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Exam FM · Outstanding Balance Methods · Tutorial 253 of 998

Timelines for Loan Balance Calculations

Use a clear timeline to identify which advances and payments affect a loan balance and when each transaction occurs.

Intermediate 11 min read

What You'll Learn

  • Mark loan advances, payments, and balance dates on a common time scale.
  • Distinguish a balance immediately before a payment from one immediately after it.
  • Apply interest only over the interval between dated transactions.
  • Handle multiple transactions on the same date in their stated order.
  • Track payments made at the beginning of a period without shifting their dates.
  • Check a timeline-based balance calculation using the transaction sequence.

Why Timing Belongs on the Timeline

A loan balance is measured at a particular time, but the answer can also depend on where that instant falls in the sequence of transactions. If a scheduled payment is due on the balance date, a balance immediately before that payment includes the amount still owed before the payment is applied. A balance immediately after it reflects the payment. Those are different amounts, even though they share the same calendar date.

A timeline makes these distinctions visible. It records when the lender advances money, when the borrower pays, when interest is applied, and whether the balance is measured before or after any transaction on that date. The aim here is to represent the loan correctly before calculating; a misplaced payment date can change the result even when the arithmetic is flawless. The timeline approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A loan timeline is a time-ordered representation of the loan's cash flows and the date at which its balance is measured. A loan advance increases the amount owed; a borrower payment decreases it. A balance measured immediately before a payment excludes that payment, while a balance measured immediately after the payment includes it.

Build the Timeline Before Calculating

Choose a time unit that matches the interest period and the dates of the transactions. For example, if interest is 2% effective per quarter, label time in quarters. A payment at the end of quarter 2 is then at \(t=2\), not \(t=2.5\) or \(t=3\). Mark loan advances with a plus sign and payments with a minus sign, or label them clearly by type. The sign convention helps, but the date and transaction order are essential.

For a level-interest period with effective rate \(i\), let \(B_{j-1}^{+}\) be the balance just after all transactions at time \(j-1\), and let \(B_j^{-}\) be the balance immediately before transactions at time \(j\). If no other cash flows occur between those dates, interest accumulates the balance over the interval. A payment \(R_j\) at time \(j\) then reduces the balance by \(R_j\). An advance \(L_j\) increases it by \(L_j\).

$$ B_j^{-}=(1+i)B_{j-1}^{+}, \qquad B_j^{+}=B_j^{-}-R_j+L_j $$

The superscripts \(^{-}\) and \(^{+}\) mean immediately before and immediately after the transactions at that date. If more than one transaction occurs on the date, apply them in the stated order. For example, if a payment is made before a new advance, subtract the payment first and then add the advance. If the balance is requested between those two transactions, stop at that point in the sequence.

Important: A balance date is not fully specified by a calendar date alone when a payment or advance occurs on that date. State whether the measurement is immediately before or after the transaction, or specify its position in a sequence of same-date transactions.

Why the Interest-and-Payment Recurrence Holds

Between dated transactions, the loan balance grows by the effective interest rate for that interval. Therefore, if the balance just after time \(j-1\) is \(B_{j-1}^{+}\), the interest for one period is \(iB_{j-1}^{+}\). Adding that interest to the starting balance gives

$$ B_{j-1}^{+}+iB_{j-1}^{+}=(1+i)B_{j-1}^{+}=B_j^{-}. $$

This is the balance just before the time-\(j\) transactions. A payment of \(R_j\) reduces the amount owed by exactly \(R_j\), and a new advance of \(L_j\) increases it by exactly \(L_j\). Thus the balance after those transactions is \(B_j^{-}-R_j+L_j\). This recurrence assumes that the stated effective rate applies over the interval, and that no unlisted advance or payment occurs within it. If rates differ by interval, use that interval's accumulation factor instead of \(1+i\).

1
Set the time scale.
Use the period implied by the interest rate and transaction dates.
2
Mark every transaction.
Record advances, payments, and any same-date ordering shown in the problem.
3
Locate the balance instant.
Label it immediately before or after the relevant transaction.
4
Move through time in order.
Accumulate interest between transaction dates, then apply transactions in sequence.
5
Check the endpoint.
Confirm that the final balance includes exactly the transactions that occurred before the specified measurement instant.

Worked Timeline Calculations

Worked Example: Before or After the Second Installment

A lender advances $10,000 at time 0. The loan earns an effective rate of 2% per quarter. The borrower pays $3,000 at the end of each of the first two quarters. Find the balance immediately before the second payment and immediately after it.

Method: timeline-led balance recurrence. The time unit is one quarter. The transaction dates are \(t=0\), \(t=1\), and \(t=2\). The balance date \(t=2\) has two requested positions: before and after the second $3,000 payment.

At \(t=0\), the advance creates a balance of $10,000. Over the first quarter, it accumulates interest:

$$ B_1^{-}=10{,}000(1.02)=10{,}200. $$

The first payment is then applied at \(t=1\):

$$ B_1^{+}=10{,}200-3{,}000=7{,}200. $$

Accumulate this balance for the second quarter to obtain the amount immediately before the second payment:

$$ B_2^{-}=7{,}200(1.02)=7{,}344. $$

The second payment reduces that balance by $3,000:

$$ B_2^{+}=7{,}344-3{,}000=4{,}344. $$

Answer: The balance immediately before the second payment is $7,344; immediately after it, the balance is $4,344. The difference is exactly the payment, as expected. The timeline avoids the common mistake of reporting the after-payment amount when the question asks for the before-payment balance.

Worked Example: A Payment and a New Advance on the Same Date

A loan of $6,000 is advanced at time 0 and earns 5% effective annually. At time 1, the borrower makes a $1,000 payment, after which the lender advances a further $2,000. At time 2, the borrower pays $3,000. Find the balance immediately after the time-2 payment.

Method: follow the dated transactions in their stated order. The timeline is \(t=0\): initial advance; \(t=1\): one year of interest, payment, then new advance; \(t=2\): one year of interest, then payment.

The initial $6,000 accumulates for one year. Thus, just before the time-1 transactions, the balance is

$$ B_1^{-}=6{,}000(1.05)=6{,}300. $$

Apply the payment first, as the problem specifies:

$$ 6{,}300-1{,}000=5{,}300. $$

Then add the new advance. The balance immediately after both time-1 transactions is

$$ B_1^{+}=5{,}300+2{,}000=7{,}300. $$

Accumulate for the next year:

$$ B_2^{-}=7{,}300(1.05)=7{,}665. $$

Finally, subtract the time-2 payment:

$$ B_2^{+}=7{,}665-3{,}000=4{,}665. $$

Answer: The balance immediately after the time-2 payment is $4,665. The calculation uses the same-date order given in the problem; treating the new advance as if it occurred before the payment would give a different balance after the time-1 transactions and would misrepresent the stated sequence.

Worked Example: Payments at the Beginning of Each Period

A loan of $9,000 is made at time 0. The effective interest rate is 4% per half-year. The borrower pays $2,500 immediately at time 0, then another $2,500 at times 1 and 2, where each unit is one half-year. Find the balance immediately before and immediately after the payment at time 2.

Method: place the time-0 payment at its actual date, then accumulate only between dates. The first payment occurs at the same time as the loan advance, so no interest accumulates before that payment.

At time 0, apply the advance and payment:

$$ B_0^{+}=9{,}000-2{,}500=6{,}500. $$

Accumulate for one half-year to time 1, then subtract the payment due at that time:

$$ B_1^{-}=6{,}500(1.04)=6{,}760, \qquad B_1^{+}=6{,}760-2{,}500=4{,}260. $$

Accumulate that balance for the next half-year to find the amount immediately before the time-2 payment:

$$ B_2^{-}=4{,}260(1.04)=4{,}430.40. $$

Subtract the time-2 payment:

$$ B_2^{+}=4{,}430.40-2{,}500=1{,}930.40. $$

Answer: Immediately before the time-2 payment, the balance is $4,430.40; immediately after it, the balance is $1,930.40. The values are exact to the cent, so no intermediate rounding is needed. The timeline avoids the trap of shifting the time-0 payment to the end of the first half-year.

Common Timing Mistakes

The most common error is not a difficult interest calculation; it is applying the correct cash flow at the wrong point on the timeline. For example, in the first worked example, the balance immediately before the second payment is $7,344. A calculation that subtracts the $3,000 payment and reports $4,344 answers the after-payment question instead.

A related error is to give a payment a full period of interest reduction when it occurs at the period's end. In the first example, the first payment reduces the balance at \(t=1\); it does not reduce the amount accruing interest from \(t=0\) to \(t=1\). Conversely, a payment at \(t=0\) in the third example reduces the balance before any half-year of interest accrues.

Key takeaway: Write the balance date and transaction order explicitly. A payment on the measurement date is excluded from a before-payment balance and included in an after-payment balance. Interest accrues only over the intervals that actually pass between dated transactions.

Check Your Understanding

For each problem, draw a timeline and state whether the requested balance is before or after the dated transaction.

  1. A $5,000 loan earns 3% effective per half-year. The borrower pays $1,200 at time 1. Find the balance at time 2 immediately before any payment at that date. (Answer: $4,068.00)
  2. A $8,000 loan earns 4% effective annually. At time 1, after interest has accumulated, the lender advances an additional $1,500. The borrower pays $2,500 at time 2. Find the balance immediately after that payment. (Answer: $7,712.80)
  3. A $7,000 loan earns 5% effective annually. The borrower pays $1,000 at time 0 and $1,000 at time 1. Find the balance immediately after the time-1 payment. (Answer: $5,300.00)
  4. A $12,000 loan earns 1.5% effective per quarter. The borrower pays $4,000 at times 1 and 2. Find the balance immediately before the second payment. (Answer: $8,302.70)
  5. A $5,000 loan earns 6% effective annually. At time 1, the borrower pays $800 and then receives a new $1,000 advance. At time 2, the borrower pays $2,000. Find the balance immediately after the time-2 payment. (Answer: $3,830.00)

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