Questions: Annuities and Perpetuities

Question 1

A perpetuity pays $200 per year forever. If the discount rate is 4%, what is its present value?

• A. $800
• B. $5,000
• C. $4,800
• D. $200

Answer: (B)

PV = C/r = 200 / 0.04 = $5,000. This is the perpetuity formula derived by summing the infinite geometric series of discounted cash flows. Intuitively, $5,000 invested at 4% earns exactly $200 per year forever — so $5,000 today is precisely equivalent to this perpetuity. Option A ($800) might come from multiplying C × r instead of dividing; the other options reflect different confusions with the formula.

Question 2

An investor tries to apply the growing perpetuity formula PV = C/(r−g) to value a stock with dividends growing at 8% per year and a discount rate of 6%. What is the problem?

• A. The formula only applies to finite annuities, not perpetuities
• B. The growth rate (8%) exceeds the discount rate (6%), making r−g negative and the formula economically meaningless
• C. The first dividend payment must always be discounted separately before applying the formula
• D. The formula requires that g = 0; a separate formula handles growth

Answer: (B)

When g ≥ r, the denominator r−g becomes zero or negative, producing an infinite or negative present value — which has no economic meaning. The formula assumes that faster growth is partially offset by discounting, converging to a finite sum only when r > g. Economically, a cash flow stream growing faster than your discount rate forever would be worth infinite money today, which cannot exist in a finite economy. The condition r > g is not a mathematical technicality; it is the economically meaningful boundary.

Question 3

A perpetuity paying the same amount forever has a finite present value because distant cash flows are so heavily discounted they contribute negligible value.

• T. True
• F. False

Answer: True

This is correct and is the intuition behind the formula PV = C/r. Each payment C/(1+r)^t shrinks geometrically as t increases. Distant payments (t = 100, 200, …) are discounted to near-zero present value. The infinite series C/(1+r) + C/(1+r)² + … converges to a finite number C/r because the ratio 1/(1+r) is less than 1. An apparently paradoxical 'infinite payments = finite value' is resolved by the logic of compounding.

Question 4

The growing perpetuity formula PV = C/(r−g) is valid as long as the growth rate g is positive.

• T. True
• F. False

Answer: False

The formula requires r > g, not merely g > 0. A positive g that still satisfies r > g (e.g., r = 8%, g = 5%) is fine. But a growth rate exceeding the discount rate (g ≥ r) makes the denominator zero or negative, yielding an undefined or negative present value. The intuition: if payments grow faster than they are discounted, the present value of future payments never shrinks enough to converge, and the sum is infinite.

Question 5

Explain intuitively why a perpetuity — an infinite stream of payments — has a finite present value. Why does the formula give PV = C/r?

Model answer: Distant payments are discounted so heavily they become negligible. The stream C/(1+r), C/(1+r)², C/(1+r)³, … is a geometric series with ratio 1/(1+r) < 1, which converges to a finite sum. Applying the geometric series formula gives PV = [C/(1+r)] / [1 − 1/(1+r)] = C/r. Alternatively: if you invest PV = C/r today at interest rate r, you earn C each period and never touch the principal — so C/r is exactly the amount needed to replicate the perpetuity, confirming the formula.

The formula's elegance is that an infinite stream of future cash flows collapses to a single ratio. This is why perpetuity pricing appears throughout finance: preferred stock dividends, government consols, endowment payouts, and the Gordon Growth Model all rest on variants of this formula.