Single Decrement Model

The most basic decrement model is single decrement model, by its definition, it only contains a single decrement, for example, deaths, and all remaining policies survived.

The equation for single decrement model (assuming the decrement is death) is:

\[Number of Policies _{t+1} = Number of Policies _t - Number of Deaths _t\]

We will normally use abbreviation in the industry. Denote NOP_IF by number of policies survived, NO_DEATHS by number of deaths, then the equation becomes:

\[NOP\_IF _{t+1} = NOP\_IF _t - NO\_DEATHS _t ... (1)\]

The key difficulty for the above equation is how to calculate NO_DEATHS. The abbreviation of death rate is qx. The equation is given by:

\[NO\_DEATHS _t = NOP\_IF _t × q _{x+t} ... (2)\]

Hence, the equation in (1) can be transformed into the below:

\[NOP\_IF _{t+1} = NOP\_IF _t × (1 - q _{x+t}) ... (3)\]

Equation (3) is equivalent to the commonly seen equation you have learnt in exam MLC:

\[l _{x+t+1} = l _{x+t} × (1 - q _{x+t})\]

Let's go through a practical example below:

Assuming an age 15, male policy, follows the select & ultimate CSO 2001 mortality table with 60% selection factor. We would like to project the survival rate for 10 years under a death only single decrement model.

The mortality rate from policy year 1 to 10 is given as below:

The calculation for NOP_IF through year 10 is as follow:

Year 1

NOP_IFSM = NOP_IF (previous) = 1

NO_DEATHS = NOP_IFSM * qx = 1 × 0.000183 = 0.000183

NOP_IF = 1 - NOP_IFSM - NO_DEATHS = 1 - 0.000183 = 0.999817

Year 2

NOP_IFSM = NOP_IF (previous) = 0.999817

NO_DEATHS = NOP_IFSM * qx = 0.999817 × 0.000355 = 0.000355

NOP_IF = 1 - NOP_IFSM - NO_DEATHS = 0.999817 - 0.000355 = 0.999462

...

Year 10

NOP_IFSM = NOP_IF (previous) = 0.995367

NO_DEATHS = NOP_IFSM * qx = 0.995367 × 0.000630 = 0.000627

NOP_IF = 1 - NOP_IFSM - NO_DEATHS = 0.995367 - 0.000627 = 0.994740

A demonstration spreadsheet showing the calculation above can be downloaded here:

https://drive.google.com/file/d/0B4OirwHLcmE1NUpWR2QxbFlOODA/view?usp=sharing