Introduction to Decrement Model

Life contingency is an unique subject in the actuarial profession. The core component of it, the life decrement model, is well covered in some textbook such as "Actuarial Mathematics" or exam such as MLC.  However, the textbooks and the exams are sometimes too theoretical, and lack practical details for junior actuaries to apply them in their daily work.

Decrement model is used to measured the expected number of policies remaining inforce, as well as leaving due to various reasons (deaths, surrenders, etc.) as time evolves.

Assuming the simplest case, there is 10,000 insurance contract signed at 1st January 2017. The company expect the death rate for each policy is identical and equal 1%. At the end of the year (31st December 2017), the policy statistics is expected as:

1st January 2017

Number of Policies Inforce: 10,000
Number of Deaths: 0

31st December 2017

Number of Policies Inforce: 9,900
Number of Deaths: 100 ( = 10,000 * 1%)

As time evolves, the cumulative amount of deaths increase and the remaining inforce policies decrease.

Life insurance contracts are typically measured individually, instead of group based. Hence, modelling actuaries will calculate each policy (what we called a model point) independently, instead of grouping them to 10,000 policies together. Therefore, what you will see in practice is, for a single model point, the initial number of policy is 1, and as time evolves, there will be fraction number of deaths and survival policies. Although the fraction doesn't make sense, and in reality the number of deaths can only be 0 or 1, but this fraction number is the expected value and when adding all contracts together, it is approximately held due to law of large number.

Model Point #1 (qx = 1%)

1st January 2017

Number of Policies Inforce: 1
Number of Deaths: 0

31st December 2017

Number of Policies Inforce: 9.99
Number of Deaths: 0.01 ( = 1 * 1%)

Model Point #2 (qx = 2%)

1st January 2017

Number of Policies Inforce: 1
Number of Deaths: 0

31st December 2017

Number of Policies Inforce: 9.98
Number of Deaths: 0.02 ( = 1 * 2%)

...

Model Point #10,000 (qx = 3%)

1st January 2017

Number of Policies Inforce: 1
Number of Deaths: 0

31st December 2017

Number of Policies Inforce: 9.97
Number of Deaths: 0.03 ( = 1 * 3%)

Portfolio Level (Summing up 10,000 individual policies)

1st January 2017

Number of Policies Inforce: 1 + 1 + .... + 1 = 10,000
Number of Deaths: 0

31st December 2017

Number of Policies Inforce: 10,000 - 178 = 9,822
Number of Deaths: 0.01 +  0.02 + ... + 0.03  = 178