Life annuities

Common variables

For simplicity, we will use monthly constant mortality and interest rate.

INTEREST_RATE = 0.005
DEATH_PROB = 0.003

In the production actuarial cash flow models:

• mortality rates depend on the age and the sex of the policyholder,
• interest rates are a curve and vary for each future period.

It's also usual that assumptions contain yearly data that must be converted into monthly rates within the model. The examples below are simplified to focus on the calculation of the expected payment.

The common variables of the models, that we will build, are:

• survival_rate,
• expected_payment,
• actuarial_present_value.

The survival_rate and the actuarial_present_value are calculated in the same way for all types of life insurance.

@variable()
def survival_rate(t):
    if t == 0:
        return 1
    return survival_rate(t-1) * (1 - DEATH_PROB)

The survival rate is the probability that the policyholder will survive from the beginning of the projection until the t time.

@variable()
def actuarial_present_value(t):
    if t == settings["T_MAX_CALCULATION"]:
        return expected_payment(t)
    return expected_payment(t) + actuarial_present_value(t+1) * 1/(1+INTEREST_RATE)

The actuarial present value is the present value of the expected annuity payments. The discount rate is calculated as 1/(1+INTEREST_RATE).

Whole life annuity

Whole life annuity provides a policyholder with a periodic (e.g. monthly) payments as long as the policyholder lives.

import pandas as pd
from cashflower import ModelPointSet

main = ModelPointSet(data=pd.DataFrame({
    "id": [1],
    "payment": [1_000]
}))

Policy data contains the value of the monthly payment which is paid to the policyholder.

from cashflower import variable

from input import main
from settings import settings

INTEREST_RATE = 0.005
DEATH_PROB = 0.003


@variable()
def survival_rate(t):
    if t == 0:
        return 1
    return survival_rate(t-1) * (1 - DEATH_PROB)


@variable()
def expected_payment(t):
    if t == 0:
        return 0
    return survival_rate(t) * main.get("payment")


@variable()
def actuarial_present_value(t):
    if t == settings["T_MAX_CALCULATION"]:
        return expected_payment(t)
    return expected_payment(t) + actuarial_present_value(t+1) * 1/(1+INTEREST_RATE)

The policyholder will receive a payment as long as they survive.

Temporary life annuity

An n-year temporary life annuity provides a policyholder with a periodic (e.g. monthly) payments for n years.

import pandas as pd
from cashflower import Runplan, ModelPointSet

main = ModelPointSet(data=pd.DataFrame({
    "id": [1],
    "payment": [1_000],
    "remaining_term": [36],
}))

Policy data contains the value of the monthly payment and the remaining term of the annuity. Here the remaining term is expressed in months starting the valuation period (rather than the issue date).

from cashflower import variable

from input import main
from settings import settings

INTEREST_RATE = 0.005
DEATH_PROB = 0.003


@variable()
def survival_rate(t):
    if t == 0:
        return 1
    return survival_rate(t-1) * (1 - DEATH_PROB)


@variable()
def expected_payment(t):
    if t == 0 or t > main.get("remaining_term"):
        return 0
    return survival_rate(t) * main.get("payment")


@variable()
def actuarial_present_value(t):
    if t == settings["T_MAX_CALCULATION"]:
        return expected_payment(t)
    return expected_payment(t) + actuarial_present_value(t+1) * 1/(1+INTEREST_RATE)

The policyholder will receive a payment as long as they survive but no longer than n-years.

Deferred whole life annuity

An m-year deferred whole life annuity provides a policyholder with a periodic (e.g. monthly) payments as long as the policyholder lives starting m years after the issue.

import pandas as pd
from cashflower import Runplan, ModelPointSet

main = ModelPointSet(data=pd.DataFrame({
    "id": [1],
    "payment": [1_000],
    "deferral": [12],
}))

Policy data contains the value of the monthly payment which is paid to the policyholder and the deferral period. Here the deferral period is expressed in months starting from the valuation period (rather than the issue date).

from cashflower import variable

from input import main
from settings import settings

INTEREST_RATE = 0.005
DEATH_PROB = 0.003


@variable()
def survival_rate(t):
    if t == 0:
        return 1
    return survival_rate(t-1) * (1 - DEATH_PROB)


@variable()
def expected_payment(t):
    if t <= main.get("deferral"):
        return 0
    return survival_rate(t) * main.get("payment")


@variable()
def actuarial_present_value(t):
    if t == settings["T_MAX_CALCULATION"]:
        return expected_payment(t)
    return expected_payment(t) + actuarial_present_value(t+1) * 1/(1+INTEREST_RATE)

The policyholder will receive a payment as long as they survive starting m-years after the issue date.

Thank you for reading this blog post! I hope you found it helpful. Feel free to share your thoughts or ask any questions in the comments section below or on github. Your feedback is important to us. Happy reading!

Read also:

• Programming actuary
• Net premium reserves
• Model variables