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}
\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}
\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:
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