CAPM Beta (β)

Measuring market-related risk in investments

Why Beta Matters

Beta (β) is the cornerstone of the Capital Asset Pricing Model (CAPM). It quantifies how sensitive a security or portfolio is to broad-market movements. Beta is a key metric for understanding investment risk and plays a crucial role in portfolio construction and risk management.

Beta answers a simple yet essential question every investor asks:

"If the market rises (or falls) 1%, what do I expect my investment to do?"

β≈1\beta \approx 1

Moves in-line with the market

β>1\beta > 1

Amplifies market swings (more volatile)

β<1\beta < 1

Dampens market swings (less volatile)

β<0\beta < 0

Tends to move opposite the market

The CAPM Equation

Beta is derived from the fundamental CAPM regression equation:

  Ri−Rf  =  αi  +  βi (Rm−Rf)  +  εi  \boxed{\; R_i - R_f \;=\; \alpha_i \;+\; \beta_i\,(R_m - R_f) \;+\; \varepsilon_i \;}

Key Components

SymbolMeaning
RiR_iReturn of asset/portfolio i
RmR_mReturn of the market index
RfR_fRisk-free rate (e.g., T-bill yield or G-Sec yield)
αi\alpha_iJensen's Alpha – performance unexplained by the market
βi\beta_iCAPM Beta – slope coefficient measuring relative volatility
εi\varepsilon_iError term (idiosyncratic shocks)

We estimate βi\beta_i (and αi\alpha_i) via Ordinary Least Squares (OLS) regression on excess returns:

yt=(Ri,t−Rf,t)y_t = (R_{i,t}-R_{f,t}) and xt=(Rm,t−Rf,t)x_t = (R_{m,t}-R_{f,t})

OLS in Matrix Form

For the mathematically inclined, here's how beta is calculated using matrix notation:

Let

y  =  [y1y2⋮yn],X  =  [1x11x2⋮⋮1xn],θ  =  [αβ]\mathbf{y}\;=\;\begin{bmatrix}y_1\\y_2\\\vdots\\y_n\end{bmatrix},\quad \mathbf{X}\;=\;\begin{bmatrix} 1 & x_1\\ 1 & x_2\\ \vdots & \vdots\\ 1 & x_n \end{bmatrix}, \quad\boldsymbol{\theta}\;=\;\begin{bmatrix}\alpha\\\beta\end{bmatrix}

OLS solves the minimization problem:

min⁡θ  (y−Xθ) ⁣⊤(y−Xθ)\min_{\boldsymbol{\theta}}\;(\mathbf{y}-\mathbf{X}\boldsymbol{\theta})^{\!\top} (\mathbf{y}-\mathbf{X}\boldsymbol{\theta})

With the closed-form solution:

  θ^=(X ⁣⊤X)−1X ⁣⊤y  \boxed{\; \hat{\boldsymbol{\theta}} = (\mathbf{X}^{\!\top}\mathbf{X})^{-1}\mathbf{X}^{\!\top}\mathbf{y} \;}

• β^\hat{\beta} is the second element of θ^\hat{\boldsymbol{\theta}}.

• α^\hat{\alpha} is the first element (excess return unexplained by the market).

How Our Portfolio Optimizer Computes Beta

Inside our optimization backend, the following steps are performed to calculate beta:

  1. Align dates of portfolio and benchmark.

  2. Compute daily excess returns

    port_excess  = port_returns - rf_series
bench_excess = benchmark_returns - rf_series

  3. Run OLS regression (using statsmodels):

    X = sm.add_constant(bench_excess.values)   # adds intercept
model   = sm.OLS(port_excess.values, X)
result  = model.fit()
beta    = result.params[1]      # slope
alpha   = result.params[0] * 252  # annualise
p_value = result.pvalues[1]
r2      = result.rsquared

  4. Fallback to the covariance method if OLS cannot run (for tiny samples).

The system also annualizes and caps extreme betas, then stores:

  • portfolio_beta\text{portfolio\_beta} (β\beta)
  • portfolio_alpha\text{portfolio\_alpha} (α\alpha)
  • beta_pvalue\text{beta\_pvalue}, r_squared\text{r\_squared} – statistical significance & fit quality.

Beta as Relative Volatility

Because β=Cov⁡(Ri,Rm)Var⁡(Rm)\beta = \dfrac{\operatorname{Cov}(R_i,R_m)}{\operatorname{Var}(R_m)}:

  • Magnitude → how volatile the asset is relative to the market.

  • Sign → direction of co-movement (negative if moving opposite).

Practical Interpretation

BetaInterpretationTypical Suitability
β≈0\beta \approx 0Market-neutralMarket-neutral funds
0<β<10 < \beta < 1Less volatile than marketDefensive stocks, utilities
β≈1\beta \approx 1Market-likeBroad index funds
β>1\beta > 1More volatileGrowth / tech equities
β<0\beta < 0Inverse correlationHedging instruments

Example: A stock with a beta of 1.5 would be expected to rise by approximately 1.5% when the market rises by 1%, and fall by approximately 1.5% when the market falls by 1%. Such high-beta stocks typically offer higher potential returns during bull markets but also greater losses during bear markets.

Advantages vs Limitations

Advantages

  • Intuitive interpretation — Clear representation of how an asset co-moves with the market.

  • Simple calculation — Can be computed from readily available return data.

  • Theoretical foundation — Firmly grounded in modern portfolio theory and CAPM.

  • Risk assessment — Identifies systematic risk that cannot be diversified away.

  • Performance attribution — Helps distinguish between market-driven returns and alpha.

Limitations

  • Non-stationarity — Beta values drift over time and are not constant.

  • Benchmark sensitivity — Results highly dependent on the market index chosen.

  • Historical bias — Past relationships may not predict future behavior.

  • Simplified model — Ignores other factors that affect returns (size, value, momentum).

  • Assumes market efficiency — May not hold in markets with significant inefficiencies.

Good Practices

To address the limitations of beta, consider these best practices:

IssueMitigation
Non-stationarity – β drifts over timeCompute rolling betas (our API does this yearly)
Leverage effects & heteroskedasticityRobust regressions or GARCH betas
Choice of market proxyUse the most relevant benchmark (e.g., NIFTY 50 vs SENSEX)
Model simplicityMultifactor models (Fama-French, Carhart) capture size, value, momentum effects

Our portfolio optimizer addresses some of these concerns by calculating rolling betas to help you visualize how a portfolio's relationship with the market may change over different time periods. This feature is especially valuable for long-term investors who need to understand how their portfolio's risk characteristics evolve over time.

References

  • Sharpe, W. F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk." Journal of Finance, 19(3), 425–442.Access the paper

  • Lintner, J. (1965). "Security Prices, Risk, and Maximal Gains from Diversification." Journal of Finance, 20(4), 587–615.

  • Black, F. (1972). "Capital Market Equilibrium with Restricted Borrowing." Journal of Business, 45(3), 444–455.

  • Bodie, Z., Kane, A., & Marcus, A. Investments (12 ed.). McGraw-Hill, 2021 – Ch. 9 (CAPM & β estimation).

Related Topics

Capital Asset Pricing Model (CAPM)

The foundational theory behind beta, explaining the relationship between systematic risk and expected return.

Blume-Adjusted Beta

A modified beta calculation that adjusts for the tendency of betas to revert toward the market average over time.

Jensen's Alpha (α)

A risk-adjusted performance measure that represents the excess return of a portfolio over what CAPM predicts.

Rolling Beta

A time-series analysis of beta that shows how an asset's relationship with the market changes over different periods.