Cash-Flow Identity at a Balance Date

One Balance Date, Two Views of the Loan

An outstanding loan balance can be checked from either side of a chosen date. From the past, accumulate the original loan advance and subtract the payments already made, with each cash flow carried forward to the balance date. From the future, discount the payments still due back to that date. Under a consistent interest model, these are not competing definitions: they are two expressions for the same value.

This identity is useful when a payment schedule has unequal amounts or irregular dates, and when a calculated balance needs an independent check. The essential discipline is to put every cash flow at the same date before comparing values.

Definition: Let a loan advance \(L\) be made at time 0, and let \(C_j\) be the borrower’s payment at time \(t_j\). Payments are treated as positive amounts. The outstanding balance at time \(t\), immediately after any payment made at \(t\), is the value at \(t\) of the loan cash flows that remain after that payment. If the loan is fully repaid by its stated schedule, the balance can be expressed either by accumulating the initial advance and payments before \(t\), or by discounting payments after \(t\).

Write Every Cash Flow at the Same Date

First suppose interest is governed by a constant effective annual rate \(i\), and times are measured in years. A cash flow at time \(s\) has value at time \(t\) equal to the cash flow multiplied by \((1+i)^{t-s}\) when \(s\leq t\). When \(s\geq t\), its value at time \(t\) is the cash flow multiplied by \((1+i)^{-(s-t)}\). These factors use the same annual compounding convention in both directions.

Suppose payments have occurred at times \(t_j<t\), and future payments are due at times \(t_k>t\). If there is no payment at \(t\), the balance identity is

B_t = L(1+i)^t-\sum_{t_j<t} C_j(1+i)^{t-t_j} = \sum_{t_k>t} C_k(1+i)^{-(t_k-t)}.

The first expression is the retrospective value: carry the advance and past payments forward to \(t\). The second is the prospective value: discount future payments to \(t\). Each payment is counted once, on the appropriate side of the date. The identity assumes the loan cash flows and interest basis are known, there are no unmodelled fees or adjustments, and the payment schedule is consistent with repayment of the loan. It does not say that arbitrary past and future cash flows will automatically produce the same balance; if they do not, the schedule, assumptions, or calculations need review.

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

Separate the cash flows.
List the advance and payments before the date for the retrospective calculation, and payments after the date for the prospective calculation.

Move each amount to the date.
Accumulate earlier amounts and discount later amounts using the stated interest model and each cash flow’s actual time difference.

Compare the results.
Matching values support the balance calculation; a difference signals that a cash flow, date, or interest factor may have been mishandled.

Why the Two Expressions Agree

The identity follows directly from valuing the complete loan schedule at time 0 and then moving that value to time \(t\). For a fully amortizing loan, the present value of all payments equals the original advance:

L=\sum_j C_j(1+i)^{-t_j}.

Multiply both sides by \((1+i)^t\). For a payment before \(t\), the resulting factor is \((1+i)^{t-t_j}\), so that payment is accumulated to \(t\). For a payment after \(t\), the factor is \((1+i)^{-(t_j-t)}\), so that payment is discounted to \(t\). Splitting the sum at \(t\) gives

L(1+i)^t = \sum_{t_j<t} C_j(1+i)^{t-t_j} + \sum_{t_k>t} C_k(1+i)^{-(t_k-t)}.

Rearranging shows that the original advance accumulated to \(t\), less the value at \(t\) of payments already made, equals the value at \(t\) of payments still due. This derivation also explains the needed consistency: if the advance and payment schedule do not satisfy the time-0 value equation under the stated interest model, the two calculated values need not agree.

Worked Example: Checking a Balance After the Second Payment

A loan of \(10{,}000\) is advanced at time 0. Interest is \(10\%\) effective annually. The borrower pays \(4{,}000\) at the end of year 1, \(4{,}000\) at the end of year 2, and \(4{,}070\) at the end of year 3. Find the balance immediately after the year-2 payment using both sides of the identity.

Method and timing. The balance date is \(t=2\), immediately after the second payment. Thus the first two payments are past cash flows, while the year-3 payment is the only future cash flow.

Retrospective value. Accumulate the original advance for two years and the year-1 payment for one year. The year-2 payment is already made at the balance date, so it is subtracted without further accumulation:

\begin{aligned} B_2 &=10{,}000(1.10)^2-4{,}000(1.10)-4{,}000\\ &=10{,}000(1.21)-4{,}400-4{,}000\\ &=12{,}100-4{,}400-4{,}000\\ &=3{,}700. \end{aligned}

Prospective value. The \(4{,}070\) payment is one year after the balance date, so discount it one year:

B_2=\frac{4{,}070}{1.10}=3{,}700.

Both methods give a balance of \(3{,}700\). As a check, that balance accumulates to \(3{,}700(1.10)=4{,}070\) by the year-3 payment date, exactly the amount due. The common mistake avoided here is to count the year-2 payment as still outstanding after defining the balance immediately after it.

Unequal Payments and Irregular Dates

The identity does not require level installments or equally spaced payment dates. What matters is using the actual time between each cash flow and the balance date. For irregular dates under a constant effective annual rate, a payment at \(t_j\) is accumulated by \((1+i)^{t-t_j}\) if it precedes \(t\), and discounted by \((1+i)^{-(t_j-t)}\) if it follows \(t\). If rates vary by period, use the accumulation factor implied by those rates for that particular interval, and discount by its reciprocal. Do not replace an actual interval with a full year merely because the payment schedule is mostly annual.

Worked Example: Verifying a Balance with Unequal and Irregular Payments

A loan of \(12{,}000\) is advanced at time 0. Interest is \(5\%\) effective annually. The borrower pays \(3{,}000\) at time 1, \(4{,}000\) at time 3, and \(6{,}913.20\) at time 4. Find the balance at time 2, when no payment is made, and check it using future payments.

Method. Use actual time differences from \(t=2\). The year-1 payment is one year before the balance date; the other payments are one and two years after it.

Retrospective value. Accumulate the advance for two years and the first payment for one year:

\begin{aligned} B_2 &=12{,}000(1.05)^2-3{,}000(1.05)\\ &=12{,}000(1.1025)-3{,}150\\ &=13{,}230-3{,}150\\ &=10{,}080. \end{aligned}

Prospective value. Discount the payment at time 3 for one year and the payment at time 4 for two years:

\begin{aligned} B_2 &=\frac{4{,}000}{1.05}+\frac{6{,}913.20}{(1.05)^2}\\ &=3{,}809.523809\ldots+6{,}270.476190\ldots\\ &=10{,}080. \end{aligned}

The two values agree. The payment dates are not a level annual sequence, but each factor uses its distance from time 2. The common mistake avoided is discounting the time-4 payment by only one year, which would place it at time 3 rather than at the balance date.

A Payment on the Balance Date

A balance date that coincides with a payment requires an explicit timing convention. Immediately before the payment, that payment is still due; immediately after it, the payment has been made and is no longer part of the outstanding obligation. The two balances differ by the payment amount. This is not a change in the interest model: it is a change in which side of the balance date contains that cash flow.

Worked Example: Before and After a Payment

A loan of \(5{,}000\) is advanced at time 0 and earns \(10\%\) effective annual interest. Payments are \(1{,}000\) at time 1, \(2{,}000\) at time 2, and \(3{,}245\) at time 3. Find the balances immediately before and immediately after the time-2 payment, and verify the result.

Before the time-2 payment. Accumulate the advance for two years and the time-1 payment for one year. The time-2 payment has not yet been made:

B_{2^-}=5{,}000(1.10)^2-1{,}000(1.10)=6{,}050-1{,}100=4{,}950.

After the time-2 payment. Subtract the \(2{,}000\) payment made on that date:

B_{2^+}=4{,}950-2{,}000=2{,}950.

Prospective checks. Immediately after time 2, only the time-3 payment remains, so its value is \(3{,}245/1.10=2{,}950\). Immediately before time 2, the payment due at that date is included as well:

B_{2^-}=2{,}000+\frac{3{,}245}{1.10}=2{,}000+2{,}950=4{,}950.

Both timing conventions match their respective retrospective values. A common wrong calculation for the after-payment balance is \(2{,}000+3{,}245/1.10=4{,}950\). That includes the time-2 payment even though it has already been made. The corrected after-payment balance is \(3{,}245/1.10=2{,}950\).

Using the Identity as a Diagnostic

A balance check is most informative when the two calculations are set up independently. In particular, write the retrospective and prospective cash-flow lists before entering values. Agreement is a useful arithmetic and timing check, but it cannot by itself prove that the stated interest rate or payment schedule is appropriate; both sides can agree under an incorrect assumption if that assumption is used consistently.

Retrospective side: include the original advance and only payments made before the balance date. Carry each amount to that date.
Prospective side: include only payments still due after the balance date. Discount each amount back to that date.
Payment on the date: include it in the before-payment prospective value, or subtract it to move from the before-payment balance to the after-payment balance.
Different dates or rates: use the correct interval and the accumulation basis applying over that interval; do not combine factors from different conventions.

The cash-flow identity is aligned with the SOA Exam FM (Financial Mathematics) syllabus because it links the two standard ways to value an outstanding loan. Its practical value is broader than a final numerical answer: it makes the timing assumptions visible and gives a structured way to locate an inconsistency.

Check Your Understanding

For each problem, state the balance-date timing convention and use consistent accumulation or discount factors.

A loan of \(8{,}000\) earns \(5\%\) effective annual interest. A payment of \(2{,}000\) is made at time 1. Find the balance immediately after that payment by the retrospective method. If the remaining payments are \(3{,}000\) at time 2 and \(3{,}906\) at time 3, verify it prospectively. (Answer: \(6{,}400\) by both methods.)
A \(15{,}000\) loan earns \(4\%\) effective annual interest. The borrower pays \(4{,}000\) at time 1, \(5{,}000\) at time 2, and \(7{,}346.56\) at time 3. Find the balance immediately after the time-1 payment and check it using the two future payments. (Answer: \(11{,}600\) by both methods.)
A loan of \(9{,}000\) earns \(8\%\) effective annual interest. A \(2{,}000\) payment is made at time 1 and a \(1{,}500\) payment at time 2. Find the balances immediately before and after the time-2 payment, and find the time-3 payment that would repay the loan. (Answer: before \(8{,}337.60\); after \(6{,}837.60\); time-3 payment \(7{,}384.608\), or \(7{,}384.61\) to the nearest cent.)
A \(6{,}000\) loan earns \(4\%\) effective annual interest. A payment of \(1{,}000\) is made at time 1, \(2{,}000\) at time 3, and a final payment at time 4. Find the balance at time 2 from past cash flows, then find the final payment required for the prospective value at time 2 to equal that balance. (Answer: balance \(5{,}449.60\); final payment \(3{,}814.28736\), or \(3{,}814.29\) to the nearest cent.)

What You'll Learn