Beta Calculation Worked Examples: Regression, Unlevered, Relevered, and Bloomberg Adjusted

Beta measures systematic risk — how much a stock moves when the market moves. A beta of 1.0 means the stock moves with the market; a beta of 1.5 means it moves 50% more than the market; a beta of 0.5 means half as much. The formula: β = Cov(Rₛ, Rₘ) / Var(Rₘ), where Rₛ is stock return and Rₘ is market return. The practical issue is that beta depends heavily on HOW you calculate it: which time period, which return frequency (daily, weekly, monthly), which market index, and whether you adjust for forward-looking expectations. A single stock can have different betas under different methodologies, all defensible. Common beta types used in practice: 1. Raw regression beta — ordinary least squares (OLS) regression of stock returns against market returns 2. Bloomberg adjusted beta — applies Blume's adjustment: 0.67 × Raw β + 0.33 × 1.0 (assumes beta regresses to 1 over time) 3. Unlevered beta (asset beta) — strips out the effect of financial leverage 4. Relevered beta — applies the target capital structure's leverage to an unlevered industry beta 5. Pure-play beta — beta for a private company or division, derived from publicly traded comparables For WACC calculations in CFA exams or textbook DCF, the typical workflow is: find comparable public companies → unlever their betas → take the average or median → relever to your target company's capital structure → use that as your project or company beta. This content is for educational purposes only and does not constitute financial advice.

The most fundamental beta calculation uses historical returns. You regress stock returns against market index returns over a chosen time window. Standard setups: - 2 years of weekly returns (~104 data points) — Bloomberg default - 5 years of monthly returns (60 data points) — Yahoo Finance default - 3 years of daily returns (~750 data points) — rarely used due to noise Formula: β = Cov(Rₛ, Rₘ) / Var(Rₘ) Worked example: calculate beta for a stock with 4 monthly returns against S&P 500: Month | Stock R | Market R 1 | 3.0% | 2.0% 2 | -4.0% | -2.0% 3 | 6.0% | 3.0% 4 | -1.0% | 1.0% Mean stock return = (3 - 4 + 6 - 1)/4 = 1.0% Mean market return = (2 - 2 + 3 + 1)/4 = 1.0% Deviations from mean: Stock: 2.0, -5.0, 5.0, -2.0 Market: 1.0, -3.0, 2.0, 0.0 Cov = Σ(stock dev × market dev) / (n-1) Cov = (2×1 + (-5)×(-3) + 5×2 + (-2)×0) / 3 Cov = (2 + 15 + 10 + 0) / 3 = 9.0 Var(market) = Σ(market dev²) / (n-1) Var = (1² + (-3)² + 2² + 0²) / 3 = 14 / 3 = 4.67 β = 9.0 / 4.67 = 1.93 This is higher than typical firm betas because only 4 data points with a clear pattern of amplified movement. Real regressions use 60-104 data points and produce more stable estimates. In Excel, =SLOPE(stock_returns_range, market_returns_range) calculates beta directly. In Python, scipy.stats.linregress does the same. In R, lm(stock ~ market) returns beta as the coefficient. R-squared from the regression tells you how much of the stock's movement is explained by the market — typical range 0.15 to 0.60 for individual stocks. The rest is unsystematic (firm-specific) risk.

Bloomberg adjusted beta corrects raw regression beta for the empirical tendency of betas to regress toward 1.0 over time. The formula, developed by Marshall Blume (1971, Journal of Finance): Adjusted β = 0.67 × Raw β + 0.33 × 1.0 So a raw beta of 1.5 becomes adjusted beta of 0.67 × 1.5 + 0.33 × 1.0 = 1.005 + 0.33 = 1.335. A raw beta of 0.6 becomes 0.67 × 0.6 + 0.33 × 1.0 = 0.732. Why Blume's adjustment? Empirical observation shows that high-beta stocks tend to have lower future betas (mean reversion), and low-beta stocks tend to have higher future betas. The 0.67/0.33 weights were derived from historical data matching this regression to the mean. When to use adjusted vs raw: - WACC for long-term valuation: use adjusted (forward-looking emphasis) - Short-term trading: use raw (recent behavior is most relevant) - CFA curriculum: adjusted beta is the default in Level I and II - Professional valuation practice (investment banks, consulting): Bloomberg adjusted beta is standard Worked example: Company XYZ has raw regression beta of 2.1 over past 2 years (high-growth tech company). Adjusted beta = 0.67 × 2.1 + 0.33 × 1.0 = 1.407 + 0.33 = 1.737. Interpretation: the raw 2.1 reflects recent volatility but is likely to regress toward 1.0. Using 1.737 in CAPM gives a more conservative (lower) cost of equity than using 2.1. Re = Rf + β × MRP Raw: Re = 4% + 2.1 × 6% = 16.6% Adjusted: Re = 4% + 1.737 × 6% = 14.4% The ~2.2% difference in cost of equity propagates through WACC and ultimately through DCF valuations, producing materially different enterprise values.

Unlevered beta (also called asset beta or βᵤ) measures the systematic risk of the company's assets independent of its financial leverage. Two companies in the same industry can have similar asset betas but very different equity betas if their debt loads differ. Hamada equation: βᵤ = βₑ / [1 + (1 - T) × D/E] Where: - βᵤ = unlevered beta - βₑ = equity beta (regression or adjusted beta) - T = corporate tax rate - D/E = debt-to-equity ratio (market value) Worked example: Company A has equity beta 1.6, D/E of 0.8, tax rate 25%. βᵤ = 1.6 / [1 + (1 - 0.25) × 0.8] βᵤ = 1.6 / [1 + 0.60] βᵤ = 1.6 / 1.60 βᵤ = 1.00 The asset beta is 1.0 — much lower than the equity beta of 1.6. The difference (0.6) is the contribution of financial leverage to equity risk. If Company A were unlevered (all equity), its beta would be 1.0. Why unlever? Two reasons: 1. Compare companies on a pure-business-risk basis. When comparing Pepsi (high leverage) to a newer beverage company (low leverage), equity betas will differ significantly due to capital structure, not business risk. Unlevered betas let you compare pure business risk. 2. Relever to target capital structure. If you're valuing a private company or a project with different leverage than your comparables, you need to unlever the comparables' betas and relever at the target capital structure. The Hamada formula assumes: - Debt is risk-free (βd = 0) - Interest tax shield discounted at cost of debt - Constant debt-to-equity over time For high-leverage firms where debt is risky (approaching investment-grade or below), the simplified Hamada may understate unlevered beta. More advanced formulas (Miles-Ezzell, Myers' adjusted present value) handle this but are rarely tested in CFA curriculum. Common mistake: using book D/E instead of market D/E. Unlevered beta calculations should use MARKET values — market cap for equity, market value of debt for D. Book values produce misleading unlevered betas.

Once you have an unlevered beta, you can relever it to any target capital structure. The formula is the inverse of Hamada: Relevered βₑ = βᵤ × [1 + (1 - T) × (D/E)_target] Worked example: you want to value Company B, a private company with target D/E of 0.4. Its comparable public companies have an average unlevered beta of 0.9. Tax rate 25%. Relevered βₑ = 0.9 × [1 + (1 - 0.25) × 0.4] Relevered βₑ = 0.9 × [1 + 0.30] Relevered βₑ = 0.9 × 1.30 Relevered βₑ = 1.17 Company B's relevered equity beta is 1.17, reflecting its business risk (0.9 unlevered) plus its leverage. This is the STANDARD workflow for valuing: - Private companies (use comparable public companies) - Divisions of public companies (use pure-play peers) - New ventures or subsidiaries (no independent trading history) - M&A target companies (adjust for post-deal capital structure) Full workflow: 1. Identify 3-6 public companies with similar business risk (pure-play peers) 2. Get their equity betas (regression or adjusted) 3. Unlever each using their current D/E and tax rate 4. Take the average or median unlevered beta 5. Relever at the target company's target D/E and tax rate 6. Apply in CAPM to calculate cost of equity This 'pure-play method' is what CFA, valuation textbooks, and most investment banks teach. Choosing target D/E: for public companies, use current market-value D/E (unless restructuring is planned). For private companies, use the industry median or the explicit target capital structure from the owner. For projects, use the firm's target capital structure if project risk matches the firm, or the project-specific optimal structure if it's different.

Let's run through a complete pure-play beta calculation for a private software company (SoftCo) using three public software peers. Step 1 — Identify peers with similar business characteristics: Peer A: β = 1.4, D/E = 0.15, T = 25% Peer B: β = 1.8, D/E = 0.05, T = 25% Peer C: β = 1.6, D/E = 0.10, T = 25% Step 2 — Unlever each peer's beta using Hamada: Peer A: βᵤ = 1.4 / [1 + (1 - 0.25) × 0.15] = 1.4 / 1.1125 = 1.258 Peer B: βᵤ = 1.8 / [1 + (1 - 0.25) × 0.05] = 1.8 / 1.0375 = 1.735 Peer C: βᵤ = 1.6 / [1 + (1 - 0.25) × 0.10] = 1.6 / 1.075 = 1.488 Step 3 — Calculate the average unlevered beta across peers: Average βᵤ = (1.258 + 1.735 + 1.488) / 3 = 1.494 Step 4 — Determine SoftCo's target D/E. SoftCo's owner plans to operate with 20% debt and 80% equity (D/E = 0.25). Tax rate 25%. Step 5 — Relever: βₑ = 1.494 × [1 + (1 - 0.25) × 0.25] βₑ = 1.494 × [1 + 0.1875] βₑ = 1.494 × 1.1875 βₑ = 1.774 Step 6 — Apply CAPM to get cost of equity: Re = Rf + β × MRP Assume Rf = 4%, MRP = 6% Re = 4% + 1.774 × 6% = 4% + 10.64% = 14.64% This is SoftCo's cost of equity based on pure-play beta analysis — higher than any single peer's cost because SoftCo will have more leverage than Peer A and C (less than Peer B). Sanity checks: - Unlevered betas range 1.26-1.74 — reasonable for software industry where business risk varies with product maturity - Relevered beta of 1.77 is higher than peer average of 1.6, reflecting SoftCo's slightly higher leverage - Resulting Re of 14.6% is in the typical software cost-of-equity range (12-17%) If the unlevered betas had wider dispersion (e.g., 0.8 to 2.5), you'd investigate why — different growth stages, different business models, or poor peer selection. The whole point of unlevering is to isolate pure business risk, so unusual dispersion signals that peers aren't comparable enough.

Mistake 1 — Using book D/E instead of market D/E. Book D/E reflects historical accounting values; market D/E reflects current economic reality. For a company whose equity has risen 5× since IPO, book D/E overstates leverage dramatically. Always use market values: market cap for equity, market value of debt for D. Mistake 2 — Ignoring preferred stock. Preferred stock is neither debt nor common equity but affects capital structure. Technically, preferred is included in leverage for Hamada purposes, but the adjustment is usually small. For CFA problems, preferred is typically handled separately. Mistake 3 — Mismatching raw and adjusted betas. If you're using Bloomberg adjusted betas for peers, unlever them using Bloomberg adjusted betas, relever the average, and don't mix raw and adjusted in the same calculation. Mistake 4 — Wrong tax rate. Use the effective tax rate or marginal tax rate of the COMPANY being valued when relevering. Don't use the peer's tax rate — the formula uses the target's tax rate. Mistake 5 — Assuming constant debt. Hamada assumes debt-to-equity stays constant. Growing companies that will add debt over time, or deleveraging companies that will pay down debt, violate this assumption. For these cases, consider Miles-Ezzell or APV approaches (beyond CFA Level I scope). Mistake 6 — Selecting peers that aren't truly comparable. Industry classification (GICS, NAICS) is a starting point but not definitive. Ask: do these companies have similar business models, similar margins, similar revenue growth, similar customer types? If not, they may not share business risk even if classified the same. A software company selling to enterprises has different risk than one selling to consumers. Mistake 7 — Using too short a time window. Beta calculated from 6 months of data is noisy and unreliable. Standard practice uses at least 2-3 years of data. For new IPOs with less history, consider using peer group data even for the public company. Mistake 8 — Ignoring structural changes. If a company underwent major restructuring (spinoff, major acquisition, divestiture) during the beta calculation window, the historical beta may not reflect forward-looking risk. In practice, analysts re-estimate beta after structural changes or use peer group data. Judgment call: when peers disagree significantly, look for systematic differences and either narrow the peer set or weight peers by comparability. Average is a starting point, not the final answer.