Double Decrement Model

Single decrement model is unrealistic as other terminal states, like lapses and critical illness may exist. Now, let's assume lapse is coming into play.

The equation for double decrement model (assuming the decrements are death and lapse) is:

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

Using abbreviation:

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

While one may think that, the number of deaths and number of surrender can be simply calculated by start of period multiplied by death rate / lapse rate, this is not that straight forward due to the interaction between decrements. For example, assuming uniform distribution of deaths (UDD), may be after 0.5 months, 0.001 policies died, so the NOP_IF at that time is 0.999 instead of 1. Using 1 as the base to multiply the lapse rate will overestimate the NO_SURRS since you have ignored the effect of NOP reduction caused by deaths.

Under UDD assumption, the NO_SURRS, can be calculated by:

\begin{equation}
\begin{split}
NO\_SURRS _t & = w _t × \int_{0}^{1} NOP\_IF_t × (1 - s × q _{x+t}) ds \\
& = w _t × NOP\_IF _t × (1 - \frac{1}{2} × q _{x+t} ) ... (2)
\end{split}
\end{equation}

Similarly, the NO_DEATHS can be calculated by:

\begin{equation}
\begin{split}
NO\_DEATHS _t & = q _{x+t} × \int_{0}^{1} NOP\_IF_t × (1 - s × w _t) ds \\
& = q_{x+t} × NOP\_IF _t × (1 - \frac{1}{2} × w_t ) ... (3)
\end{split}
\end{equation}

By substituting (2) and (3) into (1), we have:

\begin{equation}
\begin{split}
NOP\_IF _{t+1} & = NOP\_IF _t - w _t × NOP\_IF _t × (1 - \frac{1}{2} × q _{x+t} )  - q _{x+t} × NOP\_IF _t × (1 - \frac{1}{2} × w_t ) \\
& = NOP\_IF _t × \big(1 - w _t + \frac{w _t × q _{x+t}}{2} - q _{x+t} + \frac{w _t × q _{x+t}}{2} \big) \\
& = NOP\_IF _t × \big(1 - (w _t + q _{x+t}) + w _t × q _{x+t}\big) \\
& = NOP\_IF _t × (1 - w _t) (1 - q _{x+t})
\end{split}
\end{equation}

which is expected.

Sometimes, we may assume some decrement occurs after other decrements. For example, lapses are often assumed to occur at the end of period, after deaths have occurred. The reason is that, lapses typically happen only when premiums are due, but will not occurs at the middle of the period, hence it should have no interaction with deaths.

In this case, the above formula have to be adjusted:

\begin{equation}
\begin{split}
NO\_DEATHS _t = NOP\_IF_t × q _{x+t} \\
NO\_SURRS _t = NOP\_IF _t × (1 - q _{x+t}) × w _t
\end{split}
\end{equation}

The final equation also holds, since:

\begin{equation}
\begin{split}
NOP\_IF _{t+1} & = NOP\_IF _t - NOP\_IF_t × q _{x+t} - NOP\_IF _t × (1 - q _{x+t}) × w _t \\
& = NOP\_IF _t × \big(1 - q _{x+t} - (1 - q _{x+t}) × w _t\big) \\
& = NOP\_IF _t × (1 - w _t) (1 - q _{x+t})
\end{split}
\end{equation}

This lapse timing issue is typically controlled by a switch called "lapse timing" in most actuarial software, for example, Prophet.

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. Lapse table is given below with 100% selection factor. We would like to project the survival rate for 10 years under a death and lapse only double decrement model. Deaths and lapses are assumed to occur uniformly through out the year.

The mortality rate and lapse rate from policy year 1 to 10 is given as below (after selection factor):

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 - wt/2) = 1 × 0.000183 × (1 - 10%/2) = 0.000174

NO_SURRS = NOP_IFSM * wt * (1 - qx/2) = 1 × 10% × (1- 0.000183/2) = 0.099991

NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS = 1 - 0.000174 - 0.099991 = 0.899835

Year 2

NOP_IFSM = NOP_IF (previous) = 0.899835

NO_DEATHS = NOP_IFSM * qx * (1 - wt/2) = 0.899835 × 0.000355 × (1 - 5%/2) = 0.000312

NO_SURRS = NOP_IFSM * wt * (1 - qx/2) = 0.899835 × 5% × (1- 0.000355/2) = 0.044984

NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS = 0.899835 - 0.000312 - 0.044984 = 0.854540

...

Year 10

NOP_IFSM = NOP_IF (previous) = 0.785212

NO_DEATHS = NOP_IFSM * qx * (1 - wt/2) = 0.785212 × 0.000630 × (1 - 1%/2) = 0.000492

NO_SURRS = NOP_IFSM * wt * (1 - qx/2) = 0.785212 × 1% × (1- 0.000630/2) = 0.007850

NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS = 0.785212 - 0.000492 - 0.007850 = 0.776870

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

https://drive.google.com/file/d/0B4OirwHLcmE1MGwta2ZVY1hBd3c/view?usp=sharing