Payments in Advance and in Arrears

Why Payment Timing Matters

A state-contingent benefit depends on the insured’s state, such as being eligible for a hypothetical disability benefit. But specifying the benefit amount is not enough to value it. The contract must also say when the state is tested and when the payment is made. A benefit paid at the start of a period may depend on the state then; a benefit paid at the end may depend on the state then, or it may be a payment already earned for eligibility earlier in the period. These rules produce different probabilities and discount factors.

This tutorial considers regular periods of equal length, with payments at their beginnings or ends. The valuation method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central discipline is to identify, for every instalment, its condition, its payment date, and its amount before calculating its expected present value.

Definition: A payment is made in advance for a period when it is paid at the period’s beginning. A payment is made in arrears when it is paid at the period’s end. These timing labels do not, by themselves, determine the state condition: a contract must specify whether eligibility is tested at the beginning, at the end, or under another rule.

Valuing an Instalment at Its Actual Payment Date

Assume a discrete-time state model with periods of one year, an effective annual interest rate $i$, and discount factor $v=1/(1+i)$. Let $B_k$ be the amount payable at time $k$, with $B_k=0$ when its stated condition is not met. Its expected present value is

\operatorname{EPV}(B_k)=v^k E[B_k].

If the amount is a fixed benefit $b_k$ payable only when the person is in eligible state $E$ at time $k$, then $B_k=b_k I(E_k)$, where $I(E_k)$ is 1 if the state condition holds and 0 otherwise. Consequently,

\operatorname{EPV}(B_k)=b_k\,P(E_k)\,v^k.

For multiple payment dates, add the expected present values of the individual payments. This uses linearity of expectation; it does not require the state at different dates to be independent. The probabilities must, however, match the state condition actually written into the contract.

Key rule: For each instalment, first identify the event that makes it payable and the date cash is paid. Then multiply the benefit by that event’s probability and by the discount factor for that payment date.

Worked Examples

Worked Example: Comparing State Tests at the Start and End of a Period

A hypothetical contract has two states: eligible $E$ and ineligible $N$. The insured begins in $E$. For each year, the transition probabilities are

P= \begin{pmatrix} 0.8 & 0.2\\ 0.3 & 0.7 \end{pmatrix},

where rows are the state at the beginning of the year and columns are the state at its end, in the order $E,N$. A benefit of $1,000 is payable for each of three years. Find the value if paid in advance when the insured is eligible at the start of each year, and if paid in arrears when the insured is eligible at the end of each year. Use an effective annual interest rate of 10%.

Method. Calculate the eligible-state probability at each relevant time, then apply $v^k$ at the actual payment time. For advance payments, the three payment dates are $t=0,1,2$; eligibility is tested at those dates. Since the insured starts in $E$, $P(E_0)=1$. The one-year probability is $P(E_1)=0.8$. For two years,

P(E_2)=P(E_1)P(E_2\mid E_1)+P(N_1)P(E_2\mid N_1) =0.8(0.8)+0.2(0.3)=0.70.

Thus the advance value is

1000\left(1+\frac{0.8}{1.1}+\frac{0.70}{1.1^2}\right) =1000(1+0.7272727+0.5785124) =\$2,305.79.

For arrears payments, the dates are $t=1,2,3$, and eligibility is tested at each year-end. We already have $P(E_1)=0.8$ and $P(E_2)=0.70$. The probability at year 3 is

$$ P(E_3)=0.70(0.8)+0.30(0.3)=0.56+0.09=0.65. $$

Therefore the arrears value is

1000\left(\frac{0.8}{1.1}+\frac{0.70}{1.1^2}+\frac{0.65}{1.1^3}\right) =1000(0.7272727+0.5785124+0.4883546) =\$1,794.14.

The values are rounded to the nearest cent only at the final step. The arrears value is lower here because payments occur later and the final payment has only a 0.65 probability of being triggered. The common mistake is to shift the advance value back one year by dividing by 1.1. That would give $\$2,305.79/1.1=\$2,096.17$, which is not the arrears value: that shift keeps the advance payment probabilities, whereas this contract tests eligibility at each period’s end.

Worked Example: Arrears for a Benefit Earned at the Period’s Start

Use the same two-state model and initial state as in the preceding example, but let the annual interest rate be 5%. A hypothetical benefit of $500 is earned for each of three coverage years if the insured is eligible at the start of that year. Compare payment in advance with payment in arrears, where each arrears instalment is paid at year-end for the eligibility already tested at the start of that year.

Method. The relevant start-of-year probabilities are $P(E_0)=1$, $P(E_1)=0.8$, and $P(E_2)=0.70$. In advance, payments are made at $t=0,1,2$, so

\operatorname{EPV}_{\mathrm{advance}} =500\left(1+\frac{0.8}{1.05}+\frac{0.70}{1.05^2}\right) =500(1+0.7619048+0.6349206) =\$1,198.41.

In arrears, the very same three eligibility events determine the instalments, but each payment is delayed by one year. The payments are at $t=1,2,3$, respectively, giving

\operatorname{EPV}_{\mathrm{arrears}} =500\left(\frac{1}{1.05}+\frac{0.8}{1.05^2}+\frac{0.70}{1.05^3}\right) =\$1,141.35.

Here the arrears value equals the advance value divided by $1.05$, since every payment is based on the same event and moved exactly one year later. Indeed, $\$1,198.41/1.05=\$1,141.34$ using the already-rounded advance value; using the unrounded value gives $\$1,141.3454$, or $\$1,141.35$. The common mistake is to substitute $P(E_1),P(E_2),P(E_3)$ for the start-of-year eligibility probabilities in the arrears calculation. That would change the contract’s eligibility test rather than merely delay payment.

Worked Example: Different Benefit Amounts and End-of-Year Eligibility

A hypothetical two-year contract starts with the insured in eligible state $E$. In each year the probabilities of moving from $E$ to $E$, and from $N$ to $E$, are 0.75 and 0.40, respectively; the remaining probabilities lead to $N$. The effective annual interest rate is 4%. The contract pays $800 at the beginning of year 1 if eligible then, and $1,200 at the beginning of year 2 if eligible then. Find the value of a separate arrears version that pays $800 at the end of year 1 and $1,200 at the end of year 2, with eligibility tested at each payment date.

Method. For the advance version, eligibility probabilities at the two payment dates $t=0,1$ are 1 and 0.75. Its value is

800+\frac{1200(0.75)}{1.04} =800+\frac{900}{1.04} =\$1,665.38.

For the arrears version, the eligible-state probability at $t=1$ is 0.75. At $t=2$, the two possible states at $t=1$ give

$$ P(E_2)=0.75(0.75)+0.25(0.40)=0.5625+0.10=0.6625. $$

The arrears payment at $t=1$ has expected amount $800(0.75)=600$; the payment at $t=2$ has expected amount $1200(0.6625)=795$. Thus

\operatorname{EPV}_{\mathrm{arrears}} =\frac{600}{1.04}+\frac{795}{1.04^2} =576.9231+735.0222 =\$1,311.95.

Amounts are rounded to the nearest cent at the final step. The check is that each expected payment is no greater than its maximum possible amount: $600 is no greater than $800, and $795 is no greater than $1,200. The common mistake is to use the advance version’s probabilities and amounts, then merely move each payment date. The arrears version instead tests eligibility at the end of each year, so its second payment uses 0.6625, not 0.75.

Why the Timing Formula Holds

For a payment of $b$ at time $k$ made only when an event $C_k$ occurs, its random present value is $b v^k I(C_k)$. Taking expectations gives

E\left[bv^k I(C_k)\right] =bv^k E[I(C_k)] =bv^kP(C_k).

The indicator has expectation equal to the probability of its event, which justifies the instalment formula. If a payment is instead made one year later for the same event and amount, its present value is $bv^{k+1}I(C_k)=v\bigl(bv^k I(C_k)\bigr)$. Therefore its expected present value is exactly $v$ times the earlier value. This result does not apply if delaying payment also changes the event that triggers it; in that case the probability may change as well.

Write the payment rule.
For each instalment, state the amount, the payment date, and the exact state condition. In particular, distinguish “eligible at the start” from “eligible at the end.”

Find the matching probability.
Use the initial state and the transition model to calculate the probability of the specified condition at the specified testing time.

Discount to the valuation date.
For a payment at time $k$, use $v^k$, where $k$ counts periods from the valuation date. Do not discount by the number of eligibility tests; discount by the payment date.

Add and check.
Sum the instalment values and check that each expected payment is consistent with its benefit amount and triggering probability.

Common Timing Traps

Using the wrong state probability. An end-of-year eligibility rule needs the probability of eligibility at the end, not automatically the probability at the beginning.
Discounting at the eligibility-test date. Discount according to when cash is paid. A benefit earned at the start but paid at year-end is discounted to year-end.
Assuming arrears always equal advance times $v$. This holds only when the same event and amount are simply shifted one period later. If the state test changes, so may the probability.
Omitting an initial advance payment. If the insured is eligible at time 0 and the contract pays immediately, that instalment has discount factor 1, not $v$.
Applying a payment rule to a whole year without reading its wording. “Paid at year-end if eligible then” and “paid at year-end for eligibility at the start” are different contingent cash flows.

The difference matters in disability income, long-term care, and other hypothetical state-contingent contracts because payment timing affects both the present value and, when the eligibility test changes, the probability of payment. In an exam valuation, write the date and condition beside every instalment before combining values.

Key takeaway: Payment timing has two separate components: the date used for discounting and the state condition used to determine whether the instalment is paid. Keep them separate, and use the contract wording to identify each one.

Check Your Understanding

For each problem, identify the relevant payment dates and state probabilities before calculating.

A hypothetical benefit of $200 is paid at $t=0$ and $t=1$ if the insured is eligible at the beginning of each year. The insured starts eligible, $P(E_1)=0.9$, and $i=10\%$. Find the expected present value. (Answer: $363.64)
A hypothetical arrears benefit of $400 is paid at $t=1$ and $t=2$ if eligible at those respective dates. The probabilities are $P(E_1)=0.7$ and $P(E_2)=0.5$, and $i=5\%$. Find the expected present value. (Answer: $448.07)
A hypothetical advance benefit of $300 is paid at $t=0,1,2$ if eligible at the start of each year. The probabilities at those dates are 1, 0.8, and 0.64, and $i=4\%$. Find the expected present value. (Answer: $708.28)
A hypothetical benefit of $1,000 is earned if eligible at the start of each of two years, with probabilities 0.75 and 0.50. Each instalment is paid at that year’s end. If $i=8\%$, find the expected present value. (Answer: $1,123.11)
A hypothetical contract pays $1,000 at $t=1$ if eligible then, and $1,000 at $t=2$ if eligible then. The probabilities are 0.8 and 0.6, and $i=10\%$. Find the value. (Answer: $1,223.14)

What You'll Learn