Notes: Perpetuities and Annuities (Chapter 3)

Perpetuities and Growing Perpetuities

• Perpetuities are cash-flow streams that last forever (i.e., payments occur forever). If the discount rate r is constant and cash flows are constant or grow at a constant rate, shortcut present-value (PV) formulas apply.
• Perpetuities and growing perpetuities are useful quick tools for back-of-the-envelope estimates and for understanding the economics of corporate growth.

The Simple Perpetuity Formula

• Definition: A perpetuity with cash flow C1 received beginning next period (time 1) and discounted at a constant rate r has present value
  PV0 = \frac{C1}{r}.
• Key point: The first cash flow occurs in year 1, not year 0.
• Example (constant payments): A perpetuity paying $2 forever with r = 10\% per year:
  PV0 = \frac{C1}{r} = \frac{2}{0.10} = 20.
• Convergence intuition: The individual payments stay at the same nominal amount, but their present values shrink with time; the sum converges to a finite number (the PV).
• Numerical sanity check: If you sum the first 50 terms you get about $19.83; after 100 terms about $19.9986; the infinite sum converges to $20.

The Growing Perpetuity Formula

• Definition: A growing perpetuity has cash flows that grow by a constant rate g each period forever, starting next period, discounted at rate r (with r > g).
• First cash flow occurring next period is C1 (the cash flow in period 1).
• PV formula:
  PV0 = \frac{C1}{r - g}, \quad r > g.
• Example (growing perpetuity): C1 = $2, growth g = 5\%, discount r = 10\%.
  PV_0 = \frac{2}{0.10 - 0.05} = \frac{2}{0.05} = 40.
• Important restriction: If g ≥ r, the PV formula yields infinity or nonsensical results (the value diverges); in practice the model breaks down because cash flows would, in present value terms, not converge.
• Visual intuition: The cash flows rise over time, but each is discounted; the sum remains finite only when growth is slower than discount (g < r).
• Practical note: In many real cases, g is interpreted as inflation or a steady state growth rate; the model is an approximation.

The Gordon Growth Model (Constant Eternal Growth in Dividends/Earnings)

• If dividends (or earnings treated as dividends) grow at rate g forever and the cost of capital (discount rate) is r, the value today is
  P0 = \frac{D1}{r - g}.
• Connection: This is the growing perpetuity formula applied to dividends, often called the Gordon Growth Model.
• Example (Walmart-like intuition): If next year’s dividend D1 is $10 and r − g ≈ 0.10 − 0.05 = 0.05, then P0 ≈ $10 / 0.05 = $200.
• Interpreting r from market data: If a stock yields a forward dividend yield D1/P0 of, say, 2.2% and growth g is expected to be 0%, then r ≈ 2.2% (rough intuition). If g is positive, r − g reflects the required return net of growth.
• Important caveat: Growth, cost of capital, and cash-flow dynamics are rarely constant forever; use as an approximation or for quick intuition.

The Four Payoff Streams and Their Present Values (Summary of Fig. 3.3)

• Perpetuity (constant cash flow forever, starting next period):
  • PV: PV0 = \frac{C1}{r}.
• Growing Perpetuity (cash flows grow at rate g forever):
  • PV: PV0 = \frac{C1}{r - g}, \quad r > g.
• Annuity (equal cash flows for T periods, starting next period):
  • PV: PV0 = C1 \cdot \frac{1 - \left( \frac{1}{1+r} \right)^T}{r}.
• Growing Annuity (cash flows grow by g for T periods, then stop):
  • PV: PV0 = C1 \cdot \frac{1 - \left( \frac{1+g}{1+r} \right)^T}{r - g}, \quad r \neq g.

Annuities

• An annuity is a stream of equal cash flows for a fixed number of periods T, with the first payment occurring one period from today.
• Slow demonstration (3-period example, r = 10%): PV = $5 / 1.10 + $5 / 1.10^2 + $5 / 1.10^3 \approx 12.4343.$
• Shortcut formula for an ordinary annuity:
  PV0 = C1 \cdot \frac{1 - \left( \frac{1}{1+r} \right)^T}{r}.
• Quick mortgage example (30-year fixed-rate). Common 30-year mortgage with monthly payments is a 360-period annuity:
  • Monthly rate: rm = \frac{\text{quoted annual rate}}{12}. (Example: 7.5% p.a. → rm = 0.075/12 = 0.00625.)
  • PV equation for loan amount L:
    L = C1 \cdot \frac{1 - (1 + rm)^{-360}}{r_m}.
  • Example outcome (illustrative): For a $500,000 loan at 7.5% p.a. with monthly payments, solve for C1 (monthly payment) to satisfy the PV equation; result ≈ $3,496.07 per month for 360 months.
• Mortgage payment decomposition: each month contains interest and principal components.
  • Month 1 interest payment: I1 = rm \cdot \text{Outstanding principal}.
  • Principal repayment in month 1: ext{Pmt} - I_1.
  • Remaining balance updates accordingly.

Level-Coupon Bonds and Coupon Bond Valuation (an overview from the example)

• A bond with regular coupon payments and a final principal repayment is a coupon bond.
• Example structure: a bond paying $1,500 every half-year for 5 years (i.e., 10 coupons) plus $100,000 principal at the end (
  • Coupon rate in nominal terms: 3% per year on the $100,000 principal ⇒ $3,000 per year total, i.e., $1,500 every half-year).
  • The bond’s price is the PV of all coupon payments plus PV of principal:
    PV = \sum_{t=1}^{10} \frac{C}{(1+r)^{t}} + \frac{F}{(1+r)^{10}},
    where C = $1,500 per half-year, F = $100,000, and r is the per-half-year discount rate.
• Fast method (annuity formula for coupons) with semiannual payments:
  PV_{coupons} = C \cdot \frac{1 - (1+r)^{-T}}{r}.
• Example outcome (illustrative): Using a per-period discount rate of about 2.47% for the first 10 semiannual periods and appropriate PV for the final principal, the bond’s value can be computed as ≈ $91,501.42, which is below the $100,000 face value (discount bond).
• Quick reminder: The “coupon rate” label on a bond is a payout pattern designation, not the actual yield. The yield depends on prevailing rates and the price you pay.
• Later, the same example shows the alternative full PV calculation: coupon PV plus principal PV yield the same total, e.g., ≈ $91,501.42.

Important Reminders: Quotes versus Returns

• The coupon rate on a bond is not necessarily the yield or discount rate used to price it.
• Market prices move; the coupon payments are fixed by the bond’s schedule, but the discount rate (cost of capital) can change over time.
• A bond can trade at a premium (price > face) if coupon payments are high relative to prevailing rates; or at a discount if they are low.

The Formulas Summarized (Key Takeaways)

• Growing Perpetuity: PV0 = \frac{C1}{r - g}, with r > g.
• Simple Perpetuity: PV0 = \frac{C1}{r}.
• Simple Annuity: PV0 = C1 \cdot \frac{1 - \left( \frac{1}{1+r} \right)^T}{r}.
• Growing Annuity: PV0 = C1 \cdot \frac{1 - \left( \frac{1+g}{1+r} \right)^T}{r - g}, \quad r \neq g.
• Gordon Growth Model (stock valuation with eternal growth): P0 = \frac{D1}{r - g}.
• Bond valuation (coupon bonds in general):
  • PV of coupons: PV_{coupons} = C \cdot \frac{1 - (1 + r)^{-T}}{r},
  • PV of principal: PV_{principal} = \frac{F}{(1+r)^{T}},
  • Total PV: PV = PV{coupons} + PV{principal}.
• Relationship to mortgages: A fixed-rate mortgage is an annuity; the monthly payment C1 is determined by solving the annuity PV equation for the loan amount L.

Practice Questions and Quick Answers (AQs)

• AQ 3.1: Perpetuity formula. First cash flow occurs in period 1.
  • Answer: PV0 = \frac{C1}{r}.
• AQ 3.2: Perpetuity paying $5 each month, starting next month; monthly rate i = 0.5\% = 0.005.
  • Answer: PV0 = \frac{C1}{i} = \frac{5}{0.005} = 1000.
• AQ 3.3: Perpetuity paying $15 each month, starting next month; effective annual rate 12.68\%.
  • Monthly rate: im = (1+0.1268)^{1/12} - 1 \approx 0.01, so PV0 = \frac{15}{0.01} = 1500.
• AQ 3.4: Perpetuity paying $2M per year forever vs $40M one-time. When is it better to take perpetuity?
  • Solve for r where PV(perpetuity) = 40M: \frac{2}{r} = 40 \Rightarrow r = 0.05.
  • Therefore, at r = 5% you’re indifferent; if r > 5%, take the one-time; if r < 5%, take the perpetuity.
• AQ 3.5: Consol value with $2,000 per year and r = 4%.
  • Answer: PV_0 = \frac{2000}{0.04} = 50{,}000.
• AQ 3.6: Growing perpetuity formula.
  • Answer: PV0 = \frac{C1}{r - g}.
• AQ 3.7: Perpetuity with $2 starting next period, growth g = 0.1\%/period, monthly rate r = 0.5\%/period.
  • Answer (illustrative): PV0 = \frac{C1}{r - g} = \frac{2.00}{0.005 - 0.001} = 1251.25, total value including first period etc. (text uses $1{,}251.25$ as the growing-perpetuity value under given numbers).
• AQ 3.8: If g ≥ r, the growing perpetuity value is undefined (infinite) — nonsensical; keep g < r.
• AQ 3.9: Firm valuation with a 3-year fast-growth phase followed by slower growth and a terminal value.
  • Rough result in text: PV ≈ $5.42\text{ billion}$ (with present value of explicit cash flows plus a terminal value at year 7). Terminal value computed using the growing perpetuity formula at the long-run rate, discounted back to today.
• AQ 3.10: Value of a perpetual patent contract paying $1.5M next year with growth at 2% and discount rate 14%.
  • Answer: PV = \frac{1.5}{0.14 - 0.02} = 12.5\text{ million}.
• AQ 3.11: If the first payment occurs immediately instead of next year, value changes accordingly (increase). The text notes the direct adjustment via the perpetuity/level shift convention.
• AQ 3.12: Quarterly dividend example. One quarter ahead dividend D1, next-quarter growth 0.5% per quarter, r = 9% per year.
  • Answer (summary): compute i_m = (1+0.09)^{1/4}-1 ≈ 2.1778% per quarter; value today is about $302.32 when discounting the month-ahead value back to today.
• AQ 3.13: If $100 stock has earnings of $5 per year and equity cost of capital r = 12%, what is the implied perpetual growth rate for earnings (g) under the Gordon framework?
  • Concept: If using earnings as cash flow proxy, r ≈ g + (E/P). The text notes a rough approach: g ≈ E/P, and r ≈ (E/P) + g in some heuristic; exact value depends on the chosen proxy.
• AQ 3.14: Compare annuity vs perpetuity to reach 3/4 of the value. Solve for t from
  • Formula: 1 - \frac{1}{(1+r)^t} = \frac{3}{4} \Rightarrow \frac{1}{(1+r)^t} = \frac{1}{4} \ (1+r)^t = 4 \ t = \frac{\ln 4}{\ln(1+r)}.
  • For r = 5\%: t ≈ 28.41 years.
• AQ 3.15: Reiterate annuity formula: PV0 = C1 \cdot \frac{1 - (1/(1+r))^T}{r}.
• AQ 3.16: 360-month (30-year) monthly annuity with $5 monthly payment; r = 0.5\% per month.
  • Answer: PV_0 = 5 \cdot \frac{1 - (1+0.005)^{-360}}{0.005} \approx 833.96.
• AQ 3.17–3.24: Numerous historical and advanced extensions (Fibonacci problem, partials about quarter vs annual pay, 6-month vs 12-month rates, terminal values, and rates). Use the standard formulas above to compare cash-flow streams across different compounding bases and time horizons; the text provides worked numbers for each scenario.
• AQ 3.25–3.37: End-of-chapter problems on perpetual subscriptions, mortgage payments, bond pricing, and mixed cash-flow streams. Examples include:
  • Perpetual Starbucks subscription value: use the perpetuity formula with daily/annual discount interpretation; for a daily payment of $1.85 with a 6% rate, PV ≈ $1.85 / i_{daily} (convert annual rate to daily).
  • Mortgage payments for a given loan and rate: solve the annuity PV for C1 given L and r; example results depend on which rate convention (APR vs effective) is used.
  • Bond valuations with coupons: compute coupon PV plus principal PV; observe that discounts, periods, and yields change with market rates.
  • Terminal value: use the growing perpetuity to estimate the value after a finite growth phase, then discount back to present at the appropriate rate.

Quick Practical Takeaways

• Always identify whether cash flows are perpetuity, growing perpetuity, annuity, or growing annuity before applying a formula.
• The key formulas, in compact form, are:
  • Perpetuity: PV0 = \frac{C1}{r}.
  • Growing Perpetuity: PV0 = \frac{C1}{r - g}, \quad r > g.
  • Annuity: PV0 = C1 \cdot \frac{1 - (1/(1+r))^T}{r}.
  • Growing Annuity: PV0 = C1 \cdot \frac{1 - \left( \frac{1+g}{1+r} \right)^T}{r - g}, \quad r \neq g.
  • Gordon Growth Model (stocks): P0 = \frac{D1}{r - g}.
• For loans and mortgages, the practical workflow is:
  1) Use the annuity formula to solve for the payment given L, r, and T.
  2) Break the payment into interest and principal in the first period, then update the remaining balance.
• When g ≥ r in growing perpetuities, the model becomes invalid (infinite PV); recognize the boundary condition.
• The distinction between quotes (coupon rate) and the actual return (yield) matters for bonds; do not confuse the two.

Appendix: Contextual Notes and Real-World Connections

• The Gordon Growth Model is widely used for quick, rough stock valuations and for estimating the cost of capital from dividend yield and growth expectations.
• Perpetuities are rarely perfect in practice (firms don’t last forever), but they provide useful upper bounds and intuition for corporate growth limits.
• Terminal values and growing perpetuities are essential when modeling pro forma statements and long-term business valuations.
• The equivalent-annual-cost (EAC) concept is mentioned as a tool for comparing projects with different lengths; see the companion appendix.

Notation Recap (Key symbols)

• C1: Cash flow in period 1 (or next period’s cash flow in the perpetuity context).
• F: Face value or principal (bond context).
• r: per-period discount rate (cost of capital).
• g: growth rate of cash flows per period.
• T: number of periods in an annuity or growing annuity.
• P0 or PV0: present value today.
• D1: Next period dividend (for Gordon Growth Model).
• i: per-period rate (intermediate notation used in some examples; equivalent to r when appropriate).