Beta Estimation and CAPM for Asset Valuation

Beta measures the systematic risk of an asset relative to the market portfolio. In the Capital Asset Pricing Model (CAPM), beta is the single factor driving expected returns above the risk-free rate. Key Formulas: Beta from covariance: $\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2}$ Beta from correlation: $\beta_i = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m}$ Hamada equation — unlever beta: $\beta_U = \frac{\beta_L}{1 + (1 - t) \cdot \frac{D}{E}}$ Hamada equation — relever beta: $\beta_L = \beta_U \cdot \left[1 + (1 - t) \cdot \frac{D}{E}\right]$ CAPM expected return: $E(R_i) = R_f + \beta_i \cdot \left[E(R_m) - R_f\right]$ Gordon Growth Model (terminal value): $V_0 = \frac{D_1}{r - g}$ where $r$ is the CAPM-derived required return and $g$ is the perpetual dividend growth rate. Why this matters: Private equity exits require valuing a firm that may carry different leverage than publicly traded comparables. A practitioner must unlever a comparable's beta, then relever it to match the target's capital structure before applying CAPM to derive the appropriate discount rate and terminal equity value.

Scenario: A private equity firm holds a controlling stake in NorthBridge Industrial Holdings, an unlisted manufacturer. The firm plans to exit its investment at the end of Year 3 by selling NorthBridge to a strategic buyer. To price the exit, the analyst must estimate NorthBridge's required return on equity using CAPM, then apply a dividend discount framework to determine terminal equity value. Comparable Public Company — Castleton Fabricators (CF): The analyst identifies Castleton Fabricators as the best publicly traded comparable. The following data are available for CF:

Item	Value
Observed (levered) beta of CF	1.45
CF debt-to-equity ratio (D/E)	0.60
CF marginal corporate tax rate	30%

NorthBridge Industrial Holdings — Capital Structure and Market Data:

Item	Value
NorthBridge target D/E ratio at exit	0.40
NorthBridge marginal corporate tax rate	25%
Risk-free rate	3.50%
Equity risk premium (market)	6.00%

NorthBridge Projected Financials at Exit (End of Year 3):

Item	Value
Expected dividend to be paid at end of Year 4 ($D_1$ in terminal model)	$8.40 per share
Perpetual dividend growth rate	3.00%

Additional verification data (used only in Part 1): A regression of NorthBridge's monthly returns against a market index over a 36-month back-test period (pre-acquisition, when NorthBridge was briefly publicly listed) produced the following sample statistics:

Statistic	Value
Correlation of NorthBridge returns with market ($\rho$)	0.72
Standard deviation of NorthBridge returns ($\sigma_i$)	22.0% per year
Standard deviation of market returns ($\sigma_m$)	14.0% per year
NorthBridge D/E ratio at time of listing (historical)	0.80
NorthBridge marginal tax rate at time of listing	25%

Tasks:

1. Using the historical regression data for NorthBridge, calculate NorthBridge's historical levered beta and then derive its unlevered (asset) beta using the Hamada equation.
2. Using Castleton Fabricators as the comparable, calculate CF's unlevered beta.
3. Average the two unlevered betas from Parts 1 and 2. Using this average unlevered beta, relever it to NorthBridge's target exit capital structure to obtain NorthBridge's relevered (exit) beta.
4. Apply CAPM using the relevered beta from Part 3 to determine NorthBridge's required return on equity at exit.
Using the Gordon Growth Model, calculate NorthBridge's terminal equity value per share at the exit date (end of Year 3).

Part 1: NorthBridge Historical Levered Beta and Unlevered Beta

Step 1a — Compute historical levered beta from regression statistics: $\beta_L^{NB,\text{hist}} = \rho_{i,m} \cdot \frac{\sigma_i}{\sigma_m}$ $\beta_L^{NB,\text{hist}} = 0.72 \cdot \frac{0.22}{0.14}$ $\beta_L^{NB,\text{hist}} = 0.72 \cdot 1.5714 = 1.1314$ Step 1b — Unlever using Hamada equation at NorthBridge's historical D/E = 0.80, tax rate = 25\%: $\beta_U^{NB} = \frac{\beta_L^{NB,\text{hist}}}{1 + (1 - t) \cdot \frac{D}{E}}$ $\beta_U^{NB} = \frac{1.1314}{1 + (1 - 0.25) \cdot 0.80}$ $\beta_U^{NB} = \frac{1.1314}{1 + 0.75 \cdot 0.80} = \frac{1.1314}{1 + 0.60} = \frac{1.1314}{1.60}$ $\beta_U^{NB} = 0.7071$

Part 2: Castleton Fabricators Unlevered Beta

Using CF's observed levered beta of 1.45, D/E = 0.60, tax rate = 30\%: $\beta_U^{CF} = \frac{\beta_L^{CF}}{1 + (1 - t) \cdot \frac{D}{E}}$ $\beta_U^{CF} = \frac{1.45}{1 + (1 - 0.30) \cdot 0.60}$ $\beta_U^{CF} = \frac{1.45}{1 + 0.70 \cdot 0.60} = \frac{1.45}{1 + 0.42} = \frac{1.45}{1.42}$ $\beta_U^{CF} = 1.0211$

Part 3: Average Unlevered Beta and Relevered Exit Beta

Step 3a — Average the two unlevered betas: $\beta_U^{\text{avg}} = \frac{\beta_U^{NB} + \beta_U^{CF}}{2} = \frac{0.7071 + 1.0211}{2} = \frac{1.7282}{2} = 0.8641$ Step 3b — Relever to NorthBridge's exit capital structure: D/E = 0.40, tax rate = 25\%: $\beta_L^{NB,\text{exit}} = \beta_U^{\text{avg}} \cdot \left[1 + (1 - t) \cdot \frac{D}{E}\right]$ $\beta_L^{NB,\text{exit}} = 0.8641 \cdot \left[1 + (1 - 0.25) \cdot 0.40\right]$ $\beta_L^{NB,\text{exit}} = 0.8641 \cdot \left[1 + 0.75 \cdot 0.40\right] = 0.8641 \cdot \left[1 + 0.30\right]$ $\beta_L^{NB,\text{exit}} = 0.8641 \cdot 1.30 = 1.1233$

Part 4: CAPM Required Return on Equity

$E(R_{NB}) = R_f + \beta_L^{NB,\text{exit}} \cdot \text{ERP}$ $E(R_{NB}) = 3.50\% + 1.1233 \cdot 6.00\%$ $E(R_{NB}) = 3.50\% + 6.74\% = 10.24\%$

Part 5: Terminal Equity Value per Share (Gordon Growth Model)

At the exit date (end of Year 3), the terminal value is the present value of all future dividends growing at $g = 3.00\%$ forever, with $D_1 = \$8.40$ paid one year later (end of Year 4): $V_{\text{exit}} = \frac{D_1}{r - g} = \frac{\$8.40}{0.1024 - 0.03} = \frac{\$8.40}{0.0724}$ $V_{\text{exit}} = \$116.02 \text{ per share}$ Summary of Results:

Calculation Step	Result
NorthBridge historical levered beta	1.1314
NorthBridge unlevered beta	0.7071
Castleton Fabricators unlevered beta	1.0211
Average unlevered beta	0.8641
NorthBridge relevered exit beta	1.1233
Required return on equity (CAPM)	10.24%
Terminal equity value per share	$116.02