Title: Jensen’s Measure (Jensen’s Alpha) — What it Is, How to Calculate It, and How to Use It
Key takeaway
– Jensen’s measure (commonly called Jensen’s alpha) is the intercept from a CAPM-based regression that quantifies a portfolio’s risk-adjusted excess return versus a benchmark. A positive alpha implies the portfolio earned returns above what CAPM predicts for its level of market risk (beta); a negative alpha implies underperformance.
What Jensen’s Measure Is
– Definition: Jensen’s alpha (α) estimates the abnormal return of an investment after adjusting for systematic market risk, using the Capital Asset Pricing Model (CAPM). It answers: did a portfolio manager generate returns beyond what would be expected for the portfolio’s beta?
– Conceptual model:
Rp = Rf + βp (Rm − Rf) + α + ε
where:
– Rp = portfolio (or asset) return
– Rm = market (benchmark) return
– Rf = risk-free rate
– βp = portfolio beta (sensitivity to market)
– α = Jensen’s alpha (the intercept — abnormal return)
– ε = residual (unexplained return)
Formula and calculation
1) CAPM form:
α = Rp − [Rf + βp × (Rm − Rf)]
2) Regression approach:
Regress portfolio excess returns (Rp − Rf) on market excess returns (Rm − Rf):
(Rp − Rf) = α + β(Rm − Rf) + ε
The regression intercept is Jensen’s alpha and the slope is beta.
Fast facts
– Developed by Michael Jensen (1968) to evaluate mutual fund performance.
– Alpha can be expressed in percentage points (e.g., +1.2% per year).
– Statistical significance matters: a small positive alpha may be noise unless its t‑statistic shows it differs from zero.
– It relies on the CAPM; if CAPM is a poor model for returns, alpha may be misleading.
Step-by-step practical procedure (hands-on)
1. Choose your benchmark (Rm)
– Use a market index that matches the portfolio’s investment universe (e.g., S&P 500 for large‑cap US equity funds).
2. Choose the return frequency and sample period
– Monthly returns are common; specify start/end dates. Ensure period matches investment horizon and data availability.
3. Select risk‑free rate (Rf)
– Use a rate matching the return frequency (e.g., 1‑month T-bill yield for monthly returns).
4. Collect returns
– Obtain time series of Rp, Rm, and Rf for the chosen frequency/period.
5. Compute excess returns
– Portfolio excess: Rp,t − Rf,t
– Market excess: Rm,t − Rf,t
6. Estimate beta and alpha
Option A — Simple CAPM formula (if you have average returns and a beta):
– compute average Rp and average Rm
– alpha = Avg(Rp) − [Rf + β × (Avg(Rm) − Rf)]
(Use only if you already have β.)
Option B — Time-series regression (recommended):
– Regress (Rp − Rf) on (Rm − Rf) — intercept = α, slope = β
– In Excel: use LINEST or Data Analysis → Regression; in R/Python use OLS
7. Test significance
– Check t-statistic / p-value of α. If p < typical threshold (0.05), alpha is statistically significant.
8. Interpret results
– Positive, significant α: manager added value beyond compensation for market risk.
– Insignificant α: performance is indistinguishable from CAPM prediction (could be luck).
– Negative, significant α: underperformance relative to CAPM expectation.
9. Adjust for fees/transaction costs and other frictions
– Use net-of-fees returns when evaluating manager skill.
Worked numeric example
– Inputs: Rp = 15%, Rm = 12%, β = 1.2, Rf = 3% (annual)
– Expected return per CAPM = 3% + 1.2 × (12% − 3%) = 3% + 10.8% = 13.8%
– Jensen’s alpha = 15% − 13.8% = +1.2%
Interpretation: The fund returned 1.2 percentage points more than expected for its market risk. If this result is statistically significant, it indicates positive risk‑adjusted performance.
Practical tips and considerations
– Benchmark choice matters: use an index representing the portfolio’s exposures. A poor match biases alpha.
– Matching frequency: use the same frequency for Rf and returns. For monthly returns, use a monthly risk-free rate.
– Time‑varying beta: beta can change over time; consider rolling regressions or conditional models for dynamic betas.
– Small-sample error: short samples may produce unreliable alpha estimates; longer samples increase power but may mix different market regimes.
– Fees and transaction costs: always evaluate alpha on net-of-fees returns if assessing a fund manager.
– Multi-factor models: CAPM is single-factor. Some apparent alpha may be explained by other factors (size, value, momentum). To reduce model misspecification, consider Fama‑French or Carhart models; the intercept from those regressions is an alpha relative to the chosen factor set.
Criticism and limitations
– Dependence on CAPM: Jensen’s alpha assumes CAPM correctly captures expected returns. If CAPM is wrong or incomplete, alpha can be biased.
– Efficient Market advocates: EMH proponents argue positive alpha is likely due to luck, not skill, and that persistent positive alpha is rare.
– Survivorship and backtest bias: using surviving funds or choosing favorable periods can overstate alpha.
– Non-normal returns, illiquidity, and leverage can distort alpha estimates if not accounted for.
How to strengthen an alpha analysis
– Check statistical significance (t‑statistic, p‑value).
– Use multiple benchmarks and multi-factor regressions to see whether alpha persists.
– Perform rolling/regime analyses to test consistency.
– Adjust returns for fees, taxes and transaction costs.
– Combine with other metrics: Sharpe ratio (risk‑adjusted absolute return), Treynor ratio (reward per unit of market risk), information ratio (excess return relative to tracking error).
Alpha vs. Jensen’s alpha vs. beta
– Jensen’s alpha = Jensen’s measure = Jensen’s alpha (same concept).
– Alpha (general): the excess return of an investment vs. a benchmark, often used broadly. Jensen’s alpha is a specific alpha defined relative to the CAPM.
– Beta: measures sensitivity (volatility) of an asset’s returns to market returns. Alpha measures performance after accounting for that sensitivity.
When to use Jensen’s alpha
– Evaluating active managers’ skill adjusted for market risk.
– Comparing funds with similar market exposures.
– Complementing other performance metrics (Sharpe, information ratio, factor alphas).
Alternative/related measures
– Treynor ratio: (Rp − Rf) / β — reward per unit of market risk.
– Sharpe ratio: (Rp − Rf) / σp — reward per unit of total risk (standard deviation).
– Information ratio: active return / tracking error — measures manager’s consistency in beating benchmark.
– Factor-model alphas: intercept from Fama‑French or multi-factor regressions.
Bottom line
Jensen’s measure (Jensen’s alpha) is a well‑known, intuitive way to quantify a portfolio’s risk‑adjusted performance under CAPM. It is useful for gauging whether returns exceed those justified by market risk. However, its reliability depends on model selection (CAPM vs multi‑factor), choice of benchmark, sample period, and statistical significance. Use Jensen’s alpha alongside other metrics, test robustness, and account for fees and practical frictions before concluding that a manager has demonstrated skill.
Sources and further reading
– Investopedia, “Jensen’s Measure” (Jiaqi Zhou): https://www.investopedia.com/terms/j/jensensmeasure.asp
– Jensen, M. C. (1968). “The Performance of Mutual Funds in the Period 1945–1964.” Journal of Finance, 23(2), 389–416.
– Fama, E. F. (1970). “Efficient Capital Markets: A Review of Theory and Empirical Work.” Journal of Finance.
– Quantified Strategies, “Jensen Ratio” (for related ratio definitions)
If you’d like, I can:
– Build an Excel template or Google Sheets steps for computing Jensen’s alpha from a return series.
– Provide Python or R code to run time‑series regressions and output rolling alphas and t‑stats.