Notes on Perpetuities, Growing Perpetuities, Annuities, and Bond/Mortgage Valuation (Chapter 3)

Perpetuities and Growing Perpetuities

Central idea: The present value (PV) formula is the main tool for valuing cash flows, but shortcut formulas exist for special patterns like perpetuities and annuities.
Perpetuities: cash flows continue forever at a constant amount or with constant growth; PV can be computed quickly when the discount rate is constant.
Annuities: cash flows are constant for a finite number of periods; PV is also given by a quick formula.
These formulas help build intuition about corporate growth and quick rule-of-thumb estimates.

The Perpetuity Concept

Perpetuity: a stream of cash flows that lasts forever with a constant amount each period.
When the cost of capital (discount rate) r is constant and cash flow C1 is constant each period, the perpetuity PV is:
$PV0 = \frac{C1}{r}$
Example: a perpetuity paying $2 each year forever with r = 10%:
$PV_0 = \frac{2}{0.10} = 20.$
The first cash flow occurs next year (time 1). There is no payment at time 0.
The PV diagram shows cash flows of $2 each year, discounted to smaller present values as time increases, and their sum converges to a finite PV.
If you compute a finite-term approximation (e.g., first 50, then 100 terms), the PV approaches the same $20 in the limit.

The Growing Perpetuity

Growing perpetuity: cash flows grow at a constant rate g forever, with discount rate r > g.
Cash flows: C1 in period 1, then C2 = C1(1+g), C3 = C1(1+g)^2, …
PV formula (with first cash flow in period 1):
$PV0 = \frac{C1}{r - g}$
Important: the “1” subscript reminds you the first cash flow occurs one period from now (time 1).
Condition: r > g; if g ≥ r, the PV is nonsensical (diverges to infinity when g > r, or undefined for g = r).
Example: C1 = 2, r = 10%, g = 5% → $PV_0 = \frac{2}{0.10 - 0.05} = 40.$
If g = 0, this reduces to the simple perpetuity formula (C1 / r).
Implication: growing perpetuities are often used when cash flows reflect inflation, i.e., g ≈ inflation, and are used for quick terminal-value estimates.
Note on interpretation: growing perpetuities are approximations; real firms do not last forever, and real r and g can vary over time.

The Gordon Growth Model (Stock Valuation via Growing Perpetuity)

If dividends are expected to grow at a constant rate g forever and the cost of equity (discount rate) is r, the stock price today is:
$P{Today} = \frac{D{Next\ Year}}{r - g} = \frac{D_1}{r - g}$
D1 is the next year’s dividend.
Example: If D1 = $10, r = 10%, g = 5%, then P today = 10 / (0.10 - 0.05) = 200.
The Gordon Growth Model is a specific application of the growing perpetuity to dividends. It assumes perpetual, constant growth and a constant cost of capital.
Practical use: you can infer r from D1, P today, and g, or infer g from D1, P today, and r, though real-world growth and required returns are not constant forever.
Caution: use as an approximation; earnings/dividends and discount rates can vary in reality.

Annuities

An annuity pays the same cash flow C1 for T periods, beginning next period (time 1), discounted at a constant rate r.
PV of an annuity (end-of-period payments):
$PV0 = C1 \frac{1 - (1 + r)^{-T}}{r}$
Example (10% per period, $5 per period for 3 periods):
$PV_0 = 5 \cdot \frac{1 - (1 + 0.10)^{-3}}{0.10} \approx 12.4343.$
This shortcut is particularly useful for long series (e.g., 360 monthly mortgage payments).
Important reminder: the first cash flow occurs at time 1, not at time 0.

Growing Annuities (brief note)

The growing annuity formula is used when payments grow at rate g for T periods and then stop.
Formula (when first cash flow is next period):
$PV0 = \frac{C1}{r - g} \left[1 - \left(\frac{1+g}{1+r}\right)^T\right]$
This formula is less commonly memorized, but useful in pension projections and other time-limited growth scenarios.

Quick Formulas Summary (from the chapter)

Four payoff patterns and their PV formulas:
Simple perpetuity (constant cash flow, forever):
$PV0 = \frac{C1}{r}$
Growing perpetuity (constant growth g, forever):
PV0 = \frac{C1}{r - g} \, (\text{with } r > g)
Ordinary annuity (constant cash flow for T periods, starting next period):
$PV0 = C1 \frac{1 - (1 + r)^{-T}}{r}$
Growing annuity (cash flows grow at rate g for T periods, then stop):
$PV0 = \frac{C1}{r - g} \left[1 - \left(\frac{1+g}{1+r}\right)^T\right]$
Gordon Growth Model for stock pricing (special case of growing perpetuity):
$P{Today} = \frac{D1}{r - g}$

Applications to Bonds and Mortgages (Overview)

Mortgages are annuities: a fixed-rate mortgage loan with monthly payments is a 360-payment annuity.
To price a fixed-rate mortgage: set PV of payments equal to the loan amount, solve for monthly payment C1.
Example approach: for a $500,000 loan at 7.5% annual rate, monthly rate is 7.5%/12, and you solve
$500{,}000 = C_1 \cdot \frac{1 - (1 + 0.0075/12)^{-360}}{0.0075/12}$
This yields the monthly payment, which includes both interest and principal components.
Bonds: coupon bonds pay fixed coupons over time plus a final principal repayment. Their value is the PV of all coupon payments plus PV of principal.
A 3% semiannual coupon bond example:
- Principal: $100{,}000
- Semiannual coupons: $1{,}500 every 6 months (3% annual coupon rate on $100k principal)
- Maturity typical pattern: payments every 6 months for 5 years, plus final principal at end
Price determined by discounting each payment by the appropriate discount rate for that period (e.g., 2.47% for 6 months, etc.).
The bond’s price is the sum of the present values of future coupons plus the present value of the principal.
Alternatively, the annuity formula can be used to quickly value the coupons, plus a single PV for the principal:
Coupons PV: $PV{coupons} = C1 \frac{1 - (1 + r)^{-N}}{r}$ where N is number of periods and r is the per-period rate; Principal PV: $PV_{principal} = \frac{F}{(1 + r)^N}$
Important conceptual point: the coupon rate designation (e.g., a '3% semiannual coupon bond') is a description of the payout pattern, not the required return. The market discount rate can differ from the coupon rate.

Quick Worked Examples and Key Takeaways

Perpetuity example: a perpetuity paying $2/year at r = 10% has PV = $20.
Growing perpetuity: if C1 = 2, r = 10%, g = 5%, PV = 40.
If g ≥ r, the growing perpetuity model yields nonsensical (infinite) values; the model requires r > g.
Gordon Growth Model: P Today = D1 / (r - g). If D1 = 10, r = 10%, g = 5%, P = 200.
Annuity example: $5 per year for 3 years at 10% gives PV ≈ $12.4343.
The 360-month mortgage example: solve for monthly payment C1 using the annuity formula with r = 0.625% per month and T = 360 to price a $500,000 loan.
Bond example: a $100,000, 3% semiannual coupon bond with 5-year maturity priced using a set of discount rates; coupons PV ≈ $13,148.81 and principal PV ≈ $78,352.62, totaling ≈ $91,501.42.
Important teaching point: Quotes (coupon rates) are not the actual returns; the market yield (discount rate) may differ from the coupon rate.

Selected Q&A Highlights (from the chapter’s questions and answers)

Q3.1 Memorize the perpetuity formula and specify that the first cash flow occurs at time 1.
Q3.2 PV of a perpetuity paying $5 each month, starting next month, with monthly rate $0.5 extrm{%/month}$: PV = \frac{C_1}{r} = \frac{5}{0.005} = 1000.
Q3.3 PV with an effective annual rate of 12.68% but monthly compounding: r_{monthly} = (1 + 0.1268)^{1/12} - 1 ≈ 1\% per month; PV = \frac{15}{0.01} ≈ 1500.
Q3.4 Indifference threshold between perpetuity and a one-time $40M: set \frac{C1}{r} = 40{,}000{,}000; r = \frac{C1}{40{,}000{,}000} = \frac{2000}{40{,}000{,}000} = 0.00005 = 0.005\%\text{/year}.
Q3.5 Consol value with $2,000/year at 4%: PV = \frac{2000}{0.04} = 50{,}000.
Q3.6 Memorize the growing perpetuity formula: $PV0 = \frac{C1}{r - g}$ (assuming r > g).
Q3.7 PV of perpetuity starting this month with r = 0.5%/month and g = 0.1%/month: first payment next period is C1 = 5.005; PV ≈ \frac{5.005}{0.005 - 0.001} ≈ 1251.25; plus the immediate payment C0 = 5 makes total ≈ 1256.25.
Q3.8 If g ≥ r, the growing perpetuity value is infinite or undefined (nonsense).
Q3.9 A multi-stage growing perpetuity example with 3 years at 20%, 3 years at 10%, and terminal 5% growth; using stage-specific discount rates (10%, 9%, 8%), the present value computes to about $5.4248$ billion (with a terminal value ~ $8.452$ billion discounted back).
Q3.10 Patent contract value: first payment 1.5m next year; growth at 2%; r = 14%: PV = \frac{1.5}{0.14 - 0.02} = 12.5\text{ million}.
Q3.11 If the first payment is tonight (immediate), the value is approximately 14.25 million (i.e., the immediate 1.5m plus the growing perpetuity value starting next year, approximated by scaling the 12.5m by 1.14).
Q3.12 Quarterly dividend with 0.5% quarterly growth and 9% annual rate: compute quarterly rate r_q = (1.09)^{1/4} - 1 ≈ 0.022. The value involves the growing perpetuity form adjusted to quarterly periods; results shown yield a value around $299.50 for the quarterly stream plus the immediate quarter’s dividend of 5, totaling ≈ $304.50 today.
Q3.13 If a stock with earnings of $5 next year and cost of capital 12% implies an eternal growth rate g ≈ r − E/P = 12% − 5/100 = 7%.
Q3.14 Comparing annuity vs perpetuity: t solving 1 − (1 + r)^{−t} = 3/4 gives t ≈ 28.41 years for r = 5%.
Q3.15 Recalling the annuity formula: $PV0 = C1 \frac{1 - (1 + r)^{-T}}{r}.$
Q3.16 A 360-month annuity with monthly payments of $5 and monthly rate 0.5%: PV ≈ $833.96.
Q3.17 Fibonacci’s annuity problem demonstrates valuing different cash-flow streams with a common discount rate to compare changes in schedule.
Q3.18 Quarterly vs annual streams: a comparison showing PV of quarterly payments indexed to r and g.
Q3.19 Lease vs buy: implied rate of return from lease ≈ 0.31142% per month (≈ 3.8% per year).
Q3.20 Mortgage example: for $1,000 of loan, C1 ≈ $8.44 per month if 180 months at 6.168% per year.
Q3.21 Bond price sensitivity: if market rates rise from 5% to 6% per year, the price falls (illustrated in the coupon-bond example).
Q3.22 Bond with 20 years, 40 payments, 10% yield: bond value adjusts; example shows about $39,943.20 for a particular case.
Q3.23 Check coupon-rate/return consistency for the coupon bond example.

End-of-Chapter Problems and Quick Answers (selected)

AQ 3.1: C1/r is the perpetuity formula, with first cash flow one period ahead.
AQ 3.2: PV = $5 / 0.005 = $1{,}000.
AQ 3.3: Interest rate per month ≈ (1 + 0.1268)^{1/12} - 1 ≈ 1.0% per month; PV ≈ $15 / 0.01 ≈ $1{,}500.
AQ 3.4: Indifference threshold r ≈ 5%? (In the provided solution, the threshold is solved by setting PV perpetuity equal to $40M: r = 2000/40{,}000{,}000 = 0.00005 = 0.005% per year.)
AQ 3.5: PV = 2000 / 0.04 = 50,000.
AQ 3.6: Growing perpetuity formula: $PV0 = \frac{C1}{r - g}.$
AQ 3.7: Value ≈ $1,256.25 (illustrates immediate payment plus growing perpetuity starting next period).
AQ 3.8: If g ≥ r, the value is infinite (nonsense).
AQ 3.9: Multi-stage growth with 20% and 10% growth and terminal value; PV ≈ $5.425 billion (terminal value ≈ $8.452 billion in year 7, discounted back).
AQ 3.10: Patent value = \frac{1.5}{0.14 - 0.02} = $12.5\text{ million}.
AQ 3.11: Immediate payment changes the value (approx. $14.25\text{ million}).
AQ 3.12: PV for quarterly dividend with inflation growth involves quarterly rate; result around $304.50 today for the described setup.
AQ 3.13: g ≈ r − E/P = 0.12 − 0.05 = 0.07 = 7%.
AQ 3.14: Relationship between annuity and perpetuity values; t where the fraction equals 3/4 is t ≈ 28.41 years at r = 5%.
AQ 3.15: Annunity formula given above; AQ 3.16: 360-month annuity ≈ $833.96 per month for $5 payments; AQ 3.17–3.24: Fibonacci anecdote and related historical problems solved via NPV/time-value-of-money concepts.

Quick Connections to Foundations and Real-World Relevance

Perpetuities and growing perpetuities underpin terminal value calculations in pro forma financial models and in valuing mature firms with stable long-run growth.
Gordon Growth Model links dividend policy and stock valuation; it connects earnings/dividend growth with required return to derive an implied price.
Annuities model loans and mortgages; mortgage payments are a practical example of a long series of fixed payments.
Bond valuation illustrates the relationship between coupon rates, discount rates, and price: when market rates rise, price falls; when they fall, price rises.
The material emphasizes r > g for growing perpetuities, the importance of timing of cash flows (first payment timing), and the distinction between coupon rates and actual yields.
Real-world cautions: long-run growth and discount rates are rarely constant; use these formulas as building blocks and rough guides, not exact forecasts.

Summary Takeaways

The four basic formulas to memorize (or at least recognize) are:
Perpetuity: $PV0 = \frac{C1}{r}$
Growing perpetuity: PV0 = \frac{C1}{r - g} \quad (r > g)
Annuity: $PV0 = C1 \frac{1 - (1 + r)^{-T}}{r}$
Growing annuity: $PV0 = \frac{C1}{r - g} \left[1 - \left(\frac{1+g}{1+r}\right)^T\right]$
The Gordon Growth Model is a direct application of the growing perpetuity to stock dividends: $P{Today} = \frac{D1}{r - g}.$