Index page for Decrement

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.

Index

1. Introduction to Decrement Model
2. Types of Decrements
3. Select and Ultimate Table
4. Single Decrement Model
5. Double Decrement Model
6. Triple Decrement Model
7. Terminal and Non-terminal States
8. Generalized Decrement Model

Terminal and Non-Terminal States

So far the decrement models that we have seen contains only one non-terminal (or say, survival) state, that is, Inforce (NOP_IF). However, in reality, some insurance products, particularly health products, may have multi-survival states.

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 relationship can be better summarized in a transition matrix:

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}

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}

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:

https://drive.google.com/file/d/0B4OirwHLcmE1ZTQtWXktNlNLcFk/view?usp=sharing

Triple Decrement Model

Now we extend the model further to 3 decrements: Death, Lapse and Critical Illness.

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}

Using abbreviation:

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

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):

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:

https://drive.google.com/file/d/0B4OirwHLcmE1akE2TjFvSWMxTjQ/view?usp=sharing

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

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

Select and Ultimate Table

Insurance contracts have anti-selection risk, which means policyholders may hide important information behind their purchasing decision and take advantages from the insurance contracts. On the other hand, insurance contracts are also subjected to mortality improvement, for example, medical advancement makes people survived longer.

In light of these, a static morality curve is unlikely to make a fair prediction on reality. Hence actuaries have developed the select and ultimate table, in order to use different mortality rate after the first few years the insurance contracts have been inforced.

The graph above shows a typical select and ultimate mortality table. The SEL_1 column is used for the 1st policy year after the policy is issued, SEL_2 for 2nd policy year, and Ultimate for years thereafter.

For example, consider a contract entered at age 15, male, the death rates used during its 1st, 2nd, 3rd and 4th policy years will be respectively:

1st policy year (2017): 0.00031

2nd policy year (2018): 0.00059

3rd policy year (2019): 0.00074

4th policy year (2020): 0.00087

Another notable issue is that, when calculating the mortality rate, actuaries will used a multiplicative factor called "selection factor". It is a single percentage (eg: 80%) to be multiplied to the rate read-in from the mortality table. The reason for adding this multiplicative term is because, while the mortality table is collected via population census which represents the whole population, insurance contracts are more likely to be purchased by a specific group within the population (eg: middle class), hence using the population data directly is inappropriate and biased.

Insurance companies will perform experience analysis yearly (or more frequently) to evaluate its own actual death / lapse experiences against expected rate. After experience analysis, insurance companies will conclude an appropriate selection factor, then apply this to the mortality table.

For example, consider the same contract as above, a contract entered at age 15, male, and the selection factor applied is 60%.

1st policy year (2017): 0.00031 * 0.6 = 0.000183

2nd policy year (2018): 0.00059 * 0.6 = 0.000355

3rd policy year (2019): 0.00074 * 0.6 = 0.000522

4th policy year (2020): 0.00087 * 0.6 = 0.000564

A demonstration spreadsheet showing select & ultimate mortality table, lapse table and basic contract information can be downloaded here:

https://drive.google.com/file/d/0B4OirwHLcmE1amVEYTFqQThzLXc/view?usp=sharing

Types of Decrements

Typical life insurance policies include the following types of decrements:

1. Survival

2. Death

3. Lapse

4. Maturity

5. Critical Illness

Survival is also known as the number of policies inforce. It is the expected number of policies survived at the end of the measurement period (typically 1 year or 1 month).

Death is the number of policies that died within the measurement period. The death rate is often denoted by the symbol qx, which is the annual death rate of policy.

Most of countries will perform its population census regularly and create its own mortality table. For example, in the United States, the mortality table that the industry commonly adopted is called "CSO 2001". 

Lapse is the number of policies that quit the contract within the measurement period. Lapse shall include both surrenders (act voluntarily by the policyholder) and lapses (terminated by insurers due to various reason, for example missing premium), but in practice we called it "lapses" which include both situation. The lapse rate is often denoted by the symbol wx, which is the annual lapse rate of policy.

Lapse is very dependent on product features and customer segments, hence lapse rate often varies significantly by many dimensions such as policy year, product groups, channels, contract size, etc.

Maturity is the number of policies that survived until the end of the contract. For example, with an initial 10,000 policies inforce, 800 died and 5,200 lapsed, the number of maturity is then 10,000 - 800 - 5,200 = 4,000. Maturity is not measured by "maturity rate" and simply equal to the number of policies inforce at the end of the contract term.

Critical Illness is the number of policies that suffered from some predefined critical illness (eg: Heart Attack, Stroke, etc.) within the measurement period. While the policyholder may not die due to these diseases, the insurance contract will typically pay out a substantial benefit to the policyholder and terminates the contract afterwards. The critical illness rate is often denoted by the symbol ix, which is the annual morbidity rate of policy.

A demonstration spreadsheet showing mortality table, lapse table and basic contract information can be downloaded here:

https://drive.google.com/file/d/0B4OirwHLcmE1UkRaSWtLS2RvTHM/view?usp=sharing

There are many other possible decrements that can be added to decrement model, and tricks to play with if one wished to measure "non-terminated transition states". We will leave these advanced topic to the chapter on "Generalized Decrement Model".

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