Jensen’s Alpha Calculator

What is Jensen’s alpha?

Jensen’s alpha (also called Jensen’s measure) is a risk‑adjusted performance metric based on the Capital Asset Pricing Model (CAPM). It tells you how much a portfolio out‑ or under‑performed its expected return given:

• its beta (systematic risk),
• the risk‑free rate, and
• the benchmark / market return.

In other words, Jensen’s alpha asks: “Given the risk this portfolio took (beta), how much did it beat or lag the return predicted by CAPM?”

Jensen’s alpha formula

• \( \alpha \) – Jensen’s alpha (portfolio’s risk‑adjusted excess return)
• \( R_p \) – portfolio return over the period
• \( R_f \) – risk‑free rate over the same period
• \( R_m \) – benchmark / market return over the same period
• \( \beta_p \) – portfolio beta relative to the benchmark

All returns must be measured over the same horizon (e.g., all annual, all monthly, or all daily).

Step‑by‑step calculation

1. Compute the portfolio’s excess return over the risk‑free rate:
   \( R_p - R_f \)
2. Compute the benchmark’s excess return:
   \( R_m - R_f \)
3. Multiply the benchmark excess return by the portfolio beta:
   \( \beta_p \times (R_m - R_f) \)
4. Add back the risk‑free rate to get the CAPM‑expected portfolio return:
   \( R_f + \beta_p \times (R_m - R_f) \)
5. Subtract the expected return from the actual portfolio return:
   \( \alpha = R_p - \big[ R_f + \beta_p \times (R_m - R_f) \big] \)

Worked example

Suppose you have the following annual data:

• Portfolio return \(R_p = 10\%\)
• Risk‑free rate \(R_f = 2\%\)
• Benchmark return \(R_m = 8\%\)
• Portfolio beta \(\beta_p = 1.2\)

1. Benchmark excess return:

2. Beta‑adjusted excess return:

3. Expected portfolio return (CAPM):

4. Jensen’s alpha:

The portfolio generated 0.8 percentage points more than its CAPM‑expected return. That is a positive Jensen’s alpha, indicating outperformance on a risk‑adjusted basis.

How to interpret Jensen’s alpha

• \(\alpha > 0\): The portfolio outperformed its CAPM‑expected return.
• \(\alpha = 0\): Performance is exactly in line with CAPM expectations.
• \(\alpha < 0\): The portfolio under‑performed relative to its risk level.

The magnitude matters: a +0.2% alpha over one month can be meaningful if it is persistent, while +0.2% over ten years is negligible. For professional analysis, alpha is often evaluated together with:

• its statistical significance (t‑statistic),
• the time horizon and number of observations, and
• other risk‑adjusted metrics like Sharpe ratio and Treynor ratio.

Common pitfalls and best practices

1. Use consistent periods

If you use monthly portfolio and benchmark returns, you must also use a monthly risk‑free rate and a beta estimated from monthly data. Mixing annual and monthly figures will distort alpha.

2. Choose an appropriate benchmark

Jensen’s alpha is only as good as the benchmark you choose. For a U.S. large‑cap equity fund, an index like the S&P 500 is usually appropriate. For a global bond fund, you need a global bond index, and so on.

3. Understand what beta captures

Beta measures systematic risk relative to the benchmark. Jensen’s alpha assumes that beta fully captures the relevant risk. In reality, style tilts (value, size, momentum) and other factors can also drive returns.

FAQ

Is Jensen’s alpha annualized?

Jensen’s alpha is expressed over whatever period you use for the inputs. If you use annual returns, you get an annual alpha. If you use monthly returns, you get a monthly alpha, which you can annualize (approximately) by multiplying by 12, though a more precise approach uses compounding.

Can Jensen’s alpha be used for individual stocks?

Yes. You can treat a single stock as a “portfolio” and compute its alpha relative to a market index. However, idiosyncratic risk is high for single stocks, so alpha estimates will be more volatile and less stable over time.

How is Jensen’s alpha related to CAPM?

CAPM predicts a linear relationship between expected return and beta. Jensen’s alpha measures the vertical distance between the actual portfolio return and the CAPM line. A positive alpha means the portfolio lies above the security market line; a negative alpha lies below it.