Lesson 2.5: Perpetuities

Lesson Overview

In Lesson 2.4, you learned how to value a fixed number of equal payments (annuities). Now we take the limit case: what if the payments continue forever?

A cash-flow stream that continues indefinitely is called a perpetuity. While “forever” sounds unrealistic, perpetuity logic shows up constantly in finance: stock valuation (dividends), real estate (cap rates), and any business expected to operate long-term.

The surprising result is that perpetuities often have simple formulas. The challenging part is not the math, it’s understanding timing, assumptions, and especially the discount rate.

Learning Objectives

• Define a perpetuity and identify real-world examples.
• Calculate the present value of a level perpetuity.
• Calculate the present value of a growing perpetuity.
• Explain why the discount rate dominates perpetuity value.
• Recognize the conditions required for perpetuity formulas to work (including r > g).
• Connect perpetuity logic to cap rates and dividend valuation.

What Is a Perpetuity?

A perpetuity is an annuity that never ends: it pays the same amount every period forever. The standard perpetuity assumes:

• Payments are constant (same PMT each period)
• Payments occur at regular intervals (monthly, yearly, etc.)
• The discount rate per period is constant
• The first payment occurs one period from today (ordinary timing)

If the first payment is today (an annuity due-style perpetuity), we adjust just like we did for annuities.

Level Perpetuity: The Core Formula

For a perpetuity that pays a constant amount PMT starting one period from now, the present value is:

PV = PMT / r

• PMT = payment each period
• r = discount rate per period

This is one of the most famous results in finance. Notice what is (and isn’t) in the formula: there is no “n” because the payments never stop.

Worked Example: Valuing a Level Perpetuity

Suppose an asset pays $500 per year forever, with the first payment one year from today. If your discount rate is 8%, what is the asset worth today?

• PMT = 500
• r = 0.08

PV = 500 / 0.08 = $6,250

Interpretation: if you could earn 8% elsewhere, paying $6,250 today to receive $500 per year forever is “fair” under this model.

Why the Discount Rate Is Everything

In a perpetuity, the discount rate is the main lever of value. Small changes in r can cause big changes in PV.

Using the same $500 perpetuity:

• At 10%: PV = 500 / 0.10 = $5,000
• At 8%: PV = 500 / 0.08 = $6,250
• At 5%: PV = 500 / 0.05 = $10,000

Lower discount rate → higher value (because future cash is “penalized” less).

Timing Matters: What If the First Payment Is Today?

The standard perpetuity formula assumes the first payment arrives one period from now. If the first payment happens immediately (today), then the asset is worth one extra payment:

PV (perpetuity due) = PMT + (PMT / r)

Because you receive the first PMT right away (no discounting needed), and then the remaining stream starts one period later.

Growing Perpetuity: When Payments Increase Forever

Many real cash flows are not flat forever. Dividends, rents, and business cash flows often grow over time. A growing perpetuity assumes payments grow at a constant rate g each period forever.

If the first payment one period from today is C₁, the present value is:

PV = C₁ / (r - g)

• C₁ = cash flow in the next period
• r = discount rate per period
• g = growth rate per period

The Most Important Rule: r Must Be Greater Than g

The growing perpetuity formula only makes sense if r > g.

• If r ≤ g, the denominator becomes zero or negative.
• That implies infinite or nonsensical values because cash flows grow as fast as (or faster than) the rate you discount them.

In the real world, long-run growth cannot exceed the economy forever. This constraint keeps models grounded.

Worked Example: Growing Perpetuity

A dividend is expected to be $2.00 next year and grow at 3% per year forever. If the required return is 9%, what is the value today?

• C₁ = 2.00
• r = 0.09
• g = 0.03

PV = 2.00 / (0.09 - 0.03) = 2.00 / 0.06 = $33.33

This idea is the foundation of the dividend discount model (sometimes called the Gordon Growth Model).

Perpetuities in Real Estate: Cap Rates (Intuition)

Real estate investors often use a simplified perpetuity-like idea:

Value ≈ NOI / Cap Rate

This resembles PV = PMT / r. It’s not identical in every case (growth, financing, and risk vary), but the intuition is the same: an income stream capitalized by a required return produces an estimate of value.

How to Set Up Perpetuity Problems (Fast Checklist)

1. Confirm it’s a perpetuity: does it continue forever (or is it “effectively forever”)?
2. Identify the next payment: is the cash flow at t=1 (next period) or today?
3. Choose the right formula: level (PMT/r) or growing (C1/(r-g)).
4. Match periods: annual payments require annual r and g; monthly payments require monthly rates.
5. Sanity check: for growing perpetuity, ensure r > g.

Common Mistakes (and Quick Fixes)

• Mistake: Using C₀ (today’s cash flow) instead of C₁ (next period) in the growing perpetuity.
  Fix: The formula uses C₁ as the next payment.
• Mistake: Forgetting the timing assumption (first payment one period from now).
  Fix: If a payment happens today, add it separately.
• Mistake: Mixing annual r with monthly cash flows (or vice versa).
  Fix: Make r and g match the cash-flow period.
• Mistake: Allowing g ≥ r in a growing perpetuity.
  Fix: If r ≤ g, the model assumptions are broken, then rethink inputs.

Practice: Check Your Understanding

1. What is a perpetuity in one sentence?
2. Compute PV: $120 per year forever at a 6% discount rate (first payment in one year).
3. Compute PV: A cash flow is $5 next year and grows 2% forever; r = 8%. What is PV?
4. Why must r be greater than g in a growing perpetuity?
5. How does a lower discount rate affect perpetuity value?

Key Takeaways

• Level perpetuity: PV = PMT / r (first payment in one period).
• Growing perpetuity: PV = C₁ / (r - g) and requires r > g.
• Perpetuities are highly sensitive to the discount rate.
• These models power common valuation shortcuts (dividends, cap rates, long-lived businesses).
• Timing assumptions matterzzzzzzzzz; know whether the first payment is today or next period.

What’s Next?

In Lesson 2.6: Discounted Cash Flow (DCF), you’ll combine everything you’ve learned: valuing a stream of cash flows by discounting each one and summing them. This is one of the most important tools in finance.