Missed Payments, Arrears, and Catch-Up Payments

Why Missed Payments Require a Common Date

A missed loan installment does not disappear from the cash-flow record. It remains an amount owed, and interest may accrue on it. To measure the resulting debt, first bring every unpaid amount to the same valuation date. Then, if the loan continues, include the value at that date of installments still due in the future. Mixing amounts from different dates is one of the most common sources of error in loan problems.

This tutorial assumes that a missed installment remains unpaid and accumulates at the stated contractual rate. Unless a problem says otherwise, assume there are no late fees, penalties, or separate arrears rate. If the contract specifies a different rate for overdue amounts, use that rate for the arrears and the loan rate for the remaining scheduled debt. The methods here are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Arrears at date $T$ are the value at $T$ of installments that were due earlier but remain unpaid, accumulated using the applicable interest terms. The outstanding balance at $T$ is the total debt at that date. When a loan continues under its original schedule, it can be found by adding arrears at $T$ to the value at $T$ of the future scheduled installments.

The date $T$ must be specified precisely: for example, immediately before or immediately after a payment due at $T$. A payment made at $T$ is subtracted from the balance immediately after that payment. A missed payment due at $T$ is not subtracted; it becomes part of the unpaid debt.

Accumulating Unpaid Installments

With a constant effective annual rate $i$, an unpaid amount $C$ due at time $t$ accumulates to $C(1+i)^{T-t}$ at a later integer year $T$. With different effective annual rates in different years, multiply by each year's accumulation factor over the applicable interval. The accumulation is forward in time; discounting an arrear to an earlier date would not measure the amount owed at $T$.

\text{Arrears at }T = \sum_{\text{unpaid due dates }t<T} C_t\prod_{j=t}^{T-1}(1+i_j)

Here $C_t$ is the installment due at year $t$, and $i_j$ is the effective annual rate from year $j$ to year $j+1$. For a constant rate, every $i_j$ equals $i$, so the product reduces to $(1+i)^{T-t}$. If interest is applied over fractional years, the accumulation convention for those fractions must be stated and followed; do not silently treat a fraction of a year as a whole period.

Worked Example: Balance After a Missed Installment

A loan is scheduled to be repaid by four annual installments of $500 due at years 1 through 4. The effective annual rate is 8%. The year-1 installment is missed, while the year-2 and year-3 installments are paid on time. Find the outstanding balance immediately after the year-3 payment.

Method. Use a prospective common-date valuation at year 3. The unpaid year-1 installment accumulates to year 3; the only future scheduled installment is due at year 4 and must be discounted one year to year 3.

The year-1 installment has accumulated for two years:

$$ 500(1.08)^2=500(1.1664)=583.20. $$

The year-4 installment has value at year 3:

\frac{500}{1.08}=462.962962\ldots

Therefore, immediately after the year-3 payment, the balance is:

583.20+462.962962\ldots=1{,}046.162962\ldots

Rounded to the nearest cent, the balance is $1,046.16. The year-2 and year-3 installments are not included because they were paid. The year-3 payment has already occurred at the valuation date, so it is neither a future installment nor an arrear. This common-date method avoids treating a paid installment as unpaid.

Check. At year 3, the balance consists of exactly two obligations: $583.20 of accumulated arrears and the year-4 payment valued at $462.96. Their sum is $1,046.16.

Changing Rates and Multiple Missed Payments

When rates change, each unpaid amount follows the rate path from its own due date to the valuation date. It is not generally correct to apply the final rate to every year, or to apply one average rate without justification. The calculation is a product of accumulation factors, with the number of factors determined by the time interval.

Worked Example: Arrears Across a Rate Change

An installment of $200 due at year 1 and an installment of $150 due at year 3 are both missed. The effective annual rate is 4% from year 1 to year 3 and 6% from year 3 to year 5. Find the arrears at year 5.

Method. Accumulate each missed installment separately to year 5. The $200 installment crosses two years at 4% and then two years at 6%. The $150 installment crosses only the two years at 6%.

For the installment due at year 1:

$$ 200(1.04)^2(1.06)^2 =200(1.0816)(1.1236) =200(1.21528576) =243.057152. $$

For the installment due at year 3:

$$ 150(1.06)^2=150(1.1236)=168.54. $$

The total arrears at year 5 are:

$$ 243.057152+168.54=411.597152. $$

Rounded to the nearest cent, the arrears are $411.60. This is the arrears amount only; if there are future scheduled installments, their value at year 5 must also be included to find the total outstanding balance.

Check. The first installment uses four annual factors in total, two at each rate; the second uses two factors, both at 6%. The calculated year-5 values are therefore on the same date before they are added.

Catch-Up Payments and Revised Schedules

A catch-up payment is an amount paid to clear some or all of the accumulated arrears. If a borrower pays the current scheduled installment and also pays a separate catch-up amount, the catch-up amount is usually the arrears at that payment date. If the borrower instead makes one combined payment, identify whether that amount includes the current installment before setting up the equation.

The formula for arrears follows directly from compound accumulation. Over one year, an unpaid amount $C$ becomes $C(1+i)$. Over a second year at the same rate, it becomes $C(1+i)(1+i)=C(1+i)^2$. Repeating this one-year accumulation gives $C(1+i)^n$ after $n$ years. With changing rates, each one-year multiplication uses that year's rate, giving the product in the arrears formula above.

Worked Example: Catch-Up or Reamortization

A loan has five scheduled annual installments of $2,500 at years 1 through 5, and the effective annual rate is 5%. The year-1 installment is paid, the year-2 installment is missed, and the year-3 installment is paid. Immediately after the year-3 payment, determine (a) the separate catch-up payment that clears the arrears and leaves the original future schedule unchanged, and (b) the level annual payment at years 4, 5, and 6 that would instead retire the entire balance.

Method. First find the balance immediately after year 3 by adding the missed year-2 installment accumulated to year 3 to the value of the original year-4 and year-5 installments at year 3. For part (a), pay the arrears only. For part (b), replace all remaining obligations with three equal payments.

The arrears due to the missed year-2 installment are:

$$ 2{,}500(1.05)=2{,}625. $$

At year 3, the original year-4 and year-5 installments have present value:

\frac{2{,}500}{1.05}+\frac{2{,}500}{(1.05)^2} =2{,}380.952381\ldots+2{,}267.573696\ldots =4{,}648.526077\ldots

Thus the balance immediately after the year-3 payment is:

2{,}625+4{,}648.526077\ldots=7{,}273.526077\ldots

For part (a), the separate catch-up payment is $2,625.00. Paying that amount clears the arrears; the original year-4 and year-5 installments remain due. The catch-up is not the entire outstanding balance.

For part (b), let $X$ be the replacement annual payment at years 4, 5, and 6. At year 3, those three payments have value $X a_{\overline{3}|}$, where the annual effective rate is 5%. Calculate the annuity factor:

a_{\overline{3}|} =\frac{1-(1.05)^{-3}}{0.05} =\frac{1-0.863837599}{0.05} =2.72324802.

Equate the value of the revised schedule to the year-3 balance:

X(2.72324802)=7{,}273.526077\ldots, \qquad X=2{,}670.901\ldots

Rounded to the nearest cent, the revised annual payment is $2,670.90. This revised payment covers the arrears and replaces the future installments, so adding the $2,625 catch-up to it would count the arrears twice.

Check. The revised payments have value at year 3 of approximately $2{,}670.901(2.72324802)=7{,}273.526$, matching the calculated balance. In the catch-up option, $2,625 clears the arrears while the original $2,500 payments at years 4 and 5 are still required.

A Reliable Ledger and Common Traps

A common-date ledger makes the timing visible. List each relevant amount, state whether it is paid, missed, or still future, and move each amount to the chosen date exactly once. The distinction between arrears and total balance is particularly important: arrears are past-due amounts accumulated forward, while the total balance may also include future contractual payments.

Fix the valuation date.
Specify whether the balance is measured immediately before or immediately after any payment due on that date.

Classify each installment.
Mark it as paid, missed and still unpaid, or due in the future. Do not include an installment already paid.

Move amounts to the date.
Accumulate arrears forward and discount future payments back. Apply each rate only over its stated interval.

Interpret the requested amount.
Use arrears for a catch-up that clears past-due installments; use the full balance when replacing all remaining obligations.

The following mistakes are worth checking explicitly:

Using the wrong number of periods. A payment due at year 1 accumulates for two years to year 3, not three years. Count the intervals between dates.
Valuing cash flows on different dates. Add amounts only after each has been accumulated or discounted to the same date.
Using the latest rate for the entire history. A rate change affects only the years in which that rate applies.
Confusing arrears with the outstanding balance. If future payments remain, their value must be included in the full balance.
Counting a catch-up twice. If a revised payment schedule is set to amortize the entire balance, do not add the arrears again unless the revised schedule explicitly excludes them.

For the four-installment loan in the first example, a frequent wrong calculation is to report $500/1.08+500/(1.08)^2+500(1.08)$ as the balance after year 3. That expression includes the year-2 and year-3 installments as though they were still unpaid, even though both were paid. The correct year-3 balance is $500(1.08)^2+500/1.08=$1,046.16$: the missed year-1 installment accumulated to year 3 plus the future year-4 installment discounted to year 3.

Key takeaway: Choose one valuation date, account for every installment according to its payment status, and apply the correct accumulation or discount factors. A catch-up payment clears arrears; a revised schedule must be valued against the entire balance it is intended to replace.

Check Your Understanding

For each problem, identify the valuation date and keep arrears separate from future payments where relevant.

A $840 installment due at year 2 is missed. The effective annual rate is 6%. Find the arrears at year 5. (Answer: $1,000.45)
Installments of $300 at year 1 and $400 at year 2 are missed. The effective annual rate is 5% throughout. Find the arrears at year 4. (Answer: $788.29)
A $900 installment due at year 2 remains unpaid. The effective annual rate is 4%. What separate catch-up payment at year 5 clears it? (Answer: $1,012.38)
A $600 installment due at year 1 and a $700 installment due at year 3 are missed. The effective annual rate is 3% from year 1 to year 2 and 5% from year 2 to year 4. Find the arrears at year 4. (Answer: $1,416.35)
A $1,200 installment due at year 2 is missed, and the annual effective rate is 5%. The borrower will pay the arrears separately at year 4 and continue making the regular scheduled installments. Find the separate catch-up amount at year 4. (Answer: $1,323.00)

What You'll Learn