Why the Date of the Balance Matters
A loan balance is tied to a particular date. Immediately after an installment, the borrower owes less than immediately before that same installment. Between scheduled installments, no payment changes the balance, but interest continues to accrue. A calculation that uses the right amount on the wrong side of a payment date can therefore be wrong even when its interest arithmetic is correct.
This tutorial considers a balance date strictly between two scheduled installments. The main method is to start with the balance immediately after the earlier installment and accumulate it to the requested date. The result can be checked by valuing the payments still due after that date. The interest convention must be specified: an annual effective rate, a force of interest, or another stated accumulation rule may give different results.
Carrying the Balance Forward
Write \(g(s,t)\) for the accumulation factor from time \(s\) to time \(t\) under the stated loan interest model. If the balance immediately after a payment is \(B_k\), and there are no intervening payments, then
For example, if the loan uses a constant annual effective rate \(i\) and the contract applies compound accumulation over elapsed time measured in years, then \(g(s,t)=(1+i)^{t-s}\). If the model instead specifies a constant force of interest \(\delta\), then \(g(s,t)=e^{\delta(t-s)}\). These are different conventions; use the one given for the loan. Do not assume that an annual rate can be prorated in an arbitrary way.
To find the balance immediately before the next payment at \(t_{k+1}\), accumulate all the way to \(t_{k+1}\). If the scheduled payment there is \(R_{k+1}\), the balance immediately after that payment is the pre-payment balance less \(R_{k+1}\). The payment is not subtracted when the requested date is strictly before it.
Identify the last payment date before the requested date and whether the requested balance is before or after any payment on that date.
Use the balance immediately after the last payment, not the balance immediately before it.
Accumulate over the actual elapsed time to the balance date. If a payment occurs before the target date, account for it at its payment date and continue from the resulting balance.
If useful, compare the result with the present value at the balance date of the payments still due after that date.
Why Accumulation and Future-Payment Valuation Agree
The forward calculation and the prospective calculation are two ways to express the same loan cash flows. Suppose the balance immediately after payment \(k\) equals the value at \(t_k\) of all later payments \(R_j\), occurring at dates \(t_j\). Under a consistent interest model,
For a balance date \(t\) between \(t_k\) and \(t_{k+1}\), multiply both sides by \(g(t_k,t)\). Accumulation factors are consistent across dates, so \(g(t_k,t)g(t,t_j)=g(t_k,t_j)\). Therefore,
The left side is the balance carried forward from the last post-payment date. The right side is the value at \(t\) of all payments still due, including the next payment if \(t\) is strictly before its due date. This check applies only when the original balance and future-payment schedule use the same interest model and payment assumptions.
Worked Example: Carrying a Balance to a Date Before the Next Installment
Immediately after a payment at time \(t=2\), a loan balance is $4,000. The annual effective interest rate is 4%, and the loan uses compound fractional-year accumulation from \(t=2\) to \(t=3\). Find the balance at \(t=2.25\), before the next installment. Then find the balance immediately before and immediately after a $900 installment at \(t=3\).
Method. Carry the post-payment balance forward using the specified annual effective rate. The elapsed time from \(t=2\) to \(t=2.25\) is \(0.25\) years, so
Rounded to the nearest cent, the balance at \(t=2.25\) is $4,039.41. No installment is due at \(t=2.25\), so there is no payment to subtract.
From \(t=2\) to \(t=3\), one full year of interest accrues. Immediately before the installment, the balance is
Immediately after the $900 installment, it is \(4160-900=\) $3,260.00. As a check, the \(t=2.25\) balance carried for the remaining \(0.75\) years gives \(4039.4136\ldots(1.04)^{0.75}=4160.00\) before the payment. This example avoids the common mistake of rounding or subtracting the installment at the wrong date.
Payments Between the Balance Date and a Later Target
If the target date is after another scheduled payment, carrying the original balance straight to the target would incorrectly treat that payment as if it had not occurred. Instead, accumulate to the payment date, subtract the payment, and then accumulate the reduced balance to the target. This is simply the same balance rule applied in stages.
Worked Example: Updating the Balance Across a Scheduled Payment
Immediately after a payment at time \(t=6\), a loan balance is $7,500. The loan uses a constant force of interest \(\delta=0.08\) per year. A $1,000 payment is due at \(t=7\). Find the balance at \(t=6.4\), and the balance immediately after the payment at \(t=7\).
Method. Under a constant force of interest, the accumulation factor over \(u\) years is \(e^{0.08u}\). First, the balance at \(t=6.4\) is
At \(t=7\), immediately before the payment, the balance is
Subtract the payment at \(t=7\), not earlier: \(B(7^+)=8124.65-1000=\) $7,124.65. As a check on the staged calculation, \(7743.88e^{0.08(0.6)}\) gives approximately $8,124.65 before payment; subtracting $1,000 gives the same post-payment balance. The calculation assumes the stated force applies over both parts of the year and that no other cash flow occurs.
Checking an Interim Balance from Future Payments
A prospective check is particularly useful when the payment schedule is known. At a date strictly between installments, discount each payment that remains due to that date using the same accumulation model. Do not include payments that have already been made. If the date is immediately before an installment, that installment is still included; if it is immediately after, it is not.
Worked Example: Verifying an Interim Balance Prospectively
A loan has an annual effective rate of 10% and uses compound fractional-year accumulation. Immediately after the payment at \(t=2\), exactly two payments of $1,000 remain, due at \(t=3\) and \(t=4\). Find the balance at \(t=2.5\) by carrying the balance forward, then verify it from the remaining payments.
Method. First calculate the balance immediately after the payment at \(t=2\) as the value of the two remaining payments:
The balance date is half a year later. Carrying this balance forward gives
Now check from the future payments. At \(t=2.5\), the payment at \(t=3\) is \(0.5\) years away and the payment at \(t=4\) is \(1.5\) years away:
Both methods give $1,820.25 to the nearest cent. The agreement checks both the date and the interest convention. In particular, including a payment already made at \(t=2\), or omitting the payment due at \(t=3\), would not represent the remaining loan cash flows.
Common Traps and Limits of the Method
The carry-forward method is direct, but its result depends on precise timing and a consistent model. Check the following before accepting a balance:
- Using the wrong side of a payment date. A pre-payment balance has not yet had the installment subtracted. A post-payment balance has.
- Using the wrong starting balance. The amount to carry forward is the balance immediately after the last payment, not the initial loan amount or the balance immediately before that payment.
- Applying an unstated fractional-time rule. The formula \( (1+i)^{t-s} \) is appropriate only when the loan’s stated model uses compound accumulation at the annual effective rate over elapsed years. A different contract convention requires its own accumulation factor.
- Ignoring an intervening payment. If a scheduled installment occurs before the target date, first accumulate to that payment date, subtract the installment, and then continue.
- Mixing interest models in a check. Forward accumulation and prospective discounting agree only when they use the same rate structure and timing assumptions.
This method gives a balance only under the stated interest and payment assumptions. Fees, changes in contractual rates, or other cash flows must be included if they are part of the loan model. For a standard loan with no intervening cash flow, the central calculation remains simple: carry the last post-payment balance to the requested date, and verify it against the payments still due.
Check Your Understanding
For each problem, identify whether the requested balance is before or after any stated payment and use the specified interest convention.
- Immediately after a payment at \(t=3\), a balance is $5,200. The annual effective rate is 5%, with compound fractional-year accumulation. Find the balance at \(t=3.4\). (Answer: $5,302.48)
- Immediately after a payment at \(t=1.5\), a balance is $9,000. The force of interest is constant at \(\delta=0.06\) per year. A $1,500 payment is due at \(t=2.25\). Find the balance immediately after that payment. (Answer: $7,914.25)
- Immediately after a payment at \(t=4\), a balance is $3,200. The annual effective rate is 8%, with compound fractional-year accumulation. A $700 payment is due at \(t=5\). Find the balance at \(t=4.5\) and immediately after the payment at \(t=5\). (Answer: $3,325.54 at \(t=4.5\); $2,756.00 after the payment at \(t=5\))
- A loan uses an annual effective rate of 4% with compound fractional-year accumulation. Immediately after the payment at \(t=2\), the only remaining payments are $800 at \(t=3\) and $1,200 at \(t=4\). Find the balance at \(t=2.5\) by valuing those payments. (Answer: $1,915.91)
- A loan uses an annual effective rate of 5% with compound fractional-year accumulation. Immediately after the payment at \(t=5\), the remaining payments are $900 at \(t=6\) and $1,100 at \(t=7\). Find the balance at \(t=5.6\) by first finding the post-payment balance at \(t=5\), then carrying it forward. (Answer: $1,909.98)