Lecture 8: Advanced Cash Flow Valuation

Introduction to Advanced Cash Flows

Building on Previous Knowledge: We learned about simple cash flows and annuities in Lecture 7. Now we explore more complex cash flow patterns used in real-world finance.

Five Types of Cash Flows:

1. Simple Cash Flow: Single payment at one point in time
2. Annuity: Constant payments over a fixed period
3. Growing Annuity: Payments that grow at a constant rate
4. Perpetuity: Constant payments forever
5. Growing Perpetuity: Payments that grow forever

Key Insight: By combining these basic cash flow types, we can value almost any financial security or investment project.

Simple Analogy: Like building blocks - with these five basic types, we can construct complex financial structures.

Future Value of Annuities

Definition

Future Value of Annuity: The value at the end of the period of all regular payments plus accumulated interest.

Key Difference from Present Value:

• Present Value: What are the payments worth today?
• Future Value: What will the payments be worth at the end?

Example: Savings Account

Scenario: You deposit $2,000 at the end of each year for 3 years

• Interest Rate: 8%
• Question: How much will you have at the end of 3 years?

Step-by-Step Calculation:

• Year 1: $2,000 deposited, earns interest for 2 years
• Year 2: $2,000 deposited, earns interest for 1 year
• Year 3: $2,000 deposited, no interest earned

Calculation:

• Year 1: $2{,}000 \times (1.08)^2 = 2{,}332.80$
• Year 2: $2{,}000 \times (1.08)^1 = 2{,}160.00$
• Year 3: $2{,}000 \times (1.08)^0 = 2{,}000.00$
• Total: $6{,}492.80$

Future Value of Annuity

$$\text{FV}_{\text{annuity}} = \text{Payment} \times \frac{(1 + r)^n - 1}{r}$$

Where:

• Payment = Regular payment amount
• r = Interest rate per period
• n = Number of periods

Using the Formula:

• $\text{FV} = 2,000 \times \frac{(1.08)^3 - 1}{0.08}$
• $\text{FV} = 2,000 \times \frac{1.2597 - 1}{0.08}$
• $\text{FV} = 2,000 \times \frac{0.2597}{0.08}$
• $\text{FV} = 2,000 \times 3.246$
• $\text{FV} = 6,492$

Growing Annuities

Definition

Growing Annuity: A series of payments that increase at a constant rate over a fixed period.

Real-World Examples:

• Salary increases over time
• Rent escalations
• Dividend growth
• Revenue growth projections

Present Value of Growing Annuity

$$\text{PV}_{\text{growing annuity}} = \text{Payment} \times \frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g}$$

Where:

• Payment = First payment amount
• g = Growth rate
• r = Discount rate
• n = Number of periods

Important: r must be greater than g for the formula to work.

Practical Example: Gold Mine Valuation

Scenario: You own a gold mine for 20 years

• Annual Production: 5,000 ounces
• Current Gold Price: $300 per ounce
• Price Growth: 3% per year
• Required Return: 10%

Step 1: Calculate Initial Cash Flow

• Year 0: $5{,}000 \times 300 = 1{,}500{,}000$

Step 2: Calculate Future Cash Flows

• Year 1: $1{,}500{,}000 \times 1.03 = 1{,}545{,}000$
• Year 2: $1{,}545{,}000 \times 1.03 = 1{,}591{,}350$
• And so on...

Step 3: Calculate Present Value Using the growing annuity formula:

• $\text{PV} = 1,545,000 \times \frac{1 - \left(\frac{1.03}{1.10}\right)^{20}}{0.10 - 0.03}$
• $\text{PV} = 1,545,000 \times \frac{1 - 0.9367^{20}}{0.07}$
• $\text{PV} = 1,545,000 \times \frac{1 - 0.4564}{0.07}$
• $\text{PV} = 1,545,000 \times \frac{0.5436}{0.07}$
• $\text{PV} = 1,545,000 \times 7.766$
• $\text{PV} = 12,000,000$

Decision: This is the fair price of the mine. Paying more would result in negative NPV.

Perpetuities

Definition

Perpetuity: A series of constant payments that continue forever.

Historical Context: Used since the 17th-18th centuries by governments to finance wars and projects.

Modern Examples:

• British Consol bonds (perpetual government bonds)
• Preferred stock with fixed dividends
• Some types of annuities

Present Value of Perpetuity

$$\text{PV}_{\text{perpetuity}} = \frac{\text{Payment}}{r}$$

Where:

• Payment = Constant payment amount
• r = Required rate of return

Key Insight: The formula is surprisingly simple - just divide the payment by the interest rate.

Example: British Consol Bond

Scenario: British government bond paying £60 forever

• Required Return: 9%
• Question: What is the bond worth today?

Calculation:

• $\text{PV} = \frac{\pounds 60}{0.09}$
• $\text{PV} = \pounds 666.67$

Verification: If we calculate 200 years of payments, we get approximately the same result, proving that payments beyond 50-100 years contribute almost nothing to present value.

Growing Perpetuities

Definition

Growing Perpetuity: A series of payments that grow at a constant rate forever.

Why Important: This is the foundation of stock valuation models.

Present Value of Growing Perpetuity

$$\text{PV}_{\text{growing perpetuity}} = \frac{\text{Payment}}{r - g}$$

Where:

• Payment = First payment amount
• r = Required rate of return
• g = Growth rate

Important: r must be greater than g for the formula to work.

Example: Growing Dividend Stock

Scenario: Stock with growing dividends

• Current Dividend: $60
• Growth Rate: 3%
• Required Return: 9%

Calculation:

• $\text{PV} = \frac{60}{0.09 - 0.03}$
• $\text{PV} = \frac{60}{0.06}$
• $\text{PV} = 1,000$

Comparison with Constant Perpetuity:

• Constant perpetuity: $\frac{60}{0.09} = 666.67$
• Growing perpetuity: $\frac{60}{0.06} = 1{,}000$
• Difference: $333.33$ (50% higher value due to growth)

Stock Valuation Models

Dividend Discount Model (DDM)

Basic Idea: A stock's value equals the present value of all future dividends.

Why Dividends Matter:

• Dividend Payments: Direct cash returns to shareholders
• Capital Gains: Stock price appreciation based on expected future dividends
• Key Insight: Everything comes back to dividends eventually

Constant Growth Model (Gordon Growth Model)

Formula:

$$\text{Stock Price} = \frac{D_1}{r - g}$$

Where:

• D₁ = Next year's expected dividend
• r = Required rate of return
• g = Constant growth rate

Practical Example: Southwest Airlines (1992)

Data:

• Last Dividend Paid: $2.73
• Historical Growth Rate: 6%
• Required Return: 12.23%

Step 1: Calculate Next Year's Dividend

• $D_1 = 2.73 \times 1.06 = 2.89$

Step 2: Apply Gordon Growth Model

• $\text{Stock Price} = \frac{2.89}{0.1223 - 0.06}$
• $\text{Stock Price} = \frac{2.89}{0.0623}$
• $\text{Stock Price} = 46.45$

Result: According to the model, Southwest Airlines should trade at $46.45 per share.

Common Mistakes to Avoid

1. Using Current Dividend Instead of Next Year's: Always use D₁, not D₀
2. Growth Rate Assumptions: Be realistic about long-term growth rates
3. Required Return: Must be greater than growth rate
4. Dividend Timing: Understand when dividends are paid

Practical Examples

Example 1: Retirement Planning with Growing Annuities

Scenario: 30-year-old planning retirement

• Annual Savings: $5,000 (grows 3% per year)
• Time Horizon: 35 years
• Expected Return: 8%

Analysis: Calculate future value of growing annuity to determine retirement fund size.

Example 2: Real Estate Investment

Scenario: Apartment building with growing rent

• Current Annual Rent: $100,000
• Rent Growth: 2% per year
• Required Return: 10%
• Holding Period: 20 years

Analysis: Use growing annuity formula to value the rental income stream.

Example 3: Company Valuation

Scenario: Tech startup with growing revenues

• Current Revenue: $1 million
• Growth Rate: 15% per year
• Required Return: 20%
• Question: What is the company worth?

Analysis: Apply growing perpetuity model (assuming company continues forever).

Example 4: Bond vs Stock Comparison

Scenario: Choosing between government bond and dividend stock

• Bond: Pays $100 forever (perpetuity)
• Stock: Pays $100 growing 2% forever (growing perpetuity)
• Required Return: 8%

Analysis:

• Bond Value: $\frac{100}{0.08} = 1,250$
• Stock Value: $\frac{100}{0.08 - 0.02} = 1,667$
• Decision: Stock is more valuable due to growth

Key Takeaways

1. Five Cash Flow Types: Simple, annuity, growing annuity, perpetuity, growing perpetuity

2. Future Value of Annuities: Value at the end of the period, calculated using compounding

3. Growing Annuities: Payments that increase at a constant rate over time

4. Perpetuities: Constant payments forever, valued using simple formula (Payment / r)

5. Growing Perpetuities: Growing payments forever, foundation of stock valuation

6. Stock Valuation: Gordon Growth Model uses growing perpetuity concept

7. Key Formulas:

   • Future Value of Annuity: $\text{FV} = \text{Payment} \times \frac{(1 + r)^n - 1}{r}$
   • Growing Annuity PV: $\text{PV} = \text{Payment} \times \frac{1 - \left(\frac{1 + g}{1 + r}\right)^n}{r - g}$
   • Perpetuity PV: $\text{PV} = \frac{\text{Payment}}{r}$
   • Growing Perpetuity PV: $\text{PV} = \frac{\text{Payment}}{r - g}$
   • Stock Price: $\text{Price} = \frac{D_1}{r - g}$

8. Important Rules:

   • r must be greater than g for growing formulas
   • Use D₁ (next year's dividend) for stock valuation
   • Present value of distant payments approaches zero

9. Practical Applications: Retirement planning, real estate, company valuation, investment analysis

10. Real-World Limitations: Models assume constant growth and required returns, which may not hold in practice

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