Actuarial Science: Mortality Profits and DSAR

Have you ever wondered how the companies that provide life insurance cover to its’ policyholders protect their own lives?

For a company, in order to survive in the market, it must earn enough revenues to make considerable profits.

Do you know what is the major source of income for these insurance companies?

The premium paid by its policyholders/ customers. Apart from the premiums, it does invest a small portion of the premiums and makes income from the return on these investments. According to the Insurance Bureau of Canada, $0.55 of every dollar collected as a part of the premium goes to pay for the insurance claims.

Another $0.21 goes toward operating costs such as 24 hours’ customer service claims lines, the maintenance of records and making sure websites are up-to-date, while another $0.16 is paid to the government in the form of taxes. Insurance companies only earn $0.08 out of every $1 in profit and this profit margin was consistent over the last seven years, from 2007 to 2013.

Every year the insurance company collects premium from its’ customers which after deducting the aforesaid expenses are invested in the form of reserves which are then used to cover the assurance payouts upon death. For those policyholders who have survived their respective reserve amounts are then accumulated.

The business of these companies become so simple if everything worked in the above way. At times, these companies are burdened with a large number of claims than what they initially expected. In such a scenario, all the estimates of profit go in vain. The other times, it may happen that benefit that the company is liable to pay in the event of the death of its’ policyholders exceeds the amount that it holds in the form of reserves.

Yes, the above two situations can occur while running the business. Then what is the profit that the insurance company will make in such scenarios?

Let me introduce you the second situation in actuarial terms.

DSAR i.e. Death Strain at Risk. To put into words, DSAR in a policy year represents the excess sum that needs to be found at the end of the year to fund the death benefit, over the sum being held as a reserve. It represents the excess cost of the policy becoming a death claim.

DSAR = Sum Assured in the event of death – Reserves at the end of that year.

How does DSAR affect the profit?
For understanding this, we must know two important concepts.

  • Expected DSAR
  • Actual DSAR

Expected Death Strain at Risk is the expected amount of death strain that the life insurance company expects to pay in addition to the year-end reserve.

This is calculated as:

EDSAR= Death Strain at Risk * number of people alive at the start of the year * Probability of people dying during the year.

Actual Death Strain at Risk, on the other hand, is calculated in respect of the policies where the death has occurred during the year. It is the actual amount of benefit paid on death.

Formulating it: ADSAR= Death Strain at Risk * number of people actually died during the year.

Now, after understanding these two concepts we can easily calculate the mortality profit.

Mortality Profit during the year = EDSAR – ADSAR

These profits are then used to cover the costs or are reinvested in the form of reserves.

In case, ADSAR > EDSAR, then the insurance company incurs a mortality loss. Let me explain the concept better by taking an example.

Ques: A 20- year special endowment assurance policy is issued to a group of lives aged 45 exact. Each policy provides a sum assured of $10,000 payable at the end of the year of death or $20,000 payable if the life survives until the maturity date. Premiums on the policy are payable annually in advance for 15 years or until earlier death. You are given the following information:

Number of deaths during the 13th policy year – 4.

Number of policies in force at the end of the 13th policy year – 195.

Calculate the profit or loss arising from mortality in the 13th policy year.

Basis: Mortality AM92 Ultimate. Interest 4% per annum. Expenses none.

STEP 1: Write down all the given information.

STEP 2: Now we have to compute the premium that should be charged.

Since this is an endowment assurance policy, the formula to calculate premium is:

Premium paid at the start of each year = P * (a due)45: 15

Death benefit = 10,000 * (Term Assurance )45: 15

Survival Benefit = 20,000 * v20 * 20p45


Premium = Death Benefit in case of death (Term Assurance)


Survival Benefit at the end of the term (Pure Endowment)

The “..” after the letter “a” signifies that the premium is paid in due i.e. at the start of the year.

STEP 3: Calculating the reserves at the end of the 13th year.

The formula is:
Reserve at the end of 13th year = 10000 * (Term Assurance) 58:7 + 20000* v^7 * 7p58 – P (a due) 58:2

STEP 4: Calculating the DSAR and Mortality Profit. The formulas mentioned above in the article are used.

As can be seen, the DSAR is negative signifying that the company already has a sufficient amount of reserves to honor its commitments.

Also, if DSAR is negative and if more people die than expected then the company makes a profit. As from the above information, it can be seen that actual deaths were higher than expected deaths i.e.

4 (Actual Deaths) > 1.12435 (Expected Deaths)

Hence, the company made a profit. This is how this process works and help the insurance companies to find their mortality profits/ losses.

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