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Stock Valuation Models

Stock Valuation Models & Formulas

Complete mathematical framework for equity valuation with detailed formulas, derivations, and practical applications

Mathematical ModelsFormula DerivationsPractical Examples

1. Zero-Growth Dividend Discount Model (DDM)

Stocks that pay stable dividends with no expected growth

Model Conditions & Assumptions

  • Company pays constant dividends indefinitely
  • No expected growth in dividends (g = 0)
  • Required return exceeds zero
  • Perpetual dividend stream

Variables:

  • • D: Annual dividend per share (constant)
  • • r: Required rate of return (decimal)
  • • V₀: Stock intrinsic value

Formulas:

V₀ = D / r

Example & Solution

Question: A utility company pays $2.80 annual dividend with 10% required return. What is the stock value?

Given Conditions:

  • • Annual dividend (D): $2.80
  • • Required rate of return (r): 10% or 0.10
  • • Zero growth assumption (g = 0)
  • • Perpetual dividend payments

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Identify variables**: - D = $2.80 (annual dividend) - r = 0.10 (required return) 2. **Apply the formula**: V₀ = D / r 3. **Calculate intrinsic value**: V₀ = $2.80 / 0.10 = $28.00 4. **Investment decision**: - If market price = $25: Stock is undervalued ($28.00 > $25.00) - If market price = $32: Stock is overvalued ($28.00 < $32.00)

Answer: $28.00

Explanation: The stock's intrinsic value is $28.00. If trading below this price, it may be undervalued; if trading above, it may be overvalued.

2. Constant-Growth Dividend Discount Model (Gordon Model)

Gordon growth model for companies with stable dividend growth rates

Model Conditions & Assumptions

  • Company maintains constant dividend growth rate
  • Growth rate is less than required return (g < r)
  • Dividends grow at constant rate indefinitely
  • Stable business model with predictable growth

Variables:

  • • D₀: Current dividend per share
  • • D₁: Next year's dividend (D₀ × (1+g))
  • • g: Constant growth rate (decimal)
  • • r: Required rate of return (decimal)
  • • V₀: Stock intrinsic value

Formulas:

V₀ = D₁ / (r - g)

Example & Solution

Question: Johnson & Johnson pays $4.24 dividend, expects 6% growth, requires 9% return. What is the value?

Given Conditions:

  • • Current dividend (D₀): $4.24
  • • Growth rate (g): 6% or 0.06
  • • Required return (r): 9% or 0.09
  • • g < r (0.06 < 0.09) ✓

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **Calculate next year's dividend**: D₁ = D₀ × (1 + g) = $4.24 × 1.06 = $4.4944 2. **Apply Gordon growth formula**: V₀ = D₁ / (r - g) = $4.4944 / (0.09 - 0.06) = $4.4944 / 0.03 = $149.81 3. **Verify convergence condition**: g = 6% < r = 9% ✓ (Model is mathematically valid) 4. **Sensitivity analysis**: - If g = 7%: V₀ = $4.4944 / (0.09 - 0.07) = $4.4944 / 0.02 = $224.72 - If g = 5%: V₀ = $4.4944 / (0.09 - 0.05) = $4.4944 / 0.04 = $112.36

Answer: $149.81

Explanation: The stock's intrinsic value is $149.81. Higher growth expectations significantly increase the valuation, as shown in the sensitivity analysis.

3. Two-Stage Dividend Discount Model

Two-stage model for companies with changing growth patterns

Model Conditions & Assumptions

  • Company experiences high growth initially
  • Growth transitions to stable rate
  • Predictable transition timing
  • Different discount rates may apply to each stage

Variables:

  • • D₀: Current dividend per share
  • • g₁: High growth rate (first n years)
  • • g₂: Stable growth rate (thereafter)
  • • r: Required rate of return (decimal)
  • • n: Years of high growth

Formulas:

V₀ = Σ[D₀(1+g₁)ᵗ/(1+r)ᵗ] + [Dₙ₊₁/(r-g₂)]/(1+r)ⁿ

Example & Solution

Question: Microsoft: 5-year 15% growth, then 4% stable growth, 10% required return, $2 dividend

Given Conditions:

  • • Current dividend (D₀): $2.00
  • • High growth rate (g₁): 15% for 5 years
  • • Stable growth rate (g₂): 4% thereafter
  • • Required return (r): 10%
  • • Growth transition at year 5

Step-by-Step Calculation:

**Step-by-Step Calculation:** 1. **High growth phase dividends**: - Year 1: $2.00 × 1.15 = $2.30 - Year 2: $2.30 × 1.15 = $2.645 - Year 3: $2.645 × 1.15 = $3.042 - Year 4: $3.042 × 1.15 = $3.498 - Year 5: $3.498 × 1.15 = $4.023 2. **PV of high growth dividends**: - Year 1: $2.30 / 1.10 = $2.091 - Year 2: $2.645 / 1.10² = $2.186 - Year 3: $3.042 / 1.10³ = $2.288 - Year 4: $3.498 / 1.10⁴ = $2.397 - Year 5: $4.023 / 1.10⁵ = $2.512 - Total PV high growth: $11.474 3. **Terminal value at year 5**: - D₆ = $4.023 × 1.04 = $4.184 - TV₅ = $4.184 / (0.10 - 0.04) = $4.184 / 0.06 = $69.733 4. **PV of terminal value**: - PV terminal = $69.733 / 1.10⁵ = $43.477 5. **Total intrinsic value**: - V₀ = $11.474 + $43.477 = $54.951

Answer: $54.95

Explanation: The two-stage model accounts for Microsoft's initial high growth followed by a transition to stable growth, providing a more realistic valuation for growth companies.

Stock Valuation Framework

Core Valuation Principles

Dividend Discount Models

Value stocks based on expected future dividends

Cash Flow Analysis

FCFE models for non-dividend paying companies

Relative Valuation

P/E ratios and comparable analysis

Investment Applications

Value Investing

Identify undervalued stocks using intrinsic value

Growth Investing

Evaluate growth potential and sustainability

Portfolio Management

Balance risk and return using valuation models

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