relativities.ratemaking

Generated from the package’s Python API. Experience and the overall rate indication, in exact arithmetic. Triangle develops losses to ultimate: factors averages its link ratios into a table to edit in pandas, and develop projects each origin with it. Trend fits or holds an annual trend and gives the factor between two points in time. RateHistory gives parallelogram on-level factors for calendar-year premium. Indication turns a projected loss ratio and expense provisions into the indicated overall change; with_credibility blends it with a complement, weighted by credibility, and explain shows each step. Every amount is a Decimal and every factor is exact to 38 significant digits; floats are refused where they would carry binary rounding into a rate. Rating relativities by class come from relativities.models.FactorModel. Develop losses

Name	What it is
`Triangle`	Cumulative losses (or counts) by origin period and development age.

Trend and on-level

Name	What it is
`Trend`	An annual exponential trend, such as 4% a year in claim severity.
`RateHistory`	Overall rate changes by effective date, such as `{date(2023, 7, 1): Decimal("0.05")}`.

Indicate a change

Name	What it is
`Indication`	The overall rate change the loss ratio method indicates.
`credibility`	Limited fluctuation credibility: `min(1, sqrt(claims / standard))`.
`full_credibility`	The expected claims for full credibility under limited fluctuation: `(z / tolerance) ** 2`, where `z` is the standard normal quantile of `(1 + probability) / 2`.

Triangle

class Triangle(
    claims: pd.DataFrame,
    *,
    origin: str,
    age: str,
    value: str,
    cumulative: bool = True,
)

Cumulative losses (or counts) by origin period and development age. Build one from a long table with one row per origin and age; origin is the accident, policy or report period and age its development age as a positive integer, such as months. Values are exact Decimals. Each origin’s ages are the first ages of the triangle in sequence, so each link ratio compares the same origin one age apart. factors averages the link ratios into one selected factor per age and develop projects each origin to ultimate with a factors table. Edit the factor column of factors() in pandas to make your own selections, then pass it to develop.

Triangle.origins

@property
def origins(self) -> list[Hashable]

The origin periods, in order.

Triangle.ages

@property
def ages(self) -> list[int]

The development ages, in order.

Triangle.to_pandas

def to_pandas(self) -> pd.DataFrame

The cumulative triangle: one row per origin and one column per age. NaN marks each age beyond an origin’s latest.

Triangle.link_ratios

def link_ratios(self) -> pd.DataFrame

Each origin’s age-to-age ratios: one column per age, the ratio from that age to the next. NaN marks a ratio beyond an origin’s latest age, or from a zero value.

Triangle.factors

def factors(
    self,
    *,
    average: Average = 'volume',
    periods: int | None = None,
    tail: Decimal | int = 1,
) -> pd.DataFrame

One selected factor per age, and the cumulative factor to ultimate. average is one of the averages of Werner and Modlin’s development exhibits:

volume: the sum of later values over the sum of earlier values.
simple: the straight average of the link ratios.
medial: the straight average after excluding the highest and the lowest ratio, when at least three ratios remain to choose from.
geometric: the nth root of the product of the n link ratios.

periods keeps only the latest origins that reach each pair of ages, such as the latest three. tail is the factor from the last age to ultimate. The last row is the tail. Columns: age, factor (from this age to the next, or to ultimate on the last row), cdf (from this age to ultimate) and origins (how many origins the average used). Edit factor to select your own; develop reads only age and factor and recalculates cdf.

Triangle.develop

def develop(self, factors: pd.DataFrame | None = None) -> pd.DataFrame

Each origin projected to ultimate with a factors table (factors() by default). Each factor develops from its age to the next age in the table, and the last row develops to ultimate. An age’s cumulative factor is the product of the factors at that age and every later age in the table, so a table selected on a longer triangle keeps its later factors and tail. The table needs one row for each age of this triangle; ages are unique. Columns: origin, age (its latest age), latest (its latest value), cdf, ultimate (latest times cdf) and unreported (ultimate less latest: IBNR for reported losses, unpaid for paid losses).

Trend

class Trend(rate: Decimal | int)

An annual exponential trend, such as 4% a year in claim severity. Trend(Decimal("0.04")) is a selected trend; Trend.fit fits one to a series by least squares on the logarithm of its values. factor(start, end) is (1 + rate) ** years between two points in time, each a date or a number of years, such as the average accident date of an experience period and of the future policy period. A date counts its months as twelfths of a year, so 1 January 2011 to 1 July 2013 is 2.5 years.

Trend.fit

@classmethod
def fit(cls, series: pd.DataFrame, *, period: str, value: str) -> Trend

Fit value = a * (1 + rate) ** period by least squares on log(value). period holds dates or numbers of years, such as each quarter’s midpoint, and value positive amounts, such as average severity or frequency. observations on the result holds each point with its fitted value.

Trend.rate

@property
def rate(self) -> Decimal

The annual rate: 0.04 is 4% a year.

Trend.observations

@property
def observations(self) -> pd.DataFrame | None

The fitted series and each point’s fitted value, or None for a selected trend.

Trend.factor

def factor(self, start: object, end: object) -> Decimal

(1 + rate) ** years from start to end, each a date or a number of years.

RateHistory

class RateHistory(changes: Mapping[object, Decimal | int])

Overall rate changes by effective date, such as {date(2023, 7, 1): Decimal("0.05")}. The rate level is 1 before the first change and is multiplied by 1 + change at each effective date. on_level finds the factors that bring each calendar year’s premium to the current level by the parallelogram method of Werner and Modlin’s Basic Ratemaking, chapter 5: policies are written evenly through the year, each change applies to policies written on or after its date, and premium is earned evenly over the term. A date counts its months as twelfths of a year, as the textbook does: 1 July is 0.5. When you have the policies, rating them again at current rates with Program.rate is the exact alternative.

RateHistory.levels

def levels(self) -> pd.DataFrame

Each change with the cumulative rate level from its effective date.

RateHistory.on_level

def on_level(
    self,
    years: Iterable[int],
    *,
    basis: Basis = 'earned',
    term_months: int = 12,
) -> pd.DataFrame

On-level factors for calendar years. basis is earned (calendar-year earned premium) or written (calendar-year written premium). term_months is the policy term. Columns: year, average_level (the year’s average rate level), current_level and factor (current_level over average_level, which multiplies the year’s premium).

Indication

class Indication(
    loss_ratio: Decimal | int,
    *,
    fixed_expense: Decimal | int = 0,
    variable_expense: Decimal | int = 0,
    profit: Decimal | int = 0,
)

The overall rate change the loss ratio method indicates. loss_ratio is the projected ultimate loss and LAE ratio: developed, trended losses over premium at current rate level. fixed_expense is the fixed expense ratio at current rates, variable_expense the variable expense ratio and profit the profit and contingencies provision, each a share of premium. The indicated change is (loss_ratio + fixed_expense) / (1 - variable_expense - profit) - 1. with_credibility blends it with a complement, and explain shows each step.

Indication.permissible_loss_ratio

@property
def permissible_loss_ratio(self) -> Decimal

The share of premium left for losses, LAE and fixed expenses: 1 - variable_expense - profit.

Indication.indicated

@property
def indicated(self) -> Decimal

The indicated change from this experience alone, before credibility.

Indication.change

@property
def change(self) -> Decimal

The indicated change after credibility: 0.05 is a 5% increase.

Indication.with_credibility

def with_credibility(
    self,
    credibility: Decimal | int,
    complement: Decimal | int,
) -> Indication

A copy whose change is credibility * indicated + (1 - credibility) * complement. credibility is between 0 and 1, such as credibility(claims). complement is the change that takes the remaining weight, such as the trended change since the last review or a larger book’s indication.

Indication.explain

def explain(self) -> pd.DataFrame

Each step of the indication: its name, value and formula.

credibility

def credibility(claims: Decimal | int, *, standard: Decimal | int = 1082) -> Decimal

Limited fluctuation credibility: min(1, sqrt(claims / standard)). standard is the claim count for full credibility, 1,082 by default; full_credibility computes it for other probability and tolerance choices.

full_credibility

def full_credibility(
    *,
    probability: Decimal | int = Decimal('0.90'),
    tolerance: Decimal | int = Decimal('0.05'),
) -> Decimal

The expected claims for full credibility under limited fluctuation: (z / tolerance) ** 2, where z is the standard normal quantile of (1 + probability) / 2. The defaults give about 1,082 claims. The quantile is solved in Decimal to the module’s 38 digits.

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Concepts

Python reference

Data reference

relativities.ratemaking

Triangle

Triangle.origins

Triangle.ages

Triangle.to_pandas

Triangle.link_ratios

Triangle.factors

Triangle.develop

Trend

Trend.fit

Trend.rate

Trend.observations

Trend.factor

RateHistory

RateHistory.levels

RateHistory.on_level

Indication

Indication.permissible_loss_ratio

Indication.indicated

Indication.change

Indication.with_credibility

Indication.explain

credibility

full_credibility

​Triangle

​Triangle.origins

​Triangle.ages

​Triangle.to_pandas

​Triangle.link_ratios

​Triangle.factors

​Triangle.develop

​Trend

​Trend.fit

​Trend.rate

​Trend.observations

​Trend.factor

​RateHistory

​RateHistory.levels

​RateHistory.on_level

​Indication

​Indication.permissible_loss_ratio

​Indication.indicated

​Indication.change

​Indication.with_credibility

​Indication.explain

​credibility

​full_credibility

Triangle

Triangle.origins

Triangle.ages

Triangle.to_pandas

Triangle.link_ratios

Triangle.factors

Triangle.develop

Trend

Trend.fit

Trend.rate

Trend.observations

Trend.factor

RateHistory

RateHistory.levels

RateHistory.on_level

Indication

Indication.permissible_loss_ratio

Indication.indicated

Indication.change

Indication.with_credibility

Indication.explain

credibility

full_credibility