📖 永续年金

📌 Topic: The Infinite Cash Flow — When Payments Never Stop

1. From Finite to Infinite: Perpetuity Intuition

L087–L092 built a complete time value of money framework: single cash flow → annuity → annuity reverse calculations.

All annuities share a common premise: payments have an endpoint. n is a finite number — 30 years, 60 months, 120 periods.

But in reality, certain financial instruments are structurally designed to never mature:

• British "Consols" issued in 1752 — still paying interest today
• Preferred Stock — pays a fixed annual dividend as long as the company exists
• Perpetual Bonds — never mature, pay interest forever
• University endowments — principal never touched, only earnings spent annually

🔑 Perpetuity: An infinite stream of equal periodic payments (PMT), continuing forever with no endpoint.

2. Core Formula: The Most Elegant Formula in CFA Level 1

2.1 Derivation

Recall the PV formula for an ordinary annuity:

$$PV_{\text{annuity}} = PMT \times \frac{1 - (1+r)^{-n}}{r}$$

Let n → ∞:

$$\lim_{n \to \infty} (1+r)^{-n} = 0 \quad (\text{since } r > 0)$$

$$\boxed{PV_{\text{perpetuity}} = \frac{PMT}{r}}$$

🧠 The beauty of the derivation: the present value of an infinite sum collapses into a three-character formula — PV = PMT / r. That is the elegance of mathematics.

2.2 Intuition

$$PV = \frac{PMT}{r}$$

In plain English: You need to invest a principal PV such that it generates r × PV in earnings each year, exactly matching the annual payment PMT.

• If you want $10,000 per year at 5% → principal = 10,000 / 5% = $200,000
• If you want $10,000 per year at 2% → principal = 10,000 / 2% = $500,000

💡 The lower the interest rate, the larger the initial principal needed to sustain the same PMT — because money grows less efficiently.

2.3 Relationship Between the Three Variables

3. Classic Applications

3.1 Preferred Stock Valuation (High-Frequency CFA Topic)

🔴 Preferred Stock = The quintessential perpetuity

Characteristics: - Pays a fixed annual dividend (e.g., $5/share) - Never matures (unless the company redeems or goes bankrupt) - Dividend does not grow (zero-growth model) - Valuation formula: $P_0 = D / r$

Example: A preferred stock pays an annual dividend of $4.50. The market's required rate of return is 9%. What is the theoretical value?

$$P_0 = \frac{4.50}{0.09} = \$50$$

• If market price = $45 → undervalued, worth buying
• If market price = $55 → overvalued, sell or wait

Extension: Same preferred stock at market price $60. Implied yield:

$$r = \frac{4.50}{60} = 7.5\%$$

💡 If your required return > 7.5%, $60 is too expensive. If you only require 6%, $60 is a bargain.

3.2 Endowment Model

Scenario: A university receives a donation requiring that the principal is never touched. Only investment earnings may be used for annual scholarships of $200,000, with an expected annual return of 5%.

$$PV = \frac{200{,}000}{0.05} = 4{,}000{,}000$$

Answer: $4 million in principal is needed.

💡 This is the foundational math behind Yale, Harvard, and other university endowments — as long as r consistently exceeds the spending rate, the fund operates in perpetuity.

3.3 Real Estate: Perpetual Rent

Scenario: A commercial property generates net annual rental income of $120,000, with a cap rate of 6%. Value it using the perpetuity model:

$$PV = \frac{120{,}000}{0.06} = 2{,}000{,}000$$

🔑 This is the mathematical essence of "direct capitalization" in real estate pricing: Value = NOI / Cap Rate.

4. Growing Perpetuity

4.1 Formula

In reality, many "perpetual" cash flows grow every year: - Corporate dividends grow over time (Gordon Growth Model) - Rents rise with inflation - Scholarships adjust with cost of living

If PMT grows at a constant rate g each period (where g < r):

$$\boxed{PV = \frac{PMT_1}{r - g}}$$

Where $PMT_1$ = the next payment (payment at t=1)

4.2 Derivation & Intuition

This is the Gordon Growth Model — the core equity valuation formula in CFA Level 1.

Replace r in the ordinary perpetuity with (r − g): a smaller denominator → larger PV, which makes intuitive sense — a "growing perpetual stream" is worth more than a "constant perpetual stream."

Example: A stock's expected dividend next year is $2.00, expected to grow perpetually at 3% annually, with a required return of 10%. Intrinsic value:

$$P_0 = \frac{2.00}{0.10 - 0.03} = \frac{2.00}{0.07} = \$28.57$$

4.3 Growing vs. Zero-Growth

💡 g = 3% may sound small, but the impact on perpetual valuation is enormous — PV jumps from $20 to $28.57, a 42.9% increase. This is the power of compounding on an infinite time horizon.

4.4 Critical Constraint: g < r

$$\text{If } g \geq r \text{, } PV = \frac{PMT}{r-g} \text{ is negative or infinite — meaningless}$$

• g = r → denominator = 0, formula explodes
• g > r → denominator negative, mathematically PV is negative (impossible in reality)

🔴 Exam trap: if a question gives g > r and asks for valuation → answer "the model is not applicable"

5. Perpetuity vs. Ordinary Annuity: Side-by-Side

5.1 PV Comparison

💡 Key insight: For n ≥ 50 years, the PV of an annuity is already extremely close to the PV of a perpetuity. In CFA Level 1 practice, annuities with n > 50–60 years are often approximated as perpetuities.

5.2 When Does "Annuity ≈ Perpetuity"?

Rules of thumb: - n ≥ 50 and r ≥ 5% → error < 3% - n ≥ 30 and r ≥ 10% → error < 5%

$$PV_{\text{annuity}}(n) = PV_{\text{perpetuity}} \times [1 - (1+r)^{-n}]$$

When n is sufficiently large, (1+r)^(−n) → 0, and the bracket approaches 1.

6. TI BA II Plus Calculator Tips

6.1 No Direct Perpetuity Key

The TI BA II Plus has no dedicated perpetuity key. However:

Example: PMT = 500, r = 8%

6.2 Growing Perpetuity

PMT₁ ÷ (r − g). Subtract first, then divide.

Example: PMT₁ = 3.00, r = 12%, g = 4%

6.3 Verification with Annuity Approximation

To verify: input an extremely large n (e.g., n = 9999), compute the annuity PV — the result should approximately equal the perpetuity PV.

7. Common Exam Traps

Trap 1: Confusing PMT₀ and PMT₁ 🔴

The perpetuity formula uses the payment at t=1, not t=0.

• ✅ PV = PMT₁ / r (payment starts next period)
• ❌ Easy mistake: using "the most recent" payment (PMT₀)

Example: A stock just paid an annual dividend of $3.00. Dividends grow perpetually at 5%, r = 10%. Find intrinsic value.

• The $3.00 just paid is D₀ (past)
• D₁ = D₀ × (1+g) = 3.00 × 1.05 = $3.15
• P₀ = 3.15 / (0.10 − 0.05) = $63.00

❌ If wrongly using 3.00 / 0.05 = $60.00 → WRONG! Used D₀ instead of D₁.

Trap 2: Perpetuities Have No FV

Any question asking for the FV of a perpetuity → answer: infinite or undefined.

$$FV_{\text{perpetuity}} = \lim_{n \to \infty} PMT \times \frac{(1+r)^n - 1}{r} = \infty$$

🔴 If you try to compute FV on a calculator with n → ∞, it will overflow with an error.

Trap 3: g ≥ r Is Invalid

• g = 5%, r = 4% → growing perpetuity formula does not apply (PV would be negative)
• In such cases, a different valuation model must be used (e.g., multi-stage model)

Trap 4: Not Every Infinite Stream Is a Perpetuity

A perpetuity requires PMT to be constant or grow at a constant rate g.

• ✅ $1,000 fixed per year → perpetuity
• ✅ 3% annual growth → growing perpetuity
• ❌ Random / irregular amounts each year → NOT a perpetuity; formulas above do not apply

8. Worked Examples

Example 1: Basic Perpetuity (Preferred Stock)

A preferred stock pays an annual dividend of $6.00 per share. The market required return is 8%. Its intrinsic value is:

A. $48 B. $60 C. $75 D. $80

Solution: PV = 6.00 / 0.08 = $75.00 → Answer C

Example 2: Finding the Required Return

A perpetual bond has a face value of $1,000, market price of $800, and pays annual interest of $64. Its current yield is closest to:

A. 6.4% B. 8.0% C. 8.5% D. 10.0%

Solution: r = PMT / PV = 64 / 800 = 8.0% → Answer B

(Perpetual bonds never mature, so YTM is not meaningful; "current yield" = PMT / Market Price)

Example 3: Growing Perpetuity (Gordon Growth Model)

A company's expected dividend next year is $2.50 per share, expected to grow perpetually at 4%, with a required return of 11%. The intrinsic value is closest to:

A. $22.73 B. $31.25 C. $35.71 D. $62.50

Solution: P₀ = 2.50 / (0.11 − 0.04) = 2.50 / 0.07 = $35.71 → Answer C

Example 4: D₀ vs D₁ Trap

A stock just paid an annual dividend of $2.00. Future dividends are expected to grow perpetually at 5%, with a discount rate of 12%. The fair value is:

A. $16.67 B. $28.57 C. $30.00 D. $40.00

Solution: - Just paid = D₀ = $2.00 - D₁ = 2.00 × 1.05 = $2.10 - P₀ = 2.10 / (0.12 − 0.05) = 2.10 / 0.07 = $30.00 → Answer C

❌ Misusing D₀: 2.00 / 0.07 = $28.57 → B (WRONG!)

Example 5: g ≥ r (Formula Breakdown)

An analyst forecasts a stock's dividends will grow perpetually at 12%, while the stock's required return is 10%. When applying the Gordon Growth Model:

A. The stock value is negative B. The model is not applicable in this case C. The stock value is positive and finite D. The stock value is zero

Solution: g (12%) > r (10%) → denominator is negative, model not applicable → Answer B

Example 6: Endowment Model

A philanthropist wants to establish a perpetual scholarship fund paying $50,000 annually, with an expected annual investment return of 4.5%. How much principal is needed?

A. $1,000,000 B. $1,111,111 C. $1,250,000 D. $2,222,222

Solution: PV = 50,000 / 0.045 = $1,111,111 → Answer B

Example 7: Mixed Annuity + Perpetuity

An investment pays $5,000 at the end of each year for the first 10 years, then $5,000 at the end of each year in perpetuity starting year 11. Discount rate = 10%. Find total PV.

Solution:

Method 1: Discount in Segments

First 10 years: ordinary annuity $$PV_1 = 5{,}000 \times PVIFA(10\%, 10) = 5{,}000 \times 6.1446 = 30{,}723$$

From year 11 onward: perpetuity, first discount to t=10 $$PV_{t=10}^{\text{perp}} = \frac{5{,}000}{0.10} = 50{,}000$$

Then discount back to t=0: $$PV_2 = 50{,}000 \times (1.10)^{-10} = 50{,}000 \times 0.3855 = 19{,}277$$

$$PV_{\text{total}} = 30{,}723 + 19{,}277 = 50{,}000$$

Method 2 (Shortcut): Notice! This question is essentially a perpetuity of $5,000/year starting from t=1!

$$PV_{\text{total}} = \frac{5{,}000}{0.10} = 50{,}000$$

💡 First 10 years + perpetuity thereafter = perpetuity from year 1. This is a classic CFA trick: "looks complex, but one glance gives the answer."

9. Key Formula Cheat Sheet

10. Practice Questions

Q1 (Perpetuity Valuation)

A perpetual bond pays annual interest of $80. The market required return is 6.4%. Its theoretical value is closest to:

A. $1,000 B. $1,125 C. $1,250 D. $1,280

Q2 (Preferred Stock Valuation)

A preferred stock has a par value of $100 and an annual dividend rate of 7%. The current market required return is 8%. Its intrinsic value is:

A. $87.50 B. $100.00 C. $107.00 D. $114.29

Q3 (Growing Perpetuity / D₀ Trap)

A company just paid an annual dividend of $1.80 per share. Future dividends are expected to grow perpetually at 6%, with a required return of 12%. The intrinsic value is closest to:

A. $15.00 B. $30.00 C. $31.80 D. $33.00

Q4 (g ≥ r Determination)

An analyst forecasts that a startup's dividends will grow perpetually at 15%, while the reasonable discount rate for the company is 12%. Under the Gordon Growth Model:

A. Intrinsic value is $0 B. Intrinsic value is negative C. The model is not applicable; switch to a multi-stage model D. Intrinsic value is positive but extremely low

Q5 (Mixed: Annuity + Perpetuity)

An investment pays $3,000 at the end of each year for the first 5 years, then $3,000 at the end of each year in perpetuity starting year 6. Discount rate = 8%. The total PV is closest to:

A. $30,000 B. $32,750 C. $37,500 D. $40,000

11. Closing Thoughts

L092 taught you to "know four, solve for one" among five variables.

L093 reveals — when n → ∞, the formula becomes remarkably simple.

P = PMT / r. Three letters. This is the shortest formula in the CFA Level 1 formula sheet, yet it appears across Equity Valuation, Fixed Income, and Alternative Investments.

It is also the "zero-growth special case" of the Gordon Growth Model. When g = 0, P = D₁ / (r − 0) = D / r.

Starting from L094, we face the final "imperfect" scenario of the TVM module: uneven cash flows — where PMT varies each period, the formula loses its elegance, and the calculator needs a mode switch…

But before that, remember the elegant simplicity of the perpetuity: the eternal, paradoxically, is the easiest to price.

📎 Answer Key

Explanations:

• Q1: PV = 80 / 0.064 = $1,250 ✅

• Q2: Annual dividend = 100 × 7% = $7. PV = 7 / 0.08 = $87.50 ✅

• Q3: D₀ = $1.80 (just paid), D₁ = 1.80 × 1.06 = $1.908. P₀ = 1.908 / (0.12 − 0.06) = 1.908 / 0.06 = $31.80. ❌ Using D₀ directly: 1.80 / 0.06 = $30 → TRAP!

• Q4: g = 15% > r = 12%, the Gordon Growth Model denominator (r−g) is negative. The model is not applicable; use a multi-stage model instead (forecast high-growth phase first, then transition to stable growth).

• Q5: First 5-year annuity PV = 3,000 × PVIFA(8%, 5) = 3,000 × 3.9927 = 11,978. Perpetuity portion PV at t=5 = 3,000 / 0.08 = 37,500, discount to t=0: 37,500 / (1.08)⁵ = 37,500 / 1.4693 = 25,522. Total = 11,978 + 25,522 = 37,500. Shortcut: the entire cash flow is simply a perpetuity of $3,000/year from t=1 → 3,000 / 0.08 = 37,500!