Payment Timing in Equity-Linked Valuations

Why Payment Timing Matters

In an equity-linked policy, the date of a cash flow determines how long it can earn investment returns and how much it is discounted at valuation. A premium invested at the beginning of a year participates in that year’s fund growth; the same premium invested at year-end does not. Similarly, a death benefit paid at death is generally worth more at the valuation date than the same benefit paid at the end of the year of death.

A reliable calculation starts by putting every cash flow on a timeline. State whether amounts are paid at the start, during, or at the end of each period; specify what condition must hold on each payment date; and apply investment accumulation or actuarial discounting from that date. The timing convention for a fund transaction is not automatically the same as the timing convention for a benefit paid by the insurer.

Definition: A cash flow at time $t$ is an amount paid or received at that point on the timeline, measured in years from the valuation date. A cash flow at the beginning of a period occurs at its starting time; one at the end occurs after the period’s investment return has been earned. A cash flow during the period uses its actual time, or a stated assumption about that time, in accumulation or discounting.

Timing in Fund Projections

Suppose the fund earns a deterministic effective annual rate $i$ over one year, and let $F_0$ be its value at the start of that year. If a premium $P$ is invested at the start, it earns the full year’s return. If a charge $C$ is deducted at the end, it earns none of that year’s return before deduction. The ending fund is therefore

$$ F_1=(F_0+P)(1+i)-C. $$

If instead the premium is invested at the end, after the existing fund has grown and the end-of-year charge has been deducted, then

$$ F_1=F_0(1+i)-C+P. $$

These formulas assume that the premium and charge are both transactions in the fund, that the stated investment return applies throughout the year to amounts present in the fund, and that no other cash flows occur. If a charge is deducted at the beginning, it reduces the amount that earns a return. If charges are taken by cancelling units, the number of units cancelled depends on the unit price at the actual deduction date.

Worked Example: Premium and Charge Dates in a Fund Projection

A hypothetical policy has a fund of $8,000 at the start of a year. The annual effective investment return is 4%. A premium of $1,000 is invested either at the start or at the end of the year. A charge of $80 is deducted either at the start or at the end. Find the ending fund for each of these two sequences: (a) premium at the start and charge at the end; (b) charge at the start, premium immediately afterward, and both amounts then remain invested for the year.

Method. Place each transaction on the timeline first, then apply the 4% return only to the amount in the fund during the year.

(a) Start-of-year premium, end-of-year charge. The fund immediately after the premium is $8{,}000+1{,}000=9{,}000$. After one year of growth, it is $9{,}000(1.04)=9{,}360$. The end-of-year charge is then deducted: $9{,}360-80=\$9{,}280.00$.

(b) Start-of-year charge, then premium. After the charge, the fund is $8{,}000-80=7{,}920$. After the premium is added, the amount invested for the full year is $7{,}920+1{,}000=8{,}920$. The ending fund is $8{,}920(1.04)=\$9{,}276.80$.

Check. In (a), the full $9,000 earns the return and the charge does not. In (b), the charge reduces the amount earning a return by $80, so the ending value is lower by $80(1.04)=\$83.20$, exactly the difference between $9,280.00 and $9,276.80. A common mistake is to deduct an end-of-year charge before applying interest; that would incorrectly make the charge lose its timing effect.

Present Values Depend on the Payment Date

For an effective annual valuation interest rate $i$, the discount factor is $v=1/(1+i)$. A certain payment $B$ at time $t$ has present value $Bv^t$. For a benefit paid only if a specified event occurs, the present value must also reflect the probability of that event and the payment date conditional on it.

For example, a fixed death benefit paid at the end of the year of death is discounted by one full year when death occurs during that year. If the benefit is paid immediately at death, the discount period varies with the time of death. The two values are not interchangeable, even when they cover the same deaths and have the same face amount.

Assumption for the within-year formula: Suppose the probability of death during a year is $q$, and, conditional on death in that year, the time of death is uniformly distributed over the year (UDD). With effective annual interest $i$, the probability density for death during the year, without conditioning on death, is $q$ at each time $t$ from 0 to 1. This assumption is used only for the within-year timing calculation; another mortality distribution can produce a different value.

Deriving the Immediate-Payment Value

Let a benefit $B$ be payable immediately on death during the year. Under UDD, death in a small interval of length $dt$ around time $t$ has probability $q\,dt$. If death occurs at $t$, the benefit is discounted by $v^t$. Summing the discounted amounts over the year gives the expected present value:

Bq\int_0^1v^t\,dt = Bq\int_0^1 e^{-\delta t}\,dt = Bq\frac{1-e^{-\delta}}{\delta} = Bq\frac{1-v}{\delta}, \qquad \delta=\ln(1+i).

The last equality uses $e^{-\delta}=v$. If the same benefit is paid at the end of the year of death, its expected present value is $Bqv$: the probability of death is still $q$, but every payment is made at time 1. Thus, with positive interest and a positive death probability, immediate payment has the higher value because $v^t$ is at least $v$ for $0\leq t\leq1$, and is strictly greater when $t<1$.

Worked Example: Death Benefit Paid at Death or at Year-End

A hypothetical policy pays $10,000 if the insured dies during the next year. The annual death probability is 0.10, and deaths are uniformly distributed within the year conditional on death. The effective annual valuation interest rate is $i=e^{0.1}-1$, approximately 10.52%, so $\delta=\ln(1+i)=0.1$. Find the expected present value if the benefit is paid (a) immediately at death and (b) at the end of the year of death.

Method. Use the UDD integral for a benefit paid at death and a one-year discount factor for a benefit paid at year-end.

(a) Paid immediately. Here $Bq=10{,}000(0.10)=1{,}000$, and $e^{-0.1}=0.9048374$. Therefore,

1{,}000\frac{1-e^{-0.1}}{0.1} = 1{,}000\frac{1-0.9048374}{0.1} = 1{,}000(0.9516258) = \$951.63.

(b) Paid at year-end. The payment is discounted for one year, so

1{,}000v = 1{,}000e^{-0.1} = \$904.84.

Check. The immediate-payment value exceeds the year-end value, as expected because some deaths occur before the end of the year. A common mistake is to discount every immediate payment by a full year; that produces the year-end value, not the value of payment at death.

Survival-Contingent Premiums at Different Dates

A premium’s date can also affect its probability of being paid. A premium payable at the start of a year may be due if the policyholder is alive at that date; a premium payable at year-end may require survival through the year. With constant annual death probability $q$, write $p=1-q$ for the one-year survival probability. If premiums are payable at the beginning of years 1 through $n$, and the first is payable immediately, the present value under annual payment dates is

P\sum_{k=0}^{n-1}v^k\,{}_kp_x.

Here ${}_kp_x$ is the probability of surviving $k$ years from age $x$. If instead each premium is payable at the end of the year only if the insured is alive then, the corresponding value is

P\sum_{k=1}^{n}v^k\,{}_kp_x.

These expressions assume that each payment is conditional on survival to its stated date and that premiums cease after death. The first formula includes an immediate payment with ${}_0p_x=1$; the second does not. A problem can specify a different premium condition, so the survival probability must match the actual payment date and contract wording.

Worked Example: Beginning- versus End-of-Year Premiums

A hypothetical contract requires a premium of $1,200 for three years while the policyholder is alive. The annual death probability is 0.10 in each year, independently of earlier survival, and the effective annual valuation interest rate is 5%. Compare the present value if premiums are due at the beginning of each year, including an immediate first premium, with the present value if each premium is due at year-end only when the policyholder is alive at that year-end.

Method. The annual survival probability is $p=0.90$, and $v=1/1.05=0.95238095$. Use the survival probability to each premium date as well as the discount factor to that date.

Beginning of each year. The three possible payment dates are $t=0,1,2$. The survival probabilities to those dates are $1,0.90,0.90^2=0.81$. The present value is

1{,}200\left[1+(0.90)(0.95238095)+(0.81)(0.95238095)^2\right] = 1{,}200(1+0.85714286+0.73469388) = \$3{,}110.20.

End of each year. The payment dates are $t=1,2,3$, with survival probabilities $0.90,0.81,0.729$. The present value is

1{,}200\left[(0.90)(0.95238095)+(0.81)(0.95238095)^2+(0.729)(0.95238095)^3\right]

Since $0.729(0.95238095)^3=0.62973761$, this equals

1{,}200(0.85714286+0.73469388+0.62973761) = \$2{,}665.89.

Check. The beginning-of-year schedule includes a certain immediate premium and places the other premiums one year earlier than the corresponding end-of-year payments. Its value is therefore higher. The common mistake is to use the same survival probability for payments at times 0 and 1, or to discount a time-0 premium by one year.

A Practical Timeline for Valuation

For a mixed equity-linked valuation, keep the fund projection and the present-value calculation as distinct steps. The fund projection determines the amount available at a transaction date under the stated investment returns and charges. The actuarial valuation then discounts contractual payments and weights them by the probabilities of the events that trigger them. A payment taken from the fund may reduce the fund without being an additional insurer benefit; avoid counting the same amount twice.

Mark every date.
Record the valuation date, premium dates, charge dates, return-crediting dates, death or maturity date, and benefit payment date.

Project amounts to the transaction date.
Apply investment returns only over periods in which the amount is in the fund, and apply deductions at their specified dates.

Apply the event condition.
For each premium or benefit, identify whether it requires survival to a date, death in an interval, or survival to maturity.

Discount from the payment date.
Use $v^t$ for a payment at time $t$, or integrate over the payment time when it varies within a period under a stated timing distribution.

Check direction and units.
An earlier payment has a larger present value at positive interest, all else equal. Confirm that probabilities and time periods use the same annual basis as the interest rate.

Common Timing Traps

Applying a full year of return to an end-of-year premium. A premium first invested at year-end does not earn the return for the year that has just ended.
Charging at the wrong point in the projection. A beginning-of-year charge reduces the amount earning investment return; an end-of-year charge does not reduce that year’s growth.
Using annual death probability as though all deaths occur at year-end. That is appropriate for a year-end benefit, not for a benefit paid at death under a within-year death assumption.
Discounting without checking survival conditions. A premium payable at the beginning of a year and one payable at its end can have different probabilities of being due.
Confusing investment return with valuation interest. The rate used to project a policy fund and the rate used to discount expected cash flows have different roles. Use the basis specified for each calculation.

Key takeaway: Payment timing affects both the amount available in an equity-linked fund and the present value of a contingent cash flow. Put the transaction on a timeline, apply growth and discounting over the correct interval, and match each event probability to the date when payment is actually due. This timing discipline is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the relevant payment dates before calculating. Round dollar answers to the nearest cent.

A fund of $5,000 receives a $600 premium at the start of the year, earns an effective annual return of 4%, and has a $40 charge deducted at year-end. Find the ending fund. (Answer: $5,784.00)
A fund of $5,000 earns 4% for the year. At year-end, a $40 charge is deducted and then a $600 premium is added. Find the ending fund. (Answer: $5,760.00)
A hypothetical $5,000 death benefit is payable on death during the next year. The annual death probability is 0.20, deaths are uniform within the year conditional on death, and $i=e^{0.1}-1$. Find its expected present value if paid immediately at death and if paid at year-end. (Answer: Immediately, $951.63; year-end, $904.84)
A premium of $1,000 is payable for two years while the policyholder is alive. Annual death probability is 0.20, and the effective annual valuation interest rate is 5%. Find the present value when premiums are due at times 0 and 1, and when instead they are due at times 1 and 2 only if alive on each payment date. (Answer: Beginning-of-year schedule, $1,761.90; end-of-year schedule, $1,342.40)
A guaranteed maturity benefit of $10,000 is paid either at time 2 or, under a delayed-payment term, at time 3. At an effective annual valuation rate of 10%, find both present values and the reduction caused by the delay. (Answer: At time 2, $8,264.46; at time 3, $7,513.15; reduction, $751.31)

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