Time

The time variable (t) is a system variable in actuarial cash flow models. The t variable represents monthly periods starting from zero.

The model is run for the valuation date chosen by the actuary. The valuation date reflects the start of the model (t=0). The information on the valuation date can be stored in the runplan.

Below you can find model formulas for time-related variables that might be helpful in the model’s development.

Solution

Formulas:

model.py

import math
from cashflower import variable
from input import main, runplan


@variable()
def cal_month(t):
    if t == 0:
        return runplan.get("valuation_month")
    if cal_month(t-1) == 12:
        return 1
    else:
        return cal_month(t-1) + 1


@variable()
def cal_year(t):
    if t == 0:
        return runplan.get("valuation_year")
    if cal_month(t-1) == 12:
        return cal_year(t-1) + 1
    else:
        return cal_year(t-1)


@variable()
def elapsed_months():
    issue_year = main.get("issue_year")
    issue_month = main.get("issue_month")
    valuation_year = runplan.get("valuation_year")
    valuation_month = runplan.get("valuation_month")
    return (valuation_year - issue_year) * 12 + (valuation_month - issue_month)


@variable()
def pol_month(t):
    if t == 0:
        mnth = elapsed_months() % 12
        mnth = 12 if mnth == 0 else mnth
        return mnth
    if pol_month(t-1) == 12:
        return 1
    return pol_month(t-1) + 1


@variable()
def pol_year(t):
    if t == 0:
        return math.floor(elapsed_months() / 12)
    if pol_month(t) == 1:
        return pol_year(t-1) + 1
    return pol_year(t-1)

Input:

input.py

import pandas as pd
from cashflower import Runplan, ModelPointSet


runplan = Runplan(data=pd.DataFrame({
    "version": [1],
    "valuation_year": [2022],
    "valuation_month": [12]
}))


main = ModelPointSet(data=pd.DataFrame({
    "id": [1],
    "issue_year": [2020],
    "issue_month": [6],
}))

Description

Input

Model uses runplan to store the information on the valuation date.

input.py

import pandas as pd
from cashflower import Runplan, ModelPointSet


runplan = Runplan(data=pd.DataFrame({
    "version": [1],
    "valuation_year": [2022],
    "valuation_month": [12]
}))

The policyholder has a policy that was issued in June 2020.

input.py

main = ModelPointSet(data=pd.DataFrame({
    "id": [1],
    "issue_year": [2020],
    "issue_month": [6],
}))

Model

Calendar year and month

Knowing the valuation date, we can calculate actual calendar years and months. The valuation year and month can be read from the runplan.

model.py

@variable()
def cal_month(t):
    if t == 0:
        return runplan.get("valuation_month")
    if cal_month(t-1) == 12:
        return 1
    else:
        return cal_month(t-1) + 1


@variable()
def cal_year(t):
    if t == 0:
        return runplan.get("valuation_year")
    if cal_month(t-1) == 12:
        return cal_year(t-1) + 1
    else:
        return cal_year(t-1)

Elapsed months

Each policy starts at a different moment. The policy’s issue date might be part of the model points. Elapsed months reflect how many months have passed between the policy’s issue and the valuation date.

model.py

@variable()
def elapsed_months():
    issue_year = main.get("issue_year")
    issue_month = main.get("issue_month")
    valuation_year = runplan.get("valuation_year")
    valuation_month = runplan.get("valuation_month")
    return (valuation_year - issue_year) * 12 + (valuation_month - issue_month)

Policy year and month

Policy year and month reflect the duration of the given policy.

model.py

@variable()
def pol_month(t):
    if t == 0:
        mnth = elapsed_months() % 12
        mnth = 12 if mnth == 0 else mnth
        return mnth
    if pol_month(t-1) == 12:
        return 1
    return pol_month(t-1) + 1


@variable()
def pol_year(t):
    if t == 0:
        return math.floor(elapsed_months() / 12)
    if pol_month(t) == 1:
        return pol_year(t-1) + 1
    return pol_year(t-1)

Results

The result for the first 13 months.

<timestamp>_output.csv

t,cal_year,cal_month,elapsed_months,pol_year,pol_month
0,2022,12,30,2,6
1,2023,1,30,2,7
2,2023,2,30,2,8
3,2023,3,30,2,9
4,2023,4,30,2,10
5,2023,5,30,2,11
6,2023,6,30,2,12
7,2023,7,30,3,1
8,2023,8,30,3,2
9,2023,9,30,3,3
10,2023,10,30,3,4
11,2023,11,30,3,5
12,2023,12,30,3,6
13,2024,1,30,3,7

Notes:

• cal_month, cal_year - starts with valuation date,

• elapsed_months - number of months between issue of the policy (2020-06) and the valuation date (2022-12),

• pol_month, pol_year - the policy was already 2 years and 6 months “old” at the valuation date.