GRNAME, LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet, CaseIncurLoss, PaidToInc]

Note that this triangle has only 1 row; however, all of the columns would exist, but only the youngest age would have values.

We may want to extract particular diagonals from our triangles using its valuation vector.

clrd[(clrd.valuation >= "1994") & (clrd.valuation < "1995")]["CumPaidLoss"].sum()

	12	24	36	48	60	72	84
1988							10,994,014
1989						12,118,790
1990					12,878,545
1991				12,409,592
1992			12,027,983
1993		10,599,423
1994	6,246,447

We may even want to slice particular development periods to explore aspects of our data by development age. For example, we can look at the development factors between ages 24 and 36.

clrd[(clrd.development > 12) & (clrd.development <= 36)]["CumPaidLoss"].sum().link_ratio

	24-36
1988	1.2502
1989	1.2477
1990	1.2471
1991	1.2446
1992	1.2327
1993	1.2221
1994	1.2151
1995	1.2065
1996
1997

Moving Back to Pandas¶

When the shape of a Triangle object can be expressed as a 2D structure (i.e. two of its four axes have a length of 1) or less, you can use the to_frame method to convert your data into a pandas.DataFrame. Let’s pick only one financial field, CumPaidLoss and only the latest diagonal, with latest_diagonal. We are now left with LOBs and origin period as our 2 axes.

clrd.groupby("LOB").sum().latest_diagonal["CumPaidLoss"]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(6, 1, 10, 1)
Index:	[LOB]
Columns:	[CumPaidLoss]

clrd.groupby("LOB").sum().latest_diagonal["CumPaidLoss"].to_frame(
    origin_as_datetime=True
).astype(int)

origin	1988	1989	1990	1991	1992	1993	1994	1995	1996	1997
LOB
comauto	626097	674441	718396	711762	731033	762039	768095	675166	510191	272342
medmal	217239	222707	235717	275923	267007	276235	252449	209222	107474	20361
othliab	317889	350684	361103	426085	389250	434995	402244	294332	191258	54130
ppauto	8690036	9823747	10728411	10713621	11555121	12249826	12600432	11807279	9900842	5754249
prodliab	110973	112614	121255	100276	76059	94462	111264	62018	28107	10682
wkcomp	1241715	1308706	1394675	1414747	1328801	1187581	1114842	962081	736040	340132

Note that we added origin_as_datetime=True inside of to_frame(...) as the package is under-going a deprecation cycle. You can also execute the same code without the origin_as_datetime=..., but you will get a warning.

clrd.groupby("LOB").sum().latest_diagonal["CumPaidLoss"].to_frame().astype(int)

origin	1988	1989	1990	1991	1992	1993	1994	1995	1996	1997
LOB
comauto	626097	674441	718396	711762	731033	762039	768095	675166	510191	272342
medmal	217239	222707	235717	275923	267007	276235	252449	209222	107474	20361
othliab	317889	350684	361103	426085	389250	434995	402244	294332	191258	54130
ppauto	8690036	9823747	10728411	10713621	11555121	12249826	12600432	11807279	9900842	5754249
prodliab	110973	112614	121255	100276	76059	94462	111264	62018	28107	10682
wkcomp	1241715	1308706	1394675	1414747	1328801	1187581	1114842	962081	736040	340132

We can also aggregate process away 3 dimensions, then use to_frame.

clrd[clrd.origin == "1990"].groupby("LOB").sum().latest_diagonal[
    "CumPaidLoss"
].to_frame(origin_as_datetime=True).astype(int)

LOB
comauto       718396
medmal        235717
othliab       361103
ppauto      10728411
prodliab      121255
wkcomp       1394675
dtype: int64

Exercises¶

Use the clrd dataset for all of the exercises.

How do we create a new column named “NetPaidLossRatio” in the triangle using the existing columns?

clrd["NetPaidLossRatio"] = clrd["CumPaidLoss"] / clrd["EarnedPremNet"]

What is the highest paid loss ratio for across all segments for origin 1997 at age 12?

clrd[clrd.origin == "1997"][clrd.development == 12]["NetPaidLossRatio"].max()

4.769123134328358

How do we subset the overall triangle to just include “Alaska Nat Ins Co”?

clrd[clrd["GRNAME"] == "Alaska Nat Ins Co"]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(4, 9, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet, CaseIncurLoss, PaidToInc, NetPaidLossRatio]

How do we create a triangle subset that includes all triangles for companies with names starting with the letter “B”?

clrd[clrd["GRNAME"].str[0] == "B"]

	Triangle Summary
Valuation:	1997-12
Grain:	OYDY
Shape:	(31, 9, 10, 10)
Index:	[GRNAME, LOB]
Columns:	[IncurLoss, CumPaidLoss, BulkLoss, EarnedPremDIR, EarnedPremCeded, EarnedPremNet, CaseIncurLoss, PaidToInc, NetPaidLossRatio]

Which are the top 5 companies by net premium share for in 1990?

clrd.latest_diagonal.groupby("GRNAME").sum()[clrd.origin == "1990"][
    "EarnedPremNet"
].to_frame(origin_as_datetime=True).sort_values(ascending=False).iloc[0:5]

GRNAME
State Farm Mut Grp                    10532675.0
United Services Automobile Asn Grp     1378791.0
Federal Ins Co Grp                      477150.0
New Jersey Manufacturers Grp            383870.0
FL Farm Bureau Grp                      342036.0
dtype: float64

Initializing a Triangle With Your Own Data¶

The chainladder.Triangle class is designed to ingest pandas.DataFrame objects. However, you do not need to worry about shaping the dataframe into triangle format yourself. This happens at the time you ingest the data.

Let’s look at the initialization signature and its docstring.

cl.Triangle?

Let’s use a new dataset prism to construct our triangles.

prism_df = pd.read_csv(
    "https://raw.githubusercontent.com/casact/chainladder-python/master/chainladder/utils/data/prism.csv"
)
prism_df.head()

	ClaimNo	AccidentDate	ReportDate	Line	Type	ClaimLiability	Limit	Deductible	TotalPayment	PaymentDate	CloseDate	Status	reportedCount	closedPaidCount	closedUnPaidCount	Paid	Incurred
0	1	2008-01-22	4/19/2010	Home	Dwelling	False	300000.0	20000	0.00000	2010-10-08	10/8/2010	CLOSED	1	0	1	0.00000	0.00000
1	2	2008-01-02	4/20/2010	Home	Dwelling	False	200000.0	20000	0.00000	2010-11-30	11/30/2010	CLOSED	1	0	1	0.00000	0.00000
2	3	2008-01-01	9/23/2009	Home	Dwelling	True	200000.0	20000	115744.77370	2010-02-17	2/17/2010	CLOSED	1	1	0	115744.77370	115744.77370
3	4	2008-01-02	7/25/2009	Home	Dwelling	True	200000.0	20000	63678.87713	2009-11-18	11/18/2009	CLOSED	1	1	0	63678.87713	63678.87713
4	5	2008-01-16	12/7/2009	Home	Dwelling	True	200000.0	20000	112175.55590	2010-04-30	4/30/2010	CLOSED	1	1	0	112175.55590	112175.55590

prism_df.dtypes

ClaimNo                int64
AccidentDate          object
ReportDate            object
Line                  object
Type                  object
ClaimLiability          bool
Limit                float64
Deductible             int64
TotalPayment         float64
PaymentDate           object
CloseDate             object
Status                object
reportedCount          int64
closedPaidCount        int64
closedUnPaidCount      int64
openCount              int64
Paid                 float64
Outstanding            int64
Incurred             float64
dtype: object

We must specify the origin, development, columns, and cumulative to create a triangle object. By limiting our columns to one measure and not specifying an index, we can create a single triangle. For example, if we are only interested in the paid triangle. Because the data we have is transactional data, the cumulative variable should be set to False.

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    development="PaymentDate",
    columns="Paid",
    cumulative=False,
)
prism

	1	2	3	4	5	6	7	8	9	10	...	111	112	113	114	115	116	117	118	119	120
2008-01			46,915	19,899	57,216	89,783	18,302	95,054	10,332	91,772	...
2008-02		28,749	22,109	79,033	63,455	59,993	64,683	68,502	33,695	71,670	...
2008-03		48,806	27,949	90,413	54,557	83,507	12,591	72,035	70,353	35,223	...
2008-04		30,758	17,763	70,872	30,233	39,494	66,729	122,100	20,998	38,218	...
2008-05		38,672	86,974	20,483	58,400	112,015	31,354	22,457	53,778	93,303	...
2008-06		56,789	73,351	97,840	64,816	61,083	41,685	87,172	61,418	63,097	...
2008-07		27,867	45,804	61,495	73,279	71,591	111,807	80,249	40,632	32,434	...
2008-08	4,832	23,831	52,511	62,649	45,955	133,133	81,733	39,709	112,115	21,367	...
2008-09		43,464	86,157	52,664	89,928	88,738	55,617	26,599	69,522	46,788	...
2008-10		12,488	21,939	53,388	52,415	92,086	45,781	72,242	83,359	47,594	...
2008-11		45,473	9,416	78,967	84,826	39,549	41,011	74,000	33,257	35,802	...
2008-12	3,640	9,905	54,792	87,057	70,686	54,775	31,412	44,057	59,448	31,643	...
2009-01		14,021	45,350	51,208	35,520	81,655	54,275	41,538	74,129	47,962	...
2009-02	10,305	35,026	37,289	58,773	67,579	57,589	115,083	57,142	65,315	55,019	...
2009-03		48,360	40,502	74,342	64,538	49,778	85,591	82,582	56,449	88,307	...
2009-04		40,943	47,517	64,024	27,573	62,634	54,051	63,305	61,339	70,406	...
2009-05		16,944	74,478	63,856	59,266	79,707	63,857	59,326	88,817	50,656	...
2009-06	9,655	43,720	12,805	79,404	46,219	46,868	36,234	35,654	24,676	26,641	...
2009-07		14,828	48,605	90,527	54,508	57,881	64,905	16,717	46,880	89,721	...
2009-08		42,061	32,247	64,174	63,443	65,428	95,432	91,809	131,904	67,284	...
2009-09	21,000	27,607	34,679	37,523	56,305	42,851	72,404	54,366	67,543	71,032	...
2009-10	14,000	62,040	45,134	97,016	54,356	20,848	63,521	78,494	67,441	22,107	...
2009-11		22,027	64,189	20,836	33,303	50,594	79,852	54,870	111,592	70,473	...
2009-12		38,594	17,902	100,470	43,866	102,051	56,901	33,876	40,416	54,734	...
2010-01		15,192	53,338	57,394	56,985	62,175	84,624	84,959	102,734	26,495	...
2010-02	7,000	31,464	32,942	38,108	28,707	100,091	97,795	98,314	53,259	96,477	...
2010-03		36,937	64,551	61,774	39,580	81,253	35,751	87,788	41,076	117,835	...
2010-04	21,026	32,422	46,661	103,442	89,860	96,976	51,116	89,352	50,595	13,981	...
2010-05	8,668	8,920	116,376	61,586	148,647	14,288	46,840	51,205	82,714	4,240	...
2010-06		12,578	92,156	44,495	112,868	79,732	73,743	61,814	73,713	23,724	...
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2015-07	5,872	61,431	113,522	96,030	133,267	130,305	64,905	58,480	94,985	72,184	...
2015-08	12,276	73,666	69,772	156,546	96,829	122,893	77,525	120,597	42,213	81,603	...
2015-09	29,402	45,522	76,968	146,840	171,674	109,429	86,091	98,514	83,465	61,289	...
2015-10	14,000	18,824	64,809	86,112	33,952	94,798	41,473	114,279	95,470	95,322	...
2015-11		24,866	74,507	52,944	109,360	98,136	160,381	46,511	206,774	128,625	...
2015-12	14,000	34,498	85,505	113,846	118,239	121,277	179,033	131,330	130,357	89,312	...
2016-01	3,214	16,521	48,605	63,442	64,330	156,536	70,536	90,186	88,647	87,278	...
2016-02	13,892	20,722	112,166	154,319	93,667	117,578	114,468	77,207	79,020	85,925	...
2016-03	18,354	48,495	70,675	115,886	83,612	45,276	123,736	114,339	79,022	96,179	...
2016-04	14,781	35,741	51,818	124,195	76,220	168,821	117,505	82,978	99,148	112,664	...
2016-05		32,921	80,574	128,677	98,491	138,236	110,125	110,887	150,281	74,849	...
2016-06	17,959	9,031	35,721	163,644	172,139	157,653	154,168	120,193	100,214	62,572	...
2016-07	36,454	24,783	105,234	116,616	64,114	118,426	107,283	101,315	136,164	95,888	...
2016-08	45,280	14,224	93,541	147,758	88,370	195,297	117,494	57,656	132,907	100,460	...
2016-09	13,544	56,468	84,514	106,896	127,088	89,314	132,691	153,691	62,591	133,208	...
2016-10	1,766	61,196	80,437	123,346	82,947	158,137	168,589	110,019	86,504	162,549	...
2016-11		55,837	111,442	110,741	111,772	44,847	164,147	180,126	90,889	71,973	...
2016-12		52,806	55,386	116,175	107,611	109,759	74,635	145,938	100,703	114,897	...
2017-01	8,338	26,000	121,221	104,138	113,300	86,086	139,369	127,013	66,155	111,713	...
2017-02		50,237	101,651	103,479	94,798	94,285	160,239	141,802	131,585	94,257	...
2017-03	4,476	20,775	74,392	72,250	143,323	56,307	210,407	117,054	101,546	93,077	...
2017-04		30,751	126,808	154,839	157,090	72,106	102,587	91,454	142,689		...
2017-05	3,022	36,811	54,334	80,437	81,315	160,838	180,652	88,441			...
2017-06		31,858	37,911	81,229	121,619	85,185	110,103				...
2017-07	28,827	30,987	126,230	72,826	80,075	119,390					...
2017-08		14,000	105,952	81,991	174,870						...
2017-09	18,589	33,317	86,057	152,144							...
2017-10		35,037	104,444								...
2017-11	4,088	10,599									...
2017-12	10,748										...

Note that the lowest (most-detailed) grain supported is the monthly grain, so the triangle above is aggregated to the OMDM level.

prism.origin_grain

'M'

prism.development_grain

'M'

If we want to include more columns or indices we can certainly do so. Note that as we do this, we move into the 4D arena changing the display of the overall object.

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    development="PaymentDate",
    columns=["Paid", "Incurred"],
    cumulative=False,
)
prism

	Triangle Summary
Valuation:	2017-12
Grain:	OMDM
Shape:	(1, 2, 120, 120)
Index:	[Total]
Columns:	[Paid, Incurred]

Pandas has wonderful datetime inference functionality that the Triangle heavily uses to infer origin and development granularity. Even still, there are occassions where date format inferences can fail. It is often better to explicitly tell the triangle the date format, and is usually good pratice to explicitly state the date format instead.

prism_df.head()

	ClaimNo	AccidentDate	ReportDate	Line	Type	ClaimLiability	Limit	Deductible	TotalPayment	PaymentDate	CloseDate	Status	reportedCount	closedPaidCount	closedUnPaidCount	Paid	Incurred
0	1	2008-01-22	4/19/2010	Home	Dwelling	False	300000.0	20000	0.00000	2010-10-08	10/8/2010	CLOSED	1	0	1	0.00000	0.00000
1	2	2008-01-02	4/20/2010	Home	Dwelling	False	200000.0	20000	0.00000	2010-11-30	11/30/2010	CLOSED	1	0	1	0.00000	0.00000
2	3	2008-01-01	9/23/2009	Home	Dwelling	True	200000.0	20000	115744.77370	2010-02-17	2/17/2010	CLOSED	1	1	0	115744.77370	115744.77370
3	4	2008-01-02	7/25/2009	Home	Dwelling	True	200000.0	20000	63678.87713	2009-11-18	11/18/2009	CLOSED	1	1	0	63678.87713	63678.87713
4	5	2008-01-16	12/7/2009	Home	Dwelling	True	200000.0	20000	112175.55590	2010-04-30	4/30/2010	CLOSED	1	1	0	112175.55590	112175.55590

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    origin_format="%Y-%m-%d",
    development="PaymentDate",
    development_format="%Y-%m-%d",
    columns=["Paid", "Incurred"],
    cumulative=False,
)
prism

	Triangle Summary
Valuation:	2017-12
Grain:	OMDM
Shape:	(1, 2, 120, 120)
Index:	[Total]
Columns:	[Paid, Incurred]

Up until now, we’ve been playing with symmetric triangles (i.e. orgin and development periods have the same grain). However, nothing precludes us from having a different grain. Often times in practice the development axis is more granular than the origin axis. All the functionality available to symmetric triangles works equally well for asymmetric triangles.

prism_df["AccYr"] = prism_df["AccidentDate"].str[:4]

prism = cl.Triangle(
    data=prism_df,
    origin="AccYr",
    origin_format="%Y",
    development="PaymentDate",
    development_format="%Y-%m-%d",
    columns=["Paid", "Incurred"],
    cumulative=False,
)
prism["Paid"]

	1	2	3	4	5	6	7	8	9	10	...	114
2008			75,664	90,814	194,956	300,085	347,487	400,011	356,795	598,532	...	7,000
2009		24,327	80,376	136,857	175,737	297,690	358,624	325,462	529,832	540,217	...
2010		22,192	84,802	148,299	200,733	208,237	456,692	519,691	604,946	484,188	...
2011		24,934	60,404	107,198	233,346	268,904	401,595	530,490	501,836	513,133	...
2012		18,794	98,320	165,689	296,865	335,643	320,954	529,733	592,985	686,689	...
2013		70,959	99,628	238,991	180,138	302,213	477,224	585,963	648,734	637,203	...
2014	18,194	32,772	124,801	276,336	344,635	425,875	449,242	541,203	587,789	819,083	...
2015		31,221	86,418	242,766	276,598	418,870	575,634	620,098	858,771	787,203	...
2016	3,214	30,412	87,680	238,884	325,065	468,785	521,981	560,611	853,548	976,906	...
2017	8,338	26,000	175,935	226,565	324,943	416,752	646,836	649,985	811,832	957,490	...

While exposure triangles make sense for auditable lines such as workers’ compensation lines, most other lines of business’ exposures can be expressed as a 1D array (along origin period) as exposures do not develop over time. chainladder arithmetic requires that operations happen between a triangle and either an int, float, or another Triangle. To create a 1D exposure array, simply omit the development argument at initialization.

The prism data does not consist of exposure data, but we can contrive one. Let’s assume that the premium is thrice the incurred amount.

prism_df["Premium"] = 3 * prism_df["Incurred"]

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    origin_format="%Y-%m-%d",
    columns="Premium",
    cumulative=False,
)

prism

	2017-12
2008-01	41,872,294
2008-02	43,542,684
2008-03	35,920,802
2008-04	30,521,592
2008-05	46,807,611
2008-06	35,926,236
2008-07	33,305,612
2008-08	35,667,952
2008-09	36,773,174
2008-10	36,858,772
2008-11	41,939,041
2008-12	34,716,883
2009-01	40,435,087
2009-02	51,803,670
2009-03	38,273,957
2009-04	44,045,041
2009-05	38,573,516
2009-06	34,674,998
2009-07	39,372,633
2009-08	41,936,689
2009-09	39,120,076
2009-10	33,485,416
2009-11	35,884,763
2009-12	35,640,359
2010-01	39,187,508
2010-02	36,408,113
2010-03	46,100,329
2010-04	39,311,907
2010-05	38,617,436
2010-06	38,204,654
...	...
2015-07	18,075,734
2015-08	12,204,463
2015-09	11,209,737
2015-10	7,647,748
2015-11	6,591,418
2015-12	7,948,603
2016-01	5,502,819
2016-02	5,677,216
2016-03	4,541,502
2016-04	4,938,965
2016-05	4,522,476
2016-06	4,664,779
2016-07	5,134,766
2016-08	4,554,252
2016-09	4,406,520
2016-10	4,278,755
2016-11	3,955,199
2016-12	3,586,924
2017-01	3,446,326
2017-02	3,095,700
2017-03	2,680,826
2017-04	2,634,972
2017-05	2,057,550
2017-06	1,403,713
2017-07	1,375,008
2017-08	1,130,442
2017-09	870,320
2017-10	418,445
2017-11	44,062
2017-12	32,244

Let’s seperate our data by segments using index:

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    origin_format="%Y-%m-%d",
    development="PaymentDate",
    development_format="%Y-%m-%d",
    columns=["Paid", "Incurred"],
    index="Line",
    cumulative=False,
)
prism

	Triangle Summary
Valuation:	2017-12
Grain:	OMDM
Shape:	(2, 2, 120, 120)
Index:	[Line]
Columns:	[Paid, Incurred]

We can futher index by coverages, or sublines:

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    origin_format="%Y-%m-%d",
    development="PaymentDate",
    development_format="%Y-%m-%d",
    columns=["Paid", "Incurred"],
    index=["Line", "Type"],
    cumulative=False,
)
prism

	Triangle Summary
Valuation:	2017-12
Grain:	OMDM
Shape:	(2, 2, 120, 120)
Index:	[Line, Type]
Columns:	[Paid, Incurred]

Triangle Methods not Available in Pandas¶

Up until now, we’ve kept pretty close to the pandas API for triangle manipulation. However, there are data transformations commonly applied to triangles that don’t have a nice pandas analogy.

For example, we often want to convert a triangle from an incremental view into a cumulative view and vice versa. This can be accomplished with the incr_to_cum and cum_to_incr methods.

prism["Paid"].sum()

	1	2	3	4	5	6	7	8	9	10	...	111	112	113	114	115	116	117	118	119	120
2008-01			46,915	19,899	57,216	89,783	18,302	95,054	10,332	91,772	...
2008-02		28,749	22,109	79,033	63,455	59,993	64,683	68,502	33,695	71,670	...
2008-03		48,806	27,949	90,413	54,557	83,507	12,591	72,035	70,353	35,223	...
2008-04		30,758	17,763	70,872	30,233	39,494	66,729	122,100	20,998	38,218	...
2008-05		38,672	86,974	20,483	58,400	112,015	31,354	22,457	53,778	93,303	...
2008-06		56,789	73,351	97,840	64,816	61,083	41,685	87,172	61,418	63,097	...
2008-07		27,867	45,804	61,495	73,279	71,591	111,807	80,249	40,632	32,434	...
2008-08	4,832	23,831	52,511	62,649	45,955	133,133	81,733	39,709	112,115	21,367	...
2008-09		43,464	86,157	52,664	89,928	88,738	55,617	26,599	69,522	46,788	...
2008-10		12,488	21,939	53,388	52,415	92,086	45,781	72,242	83,359	47,594	...
2008-11		45,473	9,416	78,967	84,826	39,549	41,011	74,000	33,257	35,802	...
2008-12	3,640	9,905	54,792	87,057	70,686	54,775	31,412	44,057	59,448	31,643	...
2009-01		14,021	45,350	51,208	35,520	81,655	54,275	41,538	74,129	47,962	...
2009-02	10,305	35,026	37,289	58,773	67,579	57,589	115,083	57,142	65,315	55,019	...
2009-03		48,360	40,502	74,342	64,538	49,778	85,591	82,582	56,449	88,307	...
2009-04		40,943	47,517	64,024	27,573	62,634	54,051	63,305	61,339	70,406	...
2009-05		16,944	74,478	63,856	59,266	79,707	63,857	59,326	88,817	50,656	...
2009-06	9,655	43,720	12,805	79,404	46,219	46,868	36,234	35,654	24,676	26,641	...
2009-07		14,828	48,605	90,527	54,508	57,881	64,905	16,717	46,880	89,721	...
2009-08		42,061	32,247	64,174	63,443	65,428	95,432	91,809	131,904	67,284	...
2009-09	21,000	27,607	34,679	37,523	56,305	42,851	72,404	54,366	67,543	71,032	...
2009-10	14,000	62,040	45,134	97,016	54,356	20,848	63,521	78,494	67,441	22,107	...
2009-11		22,027	64,189	20,836	33,303	50,594	79,852	54,870	111,592	70,473	...
2009-12		38,594	17,902	100,470	43,866	102,051	56,901	33,876	40,416	54,734	...
2010-01		15,192	53,338	57,394	56,985	62,175	84,624	84,959	102,734	26,495	...
2010-02	7,000	31,464	32,942	38,108	28,707	100,091	97,795	98,314	53,259	96,477	...
2010-03		36,937	64,551	61,774	39,580	81,253	35,751	87,788	41,076	117,835	...
2010-04	21,026	32,422	46,661	103,442	89,860	96,976	51,116	89,352	50,595	13,981	...
2010-05	8,668	8,920	116,376	61,586	148,647	14,288	46,840	51,205	82,714	4,240	...
2010-06		12,578	92,156	44,495	112,868	79,732	73,743	61,814	73,713	23,724	...
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2015-07	5,872	61,431	113,522	96,030	133,267	130,305	64,905	58,480	94,985	72,184	...
2015-08	12,276	73,666	69,772	156,546	96,829	122,893	77,525	120,597	42,213	81,603	...
2015-09	29,402	45,522	76,968	146,840	171,674	109,429	86,091	98,514	83,465	61,289	...
2015-10	14,000	18,824	64,809	86,112	33,952	94,798	41,473	114,279	95,470	95,322	...
2015-11		24,866	74,507	52,944	109,360	98,136	160,381	46,511	206,774	128,625	...
2015-12	14,000	34,498	85,505	113,846	118,239	121,277	179,033	131,330	130,357	89,312	...
2016-01	3,214	16,521	48,605	63,442	64,330	156,536	70,536	90,186	88,647	87,278	...
2016-02	13,892	20,722	112,166	154,319	93,667	117,578	114,468	77,207	79,020	85,925	...
2016-03	18,354	48,495	70,675	115,886	83,612	45,276	123,736	114,339	79,022	96,179	...
2016-04	14,781	35,741	51,818	124,195	76,220	168,821	117,505	82,978	99,148	112,664	...
2016-05		32,921	80,574	128,677	98,491	138,236	110,125	110,887	150,281	74,849	...
2016-06	17,959	9,031	35,721	163,644	172,139	157,653	154,168	120,193	100,214	62,572	...
2016-07	36,454	24,783	105,234	116,616	64,114	118,426	107,283	101,315	136,164	95,888	...
2016-08	45,280	14,224	93,541	147,758	88,370	195,297	117,494	57,656	132,907	100,460	...
2016-09	13,544	56,468	84,514	106,896	127,088	89,314	132,691	153,691	62,591	133,208	...
2016-10	1,766	61,196	80,437	123,346	82,947	158,137	168,589	110,019	86,504	162,549	...
2016-11		55,837	111,442	110,741	111,772	44,847	164,147	180,126	90,889	71,973	...
2016-12		52,806	55,386	116,175	107,611	109,759	74,635	145,938	100,703	114,897	...
2017-01	8,338	26,000	121,221	104,138	113,300	86,086	139,369	127,013	66,155	111,713	...
2017-02		50,237	101,651	103,479	94,798	94,285	160,239	141,802	131,585	94,257	...
2017-03	4,476	20,775	74,392	72,250	143,323	56,307	210,407	117,054	101,546	93,077	...
2017-04		30,751	126,808	154,839	157,090	72,106	102,587	91,454	142,689		...
2017-05	3,022	36,811	54,334	80,437	81,315	160,838	180,652	88,441			...
2017-06		31,858	37,911	81,229	121,619	85,185	110,103				...
2017-07	28,827	30,987	126,230	72,826	80,075	119,390					...
2017-08		14,000	105,952	81,991	174,870						...
2017-09	18,589	33,317	86,057	152,144							...
2017-10		35,037	104,444								...
2017-11	4,088	10,599									...
2017-12	10,748										...

prism_cum = prism.incr_to_cum()
prism_cum["Paid"].sum()

	1	2	3	4	5	6	7	8	9	10	...	111	112	113	114	115	116	117	118	119	120
2008-01			46,915	66,814	124,030	213,813	232,115	327,169	337,502	429,274	...	13,957,431	13,957,431	13,957,431	13,957,431	13,957,431	13,957,431	13,957,431	13,957,431	13,957,431	13,957,431
2008-02		28,749	50,859	129,891	193,346	253,339	318,022	386,523	420,218	491,888	...	14,514,228	14,514,228	14,514,228	14,514,228	14,514,228	14,514,228	14,514,228	14,514,228	14,514,228
2008-03		48,806	76,755	167,168	221,724	305,232	317,823	389,858	460,211	495,434	...	11,973,601	11,973,601	11,973,601	11,973,601	11,973,601	11,973,601	11,973,601	11,973,601
2008-04		30,758	48,521	119,393	149,626	189,120	255,849	377,949	398,947	437,165	...	10,173,864	10,173,864	10,173,864	10,173,864	10,173,864	10,173,864	10,173,864
2008-05		38,672	125,646	146,129	204,529	316,543	347,897	370,354	424,131	517,435	...	15,602,537	15,602,537	15,602,537	15,602,537	15,602,537	15,602,537
2008-06		56,789	130,140	227,980	292,796	353,879	395,565	482,736	544,154	607,251	...	11,975,412	11,975,412	11,975,412	11,975,412	11,975,412
2008-07		27,867	73,671	135,167	208,446	280,037	391,844	472,093	512,725	545,159	...	11,101,871	11,101,871	11,101,871	11,101,871
2008-08	4,832	28,663	81,174	143,824	189,779	322,912	404,645	444,354	556,469	577,837	...	11,889,317	11,889,317	11,889,317
2008-09		43,464	129,621	182,285	272,213	360,952	416,569	443,168	512,690	559,478	...	12,257,725	12,257,725
2008-10		12,488	34,427	87,815	140,229	232,315	278,096	350,338	433,697	481,291	...	12,286,257
2008-11		45,473	54,888	133,855	218,682	258,231	299,241	373,241	406,498	442,300	...
2008-12	3,640	13,544	68,336	155,393	226,080	280,855	312,267	356,324	415,772	447,415	...
2009-01		14,021	59,371	110,579	146,099	227,754	282,028	323,566	397,696	445,657	...
2009-02	10,305	45,331	82,621	141,394	208,972	266,561	381,645	438,786	504,101	559,120	...
2009-03		48,360	88,861	163,203	227,741	277,519	363,110	445,692	502,141	590,448	...
2009-04		40,943	88,459	152,483	180,056	242,690	296,742	360,047	421,386	491,791	...
2009-05		16,944	91,422	155,278	214,543	294,250	358,107	417,433	506,250	556,906	...
2009-06	9,655	53,374	66,180	145,584	191,803	238,671	274,905	310,559	335,235	361,875	...
2009-07		14,828	63,434	153,961	208,469	266,350	331,255	347,971	394,851	484,572	...
2009-08		42,061	74,308	138,483	201,926	267,354	362,786	454,595	586,499	653,783	...
2009-09	21,000	48,607	83,286	120,808	177,113	219,965	292,369	346,735	414,277	485,309	...
2009-10	14,000	76,040	121,174	218,190	272,546	293,395	356,915	435,409	502,850	524,957	...
2009-11		22,027	86,215	107,052	140,355	190,949	270,801	325,670	437,262	507,735	...
2009-12		38,594	56,496	156,966	200,832	302,883	359,784	393,660	434,076	488,811	...
2010-01		15,192	68,531	125,925	182,910	245,085	329,709	414,668	517,402	543,897	...
2010-02	7,000	38,464	71,406	109,514	138,221	238,312	336,107	434,422	487,681	584,158	...
2010-03		36,937	101,487	163,262	202,842	284,095	319,846	407,635	448,711	566,546	...
2010-04	21,026	53,448	100,108	203,550	293,410	390,386	441,502	530,854	581,449	595,430	...
2010-05	8,668	17,587	133,963	195,549	344,196	358,484	405,324	456,529	539,244	543,483	...
2010-06		12,578	104,735	149,230	262,098	341,830	415,573	477,388	551,101	574,825	...
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2015-07	5,872	67,304	180,826	276,856	410,123	540,428	605,334	663,814	758,798	830,982	...
2015-08	12,276	85,942	155,713	312,260	409,088	531,981	609,506	730,103	772,317	853,920	...
2015-09	29,402	74,924	151,892	298,732	470,405	579,835	665,925	764,440	847,904	909,194	...
2015-10	14,000	32,824	97,632	183,745	217,697	312,495	353,968	468,248	563,718	659,040	...
2015-11		24,866	99,373	152,317	261,677	359,813	520,193	566,705	773,479	902,104	...
2015-12	14,000	48,498	134,004	247,850	366,089	487,366	666,399	797,729	928,086	1,017,398	...
2016-01	3,214	19,734	68,339	131,782	196,112	352,647	423,183	513,369	602,016	689,294	...
2016-02	13,892	34,613	146,780	301,098	394,766	512,344	626,812	704,019	783,039	868,964	...
2016-03	18,354	66,848	137,523	253,408	337,021	382,297	506,033	620,372	699,394	795,573	...
2016-04	14,781	50,522	102,340	226,534	302,754	471,575	589,080	672,058	771,207	883,870	...
2016-05		32,921	113,494	242,172	340,663	478,898	589,024	699,911	850,192	925,041	...
2016-06	17,959	26,990	62,711	226,354	398,493	556,146	710,314	830,507	930,721	993,293	...
2016-07	36,454	61,237	166,471	283,087	347,202	465,628	572,910	674,225	810,389	906,277	...
2016-08	45,280	59,505	153,045	300,803	389,173	584,470	701,964	759,620	892,527	992,987	...
2016-09	13,544	70,012	154,527	261,422	388,510	477,824	610,516	764,207	826,799	960,007	...
2016-10	1,766	62,962	143,398	266,745	349,691	507,828	676,418	786,436	872,940	1,035,489	...
2016-11		55,837	167,279	278,020	389,792	434,640	598,787	778,913	869,802	941,776	...
2016-12		52,806	108,192	224,367	331,978	441,737	516,372	662,311	763,014	877,911	...
2017-01	8,338	34,338	155,559	259,698	372,997	459,083	598,452	725,466	791,620	903,333	...
2017-02		50,237	151,889	255,367	350,165	444,450	604,689	746,491	878,076	972,333	...
2017-03	4,476	25,251	99,644	171,894	315,217	371,524	581,932	698,986	800,532	893,609	...
2017-04		30,751	157,559	312,397	469,488	541,593	644,181	735,635	878,324		...
2017-05	3,022	39,834	94,168	174,605	255,919	416,757	597,409	685,850			...
2017-06		31,858	69,769	150,997	272,616	357,801	467,904				...
2017-07	28,827	59,815	186,044	258,870	338,946	458,336					...
2017-08		14,000	119,952	201,944	376,814						...
2017-09	18,589	51,906	137,963	290,107							...
2017-10		35,037	139,482								...
2017-11	4,088	14,687									...
2017-12	10,748										...

prism_incr = prism_cum.cum_to_incr()
prism_incr["Paid"].sum()

	1	2	3	4	5	6	7	8	9	10	...	111	112	113	114	115	116	117	118	119	120
2008-01			46,915	19,899	57,216	89,783	18,302	95,054	10,332	91,772	...
2008-02		28,749	22,109	79,033	63,455	59,993	64,683	68,502	33,695	71,670	...
2008-03		48,806	27,949	90,413	54,557	83,507	12,591	72,035	70,353	35,223	...
2008-04		30,758	17,763	70,872	30,233	39,494	66,729	122,100	20,998	38,218	...
2008-05		38,672	86,974	20,483	58,400	112,015	31,354	22,457	53,778	93,303	...
2008-06		56,789	73,351	97,840	64,816	61,083	41,685	87,172	61,418	63,097	...
2008-07		27,867	45,804	61,495	73,279	71,591	111,807	80,249	40,632	32,434	...
2008-08	4,832	23,831	52,511	62,649	45,955	133,133	81,733	39,709	112,115	21,367	...
2008-09		43,464	86,157	52,664	89,928	88,738	55,617	26,599	69,522	46,788	...
2008-10		12,488	21,939	53,388	52,415	92,086	45,781	72,242	83,359	47,594	...
2008-11		45,473	9,416	78,967	84,826	39,549	41,011	74,000	33,257	35,802	...
2008-12	3,640	9,905	54,792	87,057	70,686	54,775	31,412	44,057	59,448	31,643	...
2009-01		14,021	45,350	51,208	35,520	81,655	54,275	41,538	74,129	47,962	...
2009-02	10,305	35,026	37,289	58,773	67,579	57,589	115,083	57,142	65,315	55,019	...
2009-03		48,360	40,502	74,342	64,538	49,778	85,591	82,582	56,449	88,307	...
2009-04		40,943	47,517	64,024	27,573	62,634	54,051	63,305	61,339	70,406	...
2009-05		16,944	74,478	63,856	59,266	79,707	63,857	59,326	88,817	50,656	...
2009-06	9,655	43,720	12,805	79,404	46,219	46,868	36,234	35,654	24,676	26,641	...
2009-07		14,828	48,605	90,527	54,508	57,881	64,905	16,717	46,880	89,721	...
2009-08		42,061	32,247	64,174	63,443	65,428	95,432	91,809	131,904	67,284	...
2009-09	21,000	27,607	34,679	37,523	56,305	42,851	72,404	54,366	67,543	71,032	...
2009-10	14,000	62,040	45,134	97,016	54,356	20,848	63,521	78,494	67,441	22,107	...
2009-11		22,027	64,189	20,836	33,303	50,594	79,852	54,870	111,592	70,473	...
2009-12		38,594	17,902	100,470	43,866	102,051	56,901	33,876	40,416	54,734	...
2010-01		15,192	53,338	57,394	56,985	62,175	84,624	84,959	102,734	26,495	...
2010-02	7,000	31,464	32,942	38,108	28,707	100,091	97,795	98,314	53,259	96,477	...
2010-03		36,937	64,551	61,774	39,580	81,253	35,751	87,788	41,076	117,835	...
2010-04	21,026	32,422	46,661	103,442	89,860	96,976	51,116	89,352	50,595	13,981	...
2010-05	8,668	8,920	116,376	61,586	148,647	14,288	46,840	51,205	82,714	4,240	...
2010-06		12,578	92,156	44,495	112,868	79,732	73,743	61,814	73,713	23,724	...
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
2015-07	5,872	61,431	113,522	96,030	133,267	130,305	64,905	58,480	94,985	72,184	...
2015-08	12,276	73,666	69,772	156,546	96,829	122,893	77,525	120,597	42,213	81,603	...
2015-09	29,402	45,522	76,968	146,840	171,674	109,429	86,091	98,514	83,465	61,289	...
2015-10	14,000	18,824	64,809	86,112	33,952	94,798	41,473	114,279	95,470	95,322	...
2015-11		24,866	74,507	52,944	109,360	98,136	160,381	46,511	206,774	128,625	...
2015-12	14,000	34,498	85,505	113,846	118,239	121,277	179,033	131,330	130,357	89,312	...
2016-01	3,214	16,521	48,605	63,442	64,330	156,536	70,536	90,186	88,647	87,278	...
2016-02	13,892	20,722	112,166	154,319	93,667	117,578	114,468	77,207	79,020	85,925	...
2016-03	18,354	48,495	70,675	115,886	83,612	45,276	123,736	114,339	79,022	96,179	...
2016-04	14,781	35,741	51,818	124,195	76,220	168,821	117,505	82,978	99,148	112,664	...
2016-05		32,921	80,574	128,677	98,491	138,236	110,125	110,887	150,281	74,849	...
2016-06	17,959	9,031	35,721	163,644	172,139	157,653	154,168	120,193	100,214	62,572	...
2016-07	36,454	24,783	105,234	116,616	64,114	118,426	107,283	101,315	136,164	95,888	...
2016-08	45,280	14,224	93,541	147,758	88,370	195,297	117,494	57,656	132,907	100,460	...
2016-09	13,544	56,468	84,514	106,896	127,088	89,314	132,691	153,691	62,591	133,208	...
2016-10	1,766	61,196	80,437	123,346	82,947	158,137	168,589	110,019	86,504	162,549	...
2016-11		55,837	111,442	110,741	111,772	44,847	164,147	180,126	90,889	71,973	...
2016-12		52,806	55,386	116,175	107,611	109,759	74,635	145,938	100,703	114,897	...
2017-01	8,338	26,000	121,221	104,138	113,300	86,086	139,369	127,013	66,155	111,713	...
2017-02		50,237	101,651	103,479	94,798	94,285	160,239	141,802	131,585	94,257	...
2017-03	4,476	20,775	74,392	72,250	143,323	56,307	210,407	117,054	101,546	93,077	...
2017-04		30,751	126,808	154,839	157,090	72,106	102,587	91,454	142,689		...
2017-05	3,022	36,811	54,334	80,437	81,315	160,838	180,652	88,441			...
2017-06		31,858	37,911	81,229	121,619	85,185	110,103				...
2017-07	28,827	30,987	126,230	72,826	80,075	119,390					...
2017-08		14,000	105,952	81,991	174,870						...
2017-09	18,589	33,317	86,057	152,144							...
2017-10		35,037	104,444								...
2017-11	4,088	10,599									...
2017-12	10,748										...

By default (and in concert with the pandas philosophy), the methods associated with the Triangle class strive for immutability. This means that the incr_to_cum and cum_to_incr methods will return new a new triangle object that must be assigned, or it is thrown away. Many of the chainladder.Triangle methods have an inplace argument, or alternatively you can just use variable reassignment to store the transformed triangle object.

# This works
prism.incr_to_cum(inplace=True)
# So does this
prism = prism.incr_to_cum()

When dealing with triangles that have an origin axis, development axis, or both at a monthly or quarterly grain, the triangle can be summarized to a higher grain using the grain method.

The grain to which you want your triangle converted, specified as “OxDy” where “x” and “y” can take on values of “M”, “Q”, or “Y”. For example:

grain(OYDY) for Origin Year x Development Year.
grain(OQDM) for Origin Quarter x Development Month.

prism_OYDY = prism.grain("OYDY")
prism_OYDY["Paid"].sum()

	12	24	36	48	60	72	84	96	108	120
2008	3,404,254	11,191,085	74,613,012	150,342,751	150,982,873	151,152,726	151,228,872	151,264,806	151,277,217	151,284,217
2009	3,609,385	11,002,927	80,726,352	156,970,789	157,599,460	157,697,094	157,736,386	157,743,386	157,748,735
2010	4,067,321	12,396,777	74,210,043	161,049,586	161,641,453	161,787,135	161,859,565	161,870,156
2011	4,125,232	13,183,144	81,239,771	161,412,913	162,187,629	162,417,460	162,490,681
2012	4,584,036	14,001,178	77,794,522	152,118,384	152,588,090	152,819,473
2013	4,889,623	14,607,742	84,418,503	161,110,312	161,673,036
2014	5,546,158	16,408,126	77,256,792	154,969,931
2015	5,909,029	17,427,611	80,914,580
2016	6,080,962	18,588,057
2017	6,396,536

Depending on the type of analysis being done, it may be more convenient to look at a triangle with its development axis expressed as a valuation rather than an age. This is also what the Schedule Ps look like. To do this, Triangle has two methods for toggling between a development triangle and a valuation triangle. The methods are dev_to_val and its inverse val_to_dev.

prism_OYDY_val = prism_OYDY.dev_to_val()
prism_OYDY_val["Paid"].sum()

	2008	2009	2010	2011	2012	2013	2014	2015	2016	2017
2008	3,404,254	11,191,085	74,613,012	150,342,751	150,982,873	151,152,726	151,228,872	151,264,806	151,277,217	151,284,217
2009		3,609,385	11,002,927	80,726,352	156,970,789	157,599,460	157,697,094	157,736,386	157,743,386	157,748,735
2010			4,067,321	12,396,777	74,210,043	161,049,586	161,641,453	161,787,135	161,859,565	161,870,156
2011				4,125,232	13,183,144	81,239,771	161,412,913	162,187,629	162,417,460	162,490,681
2012					4,584,036	14,001,178	77,794,522	152,118,384	152,588,090	152,819,473
2013						4,889,623	14,607,742	84,418,503	161,110,312	161,673,036
2014							5,546,158	16,408,126	77,256,792	154,969,931
2015								5,909,029	17,427,611	80,914,580
2016									6,080,962	18,588,057
2017										6,396,536

When working with real-world data, the triangles can have holes, such as missing evluation(s), or no losses in certain origin(s). In these cases, it doesn’t make sense to include empty accident periods or development ages. For example, in the prism dataset, the “Home” line has its latest accidents through 2016, and have no payments in development age ‘12’. Sometimes, dropping the non-applicable fields is usefule with the dropna() method.

prism_OYDY.loc["Home"]["Paid"]

	24	36	48	60	72	84	96	108	120
2008	1,129,305	61,658,874	136,520,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554
2009	187,292	67,134,599	142,267,946	142,547,946	142,547,946	142,547,946	142,547,946	142,547,946
2010	620,603	59,082,456	144,757,200	144,757,200	144,757,200	144,757,200	144,757,200
2011	503,296	65,048,051	143,971,618	144,251,618	144,251,618	144,251,618
2012	599,277	60,536,642	133,412,416	133,412,416	133,412,416
2013	536,303	66,443,157	141,645,782	141,645,782
2014	965,973	57,394,263	133,542,848
2015	371,015	59,047,107
2016	640,179
2017

Let’s see what happens if we have no data for 2011.

prism_df_2011 = prism_df.copy()
prism_df_2011.loc[
    (prism_df_2011["AccidentDate"] >= "2011-01-01")
    & (prism_df_2011["AccidentDate"] < "2012-01-01"),
    "Paid",
] = None  # Let's assume we hav no payments for losses occurred in 2011
prism_2011 = cl.Triangle(
    data=prism_df_2011,
    origin="AccidentDate",
    origin_format="%Y-%m-%d",
    development="PaymentDate",
    development_format="%Y-%m-%d",
    columns=["Paid", "Incurred"],
    index=["Line", "Type"],
    cumulative=False,
)
prism_2011.incr_to_cum(inplace=True)
prism_2011_OYDY = prism_2011.grain("OYDY")
prism_2011_OYDY.loc["Home"]["Paid"]

	24	36	48	60	72	84	96	108	120
2008	1,129,305	61,658,874	136,520,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554
2009	187,292	67,134,599	142,267,946	142,547,946	142,547,946	142,547,946	142,547,946	142,547,946
2010	620,603	59,082,456	144,757,200	144,757,200	144,757,200	144,757,200	144,757,200
2011
2012	599,277	60,536,642	133,412,416	133,412,416	133,412,416
2013	536,303	66,443,157	141,645,782	141,645,782
2014	965,973	57,394,263	133,542,848
2015	371,015	59,047,107
2016	640,179
2017

Note that the dropna() method will retain empty periods if they are surrounded by non-empty periods with valid data.

prism_2011_OYDY_droppedna = prism_2011_OYDY.loc["Home"].dropna()
prism_2011_OYDY_droppedna.loc["Home"]["Paid"]

	24	36	48	60	72	84	96	108	120
2008	1,129,305	61,658,874	136,520,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554
2009	187,292	67,134,599	142,267,946	142,547,946	142,547,946	142,547,946	142,547,946	142,547,946
2010	620,603	59,082,456	144,757,200	144,757,200	144,757,200	144,757,200	144,757,200
2011
2012	599,277	60,536,642	133,412,416	133,412,416	133,412,416
2013	536,303	66,443,157	141,645,782	141,645,782
2014	965,973	57,394,263	133,542,848
2015	371,015	59,047,107
2016	640,179

Commutative Properties of Triangle Methods¶

Where it makes sense, which is in most cases, the methods described above are commutative and can be applied in any order.

print("Commutative?", prism.sum().latest_diagonal == prism.latest_diagonal.sum())
print("Commutative?", prism.loc["Home"].link_ratio == prism.link_ratio.loc["Home"])

Commutative? True
Commutative? True

Triangle Imports and Exports¶

To the extent the Triangle can be expressed as a pandas.DataFrame, you can use any of the pandas IO to send the data in and out. Note that converting to pandas is a one-way ticket with no inverse functions.

Another useful function is to copy the triangle and put it in the clipboard so we can paste it elsewhere (such as Excel):

prism_OYDY.loc["Home", "Paid"]

	24	36	48	60	72	84	96	108	120
2008	1,129,305	61,658,874	136,520,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554	136,800,554
2009	187,292	67,134,599	142,267,946	142,547,946	142,547,946	142,547,946	142,547,946	142,547,946
2010	620,603	59,082,456	144,757,200	144,757,200	144,757,200	144,757,200	144,757,200
2011	503,296	65,048,051	143,971,618	144,251,618	144,251,618	144,251,618
2012	599,277	60,536,642	133,412,416	133,412,416	133,412,416
2013	536,303	66,443,157	141,645,782	141,645,782
2014	965,973	57,394,263	133,542,848
2015	371,015	59,047,107
2016	640,179
2017

prism_OYDY.loc["Home", "Paid"].to_clipboard()

Try to paste it elsewhere.

Alternatively, if you want to store the triangle elsewhere but be able to reconstitute a triangle out of it later, then you can use:

Triangle.to_json and its inverse cl.read_json for json format
Triangle.to_pickle and its inverse cl.read_pickle for pickle format

These have the added benefit of working on multi-dimensional triangles that don’t fit into a DataFrame.

Exercises¶

prism = cl.Triangle(
    data=prism_df,
    origin="AccidentDate",
    development="PaymentDate",
    columns=["Paid", "Incurred"],
    index=["Line", "Type"],  # multiple indices
    cumulative=False,
).incr_to_cum()
prism

	Triangle Summary
Valuation:	2017-12
Grain:	OMDM
Shape:	(2, 2, 120, 120)
Index:	[Line, Type]
Columns:	[Paid, Incurred]

What is the case incurred activity for calendar periods in 2015Q2 (March, April, and May in 2015) by “Line”?

incr_by_line = prism.groupby("Line").sum().cum_to_incr()["Incurred"].dev_to_val()
incr_by_line[
    (incr_by_line.valuation >= "2015-04-01") & (incr_by_line.valuation < "2015-07-01")
].sum("origin").to_frame(origin_as_datetime=True).astype(int)

development	2015-04	2015-05	2015-06
Line
Auto	1883976	1704338	1692254
Home	10367939	10813821	14537617

For accident year 2015, what proportion of our Paid amounts come from each “Type” of claims?

prism_OYDY = prism.grain("OYDY")
prism_OYDY

	Triangle Summary
Valuation:	2017-12
Grain:	OYDY
Shape:	(2, 2, 10, 10)
Index:	[Line, Type]
Columns:	[Paid, Incurred]

by_type = (
    prism_OYDY.latest_diagonal[prism_OYDY.origin == "2015"]["Paid"]
    .groupby("Type")
    .sum()
    .to_frame(origin_as_datetime=True)
)
by_type / by_type.sum()

Type
Dwelling    0.729746
PD          0.270254
dtype: float64

Working with Triangles

Contents

Working with Triangles¶

Getting Started¶

Disclaimer¶

Working with a Triangle¶

The Triangle Structure¶

Pandas-Style Slicing¶

Pandas-Style Arithmetic¶

Accessor Methods¶

Moving Back to Pandas¶

Exercises¶

Initializing a Triangle With Your Own Data¶

Triangle Methods not Available in Pandas¶

Commutative Properties of Triangle Methods¶

Triangle Imports and Exports¶

Exercises¶