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{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:
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