Timelines for Level-Payment Loans

Why a Loan Timeline Matters

A level-payment loan can be described as a set of cash flows: the borrower receives the principal and makes equal repayments on specified dates. Before calculating any payment or balance, the timeline must show when each cash flow occurs and which interest rate applies between those dates. A correctly labeled timeline prevents a frequent error: using a rate for one time unit with payments measured in another.

This tutorial focuses on representing a fixed-rate loan with end-of-period level payments. The same valuation principles used for annuities apply, but the loan description must make the payment dates, rate conversion, and valuation date explicit. These timeline skills are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: A level-payment loan has an initial principal $L$ advanced at time 0 and a specified number $n$ of equal payments, each of amount $R$, made at regular intervals. For an end-of-period payment arrangement, the first payment is made one full payment period after the loan is advanced, and subsequent payments are made one period apart. The periodic effective interest rate $i$ is the effective rate earned or charged over one payment period.

Build the Timeline Before Valuing the Loan

Start by choosing a time unit that matches the payment interval. If payments are monthly, label the timeline in months; if they are quarterly, label it in quarters. For a loan advanced at time 0 with $n$ end-of-period payments, the payment dates are periods 1 through $n$. A timeline may be written as follows:

\begin{array}{c|cccccc} \text{Time} & 0 & 1 & 2 & \cdots & n-1 & n\\ \hline \text{Cash flow to borrower} & +L & -R & -R & \cdots & -R & -R \end{array}

The signs above use the borrower's perspective: the principal received is positive, and repayments are negative. For a present-value equation, it is often more convenient to compare the positive principal with the positive values of the repayments. Under a constant periodic effective rate $i$, let $v=1/(1+i)$. The present value at time 0 of the end-of-period payments is the sum of each payment discounted to time 0.

L=Rv+Rv^2+\cdots+Rv^n=R a_{\overline{n}|}

This equation assumes a fixed rate per payment period, payments made at the end of each period, and valuation at time 0. It does not describe payments made in advance, irregular payment dates, or a loan with rates that change between periods without further adjustments. The annuity notation $a_{\overline{n}|}$ uses the same periodic rate $i$ as the loan timeline.

Match the Rate to the Payment Period

The rate marked on the timeline must be an effective rate for one interval between payments. If an annual effective rate is given and payments are more frequent than annually, convert the annual accumulation factor to the matching period. For $m$ equal payment periods in one year, the equivalent effective rate per period is

i_{\text{period}}=(1+i_{\text{annual}})^{1/m}-1.

If instead the loan specifies a nominal annual rate $i^{(m)}$ convertible $m$ times per year, the rate per conversion period is $i^{(m)}/m$. These are different descriptions of an annual rate and should not be interchanged. In either case, the payment interval and the rate period must agree before using an annuity factor.

Worked Example: Monthly Payments at a Nominal Annual Rate

A hypothetical loan is repaid by 12 end-of-month payments of $1,000. The stated rate is a nominal annual rate of 12% convertible monthly. Find the loan principal consistent with these payments and rate. Use the borrower's timeline and report the principal to the nearest cent.

Method. Use months as the timeline unit. There are 12 payment periods, and the effective rate per month is the nominal rate divided by 12:

i=\frac{0.12}{12}=0.01,\qquad v=\frac{1}{1.01}.

The principal is at month 0. The 12 payments occur at months 1, 2, through 12; none occurs at month 0 because the payments are at the end of each month. Thus

L=1000(v+v^2+\cdots+v^{12}) =1000\frac{1-(1.01)^{-12}}{0.01}.

Using $(1.01)^{-12}\approx0.887449225$, the annuity factor is

a_{\overline{12}|} =\frac{1-0.887449225}{0.01} \approx11.2550775.

Therefore $L\approx1000(11.2550775)=\$11,255.0775$, so the principal is $11,255.08 to the nearest cent.

Check. Each payment is discounted by a positive factor less than 1, so the present value of the 12 payments must be less than their undiscounted total of $12,000. The calculated principal, $11,255.08, satisfies that check. The common mistake avoided here is treating 12% as the monthly rate; the monthly rate is 1%.

Valuing the Payments at a Date Other Than Time 0

A loan timeline can also be valued at an intermediate date. State exactly where that date falls relative to a payment: immediately before a payment, immediately after it, or between payment dates. That distinction determines whether the payment at that date is included in the value of the payments being considered.

For example, if the valuation date is immediately after payment $k$, the first remaining payment is one full period later, at time $k+1$. The value at time $k$ of the remaining payments is therefore $R a_{\overline{n-k}|}$, provided the rate per period remains $i$. Immediately before payment $k$, that payment is still due and must be included as well.

Valuation-date rule: To value a set of payments at a chosen date, accumulate payments before that date to it and discount payments after that date to it. A payment made exactly at the valuation date is included at its full amount if it is still due; it is not included if the valuation is immediately after that payment.

The rule follows directly from equivalence of values at different dates. A payment $R$ due $j$ periods after the valuation date has value $Rv^j$ at that date, because accumulating $Rv^j$ for $j$ periods gives $Rv^j(1+i)^j=R$. A payment $R$ due $j$ periods before the date accumulates to $R(1+i)^j$. Summing these date-specific values gives the value of the whole cash-flow stream at the chosen date.

Worked Example: Value Immediately Before and After a Payment

A hypothetical loan is repaid by five annual end-of-year payments of $2,300, at times 1 through 5. The effective annual interest rate is 6%. Find the value of the unpaid payments at time 2 immediately after the second payment, and immediately before the second payment. Report each value to the nearest cent.

Method. Use years as the timeline unit, so the periodic rate is $i=0.06$ and $v=1/1.06$. Immediately after the payment at time 2, only the payments at times 3, 4, and 5 remain. Relative to time 2, their payment dates are 1, 2, and 3 periods away:

V_{\text{after at }2} =2300(v+v^2+v^3) =2300a_{\overline{3}|}.

Since $1.06^3=1.191016$, the factor is

a_{\overline{3}|} =\frac{1-(1.06)^{-3}}{0.06} =\frac{1-1/1.191016}{0.06} \approx2.67301195.

Thus the value immediately after payment 2 is $2300(2.67301195)=\$6,147.9275$, or $6,147.93.

Immediately before payment 2, the $2,300 payment at time 2 is still due. It is at the valuation date and is not discounted. The other three payments have the same value as in the immediately-after calculation:

V_{\text{before at }2} =2300+2300a_{\overline{3}|} =2300+6147.9275 =\$8,447.9275.

So the value immediately before payment 2 is $8,447.93. The difference between the two values is exactly $2,300, the payment due at time 2. The common mistake avoided is omitting that payment when the valuation date is immediately before it.

Deferred Payments and Clearly Labeled Dates

A first payment need not occur one period after the loan is advanced. If the first payment is deferred, put the actual payment dates on the timeline rather than relabeling the first payment as period 1. For payments at periods $d+1,d+2,\ldots,d+n$, the value at time 0 is the value at time $d$ of an $n$-payment annuity, discounted $d$ periods:

L=R v^d a_{\overline{n}|}.

This expression assumes a constant effective rate $i$ per period throughout the deferral and payment intervals. It comes from discounting every payment by its actual number of periods: the first is discounted $d+1$ periods and the last $d+n$ periods. Factoring out $v^d$ leaves $R(v+v^2+\cdots+v^n)$. If the rate changes, the discount factors must reflect the rates applying over each interval.

Worked Example: A Three-Month Deferral

A hypothetical loan is repaid by six level monthly payments of $500, with the first payment at the end of month 4 and the last at the end of month 9. The nominal annual rate is 12% convertible monthly. Find the value of these payments at time 0. Report the result to the nearest cent.

Method. Use months as the unit. The monthly effective rate is $i=0.12/12=0.01$, so $v=1/1.01$. Because the six payments occur at months 4 through 9, their dates are three periods later than a standard six-payment stream at months 1 through 6. Therefore

V_0=500v^3a_{\overline{6}|}.

First compute the six-payment factor. Since $(1.01)^{-6}\approx0.942045235$,

a_{\overline{6}|} =\frac{1-(1.01)^{-6}}{0.01} \approx\frac{1-0.942045235}{0.01} =5.7954765.

The value at month 3 of the six payments at months 4 through 9 is $500(5.7954765)=\$2,897.73825$. Also,

v^3=(1.01)^{-3}\approx0.970590147.

Discount the month-3 value to month 0:

V_0=2897.73825(0.970590147)\approx\$2,812.52.

Check. The first payment is at month 4, not month 1, so the deferral lowers the value relative to six immediate monthly end payments. The common mistake avoided is discounting the first payment only one month; it is four months from time 0.

Common Timeline Errors

A well-labeled timeline is a practical check on the assumptions behind a loan valuation. Before calculating, verify each of the following:

Payment timing: End-of-period payments occur at times 1 through $n$, not at time 0. Payments in advance would require different timing.
Rate period: A monthly payment schedule needs an effective monthly rate. Do not use an annual rate directly as though it were monthly.
Valuation date: Mark whether the value is immediately before or immediately after a payment. The payment at that date is treated differently in the two cases.
Deferral: Record a deferred first payment at its actual date. Do not shift it to time 1 merely because it is the first payment.
Perspective and signs: From the borrower's perspective, the advance is received and repayments are paid out. When equating present values, compare the positive principal with the positive present value of repayments.

A timeline is more than a drawing. It is a compact statement of the loan model: the principal date, payment dates, rate interval, and valuation date. Once those are consistent, the relevant annuity or discounting expression follows from the dates shown.

Key takeaway: For a fixed-rate loan with end-of-period level payments, first mark the principal at time 0 and the payments at the ends of the payment periods. Convert the stated interest rate to an effective rate per payment period, then value each cash flow at the date requested.

Check Your Understanding

For each problem, label the dates and rate period before using an annuity expression.

A loan is repaid by four annual end-of-year payments of $700. The effective annual rate is 5%. Find the present value at time 0. (Answer: $700a_{\overline{4}|}=\$2,482.17$.)
A loan has 18 end-of-month payments of $250. The nominal annual rate is 6% convertible monthly. Write the present value at time 0 and identify the monthly rate. (Answer: $i=0.005$ per month and $250a_{\overline{18}|}=250[1-(1.005)^{-18}]/0.005$.)
Five semiannual end-of-period payments of $1,200 are valued at time 0. The effective annual rate is 8%. Write the present value using the correct semiannual rate. (Answer: $i=\sqrt{1.08}-1$ per half-year and $1200a_{\overline{5}|}$ at that rate.)
Four annual payments of $900 are due at times 3, 4, 5, and 6. The effective annual rate is 4%. Write the value at time 0. (Answer: $900v^2a_{\overline{4}|}$, where $v=1/1.04$.)
Eight annual end-of-year payments of $500 are due at times 1 through 8. Find the value of the payments at time 4 immediately after the fourth payment, and immediately before that payment. The effective annual rate is 6%. (Answer: Immediately after, $500a_{\overline{4}|}\approx\$1,732.55$; immediately before, $500+500a_{\overline{4}|}\approx\$2,232.55$, with $a_{\overline{4}|}$ evaluated at 6%.)

What You'll Learn