Notes on Life Insurance (Part Two)
Created on July 23, 2023
Written by Some author
Read time: 6 minutes
Summary: The text also mentions exercises based on actuarial mathematics for life contingent risks, which involve calculating EPVs and variance of benefits. Keywords: indicator variable, EPV Formulas, term insurance and whole life insurance, geometric payout.
More generic insurance payout:
We can introduce indicator variable to model if an event happens at a particular interval to translate the expectation in insurance. It also allows us to show clear interpretation of calculation.
The indicator variable takes the following form:
Then we can expressed an insurance product that pays an individual if the individual die at first years and pays if the individual die at the second years, then we can express the above as
whose EPV will be
We can also have the continuous form for indicator.
Suppose an insurance is payable instantly with amount. Then we will have
We can denote it as under actuarial notation, where the bar indicates continuous and the indicates increasing.
If the policy is termed, then we will have
Below are a couple of exercises based on Actuarial Mathematics for Life Contingent Risks. The problems are adapted from the examples presented on pages 128 to 131. However, the solutions provided here have been written by me.
. There's an insurance policy issued to an individual with age for a term of years. The policy provides a death benefit of , which will be paid at the end of the year of death if the individual dies between ages and , excluding the right endpoint, where .
(a) Calculate the Expected Present Value of the benefit using the first approach, which involves multiplying together the benefit amount, the discount factor, and the probability of payment, and then summing these values for each possible payment date.
(b) Calculate the formula for the variance of the present value of the benefit.
Note: In the provided context, refers to the age of the insured individual, and it is assumed to be a constant value throughout the calculations.
. the benefit amount here is and the discount factor is and the probability of the payment is so we will sum them up over the course of
.
In actuarial science, we can denote the above as .
If we send to infinity, then we will have
.
Let's express the payout as a random variable with curtate future lifetime random variable
Then we will have the following as our second moment:
So our variance will be
Consider a whole life insurance policy that provides an increasing death benefit, which is payable at the end of the quarter year of the insured's death. If the insured's age at the policy's inception is denoted by , the death benefit will be if the insured dies in the first year of the contract, in the second year, and so on. Now, let's derive an expression for the Expected Present Value (EPV) of the death benefit.
Here our possible pay day will be of the year or of the year or of the year of the end of the year. So our discount factor for these scenarios will be with the payout to be . The probability for each scenario will be
So we can express the as
Instead of having arithemetic payout, let's consider geometric payout.
We introduces an -year term insurance policy issued to an individual at age . The policy pays the death benefit at the end of the year in which the insured person passes away. The amount of the death benefit varies depending on the age at which death occurs. Specifically, if death occurs between ages and , the benefit is 1. If death occurs between ages and , the benefit is , and if death occurs between ages and , the benefit is , and so forth. Therefore, if death occurs between ages and , the death benefit is for . The objective is to derive a formula for the Expected Present Value (EPV) of this death benefit.
The benefit amount here is and the discount factor is and the probability of the payment is so we will sum them up over the course of
So we will have the following formula for our EPV:
So we will have the following:
and so we can express as .
We discusses an insurance policy issued to an individual at age , wherein the death benefit is determined as , where represents the time of death measured from the beginning of the policy. The policy ensures that the death benefit is paid out immediately upon the occurrence of death. The example seeks to derive the expressions for the Expected Present Value (EPV) of the death benefit for both an -year term insurance and a whole life insurance.
(a) Calculate the EPV of the death benefit for the -year term insurance.
(b) Calculate the EPV of the death benefit for the whole life insurance.
the benefit amount here is and the discount factor is and the probability of the payment is so we will sum them up over the course of
So we will have
And so our new and we also know that and so and so
we can always sending to infinity, then we will have