Catching Up After Missed Loan Payments

Bringing a Delinquent Loan Current

When installments are missed, the amount needed to bring a loan current depends on when each installment was due and on what happens to the future payment schedule. A catch-up lump sum can pay the accumulated arrears while the borrower resumes the original installments. Alternatively, the lender and borrower can agree to replace the remaining installments with a revised schedule that also accounts for those arrears.

The valuation date is essential. An installment due earlier has accumulated interest for longer than one due more recently. For a revised schedule, the cure-date value of the arrears must be combined with the value of future payments; counting arrears alone does not measure the full obligation being rescheduled.

Definition: The cure date is the date on which the delinquency is valued and the proposed cure takes effect. Arrears at the cure date are the accumulated values, on that date, of installments that remain unpaid. Unless stated otherwise, assume the loan uses a fixed contractual interest rate, interest compounds once per installment period, no fees or penalties apply, and the valuation uses that contractual rate.

For clarity, examples below place the cure date immediately after an installment due date. An installment missed on that date has accumulated zero periods of interest by the cure date. If a contract specifies default interest, fees, or a different accrual convention, those terms must be included instead; the simple formulas here do not account for them.

Accumulating Arrears to the Cure Date

Let $i$ be the effective interest rate per installment period. Suppose installments $R_k$ due at period $k$ are unpaid, and the cure date is period $c$. An installment due at $k$ accumulates for $c-k$ complete periods. The total accumulated arrears are:

A_c=\sum_{k\ \text{missed}} R_k(1+i)^{c-k}

For a constant installment $R$, this becomes $A_c=R\sum_{k\ \text{missed}}(1+i)^{c-k}$. If the quoted rate is nominal annual $i^{(m)}$, convertible $m$ times per year, and installments are monthly with $m=12$, the effective rate per month is $i^{(12)}/12$. Use the rate per payment period, not the annual rate directly.

Worked Example: Lump-Sum Cure With Two Missed Installments

A hypothetical loan has monthly installments of $500. Its contractual rate is nominal annual $i^{(12)}=6\%$, convertible monthly. The installments due at the ends of months 5 and 6 were missed. Find the lump sum needed immediately after the month-6 due date to pay the arrears, assuming the borrower will resume the original schedule at month 7. Ignore fees and penalties.

Method. Convert the rate to the monthly rate, then accumulate each missed payment to the cure date. The month-5 installment accrues for one month; the month-6 installment accrues for zero months.

i=\frac{0.06}{12}=0.005

A_6=500(1.005)^{6-5}+500(1.005)^{6-6} =500(1.005)+500 =502.50+500.00 =\$1{,}002.50

The catch-up lump sum is $1,002.50. The borrower then resumes the original installments beginning at month 7. This sum cures only the arrears; it is not the value of the entire remaining loan.

Check. The month-5 installment has one month of interest, or $2.50, and the month-6 installment has none. Thus the total is $1,000.00 of missed installments plus $2.50 interest.

Revising the Remaining Installments

A different calculation is needed if the arrears are not paid as a separate lump sum, but are instead incorporated into a new level installment schedule. At the cure date, value the arrears and the future contractual installments on the same date. If $N$ future installments of amount $R$ remain, each due at the end of a period, their cure-date present value is $R a_{\overline{N}|}$, where $a_{\overline{N}|}$ is calculated at the effective rate per period $i$.

\text{Cure-date value of obligation}=A_c+R a_{\overline{N}|}

If the replacement schedule has $N$ equal installments $R^*$, with the first one period after the cure date, its cure-date value is $R^*a_{\overline{N}|}$. Equating the two values gives:

R^*a_{\overline{N}|}=A_c+R a_{\overline{N}|} \qquad\Longrightarrow\qquad R^*=R+\frac{A_c}{a_{\overline{N}|}}

This assumes that the new installments have the same timing as the original future installments and are valued at the same contractual rate. If the first revised installment is due immediately, or the schedule uses a different number of dates or rate, its value factor must change accordingly.

Why the Value Equation Works

The formula follows from comparing two payment streams at a common date. At the cure date, the arrears have accumulated to $A_c$. Each future original installment $R$ is discounted from its due date to the cure date; the sum of those discounted values is $R a_{\overline{N}|}$. Therefore the original unpaid obligations have cure-date value $A_c+R a_{\overline{N}|}$.

A replacement stream of $N$ level installments $R^*$, due at the same future dates, has value $R^*a_{\overline{N}|}$. A value-equivalent restructuring must satisfy:

R^*a_{\overline{N}|}=A_c+R a_{\overline{N}|}

Since $a_{\overline{N}|}>0$, division by this factor gives $R^*=R+A_c/a_{\overline{N}|}$. The result also explains why the revised installment exceeds $R$ whenever the arrears are positive: those arrears must be funded by the replacement payments as well as the original future obligations.

Worked Example: Recasting Payments After a Missed Pair

A hypothetical loan has monthly installments of $400 and a monthly effective interest rate of 1%. The borrower misses the installments due at the ends of months 5 and 6. Immediately after the month-6 due date, the lender agrees to include the arrears in six equal installments due at the ends of months 7 through 12. Find the revised installment, to the nearest cent. Assume no fees or penalties.

Method. First accumulate the missed installments to month 6. Then equate the cure-date value of arrears plus the six original future installments to the value of the six revised installments.

The month-5 installment accrues for one month and the month-6 installment for zero months:

A_6=400(1.01)+400=\$804.00

The six-payment annuity factor at 1% per month is:

a_{\overline{6}|}=\frac{1-(1.01)^{-6}}{0.01}=5.795476\ldots

The original future installments have cure-date value $400(5.795476\ldots)=\$2{,}318.19$. Including arrears, the total obligation valued at month 6 is:

804.00+400(5.795476\ldots)=\$3{,}122.19

Let $R^*$ be the revised installment. Its six-payment value must equal that obligation:

R^*=\frac{804.00+400(5.795476\ldots)}{5.795476\ldots} =538.7289\ldots

Rounded to the nearest cent, the revised installment is $538.73 per month. Equivalently, the arrears add $804/5.795476\ldots=\$138.73$ per installment to the original $400.00. The value check is $538.73 multiplied by the six-payment factor, or approximately $3,122.20; the one-cent difference from the unrounded value is due to rounding the installment.

Handling Missed Payments on Different Dates

When missed installments are not consecutive, do not multiply the number of missed installments by one common accumulation factor. Each payment has its own elapsed time. A useful way to organize the calculation is to list the due date, the number of periods to the cure date, and the accumulated amount for each payment.

Worked Example: Irregular Arrears and a Revised Schedule

A hypothetical loan has monthly installments of $250, an effective monthly interest rate of 2%, and four original installments remaining after the cure date. The borrower missed payments due at the ends of months 3 and 5. The cure date is immediately after the month-6 due date. The lender will include all arrears in four equal installments due at the ends of months 7 through 10. Find the revised installment to the nearest cent. Assume no fees or penalties.

Method. Accumulate the month-3 payment for three periods and the month-5 payment for one period. Then spread the arrears over the four remaining dates using the four-payment annuity factor.

A_6=250(1.02)^3+250(1.02) =250(1.061208)+255 =265.302+255 =\$520.302

The four-payment factor is:

a_{\overline{4}|}=\frac{1-(1.02)^{-4}}{0.02}=3.807729\ldots

The cure-date value of the four original future installments is $250(3.807729\ldots)=\$951.93$. Thus the full obligation to be represented by the revised schedule is:

520.302+250(3.807729\ldots)=\$1{,}472.23

The revised installment is:

R^*=\frac{520.302+250(3.807729\ldots)}{3.807729\ldots} =386.6436\ldots

Rounded to the nearest cent, the installment is $386.64 per month. This uses three periods of accumulation for the month-3 payment, not the same one-period accumulation applied to the month-5 payment.

Common Mistake: Giving Every Arrear the Same Accrual Period

A frequent error is to count the number of missed installments and give every one the same number of interest periods. In the second example, both installments are missed, but they are not both outstanding for two months at the cure date. The month-5 payment has one month to accrue, while the month-6 payment has none.

\text{Wrong: }400\bigl((1.01)^2+1.01\bigr)=\$812.04 \qquad \text{Correct: }400(1.01)+400=\$804.00

The wrong calculation overstates arrears by $8.04. The correction is to count elapsed periods from each due date to the cure date. A related trap is confusing the lump sum needed to pay arrears with the full value of a revised obligation. The latter includes both accumulated arrears and the cure-date value of future payments.

A Reliable Valuation Sequence

Set the cure date and payment periods.
Identify exactly when each missed payment was due and when the cure is valued.

Convert the contractual rate.
Use the effective rate per installment period, consistent with the loan's compounding convention.

Accumulate each missed payment.
Apply $(1+i)^{c-k}$ separately to an installment due at period $k$, with $c$ the cure period.

Choose the cure structure.
For a lump-sum cure while original payments resume, the lump sum is the arrears. For a revised schedule, add the cure-date value of future installments.

Check dates and value.
Verify that the replacement payments occur on the stated dates and that their cure-date value matches the obligation being replaced.

These calculations are useful whenever a loan is brought current under its existing terms or its remaining installments are reset by agreement. The key discipline is to value all amounts on the same date and to keep the two cure structures distinct. These methods are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, assume no fees or penalties and use the stated effective rate per month.

A borrower misses installments of $300 due at the ends of months 4 and 5. The cure date is immediately after the month-5 due date, and the monthly effective rate is 1%. Find the lump sum that pays the arrears. (Answer: $603.00)
Installments of $250 due at the ends of months 2, 3, and 4 are missed. The cure date is immediately after the month-4 due date, and the monthly effective rate is 2%. Find the arrears at the cure date. (Answer: $765.10)
A borrower misses two $400 installments due at months 5 and 6. At the cure date immediately after month 6, the monthly effective rate is 1%, and six future original installments remain. Find the level installment for a revised six-payment schedule that includes the arrears. (Answer: $538.73 per month)
Installments of $600 due at months 7 and 9 are missed. The cure date is immediately after month 10, and the monthly effective rate is 0.5%. Find the lump sum required to pay the arrears. (Answer: $1,212.05)
Two installments of $200 due at months 3 and 4 are missed. Immediately after month 4, the monthly effective rate is 2%, and five future original installments of $200 remain. Find the level installment for a revised five-payment schedule that includes the arrears. (Answer: $285.71 per month)

What You'll Learn