Valuation-Time Cash-Flow Conventions

Why valuation-time conventions matter

A reserve is a value at a particular time, so every future cash flow must be located relative to that time. A payment due immediately at the valuation date is not discounted. A payment at the end of the next policy year is discounted for one year, even if it is described as payable “at duration 1.” The amount of a reserve can therefore change when the valuation instant moves from just before to just after a scheduled payment.

This tutorial focuses on assigning cash flows to times consistently. It does not change the contractual amount or probability of a payment; it clarifies when the payment enters the valuation. For any calculation, state the valuation instant, the cash-flow dates, and the interest convention before discounting.

Definition: Let $s$ be the valuation time, measured from the policy issue date, and let $t_j$ be the time of a future cash flow. Its elapsed time from valuation is $t_j-s$. A cash flow at $t_j=s$ is immediate and has discount factor 1. A cash flow at $t_j>s$ is future and is discounted for the elapsed time $t_j-s$. A prospective reserve is the expected present value of future outgo less the expected present value of future income, conditional on the contract information at time $s$, using a consistent valuation basis.

Policy duration and elapsed time

Policy duration is usually counted from issue. The valuation time and a payment time may both be stated as policy durations, but the discount period is their difference. For example, a payment at duration 4 valued at duration 2 is two years away, not four years away. Writing both times on a timeline prevents this common confusion.

Use the effective annual interest rate $i$ and $v=1/(1+i)$. If all cash flows occur at integer policy durations and the effective annual rate applies each year, a certain amount $C_j$ payable at duration $t_j$ has value at duration $s$

C_jv^{t_j-s}.

For fractional times, an additional convention is needed: an annual effective rate alone specifies accumulation over a full year, not every possible fraction of a year. When a problem assumes a constant force of interest $\delta=\ln(1+i)$, accumulation for $t$ years is $(1+i)^t$, and the discount factor is $(1+i)^{-t}$. State this convention before using fractional-year discounting.

Timing rule: Include only cash flows strictly after the valuation instant in a prospective value. A cash flow paid at the valuation instant belongs in the value immediately before payment, but not in the value immediately after payment. When a cash flow is income, subtract its discounted value from future outgo; when it is outgo, add its discounted value.

Expected values when payment depends on an event

If a payment is made only when an event occurs, the expected value includes both its amount and the probability of payment. Suppose possible payment dates are $t_1,\ldots,t_n$, the payment at date $t_j$ is $C_j$, and the probability of that payment, conditional on the information at valuation time $s$, is $p_j$. If at most one of these payments occurs, its expected present value is

\sum_{j=1}^n p_j C_j v^{t_j-s}.

The formula follows from the definition of expectation. Let $I_j$ equal 1 if the payment at $t_j$ occurs and 0 otherwise. The present value of that payment is $I_jC_jv^{t_j-s}$, so its expectation is $E[I_j]C_jv^{t_j-s}=p_jC_jv^{t_j-s}$. Adding the expected values across possible payment dates gives the displayed sum. If multiple payments can occur, the same addition applies to each payment, with its own probability and amount.

The probabilities must be conditional on the valuation information. For example, a reserve at duration 1 for a policy known to be in force then uses probabilities measured from that condition, not unconditional probabilities from issue. The discount period is also measured from duration 1.

Fix the valuation instant.
Record the policy duration and whether the instant is just before or just after any cash flow due then.

Place each future cash flow.
Record its policy duration and classify it as immediate, period-end, or at another specified time.

Measure elapsed time and probability.
Subtract the valuation time from each payment time, and use event probabilities conditional on the valuation information.

Discount and apply signs.
Discount for the elapsed time, add expected outgo, and subtract expected income.

Worked Example: Death Benefits at Year-End

A hypothetical three-year term policy pays $1,000 at the end of the policy year of death. The conditional annual death probabilities for a life alive at the start of each year are $q_1=0.10$, $q_2=0.20$, and $q_3=0.25$. Assume annual effective interest $i=4\%$, no other cash flows, and no fractional-year payments. Find the expected present value at issue and the reserve at duration 1, conditional on the life being alive then. The method is to assign each claim to the end of its death year and measure discounting from the valuation date.

At issue. The unconditional probabilities of death in years 1, 2, and 3 are, respectively,

0.10,\qquad (1-0.10)(0.20)=0.18,\qquad (1-0.10)(1-0.20)(0.25)=0.18.

The payments occur at durations 1, 2, and 3. Since $v=1/1.04$, the expected present value is

1000\left(\frac{0.10}{1.04} +\frac{0.18}{1.0816} +\frac{0.18}{1.124864}\right).

The three discounted probability contributions are $0.096153846154$, $0.166420118343$, and $0.160019344558$. Their sum is $0.422593309055$, so the expected present value is $1000(0.422593309055)=\$422.593309$, or $422.59 to the nearest cent.

At duration 1, given survival. From the condition of being alive at duration 1, the probability of death in year 2 is $0.20$. The probability of surviving year 2 and dying in year 3 is $(1-0.20)(0.25)=0.20$. The payment dates are one and two years after this valuation time, so

1000\left(\frac{0.20}{1.04}+\frac{0.20}{1.0816}\right) =1000(0.377218934911) =\$377.218935.

The reserve is $377.22 to the nearest cent. This conditional calculation is checked by using only probabilities after duration 1 and discount periods measured from duration 1. The common mistake is to reuse the unconditional year-2 and year-3 probabilities from issue, or to discount the remaining claims for two and three years from issue.

Payments at the valuation instant and between anniversaries

The same timeline rule applies to scheduled payments that do not depend on mortality. In particular, an immediate payment is included at full amount in the value just before it is made. Once it has been paid, it is no longer a future cash flow. A fractional-year payment is discounted for the time remaining from valuation, provided the interest convention supports fractional-year discounting.

Worked Example: An Immediate Payment and Fractional-Year Payments

A hypothetical contract has certain outgo of $250 immediately, $400 at 0.5 years, and $600 at 1.5 years, all measured from the valuation time. Assume annual effective interest $i=5\%$ and a constant force of interest $\delta=\ln(1.05)$ for fractional-year discounting. No mortality, decrements, or other cash flows are modeled. Find the value immediately before and immediately after the $250 payment. The method is to discount each later payment by its exact elapsed time and include the immediate amount only before it is paid.

The 0.5-year discount factor is $(1.05)^{-0.5}=0.975900072949$, and the 1.5-year discount factor is $(1.05)^{-1.5}=0.929428640903$. The later payments therefore have values

400(0.975900072949)=\$390.360029, \qquad 600(0.929428640903)=\$557.657185.

Their total value is $\$390.360029+\$557.657185=\$948.017214$. Immediately before the payment, the reserve is $\$250+\$948.017214=\$1,198.017214$, or $1,198.02 to the nearest cent. Immediately after payment, the remaining reserve is $\$948.017214$, or $948.02. The difference is exactly the $250 paid at that instant, before rounding. The common mistake is to discount the immediate payment or to include it again in the after-payment value.

Premiums due at a duration

Premium timing deserves particular care because a premium is income from the insurer’s perspective and reduces a prospective net liability. A premium due at the valuation instant is subtracted in the value just before it is collected. It is not a future premium in the value just after collection. The direction of the reserve change is therefore opposite to that of an immediate benefit or expense.

Worked Example: Reserve Just Before and After a Premium

A hypothetical contract has a guaranteed benefit of $1,000 at duration 3 and premiums of $200 at durations 1 and 2. The valuation time is duration 1. Assume annual effective interest $i=4\%$, the contract remains in force with certainty, and there are no expenses or other cash flows. Find the prospective reserve just before and just after the duration-1 premium. The method is to count the due premium before collection, then remove it from future income after collection.

At duration 1, the benefit is two years away, the duration-1 premium is immediate, and the duration-2 premium is one year away. With $v=1/1.04$, the value just before collection is

1000v^2-200-200v =\frac{1000}{1.0816}-200-\frac{200}{1.04}.

The benefit value is $1000/1.0816=\$924.556213$, and the future premium value is $200/1.04=\$192.307692$. Thus the reserve just before collection is $\$924.556213-\$200-\$192.307692=\$532.248521$, or $532.25 to the nearest cent.

Immediately after collection, the duration-1 premium is no longer future income. The reserve becomes

1000v^2-200v =\$924.556213-\$192.307692 =\$732.248521.

This is $732.25 to the nearest cent. The increase of $200 equals the premium just collected. The result is checked by comparing the two expressions: the only changed item is the immediate $200 premium, which is subtracted before collection and omitted afterward. The common mistake is to subtract that premium in both values, or to discount it despite its being due immediately before the valuation instant.

Common timing traps

Before finalizing a reserve calculation, check the following distinctions. They often matter more than the arithmetic:

Policy duration versus elapsed time: a payment at duration 5 valued at duration 2 is three years away.
Beginning versus end of period: a payment at the beginning of the next year is immediate at the current year-end; a payment at the end of that next year is one year away.
Before versus after payment: include a due amount at full value just before payment, and omit it just after payment.
Conditional versus unconditional probabilities: at a later valuation duration, use probabilities conditional on the contract’s information at that time.
Fractional-year assumptions: do not infer a fractional discount factor unless the interest convention specifies how interest accumulates within the year.

A reliable solution labels the valuation instant explicitly, for example, “duration 1, immediately after the premium.” Then each cash flow can be classified as already paid, immediate but unpaid, or future. This convention supports consistent prospective valuation across different contract states without changing the underlying payment terms.

Key takeaway: A cash flow’s discount period is the time from the valuation instant to its payment date. Assign its sign, amount, probability, and timing separately; only then calculate its value.

Check Your Understanding

For each problem, state the valuation time and use the timing convention specified. Round money to the nearest cent.

A certain $500 payment is due 0.5 years after valuation. Annual effective interest is 4%, with constant force $\delta=\ln(1.04)$ for fractional times. Find its value at valuation. (Answer: $490.29)
At duration 2, immediately before any payment due then, a contract has outgo of $200 due immediately and $900 at duration 4, plus income of $300 at duration 3. Assume annual effective interest of 6%, certainty of all cash flows, and no other cash flows. Find the prospective net value. (Answer: $717.98)
A hypothetical contract pays a guaranteed benefit of $1,000 at duration 3 and receives premiums of $250 at durations 1 and 2. At duration 1, assume annual effective interest of 5%, certainty of continued coverage, and no other cash flows. Find the net reserve immediately before and immediately after the duration-1 premium. (Answer: Before $418.93; after $668.93)
A three-year term policy pays $500 at the end of the year of death. Conditional annual death probabilities for a life alive at the start of each year are $0.10, 0.20, 0.25$. At duration 1, the life is known to be alive. With annual effective interest of 4%, find the reserve at duration 1. (Answer: $188.61)
A certain $300 payment is due immediately at the valuation time, and a certain $700 payment is due one year later. With annual effective interest of 4%, find the value immediately before the $300 payment. (Answer: $973.08)

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