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Numericals

Numerical Examples: Estimating Discount Rates and Betas

These examples illustrate how to calculate the cost of equity, WACC, and different types of betas.

Topic 1: Estimating Discount Rates

Example 1: Cost of Equity (CAPM)

  • Risk-Free Rate (Rf): 3% (0.03)
  • Market Risk Premium (Rm - Rf): 6% (0.06)
  • Beta (β): 1.2

Using the CAPM formula:

Ke = Rf + β * (Rm - Rf) Ke = 0.03 + 1.2 * 0.06 Ke = 0.03 + 0.072 Ke = 0.102

  • Cost of Equity = 10.2%

Example 2: WACC Calculation

  • Market Value of Equity (E): $500 million
  • Market Value of Debt (D): $250 million
  • Cost of Equity (Ke): 12% (0.12)
  • Cost of Debt (Kd): 6% (0.06)
  • Tax Rate (t): 30% (0.30)

  1. Calculate the weights of equity and debt:

    • Total Capital (V) = E + D = $500 million + $250 million = $750 million
    • Weight of Equity (We) = E / V = $500 million / $750 million = 0.6667 (66.67%)
    • Weight of Debt (Wd) = D / V = $250 million / $750 million = 0.3333 (33.33%)

  2. Calculate the WACC:

    WACC = (We * Ke) + (Wd * Kd * (1 - t)) WACC = (0.6667 * 0.12) + (0.3333 * 0.06 * (1 - 0.30)) WACC = 0.0800 + (0.3333 * 0.06 * 0.70) WACC = 0.0800 + 0.0140 WACC = 0.0940

  • WACC = 9.4%

Topic 2: Betas

Example 3: Bottom-Up Beta

A company, "GreenTech," operates in the renewable energy industry. We want to estimate its beta using the bottom-up approach.

  1. Identify Comparable Companies: We identify three comparable companies: SolarCorp, WindPower, and EnergyRenew.

  2. Find Unlevered Betas: We obtain the following information from financial data providers:

    Company Levered Beta Debt/Equity Ratio Tax Rate
    SolarCorp 1.3 0.5 0.25
    WindPower 1.1 0.3 0.25
    EnergyRenew 1.0 0.2 0.25

  3. Calculate Unlevered Betas for Comparables: Using the formula: β_unlevered = β_levered / [1 + (1 - Tax Rate) * (Debt/Equity)]

    • SolarCorp Unlevered Beta = 1.3 / [1 + (1 - 0.25) * 0.5] = 1.3 / 1.375 = 0.945
    • WindPower Unlevered Beta = 1.1 / [1 + (1 - 0.25) * 0.3] = 1.1 / 1.225 = 0.898
    • EnergyRenew Unlevered Beta = 1.0 / [1 + (1 - 0.25) * 0.2] = 1.0 / 1.15 = 0.870

  4. Calculate Average Unlevered Beta: Average Unlevered Beta = (0.945 + 0.898 + 0.870) / 3 = 0.904

  5. Calculate Levered Beta for GreenTech: GreenTech has a Debt/Equity Ratio of 0.4 and a Tax Rate of 25%.

    Using the formula: β_levered = β_unlevered * [1 + (1 - Tax Rate) * (Debt/Equity)] GreenTech Levered Beta = 0.904 * [1 + (1 - 0.25) * 0.4] = 0.904 * 1.3 = 1.175

  • GreenTech's Bottom-Up Levered Beta = 1.175

Example 4: Impact of Leverage on Beta

A company has an unlevered beta of 0.8, a tax rate of 20%, and a debt-to-equity ratio of 0.6. Calculate the levered beta.

  • Unlevered Beta: 0.8
  • Tax Rate: 20% (0.20)
  • Debt/Equity Ratio: 0.6

Using the formula: β_levered = β_unlevered * [1 + (1 - Tax Rate) * (Debt/Equity)] β_levered = 0.8 * [1 + (1 - 0.20) * 0.6] β_levered = 0.8 * [1 + 0.48] β_levered = 0.8 * 1.48 β_levered = 1.184

  • Levered Beta = 1.184

These examples illustrate the calculations involved in estimating discount rates and betas. Remember to carefully consider the assumptions and limitations of each method and to use a combination of different approaches to arrive at a reasonable estimate.

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