Cash-Flow Timelines for Amortizing Loans

Why a Loan Timeline Matters

A loan exchanges cash at different dates. The borrower receives the principal at origination and makes repayments later; the lender’s cash flows have the opposite signs. A timeline makes the dates explicit before any discounting begins. That step is important because an otherwise correct present-value formula can give the wrong result if the payment dates or rate period do not match the contract.

This tutorial considers a loan advanced in full at time 0 and repaid by equal installments at the ends of regularly spaced periods. It focuses on representing those cash flows and expressing the original principal as their present value. It does not calculate the installment from the principal; the central question here is how to value a specified stream of installments.

Definition: A level end-of-period installment loan is a loan with equal payments made at the end of each successive payment period. If the principal is advanced at time 0, the first installment is at time 1, and there are $n$ installments, the payment dates are $1,2,\ldots,n$, measured in payment periods. A loan is fully repaid by those installments when the value of the repayments at origination equals the principal advanced, subject to the stated interest and contract assumptions.

For a borrower, money received is positive and payments made are negative. For the lender, money advanced is negative and repayments received are positive. Either perspective is valid, but a present-value equation must use one perspective consistently. When calculating a positive principal amount, it is often simplest to compare the positive loan proceeds with the positive present value of the repayments.

Build the Timeline Before Valuing It

Suppose the borrower receives principal $L$ at time 0 and makes level payments $R$ at the ends of $n$ periods. The borrower’s cash flows are $+L$ at time 0 and $-R$ at each payment date. The lender’s cash flows have the opposite signs. The following table shows the borrower’s view.

Time, in payment periods	0	1	2	...	n
Borrower cash flow	+$L$	−$R$	−$R$	...	−$R$

The rate used to discount the payments must apply to the same length of time as one interval between payments. Let $j$ be the effective interest rate per payment period, and let $v_j=1/(1+j)$ be the corresponding discount factor per payment period. If $j$ is positive and constant, the installment at time $k$ has time-0 value $Rv_j^k$. Thus the present value of the $n$ installments is the sum of their discounted values:

L=R v_j+R v_j^2+\cdots+R v_j^n =R\sum_{k=1}^{n}v_j^k =R\frac{1-v_j^n}{j}.

The annuity notation $a_{\overline{n}|}$ uses the effective rate for the period between payments. If payments are annual and $i$ is the effective annual rate, the expression is $L=R a_{\overline{n}|}$ at rate $i$. If payments are quarterly, the rate used in the annuity factor must instead be the effective rate per quarter. Rate-conversion methods establish equivalent rates; the timeline identifies which interval needs a rate.

Assumptions for the level-payment present-value formula: The principal is advanced at time 0; there are exactly $n$ equal payments; each payment occurs at the end of a payment period; the rate per payment period $j$ is constant; and there are no fees, skipped payments, or other cash flows. The formula does not directly represent beginning-of-period payments, irregular payment dates, or a loan with additional charges unless those cash flows are separately included or the timing is adjusted.

Why the Annuity Factor Works

The installment present value is a finite geometric series. The first discounted payment is $Rv_j$, and each succeeding term is the preceding term multiplied by $v_j$. For $j\ne 0$, the finite geometric-series identity gives $\sum_{k=1}^{n}v_j^k=v_j(1-v_j^n)/(1-v_j)$. Because $v_j=1/(1+j)$, we have $1-v_j=jv_j$. Substitution then gives the familiar annuity factor:

\sum_{k=1}^{n}v_j^k =\frac{v_j(1-v_j^n)}{1-v_j} =\frac{v_j(1-v_j^n)}{jv_j} =\frac{1-v_j^n}{j}.

This derivation explains why the level-payment value is not $nR$: later payments are discounted for more periods than earlier ones. It also shows why the rate $j$ must be a per-payment-period rate. If $j=0$, the geometric-series expression with division by $j$ is not used; the present value is simply $nR$, since no payment is discounted.

Worked Timelines and Present Values

Worked Example: Three Annual Installments

A borrower receives a loan at time 0 and repays it with three annual end-of-year installments of $4,000. The effective annual interest rate is 6%. Find the time-0 value of the installments.

Method. The payments occur at times 1, 2, and 3, so discount them by one, two, and three annual periods. The payment interval is one year, matching the effective annual rate $i=0.06$.

\begin{array}{c|cccc} \text{Time} & 0 & 1 & 2 & 3\\ \text{Borrower cash flow} & +L & -4000 & -4000 & -4000 \end{array}

The annual discount factor is $v=1/1.06=0.9433962264$. The powers needed are $v^2=1/1.06^2=0.8899964400$ and $v^3=1/1.06^3=0.8396192830$. Therefore:

\begin{aligned} L &=4000(v+v^2+v^3)\\ &=4000(0.9433962264+0.8899964400+0.8396192830)\\ &=4000(2.6730119494)\\ &=10692.0478\ldots \end{aligned}

Rounded to the nearest cent, the time-0 present value is $10,692.05. This is the principal consistent with the specified installments and rate under the stated assumptions. A common mistake is to count three payments but discount all of them by the same power; the timeline shows that their discount periods are 1, 2, and 3, respectively.

Worked Example: Annual Payments With Semiannual Compounding

A loan is repaid by three annual end-of-year payments of $2,500. Its quoted nominal annual rate is 8% convertible semiannually. Find the present value at origination.

Method. First convert the quoted rate to an effective rate over the annual payment interval. The periodic rate is $0.08/2=0.04$ per half-year, so two half-year periods make one year:

i_{\text{annual}}=(1+0.08/2)^2-1=1.04^2-1=0.0816.

The effective annual rate is therefore 8.16%, not 8%. Each payment is one, two, or three years from origination, so the time-0 value is:

\begin{aligned} L &=2500\left(\frac{1-(1.0816)^{-3}}{0.0816}\right)\\ &=2500(2.569676\ldots)\\ &=\$6,424.19. \end{aligned}

The timeline has annual payment dates even though the quoted rate compounds semiannually. Using the half-year rate directly as if each payment were six months away would put the payments at the wrong dates. A related common error is to use 8% as the effective annual rate. That incorrect calculation would be $2500[1-(1.08)^{-3}]/0.08$, rather than the expression above. Rate conversion and payment timing are separate checks: convert to the effective rate for the payment interval, then discount each payment by the number of those intervals to its date.

Worked Example: Quarterly Payments Under Monthly Compounding

A borrower receives a loan and makes four quarterly payments of $1,000 at the ends of quarters 1 through 4. The nominal annual rate is 6% convertible monthly. Find the time-0 present value.

Method. The monthly rate is $0.06/12=0.005$. One quarter contains three months, so the effective quarterly rate is:

$$ j=(1.005)^3-1=0.015075125. $$

The four payment dates are quarter 1, quarter 2, quarter 3, and quarter 4. Discounting each payment by the corresponding number of quarterly periods gives:

\begin{aligned} L &=1000\left(\frac{1-(1.015075125)^{-4}}{0.015075125}\right)\\ &=1000(3.85368\ldots)\\ &=\$3,853.68. \end{aligned}

The result is rounded to the nearest cent. The main trap is to use the monthly rate as though the installments were monthly. The payment interval is a quarter, so either discount each payment over 3, 6, 9, and 12 months using the monthly rate, or convert to the equivalent quarterly rate and discount over 1, 2, 3, and 4 quarters. Both methods represent the same cash flows.

Delayed First Payments

A level stream does not always begin one period after the loan date. If the first of $n$ equal payments is at time $m$, followed by payments at times $m+1,\ldots,m+n-1$, the whole stream is delayed relative to the ordinary pattern. Its time-0 value is the value of the corresponding $n$-payment annuity one period before its first payment, discounted back to time 0. This requires keeping the payment count and the number of delay periods distinct.

Worked Example: Four Payments Beginning in Year 2

Four annual payments of $2,000 are due at the ends of years 2, 3, 4, and 5. The effective annual interest rate is 5%. Find their time-0 value.

Method. First value the four payments one year before the first payment, at time 1. That is an ordinary four-payment annuity at 5%. Then discount that value one year to time 0. Equivalently, discount the four individual payments for 2, 3, 4, and 5 years.

\begin{array}{c|cccccc} \text{Time} & 0 & 1 & 2 & 3 & 4 & 5\\ \text{Payment} & 0 & 0 & 2000 & 2000 & 2000 & 2000 \end{array}

\begin{aligned} L &=2000v a_{\overline{4}|}\text{ at }i=0.05\\ &=2000\left(\frac{1}{1.05}\right) \left(\frac{1-(1.05)^{-4}}{0.05}\right)\\ &=2000(0.9523809524)(3.5459505\ldots)\\ &=\$6,754.19. \end{aligned}

The same result comes from $2000(v^2+v^3+v^4+v^5)$. A common timing error is to use $2000a_{\overline{4}|}$ directly at time 0, which places the payments at years 1 through 4. The extra factor $v$ accounts for the one-year delay.

Checks That Keep a Loan Timeline Consistent

Before evaluating a loan present value, check the dates and the rate together. A payment period is the interval between consecutive installments, not necessarily a year or the compounding period named in a rate quote. If the nominal rate and payment frequency differ, convert to the effective rate over one payment period before using an annuity factor.

Check the first payment date. End-of-period installments begin at time 1. A first payment at time 0 is a different timing pattern.
Count payment intervals, not calendar labels. Four quarterly payments at quarter-ends occur at times 1 through 4 in quarterly units.
Keep signs consistent. The borrower receives the principal and pays installments; the lender advances funds and receives installments.
Separate the payment count from a delay. A delayed four-payment stream still has four payments; the delay changes the discounting, not the number of installments.
Do not treat a quoted nominal rate as an effective rate automatically. Convert it to the effective rate that matches the interval between payments.

The present-value equality is useful for interpreting the principal of a fully amortizing loan: under the assumptions stated here, the loan proceeds and the discounted repayments balance at origination. In actual contracts, fees or other cash flows can alter the borrower’s total cost. Such amounts must be shown separately on the timeline rather than silently included in the level-payment annuity.

Key takeaway: Draw the cash flows at their actual dates, identify the time between payments, and use the effective rate for that interval. For $n$ equal end-of-period payments $R$, the time-0 value is $R a_{\overline{n}|}$ at the per-payment-period rate; if the stream is delayed, discount that annuity value for the additional delay.

Check Your Understanding

For each problem, identify the payment interval and timing before evaluating the present value.

A borrower receives funds at time 0 and makes five annual end-of-year payments of $1,800. The effective annual rate is 4%. Find the principal represented by the payments, rounded to the nearest cent. (Answer: $1800[1-(1.04)^{-5}]/0.04=$$8,013.28)
Four quarterly end-of-quarter payments of $1,250 are made under a nominal annual rate of 6% convertible monthly. Find their time-0 value. (Answer: $1250[1-(1.005)^{-12}]/[(1.005)^3-1]=$$4,817.10)
Four annual payments of $3,000 are due at the ends of years 3 through 6. The effective annual rate is 5%. Write the time-0 value, making the delay explicit. (Answer: $3000v^2a_{\overline{4}|}$ at $i=0.05$, or $3000v^2(v+v^2+v^3+v^4)$)
A loan is repaid by 12 monthly end-of-month installments of $500. The effective rate per month is 1%. Find the present value at origination, rounded to the nearest cent. (Answer: $500[1-(1.01)^{-12}]/0.01=$$5,627.54)
A lender advances a loan and receives three semiannual end-of-period payments of $4,000. The effective rate per half-year is 2%. State the lender’s signed cash flows and find the advance amount. (Answer: $-\$11,535.53$ at time 0 and $+\$4,000$ at each of times 1, 2, and 3; advance amount $=4000[1-(1.02)^{-3}]/0.02=$$11,535.53)

What You'll Learn