Why Cash-Flow Timing Matters
An asset share reflects cash flows accumulated to a particular date. A premium received early in a projection earns investment return for longer than an expense paid later; a benefit paid on the valuation date earns no return after payment. Thus, the amount and the date of each cash flow both matter. Applying one annual accumulation factor to every cash flow will generally give the wrong result.
This tutorial considers accumulation at a fixed effective annual investment return. It focuses on assigning the correct accumulation period to each specified cash flow. The cash flows in the examples are deterministic, hypothetical per-policy amounts, and the policy remains in force over the stated projection. No mortality distribution is used: there are no random survival or claim events in these calculations. If cash flows are instead contingent on such events, their treatment requires additional assumptions.
Assigning an Accumulation Period
Measure every cash-flow date and the valuation date from the same time origin. If a premium is paid at time \(s\) and the asset share is measured at time \(T\), its accumulation period is \(T-s\), not necessarily one year. An outgo can be entered as a negative amount, so it is accumulated by the same rule as an inflow.
For a fixed return and cash flows \(C_j\) at times \(t_j\), the accumulated value at \(T\) is
A cash flow at \(T\) has exponent zero and is therefore included at its amount, with no investment return earned after it occurs. A cash flow before \(T\) is accumulated for the elapsed time between its date and \(T\). A cash flow after \(T\) is not included in the value at \(T\).
The fractional-year convention needs care. An effective annual rate specifies the factor over a full year. Using \((1+i)^t\) for a fraction of a year is a consistent convention that assumes the same equivalent force of interest throughout the period. It does not mean that the annual effective rate itself is divided by the number of subperiods. If a problem specifies a different within-year crediting rule, use that rule instead.
Why the Fractional-Year Formula Works
The accumulation formula follows from the composition of accumulation factors. Suppose a balance is invested first for \(t_1\) years and then for \(t_2\) years at the same fixed return convention. The first interval multiplies it by \((1+i)^{t_1}\), and the second multiplies the resulting balance by \((1+i)^{t_2}\). Therefore the combined factor is
So a cash flow at \(s\), carried through successive intervals until \(T\), accumulates by the same factor as carrying it through the total interval \(T-s\). This justifies using the single exponent \(T-s\). For example, when \(i=21\%\), the half-year factor under this convention is \((1.21)^{1/2}=1.1\), because \(1.1^2=1.21\). It is not \(1+0.21/2=1.105\).
Worked Example: Start-, Mid-, and End-Year Cash Flows
A hypothetical policy has a $1,200 premium at the start of a one-year period, a $220 expense at midyear, and a $300 scheduled benefit payment at year-end. Find the accumulated value at the end of the year. The effective annual investment return is \(i=21\%\), fixed throughout. All amounts are deterministic; assume the policy remains in force and use no mortality distribution.
Method. Accumulate each signed cash flow from its own date to time \(T=1\). The start-year premium is positive, while the expense and benefit are negative.
The premium remains invested for one year:
The expense occurs at \(t=0.5\), so it accumulates for half a year. Since \((1.21)^{0.5}=1.1\), its signed value at year-end is
The benefit is paid at the valuation date, so it has no time to accumulate:
Adding the three year-end values gives
Check. The accumulated premium is $1,452; total accumulated outgo is \(242+300=\$542\). Their difference is $910. A common mistake is to accumulate the midyear expense for a full year, which would give $266 rather than $242 and understate the answer by $24.
Accumulating Several Years of Cash Flows
The same rule applies when the valuation date is more than one year after some cash flows and less than one year after others. Keep dates on a single time scale and calculate each exponent separately. A compact ledger helps prevent timing errors.
| Cash-flow date | Signed amount | Accumulation period to \(T\) | Value at \(T\) |
|---|---|---|---|
| \(t_j\) | \(C_j\) | \(T-t_j\) | \(C_j(1+i)^{T-t_j}\) |
The row-by-row calculation is especially useful when several premiums or outgo amounts occur within the same projection. Combine cash flows only when they occur at the same time, since cash flows at different dates generally have different accumulation factors.
Worked Example: Multiple Dates and a Midyear Outgo
A hypothetical policy receives a $500 premium at \(t=0\) and a $400 renewal premium at \(t=1\). It incurs an expense of $50 at \(t=0.5\) and a scheduled benefit outgo of $100 at \(t=1.5\). Calculate the accumulated value at \(T=2\). The fixed effective annual return is \(i=21\%\). The cash flows are deterministic, the policy remains in force, and no mortality distribution is assumed.
Method. Accumulate each amount to time 2, using a negative sign for both outgo amounts.
The first premium is invested for two years:
The renewal premium at \(t=1\) is invested for one year:
The expense at \(t=0.5\) is invested for \(1.5\) years. Its factor is \(1.21^{1.5}=1.21\sqrt{1.21}=1.21(1.1)=1.331\), so its accumulated signed value is
The benefit outgo at \(t=1.5\) is invested for half a year:
Therefore,
Check. The two accumulated inflows total $1,216.05, and the two accumulated outgo amounts total \(66.55+110=\$176.55\). Their difference is $1,039.50. The midyear expense is accumulated for 1.5 years, not for the two years between the first premium and valuation date.
Timing Errors and Rate Conversion
Two errors recur in accumulation calculations. First, an amount is sometimes given a full year’s return simply because it occurs during a year. The exponent should instead equal the time from that cash flow to the valuation date. Second, an annual effective rate is sometimes prorated linearly for a fraction of a year. Under the convention in this tutorial, a half-year factor is \((1+i)^{0.5}\), not \(1+i/2\). The two factors are generally different.
Cash-flow timing can have a material effect even when the amounts and annual return are unchanged. For example, at \(i=21\%\), $100 invested from the start to the end of a year accumulates to $121. The same $100 invested only from midyear to year-end accumulates to $110. The $11 difference is the return earned during the first half-year. A payment made at year-end, in contrast, is not multiplied by either factor when valuing at that same year-end date.
Worked Example: Comparing Two Midyear Timing Treatments
A hypothetical policy receives a $1,000 premium at \(t=0\), a $600 premium at \(t=0.5\), and pays a $100 expense at \(t=0.5\). A scheduled benefit outgo of $250 is paid at \(t=1\). Find the value at \(T=1\), and compare it with two common incorrect treatments. The fixed effective annual return is \(i=21\%\); all flows are deterministic, the policy remains in force, and no mortality distribution is assumed.
Correct method. The initial premium accumulates for a full year, giving \(1{,}000(1.21)=\$1{,}210\). The two midyear cash flows each accumulate for half a year. The premium becomes \(600(1.1)=\$660\), and the expense becomes \(-100(1.1)=-\$110\). The benefit paid at \(T=1\) contributes \(-\$250\). Thus,
Incorrect treatment 1: no return on the midyear flows. If the midyear premium and expense are entered at their original amounts, the result is \(1{,}210+600-100-250=\$1{,}460\), which is $50 too low. The net midyear inflow of $500 should have earned half-year accumulation, becoming $550.
Incorrect treatment 2: prorating the annual rate. Using \(1+0.21/2=1.105\) for half a year gives a net midyear value of \(500(1.105)=\$552.50\), and an accumulated value of \(1{,}210+552.50-250=\$1{,}512.50\). This is $2.50 too high under the stated convention. The correct half-year factor is \(\sqrt{1.21}=1.1\), not 1.105.
Check. The correct net midyear cash flow is \(600-100=\$500\); accumulating it for half a year gives $550. Then \(1{,}210+550-250=\$1{,}510\), agreeing with the cash-flow-by-cash-flow calculation.
A Practical Calculation Sequence
For a fixed effective annual return, a reliable procedure is to make the timing decision explicit before doing arithmetic. In particular, record the valuation date even when it is the end of a familiar policy year: that date determines every accumulation period.
Express each cash-flow date and \(T\) in years from the same origin.
Enter premiums and other inflows as positive amounts, and expenses and benefits paid as negative amounts.
For a flow at \(t_j\), calculate \(T-t_j\). A flow at \(T\) has period zero.
For the fixed-return fractional-year convention, multiply by \((1+i)^{T-t_j}\).
Compare accumulated inflows with accumulated outgo, and verify that each exponent matches its cash-flow date.
This method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central exam skill is not merely evaluating powers: it is translating the stated timing into the correct accumulation period before calculating.
Check Your Understanding
For each problem, use signed cash flows and accumulate to the stated valuation date. Amounts are deterministic hypothetical cash flows; unless stated otherwise, assume the policy remains in force and no mortality distribution is involved.
- A policy receives $900 at \(t=0\), incurs a $50 expense at \(t=0.5\), and pays $200 at \(t=1\). Find the value at \(T=1\) at a fixed effective annual return of \(21\%\), using the fractional-year convention. (Answer: $834.00)
- At a fixed effective annual return of \(21\%\), a $400 premium is received at \(t=0\), a $300 premium at \(t=0.5\), and a $100 expense at \(t=1\). Find the accumulated value at \(T=1.5\). (Answer: $785.40)
- A $1,000 premium is received at \(t=0\), a $500 premium at \(t=1\), and a $100 expense at \(t=1.5\). At \(i=21\%\), find the value at \(T=2\). (Answer: $1,959.10)
- At \(i=21\%\), compare the value at \(T=1\) of a $100 inflow received at \(t=0\) with the value of the same inflow received at \(t=0.5\). How much greater is the first value? (Answer: $11.00)
- A policy receives $700 at \(t=0.5\) and pays $250 at \(t=1\). At \(i=21\%\), calculate the value at \(T=1.5\). (Answer: $572.00)