What is the growing annuity PV formula?

The present value formula is PV = (PMT / (r - g)) × (1 - ((1 + g) / (1 + r))^n), where PMT is the first payment, r is the discount rate, g is the growth rate, and n is the number of periods. The growth rate must not equal the discount rate — when g = r, the formula breaks and requires L'Hopital's rule.

What is the growing annuity FV formula?

The future value formula is FV = (PMT / (r - g)) × ((1 + r)^n - (1 + g)^n). Each payment compounds forward to the final period, with earlier payments benefiting from more compounding. When g = r, use FV = PMT × n × (1 + r)^(n-1).

Growing annuity vs perpetuity — what is the difference?

A growing annuity has a finite number of payments (n periods), while a growing perpetuity has infinite payments. The perpetuity formula PV = PMT / (r - g) is the limit of the annuity formula as n approaches infinity. The Gordon Growth Model for stock valuation uses the growing perpetuity formula.

What are growing annuity applications?

Common applications include: pension valuation with COLA, salary progression modeling, commercial leases with rent escalation, dividend discount models, tuition planning with rising costs, and revenue projections for growing businesses. Any scenario with payments that increase at a constant rate fits the growing annuity framework.

What is the Gordon Growth Model?

The Gordon Growth Model values a stock as a growing perpetuity: P = D1 / (r - g), where D1 is next year's dividend, r is the required return, and g is the perpetual growth rate. It is essentially a growing annuity with n → ∞. Used for mature, dividend-paying companies with stable growth.

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Growing Annuity — Smart Financial Analysis

Calculate present value and future value of annuities with growing payments. Pensions with COLA, salaries with raises, leases with rent escalation — all use the growing annuity formula.

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A growing annuity is a series of payments that increase at a constant rate each period. The future value formula is FV = (PMT / (r - g)) × ((1 + r)^n - (1 + g)^n). A growing annuity has a finite number of payments (n periods), while a growing perpetuity has infinite payments. The Gordon Growth Model values a stock as a growing perpetuity: P = D1 / (r - g), where D1 is next year.

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Growing Annuity

Annuities fundamental

Benchmark

Industry Standard

Compare your results

Proven Math

Formula Basis

Established methodology

Expert Verified

Best Practice

Professional standard

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Why: A growing annuity is a series of payments that increase at a constant rate each period. Pensions with cost-of-living adjustments (COLA), salaries with annual raises, and leases ...

How: Enter First Payment ($), Growth Rate (%), Discount Rate (%) to get instant results. Try the preset examples to see how different scenarios affect the outcome, then adjust to match your situation.

A growing annuity is a series of payments that increase at a constant rate each period.The future value formula is FV = (PMT / (r - g)) × ((1 + r)^n - (1 + g)^n).

Sources:InvestopediaFederal Reserve

Run the calculator when you are ready.

Calculate Growing AnnuityEnter your values below

📌 Examples — Click to Load

Inputs

First Payment ($)

Growth Rate (%)

Discount Rate (%)

Number of Periods

Payment Timing

growing_annuity.sh

CALCULATED

Present Value

$767,779

Future Value

$4,167,068

Growth Advantage

31.8%

Total Payments

$1,822,963

PV of Each Payment

Growing vs Level Annuity

Total PV Breakdown (Early vs Mid vs Late)

For educational purposes only — not financial advice. Consult a qualified advisor before making decisions.

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A growing annuity is a series of payments that increase at a constant rate — pensions with cost-of-living adjustments (COLA), salaries with annual raises, and leases with rent escalation clauses are all growing annuities. A $50K pension growing 3% annually for 25 years has a present value of $764K. The formula breaks when the growth rate equals the discount rate — a special case requiring L'Hopital's rule. The Gordon Growth Model for stock valuation is essentially a growing perpetuity.

📋 Key Takeaways

• Growing annuities model payments that increase each period (COLA, raises, escalation)
• PV formula: PV = (PMT / (r - g)) × (1 - ((1 + g) / (1 + r))^n)
• Growth rate must not equal discount rate — when g = r, use PV = PMT × n
• Gordon Growth Model (P = D1/(r-g)) is a growing perpetuity (n → ∞)

💡 Did You Know?

💰A $50K pension with 3% COLA for 25 years has PV of ~$764K at 7% discountSource: CFA Institute

📈A 30-year career starting at $60K with 5% raises has PV of ~$922K at 8% discountSource: Damodaran

🏢Commercial leases often include 3% annual rent escalation clausesSource: Brealey Myers Allen

💎The Gordon Growth Model values stocks as growing perpetuitiesSource: Investopedia

📖 Growing Annuity Formula Explained

Each payment grows by (1+g) and is discounted by (1+r). When g < r, the series converges. When g = r, the standard formula divides by zero — use PV = PMT × n. When g > r, the formula is invalid (payments grow faster than they are discounted).

Present Value

PV = (PMT / (r - g)) × (1 - ((1 + g) / (1 + r))^n)

PMT = first payment, r = discount rate, g = growth rate, n = number of periods

Future Value

FV = (PMT / (r - g)) × ((1 + r)^n - (1 + g)^n)

Earlier payments compound longer; later payments are larger due to growth

Step-by-Step Example

PMT = $50,000, g = 3%, r = 7%, n = 25. r - g = 0.04. (1+g)/(1+r) = 1.03/1.07 ≈ 0.9626. (0.9626)^25 ≈ 0.389. PV = (50000/0.04) × (1 - 0.389) = 1,250,000 × 0.611 ≈ $764,000.

📐 Growing Annuity vs Perpetuity

Feature	Growing Annuity	Growing Perpetuity
Number of payments	Finite (n periods)	Infinite
PV Formula	(PMT/(r-g)) × (1 - ((1+g)/(1+r))^n)	PMT / (r - g)
Use case	Pensions, leases, careers	Gordon Growth, mature dividends

🎯 Applications

Pension with COLA

Value retirement benefits that increase with inflation.

Salary Progression

Model career earnings with annual raises.

Lease Escalation

Value commercial leases with rent step-ups.

Dividend Discount

Gordon Growth Model for stock valuation.

📊 Key Stats

$764K

PV of $50K Pension (3% COLA, 25yr)

$922K

PV of 30yr Growing Salary

Typical COLA Increase

r ≠ g

Growth Rate Must Not Equal Discount

❓ FAQs

What happens when g = r?

The standard formula divides by zero. Use PV = PMT × n (or FV = PMT × n × (1+r)^(n-1)). This is a special case requiring L'Hopital's rule in the limit.

How does Gordon Growth relate?

Gordon Growth (P = D1/(r-g)) is a growing perpetuity — the limit of the growing annuity as n → ∞. Used for mature dividend stocks.

When to use growing vs level annuity?

Use growing when payments increase (COLA, raises, escalation). Use level when payments are fixed (traditional bonds, fixed rent).

📚 Sources

CFA Institute

Growing annuity formulas and applications

Damodaran (NYU)

Valuation and dividend discount models

Brealey Myers Allen

Corporate finance and annuity theory

Investopedia

Gordon Growth Model and growing perpetuity

🎯 When to Use Growing vs Level Annuity

Use Growing Annuity

• Pensions with COLA
• Salaries with annual raises
• Leases with rent escalation
• Dividend growth (finite horizon)
• Tuition with rising costs

Use Level Annuity

• Fixed mortgage payments
• Bond coupon payments
• Fixed rent (no escalation)
• Fixed pension (no COLA)
• Short-term (< 5 yr) projections

⚠️ Common Mistakes

Using g ≥ r

When growth rate equals or exceeds the discount rate, the standard formula is invalid. Use PV = PMT × n when g = r.

Mixing Nominal and Real Rates

Use all nominal or all real rates consistently. Real rate ≈ nominal - inflation.

Ignoring Payment Timing

Annuity due (beginning) requires multiplying by (1 + r). Ordinary annuity (end) uses the base formula.

Disclaimer: This calculator provides estimates for educational purposes. Growth and discount rates are assumptions — actual results may vary. Not financial advice.

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Growing Annuity — Smart Financial Analysis

📌 Examples — Click to Load

Inputs

Payment Growth Over Time

PV of Each Payment

Growing vs Level Annuity

Total PV Breakdown (Early vs Mid vs Late)

💡 Money Facts

📋 Key Takeaways

💡 Did You Know?

📖 Growing Annuity Formula Explained

Present Value

Future Value

Step-by-Step Example

📐 Growing Annuity vs Perpetuity

🎯 Applications

Pension with COLA

Salary Progression

Lease Escalation

Dividend Discount

📊 Key Stats

❓ FAQs

What happens when g = r?

How does Gordon Growth relate?

When to use growing vs level annuity?

📚 Sources

🎯 When to Use Growing vs Level Annuity

Use Growing Annuity

Use Level Annuity

⚠️ Common Mistakes

Using g ≥ r

Mixing Nominal and Real Rates

Ignoring Payment Timing

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