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:
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