The equation for triple decrement model:
\begin{equation}
\begin{split}
Number of Policies _{t+1} & = Number of Policies _t - Number of Deaths _t \\ & - Number of Lapse _t - Number of CI _t
\end{split}
\end{equation}
\begin{equation}
\begin{split}
Number of Policies _{t+1} & = Number of Policies _t - Number of Deaths _t \\ & - Number of Lapse _t - Number of CI _t
\end{split}
\end{equation}
Using abbreviation:
\[NOP\_IF _{t+1} = NOP\_IF _t - NO\_DEATHS _t - NO\_SURRS _t - NO\_CIs _t ... (1)\]
\[NOP\_IF _{t+1} = NOP\_IF _t - NO\_DEATHS _t - NO\_SURRS _t - NO\_CIs _t ... (1)\]
Similarly, under UDD assumption, NO_SURRS is given by:
\begin{equation}
\begin{split}
NO\_SURRS _t & = w _t × \int_{0}^{1} NOP\_IF_t × (1 - s × q _{x+t}) ( 1 - s × i _{x+t}) ds \\
& = w _t × NOP\_IF _t ×\big (s - \frac{s^2}{2} × (q _{x+t} + i_{x+t}) + \frac{s^3}{3} × (q _{x+t} × i_{x+t}) \big) _{0}^{1} \\
& = w _t × NOP\_IF _t × \big(1 - \frac{1}{2} × (q _{x+t} + i_{x+t}) + \frac{1}{3} × (q _{x+t} × i_{x+t}) \big) ... (2)
\end{split}
\end{equation}
NO_DEATHS is given by:
\begin{equation}
\begin{split}
NO\_DEATHS _t & = q _{x+t} × \int_{0}^{1} NOP\_IF_t × (1 - s × w _t) ( 1 - s × i _{x+t}) ds \\
& = q _{x+t} × NOP\_IF _t ×\big (s - \frac{s^2}{2} × (w _t + i_{x+t}) + \frac{s^3}{3} × (w _t × i_{x+t}) \big) _{0}^{1} \\
& = q _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + i_{x+t}) + \frac{1}{3} × (w _t × i_{x+t}) \big) ... (3)
\end{split}
\end{equation}
NO_CIs is given by:
\begin{equation}
\begin{split}
NO\_CIs _t & = i _{x+t} × \int_{0}^{1} NOP\_IF_t × (1 - s × w _t) ( 1 - s × q _{x+t}) ds \\
& = i _{x+t} × NOP\_IF _t ×\big (s - \frac{s^2}{2} × (w _t + q_{x+t}) + \frac{s^3}{3} × (w _t × q_{x+t}) \big) _{0}^{1} \\
& = i _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + q_{x+t}) + \frac{1}{3} × (w _t × q_{x+t}) \big) ... (4)
\end{split}
\end{equation}
By substituting (2), (3), (4) into (1), we have:
\begin{equation}
\begin{split}
NOP\_IF _{t+1} & = NOP\_IF _t - w _t × NOP\_IF _t × \big(1 - \frac{1}{2} × (q _{x+t} + i_{x+t}) + \frac{1}{3} × (q _{x+t} × i_{x+t}) \big) \\ & - q _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + i_{x+t}) + \frac{1}{3} × (w _t × i_{x+t}) \big) \\ & - i _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + q_{x+t}) + \frac{1}{3} × (w _t × q_{x+t}) \big) \\
& = NOP\_IF _t × \big(1 - (w _t + q _{x+t} + i _{x+t}) + \frac{w _t × q _{x+t} + q _{x+t} × i _{x+t} + i _{x+t} × w _t }{2} × 2 - \frac{w _t × q _{x+t} × i _{x+t}}{3} × 3 \big) \\
& = NOP\_IF _t × (1 - w _t) (1 - q _{x+t})(1 - i _{x+t})
\end{split}
\end{equation}
If we assume lapse to occur at the end of period, then we should adjust the formula for death and CI that:
\begin{equation}
\begin{split}
NO\_DEATHS _t = q_{x+t} × NOP\_IF _t × (1 - \frac{1}{2} × i _{x+t} ) \\
NO\_CIs _t = i _{x+t} × NOP\_IF _t × (1 - \frac{1}{2} × q _{x+t} ) \\
NO\_SURRS _t = NOP\_IF _t × (1 - q _{x+t}) ( 1 - i _{x+t}) × w _t
\end{split}
\end{equation}
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 and morbidity table is given below with 100% and 120% selection factors respectively. We would like to project the survival rate for 10 years under a death, lapse, CI triple decrement model. Death, lapse and CI are assumed to occur uniformly through out the year.
The mortality rate, lapse rate and morbidity rate from policy year 1 to 10 is given as below (after selection factor):
\begin{equation}
\begin{split}
NO\_SURRS _t & = w _t × \int_{0}^{1} NOP\_IF_t × (1 - s × q _{x+t}) ( 1 - s × i _{x+t}) ds \\
& = w _t × NOP\_IF _t ×\big (s - \frac{s^2}{2} × (q _{x+t} + i_{x+t}) + \frac{s^3}{3} × (q _{x+t} × i_{x+t}) \big) _{0}^{1} \\
& = w _t × NOP\_IF _t × \big(1 - \frac{1}{2} × (q _{x+t} + i_{x+t}) + \frac{1}{3} × (q _{x+t} × i_{x+t}) \big) ... (2)
\end{split}
\end{equation}
NO_DEATHS is given by:
\begin{equation}
\begin{split}
NO\_DEATHS _t & = q _{x+t} × \int_{0}^{1} NOP\_IF_t × (1 - s × w _t) ( 1 - s × i _{x+t}) ds \\
& = q _{x+t} × NOP\_IF _t ×\big (s - \frac{s^2}{2} × (w _t + i_{x+t}) + \frac{s^3}{3} × (w _t × i_{x+t}) \big) _{0}^{1} \\
& = q _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + i_{x+t}) + \frac{1}{3} × (w _t × i_{x+t}) \big) ... (3)
\end{split}
\end{equation}
NO_CIs is given by:
\begin{equation}
\begin{split}
NO\_CIs _t & = i _{x+t} × \int_{0}^{1} NOP\_IF_t × (1 - s × w _t) ( 1 - s × q _{x+t}) ds \\
& = i _{x+t} × NOP\_IF _t ×\big (s - \frac{s^2}{2} × (w _t + q_{x+t}) + \frac{s^3}{3} × (w _t × q_{x+t}) \big) _{0}^{1} \\
& = i _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + q_{x+t}) + \frac{1}{3} × (w _t × q_{x+t}) \big) ... (4)
\end{split}
\end{equation}
By substituting (2), (3), (4) into (1), we have:
\begin{equation}
\begin{split}
NOP\_IF _{t+1} & = NOP\_IF _t - w _t × NOP\_IF _t × \big(1 - \frac{1}{2} × (q _{x+t} + i_{x+t}) + \frac{1}{3} × (q _{x+t} × i_{x+t}) \big) \\ & - q _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + i_{x+t}) + \frac{1}{3} × (w _t × i_{x+t}) \big) \\ & - i _{x+t}× NOP\_IF _t × \big(1 - \frac{1}{2} × (w _t + q_{x+t}) + \frac{1}{3} × (w _t × q_{x+t}) \big) \\
& = NOP\_IF _t × \big(1 - (w _t + q _{x+t} + i _{x+t}) + \frac{w _t × q _{x+t} + q _{x+t} × i _{x+t} + i _{x+t} × w _t }{2} × 2 - \frac{w _t × q _{x+t} × i _{x+t}}{3} × 3 \big) \\
& = NOP\_IF _t × (1 - w _t) (1 - q _{x+t})(1 - i _{x+t})
\end{split}
\end{equation}
If we assume lapse to occur at the end of period, then we should adjust the formula for death and CI that:
\begin{equation}
\begin{split}
NO\_DEATHS _t = q_{x+t} × NOP\_IF _t × (1 - \frac{1}{2} × i _{x+t} ) \\
NO\_CIs _t = i _{x+t} × NOP\_IF _t × (1 - \frac{1}{2} × q _{x+t} ) \\
NO\_SURRS _t = NOP\_IF _t × (1 - q _{x+t}) ( 1 - i _{x+t}) × w _t
\end{split}
\end{equation}
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 and morbidity table is given below with 100% and 120% selection factors respectively. We would like to project the survival rate for 10 years under a death, lapse, CI triple decrement model. Death, lapse and CI are assumed to occur uniformly through out the year.
The mortality rate, lapse rate and morbidity 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 + ix)/2 + (wt * ix)/3) = 1 × 0.000183 × (1 - (10%+0)/2 + (10%*0)/3) = 0.000174
NO_SURRS = NOP_IFSM * wt * (1 - (qx + ix)/2 + (qx * ix)/3) = 1 × 10% × (1- (0.000183+0)/2 + (0.000183*0)/3) = 0.099991
NO_CIs = NOP_IFSM * ix * (1 - (wt+qx)/2 + (wt * qx)/3) = 1 × 0 × (1 - (10%+0.000183)/2 + (10% * 0.000183)/3) = 0
NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS - NO_CIs = 1 - 0.000174 - 0.099991 - 0 = 0.899835
Year 2
NOP_IFSM = NOP_IF (previous) = 0.899835
NO_DEATHS = NOP_IFSM * qx * (1 - (wt + ix)/2 + (wt * ix)/3) = 0.899835 × 0.000355 × (1 - (5%+0)/2 + (5%*0)/3) = 0.000312
NO_SURRS = NOP_IFSM * wt * (1 - (qx + ix)/2 + (qx * ix)/3) = 0.899835 × 5% × (1- (0.000355+0)/2 + (0.000355*0)/3) = 0.044984
NO_CIs = NOP_IFSM * ix * (1 - (wt+qx)/2 + (wt * qx)/3) = 0.899835 × 0 × (1 - (5%+0.000355)/2 + (5% * 0.000355)/3) = 0
NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS - NO_CIs = 0.899835 - 0.000312 - 0.044984 - 0 = 0.854540
...
Year 10
NOP_IFSM = NOP_IF (previous) = 0.781449
NO_DEATHS = NOP_IFSM * qx * (1 - (wt + ix)/2 + (wt * ix)/3) = 0.781449 × 0.000630 × (1 - (1%+0.00120)/2 + (1%*0.00120)/3) = 0.000490
NO_SURRS = NOP_IFSM * wt * (1 - (qx + ix)/2 + (qx * ix)/3) = 0.781449 × 1% × (1- (0.000630+0.00120)/2 + (0.000630*0.00120)/3) = 0.007807
NO_CIs = NOP_IFSM * ix * (1 - (wt+qx)/2 + (wt * qx)/3) = 0.781449 × 0.00120 × (1 - (1%+0.000630)/2 + (5% * 0.000630)/3) = 0.000933
NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS - NO_CIs = 0.781449 - 0.000490 - 0.007807 - 0.000933 = 0.772220
A demonstration spreadsheet showing the calculation above can be downloaded here:
Your tutorial is very useful, I learn so much! Thanks! Do you have plan to continue the tutorial?
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