Portfolio Management
coreJensen's Alpha
Builds onCAPM (Security Market Line) — if this page feels steep, start there.
- alpha: the return earned beyond what the risk taken can explain
- what the portfolio actually returned
- the CAPM-required return — the fair reward for the beta the fund ran
- the portfolio's market sensitivity
- the market's excess return over cash in the period
Reading the notation
Why it must be true
A fund that returned 15% in a year when the market roared 20% did not necessarily do well — and one that returned 5% when the market fell might be brilliant. Raw return is mostly a story about how much market risk was worn.
Jensen's alpha strips that story out. It asks: given the beta this fund actually ran, what did CAPM say it should have earned — and what did it earn on top? The excess is the part no amount of passive market exposure could have produced: skill, or at least something the one-factor model can't explain. Alpha is the number active managers are paid to produce, and it is defined as a residual — what's left after the market's contribution is billed.
The derivation
CAPM prescribes the fair return for the risk taken:
Whatever the portfolio ACTUALLY earned beyond that benchmark is unexplained by market exposure:
Geometrically: plot the fund at against the Security Market Line — alpha is its vertical distance above (or below) the line.
When to reach for it
Judging a manager's risk-adjusted performance against the CAPM benchmark — actual return versus the SML-required return at the fund's beta.
Listen for
Back-of-the-envelope
Estimate it in your head first — then the calculator only confirms.
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Two steps, always in order: (1) the CAPM number (premium × beta + Rf), (2) actual minus it. Compute the benchmark FIRST or the signs tangle.
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β = 1 shortcut: alpha collapses to Rp − Rm. Use it to sanity-check magnitudes — a fund near β 1 that lagged the market has negative alpha, full stop.
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Real-world scale: sustained alphas live in low single digits. A computed alpha of +9% with ordinary inputs deserves a re-check before it deserves applause.
Traps in applying it
- ✗Comparing to the raw market return instead of the beta-adjusted CAPM return — a low-beta fund SHOULD lag a bull market.
- ✗Multiplying beta by the total market return inside the benchmark (the same premium error as CAPM itself).
- ✗Getting the sign backwards: alpha is actual MINUS required, so underperformance is negative.
Limits & criticisms
Alpha is only as good as the model behind it: measured against one-factor CAPM, a fund harvesting value, size or momentum premiums shows "alpha" that a multi-factor model reprices as beta in disguise. It is also hostage to the beta estimate and the sample period, and says nothing about whether the skill (if real) survives fees, capacity and luck — a lucky year and a skilled year produce the same number.
Where it came from
Michael Jensen built the measure for his 1968 PhD study of 115 mutual funds — the first serious test of whether professional managers beat the market after adjusting for risk. His finding (on average, they did not, especially after fees) lit the fuse for index investing and remains among the most cited results in finance. Today "alpha" has escaped the formula and become the industry's word for skill itself: hedge funds sell it, allocators hunt it, and performance systems compute Jensen's version nightly against the SML.
One identity, 1 questions
The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.
Skill as a residual
The performance-attribution face: bill the market for its contribution first; whatever's left is the manager's.
On the BA II Plus
Worked example: A fund running a beta of 0.9 posted 14% for the year, against a market return of 10% and T-bills at 2%. What alpha did the manager generate relative to the CAPM requirement?
- 1.0.1 [−] 0.02 [=] [×] 0.9 [=]beta-scaled premium
- 2.[+] 0.02 [=] [STO] 1the CAPM-required return
- 3.0.14 [−] [RCL] 1 [=]actual minus required = alpha (decimal)
→ 4.8%