Prospective Balances After the kth Payment

Why the Payment Number Matters

A loan balance is measured at a particular point in its payment schedule. Immediately after the kth payment, the borrower has made k installments, but the installments after that date are still owed. The prospective method finds the balance by valuing those remaining payments at the valuation date.

The central task is not merely to insert values into a formula. First establish the payment frequency, the rate per payment period, and exactly how many payments remain. In particular, the payment made at the valuation date is not one of the future payments. This tutorial considers level installments paid at the end of each period, with a fixed rate per period and no payment rounding.

Definition: Suppose a loan is repaid by n equal installments of R, paid at the end of each payment period. The prospective balance immediately after the kth payment, denoted here by B_k, is the present value at time k of the installments still due. If the effective interest rate per payment period is j, then $$ B_k=R a_{\overline{n-k}|} =R\frac{1-(1+j)^{-(n-k)}}{j}, \qquad 0\leq k\leq n,\quad j>0. $$ Here $a_{\overline{n-k}|}$ is the present-value annuity factor for $n-k$ end-of-period payments at rate j. At k=n, no payments remain and the balance is zero.

Set the Rate and the Valuation Date

The rate in the annuity factor must match the interval between installments. If payments are annual and the effective annual rate is i, use j=i. If payments are made m times per year and the nominal annual rate convertible m times is $i^{(m)}$, the effective rate per payment period is $$ j=\frac{i^{(m)}}{m}. $$ For example, a nominal annual rate convertible semiannually of 8% gives a rate of $0.08/2=0.04$ per half-year. The factor then discounts each future semiannual installment by one half-year, not by one year.

The phrase immediately after the kth payment fixes the timeline. The payment at time k has just been made, so the first payment included in the balance occurs at time $k+1$. There are $n-k$ such payments, at times $k+1,k+2,\ldots,n$. Treating the balance as if it were measured immediately before the kth payment would include that payment and produce a different value.

Count the installments.
With n total payments and k already made, the number remaining is $n-k$.

Find the period rate.
Use the effective rate for one interval between payments, converting a nominal rate when necessary.

Value the remaining installments.
Multiply the installment amount by the ordinary-annuity present-value factor for $n-k$ periods.

Check the date and count.
Confirm that the payment just made is excluded and that the number of payments valued matches the timeline.

Why the Prospective Formula Holds

At time k, each future installment is a known cash flow due at a later payment date. Under the fixed-rate model, one unit payable one period after time k has value $v=1/(1+j)$ at time k; one unit payable r periods later has value $v^r$. Therefore the payments due at times $k+1$ through n have values $Rv, Rv^2,\ldots,Rv^{n-k}$.

Adding those present values gives a finite geometric series. For $j>0$, $v=1/(1+j)$, and

\begin{aligned} B_k &=R(v+v^2+\cdots+v^{n-k})\\ &=R\frac{v(1-v^{n-k})}{1-v}\\ &=R\frac{1-v^{n-k}}{j} =R a_{\overline{n-k}|}. \end{aligned}

The third line follows because $1-v=j/(1+j)=jv$. The formula applies when the installments are level, paid at the end of each period, and discounted at a constant effective rate per period. It does not directly describe a balance for unequal installments, changing rates, or a valuation date between scheduled payment dates; those situations require their own cash-flow timing and valuation treatment.

Worked Applications

Worked Example: Annual Installments

A loan is repaid by four annual installments of $600, each paid at year-end. The effective annual interest rate is 5%. Find the balance immediately after the second installment.

Method. Apply the prospective method. Since two of four installments have been paid, $4-2=2$ installments remain. The valuation date is immediately after year 2, so the future payments are $600 at years 3 and 4.

The rate per annual payment period is $j=0.05$. The two-period discount factor is

v^2=(1.05)^{-2}=0.9070294785.

Thus the annuity factor is

a_{\overline{2}|} =\frac{1-(1.05)^{-2}}{0.05} =\frac{1-0.9070294785}{0.05} =1.85941043.

The balance is

B_2=600(1.85941043)=1115.646258\ldots

Rounded to the nearest cent, the balance is $1,115.65. As a check, this is the sum of the discounted values of the two remaining payments: $600/1.05+600/(1.05)^2=571.428571\ldots+544.217687\ldots=1115.646258\ldots$. The common mistake avoided here is counting the second payment again: it has already been made at the balance date.

Worked Example: Semiannual Payments at a Nominal Rate

A loan is repaid by six semiannual installments of $500. The nominal annual interest rate is 8% convertible semiannually. Find the balance immediately after the second installment.

Method. Convert the nominal annual rate to the rate per half-year, then value only the installments still due. The rate per payment period is

j=\frac{0.08}{2}=0.04.

Four installments remain because $6-2=4$. The discount factor for four half-year periods is

(1.04)^{-4}=\frac{1}{1.16985856}=0.854804191\ldots

Therefore,

a_{\overline{4}|} =\frac{1-0.854804191\ldots}{0.04} =3.629895225\ldots

and

B_2=500(3.629895225\ldots)=1814.947612\ldots.

The balance is $1,814.95 to the nearest cent. The calculation uses four half-year periods, because the four remaining installments are each one half-year apart. Using 8% as the rate for each half-year would overstate the rate; using 4% as an annual rate and then counting only two periods would instead use the wrong payment timeline.

Worked Example: Monthly Installments

A borrower makes 24 monthly installments of $250 at an effective monthly interest rate of 0.5%. Find the balance immediately after the ninth installment.

Method. The stated rate is already monthly, so $j=0.005$. There are $24-9=15$ monthly installments remaining. No annual-rate conversion is needed.

First calculate the discount factor for 15 months:

(1.005)^{15}=1.0776827376\ldots, \qquad (1.005)^{-15}=0.9279169\ldots.

Then the 15-payment annuity factor is

a_{\overline{15}|} =\frac{1-(1.005)^{-15}}{0.005} =14.416622\ldots.

The prospective balance is

B_9=250(14.416622\ldots)=3604.1555\ldots.

Rounded to the nearest cent, the balance is $3,604.16. The count is 15, not 16: the ninth installment has just been paid, leaving installments 10 through 24. A useful check is that the balance is less than the undiscounted total of those future installments, $15(250)=$3,750$, as it must be for a positive interest rate.

Worked Example: Quarterly Installments

A loan is repaid by ten quarterly installments of $1,000. The effective interest rate per quarter is 1%. Find the balance immediately after the sixth installment.

Method. The rate is already expressed per payment period. There are $10-6=4$ quarterly payments remaining, so use a four-period annuity factor.

(1.01)^4=1.04060401, \qquad (1.01)^{-4}=0.960980344\ldots

Thus,

a_{\overline{4}|} =\frac{1-0.960980344\ldots}{0.01} =3.9019656\ldots

and

B_6=1000(3.9019656\ldots)=3901.9656\ldots.

The balance is $3,901.97 to the nearest cent. It is below the $4,000 total of the remaining installments because each installment is discounted to the date immediately after quarter 6. The common error avoided is using the total original term of ten periods in the annuity factor; the balance at this date concerns only the four unpaid installments.

Common Timing and Rate Traps

The prospective method is efficient because it values only the cash flows still due. Its most frequent errors are errors of interpretation, not algebra. Before calculating, write down the payment number just made and the payment numbers that remain. Then identify the rate that applies to one interval between those remaining payment dates.

Including the payment just made. Immediately after payment k, the number of future installments is $n-k$, not $n-k+1$. The latter count would include the payment at time k.
Using the wrong period rate. A nominal annual rate convertible m times is divided by m to obtain the rate per conversion period. The annuity factor and the payment count must both use that same period.
Using the original term instead of the remaining term. The factor is based on $n-k$, not n, because the balance is measured after k payments.
Rounding too early. Retain the full annuity factor and payment amount in intermediate calculations, then round the final money amount to cents. A rounded installment or factor can shift the reported balance by a cent or more.

Key takeaway: For a level-payment loan with end-of-period installments and a constant rate per payment period, the balance immediately after payment k is the present value of the $n-k$ installments still due. The timeline determines the count; the payment frequency determines the rate period.

Check Your Understanding

For each problem, identify the remaining payment count and the rate per payment period before using the prospective formula.

A loan has five annual installments of $700 at an effective annual rate of 4%. Find the balance immediately after the third installment. (Answer: $1,320.27)
A loan has eight annual installments of $1,200 at an effective annual rate of 6%. Find the balance immediately after the fifth installment. (Answer: $3,207.61)
A loan has ten semiannual installments of $800 at a nominal annual rate of 10% convertible semiannually. Find the balance immediately after the sixth installment. (Answer: $2,836.76)
A loan has twelve monthly installments of $1,000 at an effective monthly rate of 1%. Find the balance immediately after the tenth installment. (Answer: $1,970.40)

What You'll Learn