Why a Loan Payment Can Be Derived from Present Value
A fixed-rate loan advances a principal amount at one date and requires a series of payments at later dates. To find the payment that exactly repays the loan, value the payments at the loan date using the loan’s interest rate. The payment is level when the same amount is due at every scheduled payment date.
This valuation relies on a specific timeline. The first payment is made one payment period after the loan is issued, not immediately. The loan has no fees, missed payments, balloon payment, or other cash flows, and the interest rate remains constant for the entire term. Under these assumptions, the original principal equals the present value of the payments.
For example, if payments are monthly, \(i\) must be the effective monthly rate. If a stated rate is nominal and convertible at the same frequency as the payments, first divide that nominal annual rate by the conversion frequency. The number \(n\) is the total number of payments, not the number of years unless there is one payment per year.
Deriving the Level-Payment Formula
Let the loan date be time 0. The \(n\) payments of amount \(R\) occur at times 1 through \(n\), measured in payment periods. With effective rate \(i\) per period, the present value of a payment at time \(k\) is \(R v^k\), where \(v=1/(1+i)\). Add the present values of all payments and equate the result to the principal.
The factor \(a_{\overline{n}|}\) is the present value of an annuity-immediate paying 1 at the end of each of \(n\) periods. To obtain its closed form, multiply its geometric sum by \(1+i\), or equivalently use \(v=1/(1+i)\):
The geometric-sum identity follows because \((1-v)(v+v^2+\cdots+v^n)=v-v^{n+1}\). Dividing by \(1-v\) gives the sum, and \(1-v=i/(1+i)\); simplifying yields \((1-v^n)/i\). Thus the loan equation \(L=R a_{\overline{n}|}\) can be solved for \(R\).
The formula is not a rule about every loan contract. It applies when the stated payment dates and interest periods align, the rate is constant, and there are no other amounts included in the principal or repayment stream. If fees are financed, for instance, the amount being repaid may differ from the cash advanced to the borrower; the equation must then use the actual amount financed. If payments are due at the beginning of each period, the dates differ and the annuity-immediate formula does not apply unchanged.
Worked Examples
Worked Example: Two Annual Payments at the End of Each Year
A hypothetical borrower receives a loan of $10,000 and will repay it with two equal annual payments, the first due one year after the loan date. The effective annual interest rate is 5%. Find the payment to the nearest cent and verify its present value.
Method. The payment interval is one year, so the effective rate per payment period is \(i=0.05\), and \(n=2\). Because the first payment is at the end of the first year, use an annuity-immediate.
The annuity factor is \(a_{\overline{2}|}=1/1.05+1/(1.05)^2 =0.9523809524+0.9070294785 =1.8594104309\). Therefore, \(R=10{,}000/1.8594104309 =5{,}378.048780\ldots\). Rounded to the nearest cent, the scheduled payment is $5,378.05.
To check using the unrounded payment, the first payment has present value \(5{,}378.048780\ldots/1.05 =5{,}121.951220\ldots\), and the second has present value \(5{,}378.048780\ldots/1.1025 =4{,}878.048780\ldots\). Their sum is $10,000. The small difference that could arise from discounting the displayed, cent-rounded payment is only rounding.
Common mistake avoided. Using an annuity-due factor would place the first payment at time 0. For two beginning-of-year payments, the factor is \(\ddot{a}_{\overline{2}|}=1+1/1.05=1.9523809524\), giving \(10{,}000/1.9523809524=$5{,}121.95\). That amount is not the payment for the stated end-of-year dates: its actual end-of-year present value is \(5{,}121.9512\ldots \times 1.8594104309=$9{,}523.81\), not $10,000.
Worked Example: Converting a Nominal Rate for Semiannual Payments
A hypothetical loan has principal $5,000. It is repaid by four equal payments, one every six months, with the first payment due six months after the loan date. The nominal annual rate is 8% convertible semiannually. Find the payment to the nearest cent and check the value at the end of the second year.
Method. There are two conversion periods per year, and each payment occurs after one conversion period. Thus the effective rate per payment period is \(i=0.08/2=0.04\), and there are \(n=4\) payments. Using 8% directly as the rate for each six-month period would mismatch the stated nominal rate and the payment dates.
The four-period annuity factor is \(a_{\overline{4}|}=[1-(1.04)^{-4}]/0.04\). Since \((1.04)^4=1.16985856\), its reciprocal is approximately \(0.854804191\). Hence \(a_{\overline{4}|}=(1-0.854804191)/0.04=3.629895225\). The payment is \(R=5{,}000/3.629895225=1{,}377.4502\ldots\), or $1,377.45 to the nearest cent.
For a terminal-value check, four payments accumulate to time 2 with factor \(s_{\overline{4}|}=[(1.04)^4-1]/0.04=4.246464\). Using the unrounded payment, their accumulated value is approximately \(1{,}377.4502\times4.246464=$5{,}849.29\). The principal accumulated for four half-year periods is \(5{,}000(1.04)^4=$5{,}849.29\), as required.
Common mistake avoided. The number of periods is four, not two: the term is two years, but payments are semiannual. The rate and the count must both be expressed in the same six-month units.
Worked Example: The Zero-Interest Limit
A hypothetical interest-free loan advances $9,600 and requires eight equal annual payments, with the first due one year after the loan date. Find the payment and explain why direct substitution into the positive-rate formula is inappropriate.
Method. With no interest, each payment has its face amount as its present value. The equation of value is \(9{,}600=8R\), so \(R=9{,}600/8=$1{,}200\).
Check: eight payments total \(8\times$1{,}200=$9{,}600\), equal to the amount advanced. The expression \(Li/(1-v^n)\) cannot be evaluated by simply substituting \(i=0\), because \(v=1\) then makes both numerator and denominator zero. The appropriate zero-rate case follows directly from the payment values, or as the limit of the positive-rate formula.
Common mistake avoided. Do not report an undefined \(0/0\) expression or divide the principal by an annuity factor formula that also has zero in its denominator. At zero interest, the present value factor is the sum of \(n\) ones, which is \(n\).
Using the Formula Reliably
A reliable calculation begins with the dates, not the formula. Identify when the loan is advanced, when the first payment is due, how frequently payments occur, and how many payments are required. Then express the interest rate for one interval between scheduled payments. Only after those items are aligned should the payment be calculated.
Place the loan principal at time 0 and confirm that the first payment is one full payment period later. Count all payments to obtain \(n\).
Convert the stated interest basis to the effective rate for one payment period. For a nominal annual rate \(i^{(m)}\) convertible \(m\) times per year, when payments occur at each conversion date, the period rate is \(i^{(m)}/m\).
For a positive period rate, set \(L=R a_{\overline{n}|}\) and solve \(R=L/a_{\overline{n}|}\). For zero interest, use \(L=nR\).
Discount the payments back to the loan date, or accumulate them to a common later date. Their value should match the principal under the same rate and timing assumptions.
The underlying reason for the formula is the same principle used in other present-value calculations: dollars due at different dates must first be converted to a common date. Discounting each payment separately produces a geometric series because the payment is constant and each successive payment is one period farther away. The formula is therefore not an unexplained shortcut; it is a compact way to sum the discounted cash flows.
Common Timing and Rate Errors
- Using the wrong payment timing. An annuity-immediate has payments at the ends of periods. An annuity-due has payments at the beginnings. Moving the first payment from one period after the loan date to the loan date changes the present value.
- Confusing years with payment periods. A two-year loan with quarterly payments has eight payment periods, not two.
- Using a rate for the wrong interval. A nominal annual rate must be converted according to its stated compounding frequency. It is not automatically the effective rate for each loan payment period.
- Forgetting the zero-rate case. The positive-rate expression has a zero denominator at \(i=0\); use \(R=L/n\) instead.
- Rounding too early. Retain the calculator value for the annuity factor and payment until the final reporting step. Intermediate rounding may create a small discrepancy in a present-value check.
Check Your Understanding
For each problem, identify the rate per payment period and the number of payments before applying the appropriate present-value equation.
- A $2,100 loan is repaid by two equal annual payments at the ends of the years. The effective annual rate is 10%. Find the payment. (Answer: $1,210.00)
- A $1,440 loan is repaid with two equal end-of-period payments. The effective rate per period is 25%. Find the payment. (Answer: $1,000.00)
- A $3,000 loan is repaid by two equal semiannual payments, with the first in six months. The nominal annual rate is 8% convertible semiannually. Find the payment to the nearest cent. (Answer: $1,590.59)
- A $12,000 interest-free loan is repaid by six equal annual payments, the first due one year after issue. Find the payment and state the valuation equation. (Answer: $2,000.00; \(12{,}000=6R\))
- A borrower receives $10,000 and will make two equal payments at times 0 and 1. The effective annual interest rate is 5%. Find the payment to the nearest cent, and explain why the annuity-immediate payment formula is not appropriate. (Answer: $5,121.95; the first payment is due immediately, so use the annuity-due factor.)