Level-Payment Balances by the Prospective Method

Why the Prospective Formula Is Useful

For a loan with level installments, the balance immediately after a payment can be found without rebuilding the entire amortization schedule. The prospective method asks a direct question: what is the value, at the balance date, of the payments still due? When the installments are equal and occur at regular intervals, their present value is an annuity-immediate.

The key detail is the date of valuation. Immediately after payment $k$, that payment has already been made and is not part of the outstanding balance. If the loan has $n$ installments in total, $n-k$ installments remain. The rate used in the annuity factor must be effective for one interval between installments.

Definition: Consider a loan repaid by $n$ equal installments of amount $R$, made at the ends of successive payment periods. Let $i$ be the effective interest rate per payment period. The outstanding balance immediately after the $k$th installment, for $0\leq k\leq n$, is the present value at that date of the future installments. For $k<n$, $$ B_k=R a_{\overline{n-k}|} =R\frac{1-v^{\,n-k}}{i},\qquad v=\frac{1}{1+i}. $$ After the final payment, $B_n=0$. If $i=0$, the balance is $R(n-k)$.

Deriving the Level-Payment Balance

Assume installments are paid at the end of each period, the rate is constant and effective at $i$ per payment period, and there are no extra payments, fees, or payment rounding adjustments. At the instant just after installment $k$, the remaining payments fall due one, two, and up to $n-k$ periods later. Discounting each payment to that instant gives

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

This is a finite geometric series. For $i\ne 0$, $v=1/(1+i)$, so $1-v=i/(1+i)=iv$. Thus,

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

The factor $a_{\overline{m}|}$ is the present value of $m$ unit payments made at the ends of successive periods at effective rate $i$. Here $m=n-k$: the remaining term, not the original term, determines the annuity factor. When $i=0$, each remaining payment has present value $R$, giving $B_k=R(n-k)$, which is also the limit of the formula as $i$ approaches zero.

Set the balance date.
Read whether the balance is immediately before or immediately after a payment. This formula is for the instant immediately after payment $k$.

Count the future installments.
For $n$ total installments, immediately after payment $k$ there are $n-k$ installments left.

Match the rate to one payment period.
Use the effective rate for the interval between successive installments, not automatically the quoted annual rate.

Value the remaining annuity.
Calculate $R a_{\overline{n-k}|}$, using the matched rate and the number of remaining installments.

Worked Applications

Worked Example: Six Annual Payments

A loan is repaid by six annual installments of $1,200, payable at each year-end. The effective annual interest rate is 5%. Find the outstanding balance immediately after the second installment. Also calculate the original loan amount as a consistency check.

Method. Apply the prospective method. The effective annual rate is already the rate per payment period, and after the second installment there are $6-2=4$ annual payments remaining.

First, the original loan amount is the present value of all six installments:

a_{\overline{6}|} =\frac{1-(1.05)^{-6}}{0.05} =\frac{1-0.746215397}{0.05} =5.07569206.

Thus the original principal is $1{,}200(5.07569206)=\$6{,}090.83047$, or $6,090.83 to the nearest cent.

Immediately after payment 2, four installments remain. Their annuity factor is

a_{\overline{4}|} =\frac{1-(1.05)^{-4}}{0.05} =\frac{1-0.822702475}{0.05} =3.54595050.

Therefore,

B_2=1{,}200(3.54595050)=\$4{,}255.14060.

The balance is $4,255.14 to the nearest cent. The amount is less than the original principal, as expected after two payments. Its definition also checks the result: it is the present value, at the second payment date, of exactly four future installments.

Common mistake avoided: using the original six-payment factor at the second payment date would value payments that have already been made. The prospective balance includes only the remaining four.

Worked Example: Quarterly Installments and a Nominal Rate

A loan is repaid by 10 quarterly installments of $2,500. Interest is at a nominal annual rate of 8% convertible quarterly. Find the balance immediately after the eighth installment.

Method. Convert the quoted nominal rate to the effective rate per quarter. With quarterly conversion, the quarterly rate is $i=0.08/4=0.02$. After payment 8, the remaining number of installments is $10-8=2$.

The annuity factor for the two remaining quarters is

a_{\overline{2}|} =\frac{1-(1.02)^{-2}}{0.02} =\frac{1-0.961168782}{0.02} =1.94156092.

Therefore,

B_8=2{,}500(1.94156092)=\$4{,}853.90230.

The outstanding balance is $4,853.90 to the nearest cent. As a direct check, the two payments have present values $2{,}500/1.02=\$2{,}450.98$ and $2{,}500/(1.02)^2=\$2{,}402.92$; their sum is $4,853.90 after rounding.

Common mistake avoided: using the annual nominal rate of 8% as though it were the quarterly rate gives the wrong discounting. The installments are quarterly, so each payment interval uses 2%, not 8%.

Worked Example: Monthly Payments and a Monthly Rate

A hypothetical equipment loan is repaid by 24 monthly installments of $500. The nominal annual rate is 12% convertible monthly. Find the balance immediately after payment 18.

Method. The effective monthly rate is $i=0.12/12=0.01$. Immediately after the 18th payment, $24-18=6$ monthly installments remain.

Calculate the six-payment annuity factor:

a_{\overline{6}|} =\frac{1-(1.01)^{-6}}{0.01} =\frac{1-0.942045235}{0.01} =5.79547650.

Then

B_{18}=500(5.79547650)=\$2{,}897.73825.

The balance is $2,897.74 to the nearest cent. It is the value at month 18 of the six payments still due, discounted one through six months. A common counting error is to use $24-18+1=7$ payments; that would include a payment that does not remain after payment 18.

Rate and Timing Checks

A quoted annual rate does not always equal the rate per installment period. If installments are monthly and the contract gives a nominal annual rate $i^{(12)}$ convertible monthly, the effective monthly rate is $i^{(12)}/12$. If instead the contract gives an effective annual rate $i$, the equivalent monthly effective rate is $(1+i)^{1/12}-1$. Use the convention specified in the problem; these conversions are not interchangeable.

The formula also depends on the valuation instant. Immediately after payment $k$, the first remaining payment is one full payment period away, so the factor is $a_{\overline{n-k}|}$. Immediately before that payment, the payment due at that instant has not yet been made and must be treated differently. Do not shift the formula by one installment without first fixing the balance date.

Key takeaway: For a level-payment loan with regular end-of-period installments and a constant effective rate per payment period, the balance immediately after payment $k$ is the present value of the $n-k$ future installments: $B_k=R a_{\overline{n-k}|}$. Check the payment count, rate period, and valuation timing before calculating.

Check Your Understanding

For each problem, identify the remaining number of payments before applying the prospective formula. Round money to the nearest cent.

A loan has eight annual end-of-year installments of $800 at an effective annual rate of 4%. Find the balance immediately after payment 3. (Answer: $3,561.46)
A loan has eight quarterly installments of $1,500. The nominal annual rate is 6% convertible quarterly. Find the balance immediately after payment 5. (Answer: $4,368.20)
A loan is repaid by five annual installments of $1,000 at an effective annual rate of 25%. Find the balance immediately after payment 2. (Answer: $1,952.00)
A loan has 36 monthly installments of $2,000. The nominal annual rate is 12% convertible monthly. Find the balance immediately after payment 30. (Answer: $11,590.95)

What You'll Learn