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