PV & FV of Annuities — Compute PV and FV of ordinary annuities and annuities due - Finance

SECTION 1

Historical Context & Motivation

The concept of valuing a stream of equal payments stretches back centuries, well before modern spreadsheets automated the arithmetic. Whenever a lender offered a borrower a fixed repayment schedule, or a government issued bonds with periodic coupon payments, both parties implicitly relied on annuity mathematics to determine a fair price. Understanding how these formulas developed illuminates why they remain indispensable in corporate finance, personal financial planning, and capital markets today.

1613 • Stevin's Interest Tables

Simon Stevin, a Flemish mathematician, published compound-interest tables that allowed merchants in the Low Countries to price annuities and evaluate loan contracts systematically for the first time.

1725 • De Moivre's Annuity Formulas

Abraham de Moivre formalized closed-form annuity-certain formulas in his treatise on probability, providing the algebraic framework still used in finance textbooks.

1899 • Actuarial Science Matures

Life-insurance companies adopted standardized present-value annuity tables, enabling mass-market pricing of whole-life policies and pension obligations across North America and Europe.

1979 • Financial Calculators

The introduction of programmable financial calculators (HP-12C, TI BA II Plus) made annuity computations accessible to every business student and practitioner, replacing bulky printed tables.

The central question these developments address is deceptively simple: What is a series of equal future payments worth right now, and what will it grow to if invested? Whether you are evaluating a car loan, pricing a corporate bond, or projecting the balance of a 401(k) plan, the answer hinges on the precise timing and discounting of each cash flow—an exercise that the annuity formulas handle with elegant efficiency.

SECTION 2

Core Principles & Definitions

Before diving into formulas, it is essential to anchor five foundational ideas that govern every annuity calculation. An annuity is defined as a finite series of equal cash flows occurring at regular intervals. Two variants dominate practice: the ordinary annuity (payments at the end of each period) and the annuity due (payments at the beginning of each period). This seemingly small timing difference has a material impact on valuation because each payment in an annuity due earns one additional period of interest relative to its ordinary-annuity counterpart.

Time Value of Money

A dollar today is worth more than a dollar tomorrow because it can be invested to earn interest. This principle underpins all discounting and compounding operations.

Ordinary Annuity

Cash flows occur at the end of each period. Most loan payments, bond coupons, and dividend streams follow this convention.

Annuity Due

Cash flows occur at the beginning of each period. Lease payments and insurance premiums are common real-world examples.

Present Value (PV)

The current-dollar equivalent of a future stream of payments, obtained by discounting each cash flow back to time zero at the relevant interest rate.

Future Value (FV)

The accumulated value of all payments at a specified future date, found by compounding each cash flow forward at the relevant interest rate.

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KEY TAKEAWAY

Think of an ordinary annuity like collecting your paycheck at the end of each month—you work first and then get paid. An annuity due is like paying rent on the first of the month—money leaves your account before you 'use' the apartment for that period. Because annuity-due payments arrive one period earlier, every payment has an extra period to compound (for FV) or requires one less period of discounting (for PV). This single-period shift means the annuity-due value is always exactly (1 + r) times the corresponding ordinary-annuity value.

SECTION 3

Visual Explanation — Cash-Flow Timelines

A cash-flow timeline is the single most powerful tool for setting up annuity problems correctly. It plots every payment on a horizontal time axis so you can verify the timing convention (end-of-period versus beginning-of-period) and identify the target date for discounting or compounding. The diagram below compares the two annuity types side by side for a four-period stream.

In the ordinary annuity (top), the first payment occurs at period 1 and the last at period 4. In the annuity due (bottom), payments shift to the beginning of each period (0 through 3). The PV is always calculated at period 0, and the FV at period 4.

Notice that both annuity types contain the same number of payments (four), but the annuity-due payments are each shifted one period to the left. This shift means that when computing the future value, every annuity-due payment compounds for one additional period, making the FV of an annuity due larger by a factor of (1 + r). Conversely, when computing the present value, each payment is discounted for one fewer period, again resulting in a value that is (1 + r) times greater. This relationship provides a convenient shortcut: solve the ordinary-annuity version first, then multiply by (1 + r) to convert to an annuity due.

SECTION 4

Mathematical Framework

The annuity formulas can be derived from the geometric-series representation of individually discounted (or compounded) cash flows. Rather than summing n separate present-value or future-value calculations, the closed-form expressions below collapse that summation into a single equation. Four formulas cover all basic annuity valuations.

PRESENT VALUE OF AN ORDINARY ANNUITY

PV_OA = PMT × [ (1 − (1 + r)⁻ⁿ) / r ]

Where PMT = periodic payment, r = interest rate per period, and n = total number of periods. The bracketed term is often called the present-value interest factor of an annuity (PVIFA).

FUTURE VALUE OF AN ORDINARY ANNUITY

FV_OA = PMT × [ ((1 + r)ⁿ − 1) / r ]

The bracketed term is the future-value interest factor of an annuity (FVIFA). It represents the value at time n of $1 received at the end of each of n periods.

PRESENT VALUE OF AN ANNUITY DUE

PV_AD = PMT × [ (1 − (1 + r)⁻ⁿ) / r ] × (1 + r)

Because each payment occurs one period sooner, the entire ordinary-annuity PV is multiplied by (1 + r) to remove one period of discounting.

FUTURE VALUE OF AN ANNUITY DUE

FV_AD = PMT × [ ((1 + r)ⁿ − 1) / r ] × (1 + r)

Each payment compounds for one extra period, so the ordinary-annuity FV is scaled up by (1 + r).

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Derivation Insight

The PV ordinary-annuity formula is derived by factoring PMT out of the sum Σ from t = 1 to n of PMT × (1 + r)⁻ᵗ. Recognizing this as a finite geometric series with common ratio (1 + r)⁻¹ yields the compact (1 − (1 + r)⁻ⁿ) / r expression. The FV formula follows a parallel approach, summing PMT × (1 + r)ⁿ⁻ᵗ for t = 1 to n.

SECTION 5

Ordinary Annuity vs. Annuity Due — Side-by-Side

The distinction between an ordinary annuity and an annuity due reduces to a single variable: payment timing. Yet that one-period shift cascades through every valuation result. The table below contrasts the two types across several dimensions, and the subsequent diagram visualizes how the (1 + r) multiplier amplifies the annuity-due values relative to their ordinary-annuity counterparts.

Feature	Ordinary Annuity	Annuity Due
Payment Timing	End of each period	Beginning of each period
Common Examples	Mortgage payments, bond coupons, salary	Lease payments, insurance premiums, rent
PV Formula	PMT × PVIFA	PMT × PVIFA × (1 + r)
FV Formula	PMT × FVIFA	PMT × FVIFA × (1 + r)
Value Relationship	Base case	Always (1 + r) times the ordinary annuity
Calculator Mode	END mode	BGN (Begin) mode

Key differences between ordinary annuities and annuities due.

For a $1,000 payment at 6% over 5 periods, the annuity due PV and FV exceed the ordinary annuity values by exactly 6%, confirming the × (1 + r) relationship.

The visual reinforces a powerful heuristic: you never need to memorize separate annuity-due formulas. Simply compute the ordinary-annuity value and multiply by (1 + r). On a financial calculator, toggling from END to BGN mode performs exactly this adjustment automatically.

SECTION 6

Worked Example — Retirement Savings Plan

Suppose you plan to deposit $500 at the end of every quarter into a retirement account that earns 8% annual interest, compounded quarterly, for 10 years. We will compute both the future value (how much you accumulate) and the present value (the lump sum that would be equivalent today). Then we will repeat both calculations assuming deposits are made at the beginning of each quarter (annuity due).

Ordinary Annuity — FV & PV

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Step 1 — Identify Given Values

PMT = $500 per quarter, annual rate = 8%, compounding = quarterly, so the periodic rate r = 8% ÷ 4 = 2% = 0.02. Total number of periods n = 10 years × 4 = 40 quarters.
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Step 2 — Compute FV of Ordinary Annuity

FV_OA = PMT × [((1 + r)ⁿ − 1) / r] = 500 × [((1.02)⁴⁰ − 1) / 0.02]. First compute (1.02)⁴⁰ = 2.20804. Then the FVIFA = (2.20804 − 1) / 0.02 = 1.20804 / 0.02 = 60.4020. Finally, FV_OA = 500 × 60.4020.

FV_OA ≈ $30,201.00
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Step 3 — Compute PV of Ordinary Annuity

PV_OA = PMT × [(1 − (1 + r)⁻ⁿ) / r] = 500 × [(1 − (1.02)⁻⁴⁰) / 0.02]. (1.02)⁻⁴⁰ = 1 / 2.20804 = 0.45289. PVIFA = (1 − 0.45289) / 0.02 = 0.54711 / 0.02 = 27.3555. PV_OA = 500 × 27.3555.

PV_OA ≈ $13,677.77
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Step 4 — Convert to Annuity Due

Multiply each ordinary-annuity result by (1 + r) = 1.02. FV_AD = $30,201.00 × 1.02 = $30,805.02. PV_AD = $13,677.77 × 1.02 = $13,951.33.

FV_AD ≈ $30,805.02 | PV_AD ≈ $13,951.33
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Step 5 — Interpret the Results

If deposits are made at the end of each quarter, you accumulate about $30,201 after 10 years; if made at the beginning, about $30,805—an extra $604 earned simply by depositing each payment one period earlier. The present-value figures tell you that a rational investor would be indifferent between receiving $13,678 (or $13,951) today and receiving the stream of 40 quarterly $500 payments.

SECTION 7

Strengths, Limitations & Common Pitfalls

Strengths	Limitations
Closed-form formulas allow rapid valuation without summing individual cash flows.	Assumes constant payment amount—cannot handle growing or irregular streams.
Easy conversion between ordinary annuity and annuity due with the (1 + r) multiplier.	Assumes a constant discount/interest rate over all periods.
Directly applicable to pricing bonds, loans, leases, and retirement accounts.	Does not accommodate mid-period cash flows or continuous compounding without modification.
Built into every financial calculator and spreadsheet (PV, FV, PMT functions).	Ignores taxes, transaction costs, and inflation unless explicitly modeled.

Strengths and limitations of annuity formulas.

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Common Pitfall

The most frequent exam error is a period mismatch: using an annual rate with monthly periods (or vice versa). Always ensure that r and n are stated in the same time unit. If the annual rate is 12% and payments are monthly, use r = 1% per month and n = total months.

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KEY TAKEAWAY

Annuity formulas are to finance what the compound-interest formula is to a single lump sum: they transform a tedious, repetitive calculation into a single elegant expression. However, just as a wrench only works on bolts of the right size, annuity formulas only apply when payments are equal and evenly spaced. For uneven cash flows, you must revert to individual discounting (the NPV approach), which we will cover in later coursework on capital budgeting.

SECTION 8

Connection to Advanced Topics

The ordinary annuity and annuity due are special cases within a broader family of cash-flow structures. As you progress through your finance curriculum, you will encounter several extensions that build directly on the concepts mastered here. The table below maps each annuity concept to its more general counterpart.

Annuity Concept	Advanced Extension	What Changes
Finite ordinary annuity (PV)	Perpetuity	n → ∞; PV simplifies to PMT / r
Constant PMT	Growing Annuity	Payments grow at rate g each period; formula adjusts denominator to (r − g)
Single discount rate	Term-Structure Valuation	Each cash flow is discounted at a different spot rate from the yield curve
Equal periodic payments	Uneven Cash Flows (NPV/IRR)	Each cash flow can differ; no closed-form—requires iterative or spreadsheet calculation

How annuity concepts connect to advanced finance topics.

Mastering the basic annuity framework is not merely an academic exercise; it is the foundation upon which bond pricing, capital budgeting, and financial planning models are constructed. A solid grasp of PVIFA and FVIFA will make the transition to NPV analysis, lease-versus-buy decisions, and yield-to-maturity calculations substantially more intuitive, because those tools simply generalize or modify the annuity formulas you already know.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why the present value of an annuity due is always greater than the present value of an otherwise identical ordinary annuity. In your answer, refer to the timing of payments and the role of discounting.

Show answer

In an annuity due, each payment occurs one period earlier than in an ordinary annuity. Because present-value calculations discount future cash flows back to time zero, an earlier payment is discounted for one fewer period, making it worth more in today's dollars. Formally, PV_AD = PV_OA × (1 + r), so the annuity-due PV exceeds the ordinary-annuity PV by exactly the factor (1 + r). This means the annuity due always has a higher PV whenever r > 0, because receiving money sooner is always preferable under the time-value-of-money framework.

Question 2: BASIC CALCULATION

Calculate the future value of an ordinary annuity that pays $2,000 at the end of each year for 6 years, earning 5% annual interest.

Show answer

Using FV_OA = PMT × [((1 + r)ⁿ − 1) / r], we have FV_OA = 2,000 × [((1.05)⁶ − 1) / 0.05]. First, (1.05)⁶ = 1.34010. The FVIFA = (1.34010 − 1) / 0.05 = 0.34010 / 0.05 = 6.8019. Therefore FV_OA = 2,000 × 6.8019 = $13,603.83. This means that six end-of-year deposits of $2,000, compounding at 5%, will accumulate to approximately $13,604 by the end of year 6.

Question 3: INTERMEDIATE

A car lease requires monthly payments of $450, due at the beginning of each month, for 4 years. If the appropriate annual discount rate is 6% compounded monthly, what is the present value of the lease?

Show answer

Because payments are at the beginning of each month, this is an annuity due. The periodic rate is r = 6% / 12 = 0.5% = 0.005, and n = 4 × 12 = 48 months. First compute the ordinary-annuity PV: PV_OA = 450 × [(1 − (1.005)⁻⁴⁸) / 0.005]. (1.005)⁴⁸ = 1.27024, so (1.005)⁻⁴⁸ = 0.78710. PVIFA = (1 − 0.78710) / 0.005 = 0.21290 / 0.005 = 42.5803. PV_OA = 450 × 42.5803 = $19,161.14. Convert to annuity due: PV_AD = $19,161.14 × 1.005 = $19,256.94. The lease has a present value of approximately $19,257.

Question 4: APPLIED

A company is evaluating a machine that will generate cost savings of $12,000 at the end of each quarter for 5 years. Management's required return is 10% per year, compounded quarterly. What is the maximum amount the company should pay for this machine today?

Show answer

The maximum price equals the present value of the cost-savings stream—paying more would yield a negative NPV. Here PMT = $12,000, r = 10% / 4 = 2.5% = 0.025, and n = 5 × 4 = 20. PV_OA = 12,000 × [(1 − (1.025)⁻²⁰) / 0.025]. (1.025)²⁰ = 1.63862, so (1.025)⁻²⁰ = 0.61027. PVIFA = (1 − 0.61027) / 0.025 = 0.38973 / 0.025 = 15.5892. PV_OA = 12,000 × 15.5892 = $187,070.39. The company should pay no more than approximately $187,070 for the machine.

Question 5: CRITICAL THINKING

Prove algebraically that the relationship FV_OA = PV_OA × (1 + r)ⁿ holds, and then explain its intuitive financial meaning. Why does this connection make sense given the definitions of PV and FV?

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Start with PV_OA = PMT × [(1 − (1 + r)⁻ⁿ) / r]. Multiply both sides by (1 + r)ⁿ: PV_OA × (1 + r)ⁿ = PMT × [(1 − (1 + r)⁻ⁿ) / r] × (1 + r)ⁿ = PMT × [((1 + r)ⁿ − 1) / r] = FV_OA. The algebraic step leverages the fact that (1 + r)ⁿ × (1 − (1 + r)⁻ⁿ) = (1 + r)ⁿ − 1. Intuitively, PV_OA is the single lump sum at time 0 that is equivalent to the annuity stream; if you invest that lump sum at rate r for n periods, it grows to FV_OA, confirming the equivalence. This is simply the fundamental relationship between present and future value applied to the annuity context: any cash-flow stream's PV, when compounded forward to time n, must equal its FV by definition of the time-value-of-money framework.

SUMMARY

Lesson Summary

Summary

An annuity is a finite series of equal cash flows at regular intervals. The ordinary annuity places payments at the end of each period, while the annuity due places them at the beginning. The present value of an ordinary annuity is computed as PMT × [(1 − (1 + r)⁻ⁿ) / r], and its future value as PMT × [((1 + r)ⁿ − 1) / r]. Converting either result to an annuity due requires a single multiplication by (1 + r).

Correct application demands that the periodic interest rate and the number of periods share the same time unit. These formulas underpin loan amortization, bond pricing, lease valuation, and retirement planning, and they generalize into perpetuities, growing annuities, and uneven-cash-flow NPV analysis as you advance through your finance studies.