# The BELL Shaped Reserve Graph of Term Insurance Product

Introduction

A term Insurance product is a very thinly priced protection product. Under this product, the nominee of the insured gets a lump-sum amount if the insured dies during the term of the cover. In return, the policyholder keeps on paying premium for covering the risk of his/her death to the Insurance company. The Premium Paying Term can be Limited Pay, Regular Pay or Single Pay.

Objective

The credential of the policy holder whose example we have taken for the explanation is given below (amounts in INR) :-

Assumptions used

Mortality Assumption Used – AM92

Discount Rate – 10%

Modelling method

The excel file with the Term model is attached with the article.

In the tab “Inputs”, in column B & C we have the age wise mortality rates which we have used in our projections and calculations.

In column G & H, from cell G3 to H8 we have input the credentials of the policyholder which includes the Sum Assured, Annual Premium, Age, Discount Rate, Policy Term & Premium Paying Term.

In cells G9 to H10, we have Total Expected Present Value of Annual Premium & Total Expected Present Value of Benefits. These figures are calculated in tab “Calculations” and are linked from there.

We have calculated the Annual Premium in cell H8 in such a way that the Expected Present Value of Premiums is equal to the Expected Present Value of Benefits and there is no profit or loss with the help of Goal Seek.

Tab Calculations – In this tab we have done certain calculations which are very easy to follow for anyone who has read CT 5/ CM 1(Actuarial Science).

Explanation

The Column H & I of the calculations tab, we have calculated the Year Wise Expected Present Value of Annual Premium & Year Wise Expected Present Value of Benefits. Understanding these two columns is very important to understand the shape of the Reserves graph.

The Year Wise Expected Present Value of Annual Premium will always be a decreasing function of time. It is because it is dependent upon the cumulative probability of survival till the premium paying time and the discount rate. Both these factors will only decrease the Year on Year EPV of the annual premium. It is because the probability of survival decreases with age (above age 25).

The Year Wise Expected Present Value of Death Benefit will depend on the discount rate and mortality rate. If we remove the time value of money, or say, keep discount rate as 0%, then this will obviously be an increasing function of time as mortality keeps on increasing with age (above age 25). However, in real world we cannot ignore the time value of money and the discount rate will pull down this value. So the behavior of the graph of Year Wise Expected Present Value of Death Benefit will depend on the selection of the mortality rate and the discount rate. Suppose, the discount rate chosen is too high (as in our example), then this graph will be a decreasing function of time.

However, the rate of decrease in year wise EPV of annual premium will always be higher than the rate of decrease of in EPV of Benefits as the mortality factor keeps increasing the EPV of Benefits and eventually at one point (time period), the year wise EPV of Benefits will surpass the EPV of annual premiums. We have highlighted in yellow this time period in our model in “calculations” tab.

In column K & L, we have calculated the Total EPV of Benefits and Total EPV of Premiums at the start of each year for all years respectively. These are just the sum of Year Wise Expected Present Value of Annual Premium & Death Benefits from that year onwards till the end of the term of the product.

At Column J, we have calculated the Reserves at the start of each year. This is just the difference between Column K & L.

So at year 1 start, we have reserve 0 as we have set the premium in such a way that the EPV of benefits is equal to the EPV of Premiums and there is no profit or loss. The Reserves will keep on increasing each prospective year as the mortality and hence the expected death benefit cost increases with age but the premiums are level. The year wise EPV of Benefits is lower than the year wise EPV of premium in the initial years (till year 12 in our model). So on calculating the Total EPV of benefits and Total EPV of premiums (column K & L), as we are moving ahead year on year, we are deducting less values from the total EPV of Benefits then we are deducting from the total EPV of premiums in the initial years. Hence, the reserves are increasing.

However, on later years (from year 13 onwards in our model), the year wise EPV of Benefits surpasses the year wise EPV of Premiums. Hence, the reserve starts decreasing from year 13 onwards as now we will be deducting comparatively higher amount from the total EPV of benefits then we will be deducting from the total EPV of premium.

This is the reason why reserve graph of a Term Product is Bell Shaped. It Increases at a decreasing rate during the initial years as Year Wise EPV of Benefits and Year Wise EPV of Premiums are tending towards each other (Reason explained above). However, as the Year Wise EPV of Benefits surpasses the Year Wise EPV of Premiums, it will start decreasing again and it decreases at an increases rate (Reason explained above).

We hope this article was interesting and meaningful to all its readers and everyone was able to understand why the term product reserve graph is bell shaped.

Meanwhile, do let us know in the comment section what do you think how will the reserve graph differ for a policy holder age 18 for a Policy Term & Premium term of 8 years. Try to think it out without looking at its graph.

Hint: Look at the mortality pattern.

Feel free to reach out to us in case of any query.

### Mathematica-City

Mathematica-city is an online Education forum for Science students run by Kounteyo, Shreyansh and Souvik. We aim to provide articles related to Actuarial Science, Data Science, Statistics, Mathematics and their applications using different Statistical Software. We also provide freelancing services on the aforementioned topics. Feel free to reach out to us for any kind of discussion on any of the related topics.