Jensen's Alpha

What Is Jensen’s Alpha?

Imagine you’re baking a cake using a recipe that tells you exactly how high your cake should rise if you follow the instructions. Now, suppose you try your best, and your cake ends up rising even higher than the recipe predicted. That extra height is like a bonus—it shows that you did something extra well. In the world of investing, Jensen’s alpha is that "bonus" for a portfolio or investment manager. It tells us whether an investment earned more or less than what was expected, given the amount of risk it took.

Jensen’s alpha is a risk-adjusted performance measure that compares the actual return on an investment to the expected return, given its level of risk. It’s named after Michael Jensen, a finance professor at Harvard Business School who introduced the concept in the 1960s.

Jensen’s Alpha Formula

Jensen’s alpha is the difference between the actual return on an investment and the expected return. The formula is:

$$ \text{Jensen’s Alpha} = \text{Actual Return} - \text{Expected Return} $$

If Jensen’s alpha is positive, it means the investment earned more than expected for the risk it took. If it’s negative, it means the investment underperformed.

Interpreting Jensen’s Alpha

Alpha measures the extra return you get, after factoring in risk. A positive alpha means you earned more than expected relative to the risk taken, indicating a good performance. A negative alpha means you earned less than expected, which suggests poor performance. If alpha is zero, it means your returns were exactly what you’d expect for the level of risk you took. Everything's based on how well the manager performed given the risk involved.

In plain terms, Jensen's alpha tells you how well an investment did compared to what was expected given its risk.

Positive Alpha: The investment earned more than expected (good performance).
Negative Alpha: The investment earned less than expected (poor performance).
Zero Alpha: The investment performed exactly as expected.

How Do We Know What’s “Expected”?

In finance, there’s a simple formula called the Capital Asset Pricing Model (CAPM). This formula gives us the expected return for an investment based on two things:

1. A Safe Return:

Think of it as the return from a super-safe investment (like a government bond).

2. Extra Return for Taking Risk:

If you invest in something riskier (like stocks), you should earn more than the safe return.

The formula is:

$$ \text{Expected Return} = \text{Risk-Free Rate} + \beta \times (\text{Market Return} - \text{Risk-Free Rate}) $$

Here’s what each part means:

Risk-Free Rate: The return you’d get from a safe investment.
Market Return: The average return from the stock market.
Beta (β): A number that shows how much risk the investment has compared to the market. A beta of 1 means it moves exactly with the market. More than 1 means it’s riskier, less than 1 means it’s less risky.

Understanding Beta: What It Means and How It’s Used

What Is Beta?

Beta (β): measures how much an investment’s return moves in relation to the overall market.

Interpretation:

Beta = 1: Moves exactly with the market.
Beta > 1: Riskier than the market.
Beta < 1: Less risky than the market.

Why Does Beta Matter?

Risk and Return: In the CAPM formula, the expected return of an investment increases with beta because investors demand more return for taking on more risk.

Adjusting Expectations:

For a high-beta investment, a higher return is “expected” because the risk is higher.
For a low-beta investment, a lower return is acceptable because it’s less risky.

Manager Performance: Jensen’s alpha uses beta to determine what return was “expected.” If a portfolio with a high beta earns a high return, it might not be impressive if that high return was just a result of taking on extra risk. Jensen’s alpha tells you if the manager earned more than what that extra risk would normally produce.

How Do We Calculate Beta?

Using Historical Data

Beta is typically calculated using past returns. Here’s how it’s done in a nutshell:

Collect Data:
You need historical returns for both the investment (or portfolio) and a market benchmark (like a stock market index).
Excess Returns:
Often, we use “excess returns,” which are the returns above a risk-free rate (like a government bond rate).

$$ \text{Excess Return} = \text{Actual Return} - \text{Risk-Free Rate} $$
Run a Regression Analysis:
1. You plot the Excess Returns of Investment on one axis and the market’s excess returns on the other.
2. You then fit a line (using a method called linear regression) to this data.
3. The slope of this line is beta. For example, if the slope is 1.2, then on average, for every 1% change in the market’s excess return, the investment’s excess return changes by 1.2%.
Interpretation:
- A beta of 1.2 means the investment is 20% more volatile than the market.
- A beta of 0.8 means it’s 20% less volatile.

Bringing It All Together with Jensen’s Alpha

CAPM Expected Return Formula

The CAPM helps us calculate the “expected return” for an investment given its beta:

$$ \text{Expected Return} = \text{Risk-Free Rate} + \beta \times (\text{Market Return} - \text{Risk-Free Rate}) $$

Calculating Jensen’s Alpha

$$ \alpha = R_p - \left[ R_f + \beta_p \times (R_m - R_f) \right]$$ where: $$ R_p \text{: is the actual portfolio return,}$$ $$ R_f \text{: is the risk-free rate,} $$ $$ R_m \text{: is the market return, and} $$ $$ \beta_p \text{: is the portfolio beta.}$$

Suppose we have:

$$ R_f = 0.03, $$ $$\quad R_m = 0.10, $$ $$\quad \beta_p = 1.2, $$ $$\quad R_p = 0.15$$

First, calculate the expected return using CAPM:

$$ \text{Expected Return} = 0.03 + 1.2 \times (0.10 - 0.03) = 0.03 + 1.2 \times 0.07 = 0.03 + 0.084 = 0.114 $$

Then, Jensen's alpha is:

$$ \alpha = 0.15 - 0.114 = 0.036 $$

which corresponds to a 3.6% excess return. In plain lenguage means that the investment earned 3.6 percentage points more than what was expected based on its risk.

You could also see the interpretation.

Conclusion

Jensen’s alpha is a powerful tool for evaluating investment performance. It tells us whether an investment manager earned more or less than expected, given the risk they took. Positive alpha means the manager did well, negative alpha means they underperformed, and zero alpha means they did as expected.

Jensen’s alpha is based on the Capital Asset Pricing Model (CAPM), which calculates the expected return for an investment based on its beta. Beta measures how much an investment’s return moves in relation to the overall market. A high beta means the investment is more volatile than the market, while a low beta means it’s less volatile.

Jensen’s alpha is a great way to see how well an investment did compared to what was expected given its risk. It’s a key tool for evaluating investment managers and understanding how they performed.

Calculate Your Portfolio's Jensen Alpha and Beta

Want to see how your portfolio is performing? Use our Jensen Calculator to find out your portfolio's Jensen Alpha and Beta. This tool helps you understand if your investments are earning more or less than expected, given the risk involved.

Simply input your portfolio details, and our calculator will provide you with the insights you need to make informed investment decisions.

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