Why Premium Timing Matters
A premium amount alone does not determine the investment income it generates during a policy year. A premium received at the start of the year can be invested for longer than the same amount received near year-end. Consequently, even when premium income is unchanged, the annual profit attributable to a policy can depend on the receipt date.
The calculation requires a clear time convention. In this tutorial, time \(t\) is measured in years from the start of the policy year, so \(t=0\) means receipt at the start and \(t=1\) means receipt at the end. Unless stated otherwise, the annual effective interest rate is \(i\), and a balance invested for a fraction \(s\) of a year accumulates by the factor \((1+i)^s\). This fractional-year compound convention must be part of the model; an annual effective rate by itself does not prescribe every possible within-year crediting rule.
If the model specifies a constant force of interest \(\delta\) instead, the year-end accumulation factor over \(1-t\) years is \(e^{\delta(1-t)}\), and the investment income is \(P[e^{\delta(1-t)}-1]\). When \(\delta=\ln(1+i)\), the two accumulation conventions agree. A model that specifies simple interest prorated by time uses a different rule, \(P i(1-t)\); it should not be substituted for compound accumulation unless that is the stated convention.
Receipt at the Start, Middle, or End
For a fixed premium and the same interest convention, later receipt leaves less time for investment. At the start of the year, the premium earns a full year's interest. At the midpoint it earns interest for half a year. At the end it earns no investment income during that policy year. These comparisons concern investment income, not whether the premium itself is included as a receipt in the annual cash-flow calculation.
| Receipt time | Time invested | Investment income at effective annual rate \(i\) |
|---|---|---|
| Start of year, \(t=0\) | 1 year | \(P[(1+i)-1]=Pi\) |
| Middle of year, \(t=0.5\) | 0.5 year | \(P[(1+i)^{0.5}-1]\) |
| End of year, \(t=1\) | 0 years | \(P[(1+i)^0-1]=0\) |
A useful justification is to compare the premium's value immediately after receipt with its value at year-end. At receipt, the insurer has principal \(P\). After \(1-t\) years, that principal has accumulated to \(P(1+i)^{1-t}\). The increase in the invested balance is therefore
This argument assumes the premium remains invested for the entire remaining period and earns the stated accumulation. It does not account for expenses, claims, taxes, or other cash flows; those must be included separately in a profit calculation.
Worked Example: Comparing Three Receipt Dates
A hypothetical one-year policy receives a premium of $1,200 at the start, midpoint, or end of the policy year, depending on the scenario. The effective annual interest rate is 8%, with fractional-year compound accumulation. Calculate the investment income by year-end for each receipt date.
Method. Apply the premium timing formula with \(P=1{,}200\) and \(i=0.08\). Keep the receipt time \(t\) distinct from the time invested, \(1-t\).
Start of year. Here \(t=0\), so the premium is invested for one year:
Middle of year. Here \(t=0.5\), so it is invested for half a year. Since \(\sqrt{1.08}\approx1.0392305\),
End of year. Here \(t=1\), leaving no time to earn interest before year-end:
The year-end investment incomes are therefore $96.00, approximately $47.08, and $0.00. The ordering is sensible: with the same principal and rate, the earlier receipt earns more. The common mistake this calculation avoids is treating a half-year receipt as if it earned exactly half of the annual interest. Under the stipulated compound convention, the correct midpoint amount is $47.08, not \(1{,}200(0.08)(0.5)=\$48.00\).
Putting Timing into Annual Profit
To isolate the effect of premium timing, use a simplified annual cash-flow profit model: annual profit equals the premium received during the year plus investment income on that premium, less outgo paid during the year. Assume no reserve movement or other profit items in the comparison. The premium is counted once as a receipt; investment income is an additional amount earned on the invested premium. If other cash flows are present, their amounts and timing must also be specified before calculating profit.
Worked Example: Receipt Timing and Annual Profit
A hypothetical policy remains in force for the full year. Its $2,000 premium is received either at the start or at the midpoint of the year. A $600 benefit and $100 of expenses are both paid at year-end. The effective annual interest rate is 6%, with fractional-year compound accumulation. Calculate annual profit under each premium timing.
Method. Total year-end outgo is \(C=600+100=\$700\). Calculate the premium's investment income for each receipt date, then use annual profit \(=P+I_P-C\).
Start-of-year receipt. The premium earns a full year's interest:
Therefore,
Midyear receipt. The premium earns half a year's interest. Since \(\sqrt{1.06}\approx1.0295630\),
Thus,
The difference in profit is approximately \(1{,}420.00-1{,}359.13=\$60.87\), exactly the difference between the two investment-income amounts to the displayed precision. The check is that the premium and outgo are unchanged; only investment income changes. A common error would be to subtract the year-end outgo from the premium before calculating interest, even though the stated benefit and expenses are not paid until year-end. The specified timing means they do not reduce the amount invested during the year in this simplified comparison.
Receipts at Other Times and the Main Exam Trap
Receipt need not occur at one of the three benchmark dates. For any specified \(t\), first calculate the remaining investment period \(1-t\), then apply the accumulation convention. For example, a receipt at \(t=0.8\) is made after 80% of the year has elapsed, so it earns interest for the remaining 20% of the year. Do not confuse “received at 80% of the year” with “invested for 80% of the year.”
Worked Example: A Premium Received Near Year-End
A hypothetical insurer receives a $1,500 premium at time \(t=0.8\) during a policy year. The effective annual interest rate is 5%, and fractional-year compound accumulation applies. Find the investment income by year-end and the year-end value of the premium.
Method. Since receipt occurs at \(t=0.8\), the time invested is \(1-0.8=0.2\) years.
Using \((1.05)^{0.2}\approx1.0098057\), the interest is
The year-end value is principal plus interest:
The result is plausible because the premium earns interest for only one-fifth of a year, so the interest is much less than the full-year amount \(1{,}500(0.05)=\$75.00\). A timing error that uses \(0.8\) as the invested period would materially overstate the income.
Another frequent error is to apply simple prorating \(Pi(1-t)\) without checking the model. That method may be appropriate when simple interest is explicitly specified, but it is not the same as the compound fractional-year convention. At a 5% effective annual rate, for example, simple prorating for 0.2 years gives \(1{,}500(0.05)(0.2)=\$15.00\), whereas the stipulated compound result is $14.71. The method and timing convention should be stated before the calculation.
Check Your Understanding
For each problem, use the stated receipt time and interest convention. Report money to the nearest cent.
- A hypothetical $900 premium is received at the start of a policy year. The effective annual interest rate is 4%. Find its investment income by year-end. (Answer: $36.00)
- A hypothetical $2,400 premium is received at the midpoint of the year. The effective annual interest rate is 9%, with fractional-year compound accumulation. Find the year-end investment income. (Answer: $105.67)
- A hypothetical $1,800 premium is received at \(t=0.75\). The effective annual interest rate is 8%, with fractional-year compound accumulation. Find its investment income by year-end. (Answer: $34.97)
- A hypothetical policy receives a $1,300 premium at the start of the year. The effective annual interest rate is 6%, and $500 of benefit and expense outgo is paid at year-end. Under the simplified profit model in this tutorial, find annual profit. (Answer: $878.00)
- A hypothetical $1,500 premium is received at \(t=0.8\). The effective annual interest rate is 5%, with fractional-year compound accumulation. A $200 expense is paid at year-end. Under the simplified profit model, find annual profit. (Answer: $1,314.71)