Jensen's Alpha for Indian Portfolios

What Is Jensen's Alpha?

Jensen's Alpha measures the excess return an investment earns after accounting for the risk it takes relative to the market, as prescribed by the Capital Asset Pricing Model (CAPM).

Put differently, alpha tells you whether the manager (or strategy) has delivered skill-based performance beyond what β-driven market exposure explains.

Positive α → outperformance (value added)

Negative α → underperformance (value destroyed)

CAPM Refresher

Jensen's Alpha is derived from the Capital Asset Pricing Model (CAPM) regression equation:

\boxed{\;R_i - R_f = \alpha_i + \beta_i\,(R_m - R_f) + \varepsilon_i\;}

Solving that regression for $\alpha_i$ and $\beta_i$ :

\alpha_i = (R_i - R_f) - \beta_i\,(R_m - R_f)

This formula shows that alpha is the excess return of an asset or portfolio beyond what would be predicted by its market risk (beta).

Econometric Estimation (OLS Matrix Form)

Using $n$ observations of daily or monthly excess returns:

y_t = R_{i,t}-R_{f,t},\quad x_t = R_{m,t}-R_{f,t}

The regression model in matrix form is:

\underbrace{\mathbf{y}}_{n\times1} = \underbrace{\mathbf{X}}_{n\times2} \underbrace{\boldsymbol{\theta}}_{2\times1} + \boldsymbol{\varepsilon}, \qquad \mathbf{X} = \begin{bmatrix} 1 & x_1\\ 1 & x_2\\ \vdots & \vdots\\ 1 & x_n \end{bmatrix}, \; \boldsymbol{\theta} = \begin{bmatrix}\alpha\\\beta\end{bmatrix}

The Ordinary Least Squares (OLS) solution is:

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

•

\hat{\alpha}

— Jensen's Alpha (annualized in practice)

•

\hat{\beta}

— market beta

• t-stat / p-value on

\alpha

— significance of skill

Our Backend Implementation

Our portfolio optimization backend calculates Jensen's Alpha using the following approach:

X       = sm.add_constant(bench_excess.values)   # [1, x_t]
model   = sm.OLS(port_excess.values, X)
result  = model.fit()

daily_alpha   = result.params[0]
beta          = result.params[1]
portfolio_alpha = daily_alpha * 252     # annualise
beta_pvalue     = result.pvalues[1]
r_squared       = result.rsquared

Excess returns are calculated against a daily risk-free series
Alpha is annualized by multiplying the daily intercept by 252 trading days
Stored as portfolio_alpha in the API response, shown alongside β, p-value, and $R^{2}$

Interpreting Jensen's Alpha

α Value	Meaning	Practical Take-away
$\alpha > 0$	Strategy beat CAPM expectations	Indicates skill or unique edge
$\alpha = 0$	Matches risk-adjusted benchmark	Pure β exposure—no evidence of skill
$\alpha < 0$	Under-performed given its β	Destroyed value versus passive market

Always pair α with p-value or t-stat to confirm statistical significance. A high alpha that is not statistically significant might be due to chance rather than skill.

Typical Use-Cases

Use-Case	How Alpha Helps
Fund selection	Identify managers delivering true skill
Performance fees	Many hedge-fund agreements pay incentive fees only on positive α
Style drift monitoring	Persistent α turning negative suggests strategy degradation

Limitations & Best Practice

Issue	Mitigation
Model mis-specification	Use multi-factor models (Fama-French, Carhart) to separate size, value, momentum premiums
Beta Instability	Compute rolling α/β to detect regime changes
Short sample noise	Prefer ≥ 3 years monthly data or ≥ 250 daily observations for robust inference

Advantages and Limitations

Advantages

Benchmark-adjusted assessment: Evaluates performance specifically relative to a relevant benchmark, not just in absolute terms.
Risk-adjustment: Accounts for the level of systematic risk taken, enabling fair comparison between strategies with different risk levels.
Statistical validation: Can be tested for statistical significance to determine if outperformance is likely skill-based or simply due to chance.
Skill identification: Provides a clear distinction between returns generated through manager skill versus those from general market exposure.
Academically robust: Based on established financial theory and supported by decades of empirical research in portfolio performance evaluation.

Limitations

CAPM dependency: Inherits all limitations of the CAPM model, including assumptions about market efficiency and investor rationality.
Benchmark sensitivity: Results can vary dramatically based on which benchmark is chosen as the market proxy.
Single-factor limitation: Ignores other systematic risk factors that might explain returns beyond market risk alone.
Time period dependency: Alpha values can be highly sensitive to the specific time period used in the analysis.
Data requirements: Needs sufficient historical data to produce statistically meaningful results, potentially limiting usefulness for new strategies.

References

Jensen, M. C. (1968). "The Performance of Mutual Funds in the Period 1945–1964." Journal of Finance, 23(2), 389-416.
Roll, R. (1978). "Ambiguity when Performance Is Measured by the Securities Market Line." Journal of Finance, 33(4), 1051-1069.
Bodie, Kane & Marcus. Investments (12 ed.), McGraw-Hill, 2021 — Ch. 24 (Performance Evaluation).

Jensen's Alpha (α)

Measuring risk-adjusted outperformance beyond market exposure