Outstanding Balance as the Value of Unpaid Payments

Why Value the Payments That Remain?

A loan balance at a particular date is the amount that, at that date, is financially equivalent to the payments still required by the contract. For a level-payment loan with a fixed interest rate, this gives a direct valuation method: discount each unpaid payment to the balance date and add the present values. The balance is not the sum of the unpaid payments, because payments due later are worth less at the balance date when interest is positive.

The timing convention matters. If the balance is required immediately after payment $k$, that payment has already been made and is not included. The next scheduled payment is one full payment period later. Counting the unpaid payments correctly is as important as selecting the interest rate.

Definition: Suppose a loan is repaid by $n$ equal payments of $R$, each made at the end of a payment period, and the effective interest rate per payment period is $i$. The outstanding balance immediately after payment $k$, for $0 \leq k \leq n$, is the value at that time of payments $k+1$ through $n$. For $i \neq 0$, it is $$ B_k=R a_{\overline{n-k}|} =R\frac{1-v^{\,n-k}}{i}, \qquad v=\frac{1}{1+i}. $$ Here $a_{\overline{m}|}$ denotes the present value of $m$ unit payments made at the ends of successive periods, valued one period before the first payment.

Deriving the Remaining-Payments Formula

Let the valuation date be immediately after payment $k$. There are $n-k$ payments left. The first is due one period after the valuation date; the next is due two periods after it; the last is due $n-k$ periods after it. Under a fixed effective rate $i$ per payment period, a payment due $j$ periods later has value $Rv^j$ at the valuation date. Therefore the balance is the sum of those discounted payments:

B_k=Rv+Rv^2+\cdots+Rv^{n-k}.

This is a geometric series. Factoring out $R$ and summing the series gives the annuity-immediate factor. The derivation also explains why the first discount power is $v^1$, not $v^0$: the next payment is one period in the future, not due at the valuation date.

\begin{aligned} B_k &=R\sum_{j=1}^{n-k}v^j\\ &=R\frac{v(1-v^{n-k})}{1-v}\\ &=R\frac{1-v^{n-k}}{i} =R a_{\overline{n-k}|}, \end{aligned}

In the last equality, $1-v=i v$, which follows from $v=1/(1+i)$. The formula assumes level payments at the ends of regular periods and a constant effective rate per period. If the payments vary, the dates are irregular, or rates differ across periods, value each unpaid payment using the applicable discount factors rather than using one level-annuity factor.

A Reliable Calculation Process

Set the balance date.
Record whether the balance is immediately before or immediately after a scheduled payment. This tutorial values balances immediately after a payment.

Count future payments.
If the loan has $n$ payments and payment $k$ has just been made, the number remaining is $n-k$.

Match the rate to a payment period.
Use the effective interest rate for the interval between payments. Convert a nominal rate if necessary.

Discount the remaining payments.
Calculate $R a_{\overline{n-k}|}$, or add the individual present values when that is clearer.

Check the timing.
The first term should discount the next payment by one period. Confirm that the number of terms equals the number of payments still due.

Worked Applications

Worked Example: Two Monthly Payments Remain

A hypothetical loan is repaid by six end-of-month payments of $300. The effective interest rate is 0.5% per month, and the balance is required immediately after the fourth payment. Find the balance, to the nearest cent.

Method and payment count. Use the present value of the unpaid level payments. Payments 1 through 4 have been made, so $6-4=2$ payments remain. They are due one and two months after the balance date.

The periodic rate is $i=0.005$, and $v=1/1.005$. Thus

\begin{aligned} B_4 &=300a_{\overline{2}|}\\ &=300\left(\frac{1}{1.005}+\frac{1}{1.005^2}\right)\\ &=300(0.9950248756+0.9900745031)\\ &=300(1.9850993787)\\ &=595.5298136\ldots \end{aligned}

Rounded to the nearest cent, the balance is $595.53. The result is less than the $600 total of the two future payments, as expected with a positive interest rate. The method avoids the common mistake of treating a future payment as if it were due immediately.

Worked Example: Quarterly Payments and a Nominal Rate

A hypothetical loan has seven quarterly payments of $800. Interest is charged at a nominal annual rate of 8% convertible quarterly. Find the balance immediately after payment 3, to the nearest cent.

Method and rate conversion. A nominal annual rate convertible quarterly is divided by four to obtain the effective rate per quarter:

i=\frac{0.08}{4}=0.02.

After the third of seven payments, $7-3=4$ payments remain. Apply the annuity-immediate formula with four periods and a quarterly rate of 2%:

\begin{aligned} B_3 &=800a_{\overline{4}|}\\ &=800\left(\frac{1}{1.02}+\frac{1}{1.02^2} +\frac{1}{1.02^3}+\frac{1}{1.02^4}\right)\\ &=800(0.9803921569+0.9611687812 +0.9423223345+0.9238454260)\\ &=800(3.8077286986)\\ &=3046.1829589\ldots \end{aligned}

The balance is $3,046.18. As a check, four payments total $3,200, so a balance below that amount is consistent with discounting at a positive rate. The rate conversion is essential: using 8% per quarter would not match the stated nominal annual rate.

Worked Example: Three Annual Payments Remain

A hypothetical loan is repaid by ten annual payments of $2,000, made at each year-end. The effective annual interest rate is 4%. Find the balance immediately after the seventh payment, rounded to the nearest cent.

Method and timing. The valuation date is just after payment 7, so payments 8, 9, and 10 remain. Their due dates are one, two, and three years after the valuation date. The effective annual rate is already expressed per payment period.

\begin{aligned} B_7 &=2000a_{\overline{3}|}\\ &=2000\left(\frac{1}{1.04}+\frac{1}{1.04^2} +\frac{1}{1.04^3}\right)\\ &=2000(0.9615384615+0.9245562130+0.8889963587)\\ &=2000(2.7750910332)\\ &=5550.1820664\ldots \end{aligned}

The outstanding balance is $5,550.18. It is the present value, immediately after payment 7, of the three future $2,000 payments. The calculation avoids including the seventh payment again or omitting the tenth payment.

Common Timing and Rate Errors

A frequent error is using an annuity-due factor for a balance immediately after a payment. In the first example, the two remaining payments are due one and two months later. Incorrectly treating one as due immediately gives

300\left(1+\frac{1}{1.005}\right) =598.5074627\ldots \approx \$598.51.

That calculation places a $300 payment at the balance date, even though the contract says it is due at the next month-end. The correct valuation discounts both payments:

300\left(\frac{1}{1.005}+\frac{1}{1.005^2}\right) =595.5298136\ldots \approx \$595.53.

Other checks help catch related errors:

Do not count the payment just made. Immediately after payment $k$, only payments $k+1$ through $n$ remain.
Do not use the annual rate as a payment-period rate without checking. A nominal annual rate convertible $m$ times per year has rate $i^{(m)}/m$ per conversion period; the payment schedule must match that period or the rate must be converted appropriately.
Do not use $n$ payments in place of $n-k$. The original term determines how many payments were scheduled, while the balance date determines how many remain.
Do not apply the level-annuity formula to a different cash-flow pattern. If amounts or intervals vary, discount each future payment on its own terms.

For a positive rate and at least one remaining payment, the balance should be positive and less than the undiscounted total of the remaining payments. This is a useful reasonableness check, not a substitute for verifying the cash-flow dates and discount rate.

Key takeaway: Immediately after payment $k$, a fixed-rate loan with $n$ end-of-period level payments of $R$ has $n-k$ payments left. Its balance is $R a_{\overline{n-k}|}$, using the effective rate per payment period. The first remaining payment is discounted for one full period.

Check Your Understanding

For each hypothetical loan, calculate the balance immediately after the stated payment. Round money to the nearest cent.

A loan has five annual payments of $1,000 at an effective annual rate of 5%. Find the balance immediately after payment 2. (Answer: $2,723.25)
A loan has eight semiannual payments of $650 at a nominal annual rate of 6% convertible semiannually. Find the balance immediately after payment 5. (Answer: $1,838.60)
A loan has nine monthly payments of $250 at an effective monthly rate of 0.75%. Find the balance immediately after payment 6. (Answer: $738.89)
A loan has ten quarterly payments of $1,200 at a nominal annual rate of 8% convertible quarterly. Find the balance immediately after payment 7. (Answer: $3,460.66)
A loan has twelve monthly payments of $200 at a nominal annual rate of 9% convertible monthly. Find the balance immediately after payment 9. (Answer: $591.11)

What You'll Learn