Dividend on Deposit (DoD)

When policyholder received coupon benefit or cash dividend payment, they can either cash out or retain them in the insurance company. If they choose to retain them in the insurance company, this amount will become a balance, called "Dividend on Deposit (DoD)".

DoD will accumulate with interest, normally a crediting rate determined by the insurance company. Future coupon and dividend will increase the DoD balance as well. Some company will model it more accurately with DoD partial surrender, meaning that policyholders may at some time point determine to cash out the DoD in his / her account.

Hence, the formula for DoD is:

\begin{equation}
\begin{split}
DoD\_PP _t = DoD\_PP _{t-1} × (1 + DoD\_Cred\_PC _t) + Coupon\_PP _t + Dividend\_PP _t - DoD\_PartSurr\_PP _t
\end{split}
\end{equation}

In deed, a more organized formula can be rewritten as:

\begin{equation}
\begin{split}
DoD\_PP _t & = DoD\_PP _{t-1} + DoD\_Inflow\_PP _t - DoD\_Outflow\_PP _t \\
DoD\_Inflow\_PP _t & = DoD\_Cred\_Int _t + DoD\_Prem_PP _t \\
DoD\_Prem\_PP _t & = Coupon\_PP _t + Dividend\_PP _t \\
DoD\_Outflow\_PP _t & = DoD\_PartSurr\_PP _t \\
\\
DoD\_Cred\_Int _t & = DoD\_PP _{t-1} × DoD\_Cred\_PC _t \\
DoD\_PartSurr\_PP _t & = (DoD\_PP _{t-1} + DoD\_Cred\_Int _t) × DoD\_PartSurr\_PC _t
\end{split}
\end{equation}

Although the formula becomes longer, it makes clear that DoD is just composed of some cash inflow and cash outflow, where the inflow are coming from interest, coupon and dividend, and outflow are coming from partial surrender.

Some company may model it even in more detail. They have approximated the percentage of policyholder decided to leave the coupon / dividend in the DoD account (and other will cash out). We called this the "DoD_Opt_PC", the the DoD premium formula may become:

\begin{equation}
\begin{split}
DoD\_Prem\_PP _t & =  (Coupon\_PP _t + Dividend\_PP _t) × DoD\_Opt\_PC _t
\end{split}
\end{equation}

And DoD balance is simply a balance. When it will be distributed out? Upon the following conditions:

1. Death: As DoD Death Outgo

2. Surrender: As DoD Surrender Outgo

3. Maturity: As DoD Maturity Outgo

4. Partial Surrender: As DoD Partial Surrender Outgo

The outgo calculation is simple, just like a normal benefit payment:

\begin{equation}
\begin{split}
DoD\_Dth\_Outgo _t & = DoD\_PP _t × NO\_DEATHS _t \\
DoD\_Surr\_Outgo _t & = DoD\_PP _t × NO\_SURRS _t \\
DoD\_Mat\_Outgo_t & = DoD\_PP _t × NO\_MATS _t \\
DoD\_PartSurr\_Outgo _t & = DoD\_PartSurr\_PP _t × NOP\_IF _t
\end{split}
\end{equation}

Readers should note that, once you have included coupon and dividend in DoD, you should not count them in coupon outgo and dividend outgo. Otherwise the benefit will be double counted (first time as coupon / dividend outgo, second time as DoD outgo). If you have added DoD_Opt_PC, then the coupon / dividend outgo should be adjusted by ( 1 - DoD_Opt_PC ) to reflect the amount going into DoD.

Let's go through a practical example below:

We shall use the same double decrement model we have introduced in previous chapters. The detail for NOP and benefits are given below:

DoD_Opt_PC = 50%

DoD_Cred_Rate = 4%

DoD_PartSurr_PC = 10%

Year 1

DoD_Cred_Int(1) = DoD_PP(0) * DoD_Cred_Rate = 0 * 4% = 0

DoD_Prem_PP(1) = ( Coupon_PP(1) + Dividend_PP(1) ) * DoD_Opt_PC = (0+0) * 50% = 0

DoD_PartSurr_PP(1) = (DoD_PP(0) + DoD_Cred_Int(1)) * DoD_PartSurr_PC = (0+0) * 10% = 0

DoD_PP(1) = DoD_PP(0) + DoD_Cred_Int(1) + DoD_Prem_PP(1) - DoD_PartSurr_PP(1) = 0 + 0 + 0 - 0 = 0

...

Year 6

DoD_Cred_Int(6) = DoD_PP(5) * DoD_Cred_Rate = 0 * 4% = 0

DoD_Prem_PP(6) = ( Coupon_PP(6) + Dividend_PP(6) ) * DoD_Opt_PC = (4+5) * 50% = 4,5

DoD_PartSurr_PP(6) = (DoD_PP(5) + DoD_Cred_Int(6)) * DoD_PartSurr_PC = (0+0) * 10% = 0

DoD_PP(6) = DoD_PP(5) + DoD_Cred_Int(6)+ DoD_Prem_PP(6) - DoD_PartSurr_PP(6) = 0 + 0 + 4.5 - 0 = 4.5

...

Year 10

DoD_Cred_Int(10) = DoD_PP(9) * DoD_Cred_Rate = 16.34 * 4% = 0.65

DoD_Prem_PP(10) = ( Coupon_PP(10) + Dividend_PP(10) ) * DoD_Opt_PC = (4+5) * 50% = 4,5

DoD_PartSurr_PP(10) = (DoD_PP(9) + DoD_Cred_Int(10)) * DoD_PartSurr_PC = (16.34+0.65) * 10% = 1.70

DoD_PP(10) = DoD_PP(9) + DoD_Cred_Int(10)+ DoD_Prem_PP(10) - DoD_PartSurr_PP(10) = 16.34 + 0.65 + 4.5 - 1.7 = 19.80

And the DoD related outgo as follow:

Year 1

DoD_Death_Outgo(1) = DoD_PP(1) * NO_DEATHS(1) = 0 * 0.000174 = 0

DoD_Surr_Outgo(1) = DoD_PP(1) * NO_SURRS(1) = 0 * 0.099991 = 0

DoD_Mat_Outgo(1) = DoD_PP(1) * NO_MATS(1) = 0 * 0 = 0

DoD_PartSurr_Outgo(1) = DoD_PartSurr_PP(1) * NOP_IF(1) = 0 * 0.899835 = 0

...

Year 6

DoD_Death_Outgo(6) = DoD_PP(6) * NO_DEATHS(6) = 4.5 * 0.000489 = 0.00

DoD_Surr_Outgo(6) = DoD_PP(6) * NO_SURRS(6) = 4,5 * 0.008192 = 0.04

DoD_Mat_Outgo(6) = DoD_PP(6) * NO_MATS(6) = 4.5 * 0 = 0

DoD_PartSurr_Outgo(6) = DoD_PartSurr_PP(6) * NOP_IF(6) = 0 *  0.810730 = 0

...

Year 10

DoD_Death_Outgo(10) = DoD_PP(10) * NO_DEATHS(10) = 19.80 * 0.000492 = 0.01

DoD_Surr_Outgo(10) = DoD_PP(10) * NO_SURRS(10) = 19.80 * 0.007850 = 0.16

DoD_Mat_Outgo(10) = DoD_PP(10) * NO_MATS(10) = 19.80 * 0.776870 = 15.38

DoD_PartSurr_Outgo(10) = DoD_PartSurr_PP(10) * NOP_IF(10) = 1.7 *  0.776870 = 1.32

And be careful that you have adjusted the formula for coupon and dividend accordingly!

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

https://drive.google.com/file/d/0B4OirwHLcmE1ZXZBNGVlUkh1REE/view?usp=sharing

Waiver of Premium (WOP)

Waiver of premium (WOP) means that if certain situation occurs, the premium will be waived for the policyholder. A typical condition for that is total permanent disabilities (TPD).

A proper modelling approach for WOP is that, for policies falling into TPD state, the premium is set to 0, but the benefit is projected as usual.

However, given most companies do not use a generalized decrement model, they tend to use some simplified modelling approach to proxy it.

The more common approach is, measure the amount of policies fell into TPD states (typically done by projecting a new NOP), or use the NOP multiplied by some factor. Then, calculate the Present value of the future premium, and treat that as a benefit outgo.

In formula:

\begin{equation}
\begin{split}
WOP\_PP _t & = \sum _{s =  t + 1}^{BPP} PREM\_INC\_PP × (1 + i)^s \\
COST\_OF\_WOP _t & = WOP\_PP _t  × NOP\_IFSM _t × TPD\_PROXY _t \\
\end{split}
\end{equation}

where i is some discount rate chosen (either statutory, or risk discount rate).

It is a start of period cashflow since premium is payable at start of period.

Let's go through a practical example below:

We shall use the same double decrement model we have introduced in previous chapters. The annual premium is 100 and payable for 5 years. The proxy % of TPD (to inforce policies) is given as below.

Assume a 5% discount rate. We need to calculate the present value of future premium first.

WOP_PP (4) = WOP_PP (5) / (1+5%) + PREM(5) = 0/1.05 + 100 = 100

WOP_PP (3) = WOP_PP (4) / (1+5%) + PREM(4) = 100/1.05 + 100 = 195.24

WOP_PP(2) = WOP_PP(3) / (1+5%) + PREM(3) = 195.24/1.05 + 100 = 285.94

WOP_PP(1) = WOP_PP(2) / (1+5%) + PREM(2) = 285.94/1.05 + 100 = 372.32

WOP_PP(0) = WOP_PP(1) / (1+5%) + PREM(1) = 372.32/1.05 + 100 = 454.60

The cost of waiver of premium is therefore:

Year 1

COST_OF_WOP = WOP_PP * NOP_IFSM * TPD_PROXY = 454.60 * 1 * 0 = 0

Year 2

COST_OF_WOP = WOP_PP * NOP_IFSM * TPD_PROXY = 372.32 * 0.899835 * 0.01% = 0.03

...

Year 5

COST_OF_WOP = WOP_PP * NOP_IFSM * TPD_PROXY = 100 * 0.828174 * 0.025% = 0.02

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

https://drive.google.com/file/d/0B4OirwHLcmE1bTNjajFfOUtnY28/view?usp=sharing

Return of Premium (ROP)

Return of Premium (ROP), as its name stated, is returning the premium that the policyholders have paid to them. The ROP can be paid upon death / surrender / maturity as the contract specified. In some jurisdiction (eg: India), there may be statutory requirement for insurers to paid a certain portion of ROP to the policyholders upon death or surrender.

The simplest formula, or the industry widely adopted formula for ROP is:

\begin{equation}
\begin{split}
BENEFIT\_PP _t & = ANN\_PREM × POL\_YR _t  × ADJ\_PC _t\\
BENEFIT\_OUTGO _t & = BENEFIT\_PP _t × NO\_STATES _t
\end{split}
\end{equation}

For example, if it is the 3 policy year for a whole life insurance policies, and the annual premium is 100. The contract states the policyholder can get 50% of ROP upon surrender. And it is expected 0.3 people will surrender at year 3. Then the surrender outgo is calculated by:

SURR_PP = 100 * 3 * 50% = 150
SURR_OUTGO = 150 * 0.3 = 45

Since premium may not be fixed (eg: for some guarantee renewal term products the premium can change every year), the above formula using constant "ANN_PREM" is flawed in this case. Some insurers therefore adopt the modified formula for these circumstances:

\begin{equation}
\begin{split}
BENEFIT\_PP _t & = \sum_{s = 1}^{t} PREM\_INC\_PP _s × ADJ\_PC _t\\
BENEFIT\_OUTGO _t & = BENEFIT\_PP _t × NO\_STATES _t
\end{split}
\end{equation}

I personally don't like this formula as well. I would suggest a more generic formula by treating cumulative paid premium as a balance (without interest credited to it). There is, creating a new variable to save the total premium paid and use it as a base for calculation, as below:

\begin{equation}
\begin{split}
ACCM\_PREM _t & = \sum_{s = 1}^{t} PREM\_INC\_PP _s \\
BENEFIT\_PP _t & =  ACCM\_PREM _t × ADJ\_PC _t\\
BENEFIT\_OUTGO _t & = BENEFIT\_PP _t × NO\_STATES _t
\end{split}
\end{equation}

This gives more flexibility and generalized the formula for any situation even there are twist on the premium payment formula.

Let's go through a practical example below:

We shall use the same double decrement model we have introduced in previous chapters. The annual premium is 100 and payable for 5 years. Assume the contracts stated that it will pay ROP upon death, surrender and maturity with the following percentages:

Year 1

ACCM_PREM(1) = ACCM_PREM(0) + PREM_INC_PP(1) = 0 + 100 = 100

DB_PP(1) = ACCM__PREM(1) * ROP_DTH_PC(1) = 100 * 120% = 120

DTH_OUTGO = DB_PP * NO_DEATHS = 120 * 0.000174 = 0.02

GCV_PP(1) = ACCM_PREM(1) * ROP_SURR_PC(1) = 100 * 30% = 30

SURR_OUTGO = GCV_PP * NO_SURRS = 30 * 0.099991 = 3.00

MAT_PP(1) = ACCM_PREM(1) * ROP_MAT_PC(1) = 100 * 0 = 0

MAT_OUTGO = MAT_PP * NO_MATS = 0 * 0 = 0

Year 2

ACCM_PREM(2) = ACCM_PREM(1) + PREM_INC_PP(2) = 100 + 100 = 200

DB_PP(1) = ACCM__PREM(2) * ROP_DTH_PC(2) = 200 * 120% = 240

DTH_OUTGO = DB_PP * NO_DEATHS = 240 * 0.000312 = 0.07

GCV_PP(1) = ACCM_PREM(2) * ROP_SURR_PC(2) = 200 * 40% = 80

SURR_OUTGO = GCV_PP * NO_SURRS = 80 * 0.044984 = 3.60

MAT_PP(1) = ACCM_PREM(2) * ROP_MAT_PC(2) = 200 * 0 = 0

MAT_OUTGO = MAT_PP * NO_MATS = 0 * 0 = 0

...

Year 10

ACCM_PREM(10) = ACCM_PREM(9) + PREM_INC_PP(10) = 500 + 0 = 500

DB_PP(10) = ACCM__PREM(10) * ROP_DTH_PC(10) = 500 * 120% = 600

DTH_OUTGO = DB_PP * NO_DEATHS = 600 * 0.000492 = 0.30

GCV_PP(10) = ACCM_PREM(10) * ROP_SURR_PC(10) = 600 * 100% = 600

SURR_OUTGO = GCV_PP * NO_SURRS = 600 * 0.007850 = 3.92

MAT_PP(10) = ACCM_PREM(10) * ROP_MAT_PC(10) = 600 * 100% = 600

MAT_OUTGO = MAT_PP * NO_MATS = 600 * 0.776870 = 388.43

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

https://drive.google.com/file/d/0B4OirwHLcmE1c1hYQVQzU1cyNlk/view?usp=sharing

Introduction to Traditional Life (TL)

Traditional Life business including the followings:

1. Term Life
2. Whole Life
3. Critical Illness / Catastrophe cash
4. Annuity
5. Endowment
6. Participating
...

Basically, all what may not be classified as Universal Life / Unit-Linked will fall into traditional life business. The reason for grouping all these products into TL is due to their cashflow structure and reserving techniques. They usually have some predefined, guaranteed cashflows to be paid in the future (as we have introduced in previous chapter on "benefit payments"), and calculating the reserve using NPV / GPV reserving as statutory required.

Non-guarantee benefits

The last cashflow to introduce is non-guarantee benefits.

Whist all the benefits we have mentioned in (3) are guarantee benefits that insurers have the obligation to follow, there are non-guarantee benefits that the insurance company do not have a straight obligation to pay. It can either be in form of discretionary bonus, or using some ring-fence rule to pay a certain percentage of surplus to the policyholders.

Non-guarantee benefits have different variations. The simplest form is a "cash dividend", which pays policyholders a dividend if he survives to certain age. This is like a " non-guarantee" version of coupon benefit.

\begin{equation}
\begin{split}
CASH\_DIV\_PP _t & = FA × POL\_VAL\_TBL(DIV, t) × DIV\_ADJ _t \\
DIV\_OUTGO _t & = CASH\_DIV\_PP _t × NOP\_IF
\end{split}
\end{equation}

DIV_ADJ is dividend adjustment. The company may adjust the original planned dividend (which is set during pricing) using this adjustment factor in the future if there are any unexpected favorable / unfavorable events that boost up / deteriorate profits. That's why dividend is "non-guaranteed" because it is subjected to adjustment.

Except cash dividend, there are terminal dividend which is paid upon termination events, like death or surrender. The formula is similar:

\begin{equation}
\begin{split}
TB\_DTH\_PP _t & = FA × POL\_VAL\_TBL(TB\_DTH, t) × TB\_ADJ _t \\
TB\_SURR\_PP _t & = FA × POL\_VAL\_TBL(TB\_SURR, t) × TB\_ADJ _t \\
TB\_MAT\_PP _t & = FA × POL\_VAL\_TBL(TB\_MAT, t) × TB\_ADJ _t \\
\\
TB\_DTH\_OUT _t & = TB\_DTH\_PP _t × NO\_DEATHS \\
TB\_SURR\_OUT _t & = TB\_SURR\_PP _t × NO\_SURRS \\
TB\_MAT\_OUT _t & = TB\_MAT\_PP _t × NO\_MATS
\end{split}
\end{equation}

Cash dividend can be saved in the insurance company to form Dividend on Deposit (DoD). There is also another form of non-guarantee benefit called Revisionary Bonus (RB). We will introduce them in later chapters.

Using the same decrement model we used before, let's calculate the dividend outgo.

DIV_ADJ_PC = 80%

Year 1

DIV_PP = FA * POL_VAL_TBL(Div, 1) / NO_OF_UNITS * DIV_ADJ_PC = 500 * (0/1000) * 80% = 0

DIV_OUTGO = DIV_PP * NOP_IF = 0 * 0.899835 = 0

Year 2

DIV_PP = FA * POL_VAL_TBL(Div, 2) / NO_OF_UNITS * DIV_ADJ_PC = 500 * (0/1000) * 80% = 0

DIV_OUTGO = DIV_PP * NOP_IF = 0 * 0.854540 = 0

...

Year 10

DIV_PP = FA * POL_VAL_TBL(Div, 10) / NO_OF_UNITS * DIV_ADJ_PC = 500 * (50/1000) * 80% = 20

DIV_OUTGO = DIV_PP * NOP_IF = 20 * 0.776870 = 15.54

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

https://drive.google.com/file/d/0B4OirwHLcmE1ZVI4Mi15bFdPRXM/view?usp=sharing

Tax

Next we will investigate tax.

For simplicity, we will only introduce 2 taxes, namely the premium tax and profit tax.

Premium tax is a sales tax, it is considered as a expense to insurance company.

\[PREM\_TAX _t = PREM\_INC _t × PREM\_TAX\_PC\]

And profit tax, as said, is the tax on profit.

\[PROFIT\_TAX _t = GROSS\_PROFIT _t × PROFIT\_TAX\_PC\]

If some of the tax is deductible, then we can subtract the tax deductible to the tax. The base for tax deduction varies by country. Take China as an example, the investment income is partly tax deductible, by 19%.

\[TAX\_DEDUCTIBLE _t = TAX\_DEDUCT\_BASE _t × TAX\_DEDUCT\_PC × PROFIT\_TAX\_PC \]

Hence the overall tax charged is:

\[TAX _t = PROFIT\_TAX _t - TAX\_DEDUCTIBLE _t \]

Note that tax is subtracted from gross profit to give the profit (after tax).
Premium tax is not a profit tax, it should be included in the formula of gross profit and considered like an expense.

The detail of the calculation of gross profit, profit and tax will be left in later chapters.

Commission

Similar to expense, commission can also be divided into 2 parts: initial commission and renewal commission.

Initial commission is typically higher than renewal commission. Unlike expense, the formula for commission is much simpler.

\begin{equation}
\begin{split}
INIT\_COMM\_PP & = INIT\_COMM\_PC × PREM\_INC\_PP _t \\
REN\_COMM\_PP _t & = REN\_COMM\_PC _t × PREM\_INC\_PP _t
\end{split}
\end{equation}

In addition to commission, there is a special type of commission called "commission override". Agents have managers / supervisors. They will receive an additional fee on top of the commissions earned by those agents (his / her down-line). That additional fee is called "commission override".

Commission override is typically only payable at the first year. It is also a percentage of premium income.

\[COMM\_OR\_PP _t = COMM\_OR\_PC _t × PREM\_INC\_PP _t \]

The commission outgo is calculated by its per policy amount multiplied by the number of policies inforce start of period. Note that commission is a start of period cashflow.

\begin{equation}
\begin{split}
INIT\_COMM & = INIT\_COMM\_PP _t × NOP\_IFSM _t  \\
REN\_COMM _t & = REN\_COMM\_PP _t × NOP\_IFSM _t \\
COMM\_OR _t & = COMM\_OR\_PP _t × NOP\_IFSM _t
\end{split}
\end{equation}

And the total commission is therefore:

\[TOT\_COMM _t = INIT\_COMM + COMM\_OR _t + REN\_COMM _t \]

Some companies may have commission clawback. We skipped this topic first.

Let's go through a practical example below:

We use the same model as in previous chapters. And the following commission related information is given:

 

Year 1

COMM_OR_PP = COMM_OR_PC * PREM_INC_PP = 10% * 100 = 10

COMM_OR = COMM_OR_PP * NOP_IFSM = 10 * 1 = 10

INIT_COMM_PP = INIT_COMM_PC * PREM_INC_PP = 30% * 100 = 30

INIT_COMM = INIT_COMM_PP * NOP_IFSM = 30 * 1 = 30

TOT_COMM = COMM_OR + INIT_COMM + REN_COMM = 10 + 30 +  0 = 40

Year 2

REN_COMM_PP = REN_COMM_PC * PREM_INC_PP = 2% * 100 = 2

REN_COMM = REN_COMM_PP * NOP_IFSM = 2 * 0.899835 = 1.80

TOT_COMM = COMM_OR + INIT_COMM + REN_COMM = 0 + 0 + 1.80 = 1.80

Year 3

REN_COMM_PP = REN_COMM_PC * PREM_INC_PP = 1% * 100 = 1

REN_COMM = REN_COMM_PP * NOP_IFSM = 1 * 0.854540 = 0.85

TOT_COMM = COMM_OR + INIT_COMM + REN_COMM = 0 + 0 + 0.85 = 0.85

...

Year 10

REN_COMM_PP = REN_COMM_PC * PREM_INC_PP = 1% * 0 = 0

REN_COMM = REN_COMM_PP * NOP_IFSM = 0 * 0.785212 = 0

TOT_COMM = COMM_OR + INIT_COMM + REN_COMM = 0 + 0 + 0 = 0

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

https://drive.google.com/file/d/0B4OirwHLcmE1N05aNmVGa0pRZlU/view?usp=sharing

Expense

Expense can be divided into 2 parts: Initial Expense and Renewal Expense.

Initial Expense is also called acquisition expense, it represents the large expenses incurred upon the issuance of a new policy. New policy tends to be expensive because of the expense incurred for underwriting, administrative procedures, payment collection, etc.

Renewal Expense is the expense incurred for monitoring the policy, like back-end policy monitoring, actuarial valuations, claims handling, etc. These expenses are more recurrent and predictable.

Expenses (either initial or renewal) will normally be divided into two components:

1. Fixed component
2. Variable component

Fixed component is the absolute dollar amount expense that is considered to be fixed per contract, regardless of contract size. For example, workload (and salaries) of actuaries are proportional to number of contracts, hence it is a fixed amount charges per contract, regardless of its premium / face amount.

Variable component is the percentage amount expense that is considered to be varied with contract size. For example, a high-net-worth policy with large lump sum premium is expected to have a larger transaction costs via credit card payment. This expense is proportional to the premium / face amount.

Hence, we can write down the formula below:

\begin{equation}
\begin{split}
INIT\_EXP\_PP & = INIT\_FIXED\_Y + INIT\_PREM\_PC × PREM\_INC\_PP _t \\
REN\_EXP\_PP _t & = REN\_FIXED\_Y + REN\_PREM\_PC _t × PREM\_INC\_PP _t
\end{split}
\end{equation}

Expenses are subjected to inflation. The longer the time passed since initial Pricing, the higher the expense can be. In the formula above, since we will expect the premium to be adjusted with inflation upon re-pricing exercise, the left over term is the fixed component. Hence actuaries will apply inflation adjustment to the fixed term.

\begin{equation}
\begin{split}
INIT\_EXP\_PP & = INIT\_FIXED\_Y × (1 + infl)^{Proj\_Yr}+ INIT\_PREM\_PC × PREM\_INC\_PP _t \\
REN\_EXP\_PP _t & = REN\_FIXED\_Y × (1 + infl)^{Proj\_Yr} + REN\_PREM\_PC _t × PREM\_INC\_PP _t
\end{split}
\end{equation}

Note that the "Proj_Yr" here means "Projection Years", means the year passed since valuation date. Some actuaries are confused on the inflation timing adjustment and applied policy year there. It is wrong because the fixed expense component is typically updated after experience studies. That represents an amount at valuation date, not policy inception.

There is also some confusion around the split between renewal expense and fixed expense. Some company will apply renewal expense only after year 1 (i.e. Month 13), while some will apply renewal expense since inception (i.e. Month 1). The two methods is in theory identical, but this will affect the split between initial expense and renewal expense.

The graph below demonstrates the difference between method 1 and method 2.

Method 1 apply renewal expense only after year 1. Hence all expense in year 1 is absorbed by "INIT_FIXED_Y". The philosophy behind is that, it is hard to differentiate acquisition expense and break it down into initial and renewal expense. Hence we treat them all as initial expense.

Method 2 apply renewal expense from inception. Only the acquisition expense in excess of renewal expense will be allocated to INIT_FIXED_Y. The philosophy behind is that, renewal expense is expense anyway the company will incur, hence it should also exist in year 1.

In this blog, we will adopt Method 1 for expense allocation.

And remaining calculation for expense is simple, multiple the per policy value by number of policy inforce, and summing them up will give the total expense. Note that expense is in general a beginning of period cashflow.

\begin{equation}
\begin{split}
INIT\_EXP _t & = INIT\_EXP\_PP × NOP\_IFSM _t \\
REN\_EXP _t & = REN\_EXP_\_PP × NOP\_IFSM _t \\
TOT\_EXP _t & = INIT\_EXP _t + REN\_EXP _t
\end{split}
\end{equation}

In deed, there are more expense variation like percentage of Sum Assured / Face Amount, or claims expenses. They are not difficult to calculate if readers can grab the concept in this chapter hence we will not introduce them one by one.

Let's go through a practical example below:

We use the same model as in previous chapters. And the following expense related information is given:

Year 1

INIT_EXP_PP = INIT_FIXED_Y * (1 + infl) ^ (0) + INIT_PREM_PC (1) * PREM_INC_PP (1) = 100 * 1.03 ^ 0 + 5% * 100 = 105

INIT_EXP = INIT_EXP_PP * NOP_IFSM = 105 * 1 = 105

Year 2

REN_EXP_PP = REN_FIXED_Y * (1 + infl) ^ (1) + REN_PREM_PC (2) * PREM_INC_PP (2) = 20 * 1.03 ^ 1 + 2% * 100 = 22.6

REN_EXP = REN_EXP_PP * NOP_IFSM = 22.6 * 0.899835 = 20.34

Year 3

REN_EXP_PP = REN_FIXED_Y * (1 + infl) ^ (2) + REN_PREM_PC (3) * PREM_INC_PP (3) = 20 & 1.03 ^ 2 + 1% * 100 = 22.22

REN_EXP = REN_EXP_PP * NOP_IFSM = 22.22 * 0.854540 = 18.99

...

Year 10

REN_EXP_PP = REN_FIXED_Y * (1 + infl) ^ (9) + REN_PREM_PC (10) * PREM_INC_PP (10) = 20 & 1.03 ^ 9 + 1% * 0 = 26.10

REN_EXP = REN_EXP_PP * NOP_IFSM = 26.10 * 0.785212 = 20.49

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

https://drive.google.com/file/d/0B4OirwHLcmE1YTJCaTBuR2tSREk/view?usp=sharing

Policy Value Table

Regardless of type of benefit, any payment to the policyholder can be expressed as a percentage of Face Amount.

For example, the insurance contract will pay 100% of Face Amount to the policyholder upon death, 20% of Face Amount to the policyholder upon surrender at policy year 3, 5% of Face Amount as a coupon to the policyholder upon survival at policy year 6.

Hence, the "Percentage of Face Amount" information for each benefit type can be summarized in a table called "Policy Value Table".

An example of Policy Value Table:

Note that the number of unit shown above is 1,000. Hence a "1,000" on death benefit represents a 100% (1,000/1,000) payment on Face Amount.

In formula:

\[BEN\_PP _t = FA × POL\_VAL _{Type, t} \]

Using the same decrement model we used before, let's calculate the benefit payment again.

Year 1

DTH_OUTGO = FA * POL_VAL_TBL(DB, t) * NO_DEATHS = 500 * 0.000174 = 0.09

SURR_OUTGO = FA * POL_VAL_TBL(GCV, t) * NO_SURRS = 50 * 0.099991 = 5.00

COU_OUTGO = FA * POL_VAL_TBL(COU, t) * NOP_IFSM = 0 * 1 = 0

MAT_OUTGO = FA * POL_VAL_TBL(MAT, t) *  NOP_IF = 0 * 0.899835 = 0

Year 2

DTH_OUTGO = FA * POL_VAL_TBL(DB, t) * NO_DEATHS = 500 * 0.000312 = 0.16

SURR_OUTGO = FA * POL_VAL_TBL(GCV, t) * NO_SURRS = 100 * 0.044984 = 4.50

COU_OUTGO = FA * POL_VAL_TBL(COU, t) * NOP_IFSM = 0 * 0.899835 = 0

MAT_OUTGO = FA * POL_VAL_TBL(MAT, t) * NOP_IF = 0 * 0.854540 = 0

...

Year 10

DTH_OUTGO = FA * POL_VAL_TBL(DB, t) * NO_DEATHS = 500 * 0.000492 = 0.25

SURR_OUTGO = FA * POL_VAL_TBL(GCV, t) * NO_SURRS = 500 * 0.007850 = 3.92

COU_OUTGO = FA * POL_VAL_TBL(COU, t) * NOP_IFSM = 25 * 0.785212 = 19.63

MAT_OUTGO = FA * POL_VAL_TBL(MAT, t) * NOP_IF = 500 * 0.776870 = 388.43

A demonstration spreadsheet showing the calculation above can be downloaded here:
https://drive.google.com/file/d/0B4OirwHLcmE1UkZ2U0Q0ODI0MGs/view?usp=sharing

Guarantee Cash Value

Guarantee Cash Value (GCV) / Surrender Outgo is a benefit type that worth special mention, because of its complexity in formula.

For most saving products, policyholders will have a large maturity benefit payable upon maturity. Hence it is unfair to pay nothing to the customer if his / her policy has already been inforced for a long time. Meanwhile, early surrenders will be subjected to surrender penalty, since insurance is a mutual contract and insurers have the expectation that the customers will remain the contract inforce, so early surrenders will probably create losses to insurers that can only be compensated by surrender penalty.

The following graph demonstrated a simplified case (using linear interpolation) for calculating GCV:

In reality, GCV will normally calculated by Face Amount multiplied by a Surrender Penalty Factor. In formula:

\[GCV\_PP _t = FA × (1 - SURR\_PEN\_PC _t) \]

Let's go through a practical example below:

The maturity benefit at time 10 is equal to Face Amount which is 10,000 and the surrender penalty is given as below:

Year 1

GCV_PP = FA × (1 - 90%) = 1,000

Year 2

GCV_PP = FA × (1 - 80%) = 2,000

...

Year 10

GCV_PP = FA × (1 - 0%) = 10,000

Now we go more complicated by considering coupons.

Imagine that you are eligible to get the coupon at the end of year 6 for 100. If you surrendered your policy at the middle of the year 6, would you be eligible to get partial of the coupon? Say, 50?

It really depends on the company's practice. We have illustrated the two situations below:

Either case is possible. GCV_PP_1 illustrates the situation where coupon is not payable upon survival. Hence it is a "all-or-nothing" case: either you survived to the end of year and get the coupon, or you get nothing. GCV_PP_2 illustrates the situation where partial coupon is redeemable upon survival. Hence you can get the interpolated coupon value within the year that the coupon will be distributed.

We won't go into the detail for the calculation here since it includes monthly projection model that we still haven't discussed. But reader may have an understanding on the coupon effect and how it may affect the surrender value.

Benefit Payments

Benefit Payments represents the contractual obligation that the insurers needed to pay to policyholders upon some predefined events.

For example, if the policyholder died, they will receive the death benefit (DB); when they lapsed, they will receive guarantee cash value (GCV) / surrender benefit; when the policy matured, they will receive maturity benefit (MB).

To summarize, guarantee insurance benefits can be summarized in the following matrix:

Here is the introduction for each of the benefit:

Coupon Benefit: This is an annuity like, guarantee cash benefit paid to the policyholder. For some company, this benefit will combine with maturity benefit and called "Survival Benefit".

Death Benefit: This is the death benefit paid to the policyholder upon death.

Surrender Benefit / Guarantee Cash Value: This is the amount to be paid to the policyholder when he / she lapsed. It is also called "Guarantee Cash Value" because it represents the minimum value of the contract the policyholder can get, regardless of insured events.

Maturity Benefit: This is the maturity benefit paid to the policyholder when the contract mature.

Critical Illness Benefit: This is the CI benefit paid to the policyholder upon diagnosed diseases.

The calculation for benefit is defined by:

\begin{equation}
\begin{split}
Benefit\_Outgo _t & = Benefit\_PP _t × NO\_STATES _t
\end{split}
\end{equation}

While the equation is simple, devil is in the detail. The hardest part of the calculation is on the "Benefit Per Policy (Benefit_PP)" calculation. We will introduce the PP calculation for some benefit in later chapters.

A simpler example using death benefit shows how the above formula should be applied. Assume a insurance contract pays the Face Amount (FA) upon the death of the policyholder. Then the death outgo is calculated by:

\begin{equation}
\begin{split}
DTH\_OUTGO _t & = DB\_PP _t × NO\_DEATHS _t \\
& = FA × NO\_DEATHS _t
\end{split}
\end{equation}

Let's go through a practical example below:

We shall use the same double decrement model we have introduced in previous chapters. Assume the Face Amount (FA) is 500. The number of policies as well as benefit per policy amount is given in the following table:

Assume coupon benefit is a BOP cashflow and maturity benefit is a EOP cashflow.

Year 1

DTH_OUTGO = DB_PP * NO_DEATHS = 500 * 0.000174 = 0.09

SURR_OUTGO = GCV_PP * NO_SURRS = 50 * 0.099991 = 5.00

COU_OUTGO = CB_PP * NOP_IFSM = 0 * 1 = 0

MAT_OUTGO = MB_PP * NOP_IF = 0 * 0.899835 = 0

Year 2

DTH_OUTGO = DB_PP * NO_DEATHS = 500 * 0.000312 = 0.16

SURR_OUTGO = GCV_PP * NO_SURRS = 100 * 0.044984 = 4.50

COU_OUTGO = CB_PP * NOP_IFSM = 0 * 0.899835 = 0

MAT_OUTGO = MB_PP * NOP_IF = 0 * 0.854540 = 0

...

Year 10

DTH_OUTGO = DB_PP * NO_DEATHS = 500 * 0.000492 = 0.25

SURR_OUTGO = GCV_PP * NO_SURRS = 500 * 0.007850 = 3.92

COU_OUTGO = CB_PP * NOP_IFSM = 25 * 0.785212 = 19.63

MAT_OUTGO = MB_PP * NOP_IF = 500 * 0.776870 = 388.43

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

https://drive.google.com/file/d/0B4OirwHLcmE1Ti1zVDJPYjRhUm8/view?usp=sharing

Premium Income

The first cashflow to discuss is premium income.

Premium Income is the cash paid by policyholders to the insurance company. For traditional policies, Insurers normally will classify premium into two types:

1. Single Premium
2. Regular Premium

Single premium means that, the policyholder only pays 1 large premium at the beginning of the contract. Regular premium means the policyholder is regularly paying premium (say, monthly for 10 years) into the insurance contract.

For UL policies, "Top-Up Premium" may also exist, meaning the "extra premium" that the policyholder puts into the contract on top of the promised regular premium.

Premium can be paid in different frequency, namely:

1. Yearly
2. Half-yearly
3. Quarterly
4. Monthly

Insurers called this the "premium frequency" or "pay mode". Since monthly premium means that the policyholder can pay most of premiums later (compare to paying as a lump sum at the beginning of the year), insurers may loss the time value of money. Hence, some adjustment has to be made for different premium frequency to "load-up" the premium charges.

To adjust for this timing differences, insurers will multiply a "modal factor" to the annual premium. Modal factor is a multiplicative factor varies with premium frequency. A typical modal factor table is shown as below:

Assume the annual premium is 100. The calculation for other pay-mode is as below:

Half-yearly: 100 * 0.51 = 51
Quarterly: 100 * 0.26 = 26
Monthly: 100 * 0.0875 = 8.75

One can see that, the more frequent the pay-mode, the more expensive the policy is. Since:

Half-yearly: 51 * 2 = 102
Quarterly: 26 * 4 = 104
Monthly: 8.75 * 12 = 105

The calculation for premium income is simple, it is just the annual premium, multiplied by the modal factor, then multiplied by the number of policy inforce at start of period. (you won't expect someone died to pay premium right?) Beware of the cashflow timing, since premium is collected at the start of period, it is a beginning of period (BOP) cashflow.


\begin{equation}
\begin{split}
Premium\_Income\_PerPolicy _t & = Annual\_Premium × Modal\_Factor \\
Premium\_Income _t & = Premium\_Income\_PerPolicy _t × Number\_of\_Inforce\_Start _t
\end{split}
\end{equation}

Using abbreviation:

\begin{equation}
\begin{split}
PREM\_INC\_PP _t & = ANN\_PREM × MODAL\_FAC  \\
PREM\_INC _t & = PREM\_INC\_PP _t × NOP\_IFSM _t
\end{split}
\end{equation}

Let's go through a practical example below:

We shall use the same double decrement model we have introduced in previous chapters. Assume premium payment period (PPP) is 5 years, and the annual premium is 100. The pay-mode is annually.

Year 1

PREM_INC_PP = 100 * 1 = 100

PREM_INC = 100 * 1 = 100

Year 2

PREM_INC_PP = 100 * 1 = 100

PREM_INC = 100 * 0.899835 = 89.98

...

Year 5

PREM_INC_PP = 100 * 1 = 100

PREM_INC = 100 * 0.828174 = 82.82

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

https://drive.google.com/file/d/0B4OirwHLcmE1TUdUbkVKazJ3ZUU/view?usp=sharing

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