For example, an insurance company has developed a health insurance product that will pay the policyholder a lump-sum of $10,000 when he is diagnosed early stage cancer. The contract will not terminate after the diagnosis, but remain inforce until he has died, or suffer from critical illness (including late stage of the same cancer).
In this case, there are two survival states, or two non-terminal states, namely: Inforce and Early-Stage CI. In contrast, the terminal states are death, surrender and major CI.
The matrix should be familiar by those who have studied Markov Chain.
The terminal states, namely, death, surrender and CI, are absorbing states. All policies at the end, will fall into these states. Therefore, what we will observe overtime is, policies remaining in Inforce states will gradually decrease, and going into other states. The number of policies in early stage CI state will initially increase since there are lots of policies coming in from inforce state, but later on vanish as policies going to terminal states exceed those coming in.
With non-terminal states, the decrement model will become more complex,
\begin{equation}
\begin{split}
Number\_of\_Policies\_Inforce _{t+1} & = Number\_of\_Policies\_Inforce _t - Inforce\_trans\_to\_Deaths _t \\ & - Inforce\_trans\_to\_Lapse _t - Inforce\_trans\_to\_CI _t \\ & - Inforce\_trans\_to\_Early\_Stage\_CI _t \\
Number\_of Early\_Stage\_CI _{t+1} & = Number\_of\_Early\_Stage\_CI _t + Inforce\_trans\_to\_Early\_Stage\_CI _t \\ & - Early\_Stage\_CI\_trans\_to\_Deaths _t - Early\_Stage\_CI\_trans\_to\_Lapse _t \\ & - Early\_Stage\_CI\_trans\_to\_CI _t
\end{split}
\end{equation}
Using abbreviation:
\begin{equation}
\begin{split}
NOP\_IF _{t+1} & = NOP\_IF _t - IF\_DEATHS _t - IF\_SURRS _t - IF\_CIs _t - IF\_MinCIs _t \\
NOP\_MinCIs _{t+1} & = NOP\_MinCIs _t + IF\_MinCIs _t - MinCI\_DEATHS _t - MinCI\_SURRS _t - MinCI\_CIs _t
\end{split}
\end{equation}
An important concept here is, reader should be careful on the distinction between total number of deaths and transition from some states to deaths. In previous decrement models, there is only one survival states, so the total number of deaths is always equal to those transit from inforce state to death state. However, now we have two survival states, and the total number of deaths is equal to those transits from inforce to death, as well as from early stage CI to death.
\[NO\_DEATHS _t = IF\_DEATHS _t + MinCI\_DEATHS _t\]
In the actual calculation, we also have to split the calculation for inforce state and early stage CI state. Under UDD assumption, assuming lapse occurs at the end of period:
For Inforce state
\begin{equation}
\begin{split}
IF\_DEATHS _t & = q _{x+t} × NOP\_IF _t × (1 - \frac {i _{x+t} + j _{x+t}}{2} + \frac {i _{x+t} × j _{x+t}}{3}) \\
IF\_CIs _t & = i _{x+t} × NOP\_IF _t × (1 - \frac {q _{x+t} + j _{x+t}}{2} + \frac {q _{x+t} × j _{x+t}}{3}) \\
IF\_MinCIs _t & = j _{x+t} × NOP\_IF _t × (1 - \frac {q _{x+t} + i _{x+t}}{2} + \frac {q _{x+t} × i _{x+t}}{3}) \\
IF\_SURRS & = w _t × NOP\_IF _t × (1 - q _{x+t}) (1 - i _{x+t}) (1 - j _{x+t})
\end{split}
\end{equation}
For Early Stage CI state
\begin{equation}
\begin{split}
MinCI\_DEATHS _t & = q _{x+t} × NO\_MinCIs _t × (1 - \frac{i _{x+t}}{2}) \\
MinCI\_CIs _t & = i _{x+t} × NO\_MinCIs _t × (1 - \frac{q _{x+t}}{2}) \\
MinCI\_SURRS & = w _t × NO\_MinCIs _t × (1 - q _{x+t}) (1 - i _{x+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, morbidity and early stage CI table is given below with 100%, 120% and 100% selection factors respectively. We would like to project the survival rate for 10 years under a death, lapse, CI multi-states decrement model. Death, CI and early stage CI are assumed to occur uniformly through out the year, and lapse occurs at the end of year.
The mortality rate, lapse rate, morbidity rate and early stage CI rate from policy year 1 to 10 is given as below (after selection factor):
\begin{split}
NOP\_IF _{t+1} & = NOP\_IF _t - IF\_DEATHS _t - IF\_SURRS _t - IF\_CIs _t - IF\_MinCIs _t \\
NOP\_MinCIs _{t+1} & = NOP\_MinCIs _t + IF\_MinCIs _t - MinCI\_DEATHS _t - MinCI\_SURRS _t - MinCI\_CIs _t
\end{split}
\end{equation}
An important concept here is, reader should be careful on the distinction between total number of deaths and transition from some states to deaths. In previous decrement models, there is only one survival states, so the total number of deaths is always equal to those transit from inforce state to death state. However, now we have two survival states, and the total number of deaths is equal to those transits from inforce to death, as well as from early stage CI to death.
\[NO\_DEATHS _t = IF\_DEATHS _t + MinCI\_DEATHS _t\]
In the actual calculation, we also have to split the calculation for inforce state and early stage CI state. Under UDD assumption, assuming lapse occurs at the end of period:
For Inforce state
\begin{equation}
\begin{split}
IF\_DEATHS _t & = q _{x+t} × NOP\_IF _t × (1 - \frac {i _{x+t} + j _{x+t}}{2} + \frac {i _{x+t} × j _{x+t}}{3}) \\
IF\_CIs _t & = i _{x+t} × NOP\_IF _t × (1 - \frac {q _{x+t} + j _{x+t}}{2} + \frac {q _{x+t} × j _{x+t}}{3}) \\
IF\_MinCIs _t & = j _{x+t} × NOP\_IF _t × (1 - \frac {q _{x+t} + i _{x+t}}{2} + \frac {q _{x+t} × i _{x+t}}{3}) \\
IF\_SURRS & = w _t × NOP\_IF _t × (1 - q _{x+t}) (1 - i _{x+t}) (1 - j _{x+t})
\end{split}
\end{equation}
For Early Stage CI state
\begin{equation}
\begin{split}
MinCI\_DEATHS _t & = q _{x+t} × NO\_MinCIs _t × (1 - \frac{i _{x+t}}{2}) \\
MinCI\_CIs _t & = i _{x+t} × NO\_MinCIs _t × (1 - \frac{q _{x+t}}{2}) \\
MinCI\_SURRS & = w _t × NO\_MinCIs _t × (1 - q _{x+t}) (1 - i _{x+t})
\end{split}
\end{equation}
And for other terminal states
\begin{equation}
\begin{split}
NO\_DEATHS _t & = IF\_DEATHS _t + MinCI\_DEATHS _t \\
NO\_CIs _t & = IF\_CIs _t + MinCI\_CIs _t \\
NO\_SURRS _t & = IF\_SURRS _t + MinCI\_SURRS _t \\
\end{split}
\end{equation}
\begin{split}
NO\_DEATHS _t & = IF\_DEATHS _t + MinCI\_DEATHS _t \\
NO\_CIs _t & = IF\_CIs _t + MinCI\_CIs _t \\
NO\_SURRS _t & = IF\_SURRS _t + MinCI\_SURRS _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, morbidity and early stage CI table is given below with 100%, 120% and 100% selection factors respectively. We would like to project the survival rate for 10 years under a death, lapse, CI multi-states decrement model. Death, CI and early stage CI are assumed to occur uniformly through out the year, and lapse occurs at the end of year.
The mortality rate, lapse rate, morbidity rate and early stage CI 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
IF_DEATHS = NOP_IFSM * qx * (1 - (jx + ix)/2 + (jx * ix)/3) = 1 × 0.000183 × (1 - (0+0)/2 + (0*0)/3) = 0.000183
IF_SURRS = NOP_IFSM * wt * (1 - qx) (1 - ix)(1 - jx) = 1 × 10% × (1 - 0.000183)(1 - 0)(1 - 0) = 0.099982
IF_CIs = NOP_IFSM * ix * (1 - (jx+qx)/2 + (jx * qx)/3) = 1 × 0 × (1 - (0+0.000183)/2 + (0 * 0.000183)/3) = 0
IF_MinCIs = NOP_IFSM * jx * (1 - (ix+qx)/2 + (ix * qx)/3) = 1 × 0 × (1 - (0+0.000183)/2 + (0 * 0.000183)/3) = 0
NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS - NO_CIs - NO_MinCIs = 1 - 0.000183 - 0.099982 - 0 - 0 = 0.899835
NOP_MinCIsSM = NOP_MinCIs (previous) = 0
MinCI_DEATHS = NOP_MinCIs * qx * (1 - ix/2) = 0 * 0.000183 * (1 - 0/2) = 0
MinCI_SURRS = NOP_MinCIs * wt * (1 - qx)(1 - ix) = 0 * 10% * (1 - 0.000183) * (1 - 0) = 0
MinCI_CIs = NOP_MinCIs * ix * (1 - qx/2) = 0 * 0 * (1 - 0.000183/2) = 0
NOP_MinCIs = NOP_MinCIsSM + IF_MinCIs - MinCI_DEATHS - MinCI_SURRS - MinCI_CIs = 0 + 0 - 0 - 0 - 0 = 0
Year 2
NOP_IFSM = NOP_IF (previous) = 0.899835
IF_DEATHS = NOP_IFSM * qx * (1 - (jx + ix)/2 + (jx * ix)/3) = 0.899835 × 0.000355 × (1 - (0+0)/2 + (0*0)/3) = 0.000320
IF_SURRS = NOP_IFSM * wt * (1 - qx) (1 - ix)(1 - jx) = 0.899835 × 5% × (1 - 0.000355)(1 - 0)(1 - 0) = 0.044976
IF_CIs = NOP_IFSM * ix * (1 - (jx+qx)/2 + (jx * qx)/3) = 0.899835 × 0 × (1 - (0+0.000355)/2 + (0 * 0.000355)/3) = 0
IF_MinCIs = NOP_IFSM * jx * (1 - (ix+qx)/2 + (ix * qx)/3) = 0.899835 × 0 × (1 - (0+0.000355)/2 + (0 * 0.000355)/3) = 0
NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS - NO_CIs - NO_MinCIs = 0.899835 - 0.000320 - 0.044976 - 0 - 0 = 0.854540
NOP_MinCIsSM = NOP_MinCIs (previous) = 0
MinCI_DEATHS = NOP_MinCIs * qx * (1 - ix/2) = 0 * 0.000355 * (1 - 0/2) = 0
MinCI_SURRS = NOP_MinCIs * wt * (1 - qx)(1 - ix) = 0 * 5% * (1 - 0.000355) * (1 - 0) = 0
MinCI_CIs = NOP_MinCIs * ix * (1 - qx/2) = 0 * 0 * (1 - 0.000355/2) = 0
NOP_MinCIs = NOP_MinCIsSM + IF_MinCIs - MinCI_DEATHS - MinCI_SURRS - MinCI_CIs = 0 + 0 - 0 - 0 - 0 = 0
...
Year 10
NOP_IFSM = NOP_IF (previous) = 0.765937
IF_DEATHS = NOP_IFSM * qx * (1 - (jx + ix)/2 + (jx * ix)/3) = 0.765937 × 0.000630 × (1 - (0.00120+0.00500)/2 + (0.00120*0.00500)/3) = 0.000481
IF_SURRS = NOP_IFSM * wt * (1 - qx) (1 - ix)(1 - jx) = 0.765937 × 1% × (1 - 0.000630)(1 - 0.00120)(1 - 0.00500) = 0.007607
IF_CIs = NOP_IFSM * ix * (1 - (jx+qx)/2 + (jx * qx)/3) = 0.765937 × 0.00120 × (1 - (0.00500+0.000630)/2 + (0.00500 * 0.000630)/3) = 0.000917
IF_MinCIs = NOP_IFSM * jx * (1 - (ix+qx)/2 + (ix * qx)/3) = 0.765937 × 0.00500 × (1 - (0.00120+0.000630)/2 + (0.00120 * 0.000630)/3) = 0.003826
NOP_IF = 1 - NOP_IFSM - NO_DEATHS - NO_SURRS - NO_CIs - NO_MinCIs = 0.765937 - 0.000481 - 0.007607 - 0.000917 - 0.003826 = 0.753106
NOP_MinCIsSM = NOP_MinCIs (previous) = 0.015683
MinCI_DEATHS = NOP_MinCIs * qx * (1 - ix/2) = 0.015683* 0.000630 * (1 - 0.00120/2) = 0.000010
MinCI_SURRS = NOP_MinCIs * wt * (1 - qx)(1 - ix) = 0.015683 * 1% * (1 - 0.000630) * (1 - 0.00120) = 0.000157
MinCI_CIs = NOP_MinCIs * ix * (1 - qx/2) = 0.015683 * 0.00120 * (1 - 0.000630/2) = 0.000019
NOP_MinCIs = NOP_MinCIsSM + IF_MinCIs - MinCI_DEATHS - MinCI_SURRS - MinCI_CIs = 0.015683 + 0.003826 - 0.000010 - 0.000157 - 0.000019 = 0.019324
A demonstration spreadsheet showing the calculation above can be downloaded here:
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