Commutation Functions Used in Insurance

Commutation functions are used in Insurance to determine the value of benefit of premiums paid at the happening of an event that is insured under the policy contract.

Here are some guidelines for the commutation functions that can be used by a life insurance company;

Please note that 'i' is the rate of interest used

Present Value of Survivors Benefit  at age x = 1/(1+i) * Number of survivors at age x

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Present Value of Benefit Estimated as Payable at age (x+1)={ 1/(1+i) * (Number of survivors at age x * probability of death of a life aged x in a year.}


Present Value of Benefits After Age x = Present Value of Survivors benefit at age x + Present Value of benefit at age (x+1) +Present Value of benefit at age (x+2) + Present Value of benefit at age (x+3) + and so on

Present Value of Benefits Estimated as  Payable After Age x = Present Value of Benefit estimated as payable at age (x+1) + Present Value of Benefit estimated as payable at age (x+2) + Present Value of Benefit estimated as payable at age (x+3) + and so on

From here we can infer that:

1) The present value of benefit of premium payable on death after age x =
Present Value of Benefits Estimated as  Payable After Age x / Present Value of Survivors Benefit  at age x

2) The present value of benefit of premium payable on death after any age x before  age (x+n) or on age (x+n) if alive ={ (Present Value of benefits estimated as payable after age x - Present Value of benefits estimated as payable at age (x + n)+ Present Value of Survivor benefit at age x + n )} / Present Value of Survivors Benefit  at age x

3) The present value of benefit of premium payable on death at any age after x but before (x +n) = Present Value of benefits estimated as payable after age x - Present Value of benefits estimated as payable at age (x+n)/Present Value of Survivors Benefit  at age x

4) The present value of benefit of premium payable on survival at age (x+n) = Present Value of Survivors Benefit  at age x+n / Present Value of Survivors Benefit  at age x

5) The present value of premium payable every year to a life aged x as long as the life is alive ( the first payment of premium is made at the commencement of the contract) =
Present Value of Benefits After Age x/ Present Value of Survivors Benefit  at age x

6) The present value of premium payable every year to a life aged x as long as the life is alive before age (x +n)  ( the first payment of premium is made at the commencement of the contract) = {Present Value of Benefits After Age x - Present Value of Benefits at age (x+n)}/ Present Value of Survivors Benefit  at age x

7) The present value of premium payable every year to a life aged x as long as the life is alive ( the first payment of premium is made at the end of the year of commencement of the contract) =  Present Value of Benefits after age x+1/Present Value of Survivors Benefit  at age x

8) The present value of premium payable every year to a life aged x as long as the life is alive before age (x+n) ( the first payment of premium is made at the end of the year of commencement of the contract)= { Present Value of Benefits after age x+1 - Present Value of benefits at age (x+n+1)}/Present Value of Survivors Benefit  at age x

Estimating Pure Premium

To estimate pure premium in insurance products the basic equation is as follows:

• Present Value of Premium = Present Value of Benefits + Present Value of Expenses which include profit to shareholders.

Here is an example:
XYZ insurance company wants to issue a one year policy of temporary assurance contracts to persons aged 55. The insurance cover for each contract is $ 500/-. He issues 4000 contracts. The mortality rate is 1 death for 1000 such lives.
For 4000 lives:
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The Present Value of Benefits = 4(claims) * 500 = $2000
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The Present Value of Premium should be equal to the Present Value of Benefits i.e $ 2000
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Therefore the premium charge should be $0.50 per policy ($0.50 * 4000 = $2000)
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In the above example, the methodology to calculate the pure rate of premium has been given. As this is an oversimplified example, no expenses and no investment income has been included in the model. The premium will increase or decrease once we incorporate the  expenses and investment income in the basic equation for calculating the premium.

First Year Expenses and Renewal Expenses

The expense equation for an insurance company consists of:
First Year Expense + Renewal Expenses

The First Year Expenses has the following constituents:

• Initial Costs: These normally include a Commission Payment and is usually related to Premium. Other categories include advertisement and miscellaneous marketing expenses.
• An x% of premium of the first year which is the premium related expenses like expenses involved in assessing the health and/or financial status of the policyholder
• A constant expense per policy of unit sum insured. These normally include stamp duty which is the cost of stamps to be affixed on the policy which is levied by the Government.
• A constant cost which is independent of premium and sum insured but can be calculated per policy. These include administrative expenses to handle all administrative costs for preparing the insurance contract document and rendering service to the policy holder to pay premium.

Renewal Expenses have the following constituents:

• A y% of premium receivable after the first year: These include commission payments if any, advertisement, reinsurance taxes and so on.
• A constant expense per policy of unit sum insured: These normally include expected litigation costs.
• A constant cost which is independent of premium and sum insured but can be calculated per policy.These normally include settlement of claims and expenses incurred for doubts posed by the policyholders,payment of salaries to the staff and employees of the insured.

To reiterate,

The value of expenses = First Year Expenses + Renewal Expenses

The value of expenses is discounted to the date of commencement of policy. It should be noted that expenses would not occur once the contract has ended. For this purpose, the probability that the expense would be incurred would be taken into account.

Value of expenses in mathematical terms is the discounted value calculated as follows:

(The amount of expenses/benefit * The probability that the expense/benefit is payable at that point of time)

Cash Flow Mehtod For Premium Calculations

Expected income and expected outgo of an insurer determines the premium using the cash flow method.

• The net cash flow is determined by determining the expected income and the expected outgo for each policy year. 
• These estimated are made based on assumptions that the premium rates are given at every age 'x', the life table is assumed so an insurer can calculate the expected premium amounts and the expected claim amounts.Further, the insurer assumes a predetermined set of commission rates, initial expenses, renewal expenses and also investment returns.
• The present value of the net cash flow is determined at a discount rate which correlates to the rate of returns expected by shareholders. The rate at which the present value of the net cash flow is made to zero determines the expected rate of return on the policy.
• Profit margins are determined using actuarial software.
• Such exercises are done in the competitive market to determine whether the premium charged would be in the best interest of the insurer and the insured.

Premium Rating Using Formula Method

Under the formula method actuarial formulas are used. The fundamental principle in the equation of value is that:
The present value of premium (Receipts) = Present Value of Benefits (Outgo) + Present Value of Expenses (Outgo).
Actuarial formula makes the use of commutation functions which take into account the probability of death and survival along with discount rates. Usually a life table is used  in which we have the age x, the number of survivors at age x and the number of deaths from exact age x to exact age x +1. 

Premium Rating for Whole Life Contracts:

Premium payable for a life at entry age x every year as long as he is alive = (The present value of benefit payable on the death of survivor at any age after x) / ( The present value of benefits payable every year to a life aged x as long as the life is alive)

Premium payable for a limited period of n years  =  (The present value of benefit payable on the death of survivor at any age after x) / ( The present value of benefit payable every year to a life aged x as long as the life is alive before age x+n )

Term Assurance Contracts:

Premium payable every year for n years = {(The present value of benefits payable on death at any age after x but before x+n) + ( The present value of benefits payable on survival at age x+n)}(The present value of benefits payable every year to a life aged x as long as the life is alive before age x+n)

A simple premium payable only once as a lump sum at the beginning of the policy =  (The present value of benefits payable on death at any age after x but before x+n) + ( The present value of benefits payable on survival at age x+n)

Pure Endowment Contracts:
Premiums payable every year for n years =(The present value of benefits payable on survival at age x+n)/ (Present value of benefits payable every year to a life aged x as long as the life is alive before  age x+n)

A single premium payable only once as a lump sum at the beginning of the policy = The present value of benefits payable on survival at age x+n

Endowment Assurance Contracts:
Premiums payable every year for the n years of the contract = (The present value of benefit payable on death after the commencement of the policy before  n years)/( The present value of benefit payable every year to the insured life as long as the life is alive for n years of the contract.)
A single premium payable = The present value of benefit payable on death after the commencement of the policy before  n years)

Simple and Compound Re visionary Bonus

Among the various methods of bonus payments are Simple and Compound Re visionary bonus.

Simple re visionary bonus:  Bonus is declared as x% of Sum Assured or face value of the insurance contract. Alternatively, bonus may be declared as x% of premium paid.
Example: SRB of 7% of SA of $ 1000/- would amount to $ 70/- of bonus

Compound re visionary bonus: Here bonus is declared as x% of bonus declared in the preceding policy year.
Example: Bonus declared this year is $70/-. The CRB is 10% of the bonus declared in the preceding year. Then the CRB is calculated as $77/- which is (70 * 1.10 = 77). $ 77/- will be the bonus declared in the succeeding year. In this manner, the CRB is calculated for each succeeding year.

There are some terms and conditions that must be in force for payment of bonus:

• There must be a 'with profits' policy
• A minimum number of years after the policy has been in force. 
• Vested bonus are payable along with the sum assured that are payable on death or maturity.
• There is no guarantee by the insurer that the future bonus rates will be the same as the current declared rates

Use and File System

According to the 'use and file' system, insurers can design a product and use it in the market first before filing the model policy with the regulators. The main features under this method are that:

• The insurer is given freedom with restrictions
• The main aim is to provide opportunity for the insurer to innovate in the product design
• No technical analysis is done by the regulators
• The regulator will ask the insurer to stop further sales if it is felt that the product is no longer in the interests of public.
• The regulator will take action as and when he receives complaints about the product from the public.
• The main risk is that there could be a possibility of insolvency due to the product which may need to be resolved.

On the other hand, the 'file and use' system requires an insurer to fill in an application form according to the instructions of the regulator. The main purpose is to ensure that the products manufactured by the insurers are for the interest of the public society and  that the product should be financially viable and does not create insolvency for the insurance company.

Under the 'free system' there is no intervention whatsoever from the regulators. This may also be referred to as 'freedom with responsibility'.  Under this system, the philosophy is that the market will take care of the price and hence the fittest only will survive.However, the insurer will be monitored by the regulator through tests like the 'solvency test' and 'asset-liability matching' test and so on.

In all the above options available, the customer is the main focus. The customer should have a choice of product. The products should be within reach of the customer.