Annuities and Gradients

Annuities and Gradients

$Learning Objectives$

Identify ordinary annuities, annuities due, deferred series, perpetuities, arithmetic gradients, and geometric gradients from their timing patterns.
Apply the standard uniform-series factors on the correct period basis.
Explain why an arithmetic gradient has zero gradient increment at year 1 and begins changing at year 2.
Convert arithmetic and geometric gradient series to present or annual equivalents.
Decompose mixed cash-flow patterns into simpler equivalent series without double counting.

$Ordinary Annuity$

An ordinary annuity is a uniform series of equal cash flows $A$ occurring at the end of consecutive periods, with the first payment one period after the focal date.

$Payment Timing and Focal-Date Patterns$

Series	First and subsequent dates	Equivalent-value placement
Ordinary annuity	Equal end-of-period payments; first at period 1	One period before the first payment
Annuity due	Equal beginning-of-period payments; first at time 0	One period earlier than the matched ordinary annuity
Deferred annuity	Equal payments begin after one or more blank periods	Value the series one period before its first payment, then shift to the requested date
Perpetuity	Equal end-of-period payments continue indefinitely	One period before the first payment; requires $i>0$

Uniform-Series Present Worth (P/A)

Present worth one period before the first of n equal end-of-period payments.

P=A\frac{(1+i)^n-1}{i(1+i)^n}

Variables

Symbol	Description	Unit
$P$	Present equivalent one period before the first payment	-
$A$	Uniform end-of-period amount	-
$i$	Effective rate per payment period	-
$n$	Number of uniform payments	-

Capital-Recovery Factor (A/P)

Uniform end-of-period amount equivalent to a present sum.

A=P\frac{i(1+i)^n}{(1+i)^n-1}

Variables

Symbol	Description	Unit
$A$	Equivalent uniform amount	-
$P$	Present amount	-
$i$	Effective rate per period	-
$n$	Number of payments	-

Uniform-Series Future Worth (F/A)

Future equivalent at the same date as the final payment.

F=A\frac{(1+i)^n-1}{i}

Variables

Symbol	Description	Unit
$F$	Future equivalent at period n	-
$A$	Uniform end-of-period amount	-
$i$	Effective rate per period	-
$n$	Number of payments	-

Sinking-Fund Factor (A/F)

Uniform amount required to accumulate a target future sum.

A=F\frac{i}{(1+i)^n-1}

Variables

Symbol	Description	Unit
$A$	Uniform end-of-period deposit	-
$F$	Target future amount	-
$i$	Effective rate per period	-
$n$	Number of deposits	-

$Annuity Due$

An annuity due is a uniform series whose payments occur at the beginning of each period. Relative to an otherwise identical ordinary annuity, every payment is shifted one period earlier.

Annuity-Due Present Worth

Shifts an ordinary-annuity present equivalent one period earlier.

P_{\text{due}}=P_{\text{ordinary}}(1+i)

Variables

Symbol	Description	Unit
$P_{\text{due}}$	Present worth of the beginning-of-period series	-
$P_{\text{ordinary}}$	Present worth of the corresponding end-of-period series	-
$i$	Effective rate per payment period	-

$Deferred Annuity$

A deferred annuity is a uniform series that begins after one or more periods with no payments. First find the equivalent one period before the first payment, then shift that equivalent to the required focal date.

$Perpetuity$

A perpetuity is a uniform end-of-period series that is modeled as continuing indefinitely. It is an idealized economic model used only when the continuing-service assumption is defensible.

Present Worth of a Perpetuity

Present equivalent one period before an indefinite uniform series, valid for a positive periodic interest rate.

P=\frac{A}{i},\qquad i>0

Variables

Symbol	Description	Unit
$P$	Present worth immediately before the first perpetual payment	-
$A$	Uniform end-of-period amount	-
$i$	Positive effective rate per payment period	-

$Perpetuity Requires a Positive Rate$

For a nonzero uniform end-of-period series under the standard discrete model, the infinite present worth converges only when the effective rate per payment period is positive. When $i\le0$ , do not use $P=A/i$ ; revisit the rate basis or use a finite analysis horizon.

Perpetuity Applicability and Valuation

Check the indefinite uniform-series assumption and positive payment-period rate before valuing and shifting a perpetuity.

Use this screen only for a nonzero uniform end-of-period series whose continuing-service and cash-flow assumptions support an indefinite model. Convert the effective rate to the payment period. If i is not positive, do not use P = A/i; use a finite horizon or revisit the rate basis. The present worth lies one payment period before the first payment and may need a focal-date shift.

$Arithmetic Gradient$

An arithmetic gradient is a series whose cash flow changes by a constant amount $G$ each period. In the standard factor convention, the gradient component is $0$ at year 1, $G$ at year 2, $2G$ at year 3, and $(n-1)G$ at year $n$ .

$Arithmetic and Geometric Gradient Signatures$

Period	Uniform base	Arithmetic increment	Combined arithmetic series	Geometric series from $A_1$
1	$A$	$0$	$A$	$A_1$
2	$A$	$G$	$A+G$	$A_1(1+g)$
3	$A$	$2G$	$A+2G$	$A_1(1+g)^2$
$n$	$A$	$(n-1)G$	$A+(n-1)G$	$A_1(1+g)^{n-1}$

The arithmetic increment is additive and starts at zero in period 1. The geometric amount compounds from the first payment.

Arithmetic-Gradient Present Worth (P/G)

Present worth of the standard 0, G, 2G, ..., (n-1)G gradient sequence.

P=G\frac{(1+i)^n-in-1}{i^2(1+i)^n}

Variables

Symbol	Description	Unit
$P$	Present worth of the gradient component	-
$G$	Constant arithmetic change per period	-
$i$	Effective rate per period	-
$n$	Number of periods in the series	-

Arithmetic Gradient to Uniform Series (A/G)

Uniform annual equivalent of the standard arithmetic-gradient component.

A=G\left[\frac{1}{i}-\frac{n}{(1+i)^n-1}\right]

Variables

Symbol	Description	Unit
$A$	Equivalent uniform end-of-period amount	-
$G$	Constant arithmetic change per period	-
$i$	Effective rate per period	-
$n$	Number of periods in the gradient series	-

$Geometric Gradient$

A geometric gradient is a series in which each payment changes by a constant percentage $g$ from the preceding payment. If the first payment is $A_1$ at year 1, payment $t$ is $A_1(1+g)^{t-1}$ .

Geometric-Gradient Present Worth

Present worth of n payments starting with A1 at year 1 and growing at g each period.

P=\frac{A_1}{i-g}\left[1-\left(\frac{1+g}{1+i}\right)^n\right],\quad i\ne g

Variables

Symbol	Description	Unit
$P$	Present worth at time zero	-
$A_1$	First payment at the end of period 1	-
$i$	Effective discount rate per period	-
$g$	Geometric growth rate per period	-
$n$	Number of payments	-

Geometric-Gradient Limit for i = g

Special case because the general geometric-gradient expression has a zero denominator when i equals g.

P=\frac{nA_1}{1+i}

Variables

Symbol	Description	Unit
$P$	Present worth when i equals g	-
$A_1$	First payment at period 1	-
$i$	Common discount and growth rate	-
$n$	Number of payments	-

$Interactive Series Laboratory$

Use the gradient laboratory to see the actual payment sequence while changing $A$ , $G$ , $g$ , $i$ , and $n$ .

Series typeUniform/base amount, APHPArithmetic increment, GPHP/period

Discount rate per period10.00%

Number of payments8

Payment timing

Present worth at time zero

₱3,949,757

For the standard arithmetic component, the first gradient increment is zero at the first payment; G first appears at the second payment.

Timing interpretation

First payment: t = 1

Ordinary timing places the first payment at the end of period 1.

Signed cash-flow sequence

The simulator discounts the explicit dated cash flows rather than changing a factor label only. That makes timing shifts visible and prevents ordinary, due, and deferred series from being treated as interchangeable.

Amount at source periodPHP

Effective rate per period8.00%

Source periodt = 2

Focal periodt = 8

Equivalent amount at focal date

₱1,586,874.32

Compounded across 6 periods.

Transformation

Use F/P

The rate and period count both use the same period basis.

$Mixed-Series Decomposition$

A cash-flow pattern such as $500, 600, 700, 800$ may be modeled as a uniform base of $500$ plus an arithmetic gradient of $0,100,200,300$ . Decomposition is a bookkeeping device: every actual cash flow must be represented exactly once.

$Gradient-Origin Trap$

Do not assign $G$ to year 1 when using the standard arithmetic-gradient factor. The first gradient increment is zero at year 1; the first $G$ occurs at year 2. If the physical series starts differently, shift or decompose it before using the factor.

$Perpetuity Is an Assumption, Not a Physical-Life Claim$

The formula $P=A/i$ represents an indefinitely continuing economic series. Do not use it merely because an infrastructure asset is long-lived; the service, renewal, and cash-flow assumptions must support an indefinite model.

$Series Identification Workflow$

STEP-BY-STEP

Mark the first and last cash-flow dates.
Determine whether payments are equal, linearly changing, or percentage-changing.
Identify whether the first payment is at the beginning or end of the period.
Separate any uniform base from the gradient component.
Convert the rate to the payment period.
Apply the factor at its natural focal date, then shift the result if required.
Reconstruct selected periods to verify the decomposition.

Annuity and Gradient Series Selection

Classify a finite payment pattern before applying an equivalence factor, then shift and verify its focal date.

Use an effective rate per payment period. The standard arithmetic-gradient component is 0, G, 2G, …, (n−1)G. For a mixed stream, decompose its components before classifying them. This chart selects finite uniform, arithmetic-gradient, and geometric-gradient patterns; use the separate perpetuity screen for an indefinite uniform end-of-period series.

Key Takeaways

Ordinary annuities use equal end-of-period payments; annuities due are shifted one period earlier.
Deferred annuities require an additional focal-date shift after applying the uniform-series factor.
A perpetuity is an idealized indefinite uniform series with $P=A/i$ for $i>0$ .
The standard arithmetic gradient is $0,G,2G,\ldots,(n-1)G$ , so $G$ begins at year 2.
The $A/G$ factor converts the standard arithmetic-gradient component to an equivalent uniform series.
Geometric gradients change by a percentage, not by a constant currency increment.
Mixed series can be decomposed into uniform and gradient components if every cash flow is represented once.
Rate period, payment period, and factor timing must be consistent.

$References$

Engineering Economy, 18th edition — William G. Sullivan, Elin M. Wicks, and Joseph H. Wilck, Pearson, 2024; chapter 4 covers annuities, deferred series, gradients, cash-flow diagrams, and equivalence.