Jensen's Alpha: Separating Skill From Market Exposure
The single most common mistake in performance analysis is treating a return above the index as evidence of skill. Most of the time it is evidence of leverage.
1. The Beta Mirage
Suppose the market returns 10%, cash returns 4%, and a fund returns 15%. It looks like the fund beat the market by five points. But if the fund runs a beta of 1.5, the Capital Asset Pricing Model says it should have returned 4% + 1.5 × (10% − 4%) = 13% purely from carrying more market risk. The genuine outperformance is 2%, not 5%.
Run the numbers again with a beta of 1.83 and the expected return becomes 15% exactly: the entire apparent edge disappears. The fund did not pick better assets, it simply held a more volatile version of the same market — and in a downturn that same beta works in reverse.
Rp = realised return of the asset or portfolio.
Rf = risk-free rate over the same period.
Rm = realised return of the benchmark.
βp = OLS slope of the asset's returns on the benchmark's.
Units: percentage points per year. Every term is a return, which is what makes the subtraction meaningful.
2. What Dynamic Time Warping Is For
DTW finds the lowest-cost alignment between two sequences by stretching the time axis, using the dynamic-programming recurrence below. It is genuinely useful for pattern matching: correlation demands that two series move on the same day, whereas DTW recognises the same shape arriving early or late.
γ(i, j) = cumulative minimal distance to grid point (i, j).
d(xi, yj) = local distance between z-normalised points.
The tool above computes this matrix in full, with a Sakoe-Chiba band limiting how far the alignment may drift.
Why DTW is not part of the alpha formula
A DTW distance is a dimensionless measure of shape dissimilarity. An alpha is a return, measured in percentage points per year. Subtracting one from the other is a unit error — the result is not a return, not a distance, and not interpretable as either. Any screen that mixes an oscillator reading, a warping distance and a beta into one number and calls the output “alpha” is producing a number with no defined meaning. The two quantities are kept in separate panels above deliberately.
3. The Limits of Alpha
Alpha is a description of the past, computed with hindsight. Its weaknesses are well documented: it depends entirely on the benchmark chosen, a single period carries enormous sampling noise, and CAPM's single-factor view is known to be incomplete — multi-factor models attribute much of what looks like alpha to size, value, momentum and quality exposures instead.
Treat a positive alpha as a question worth investigating rather than an answer. Over short samples it is more likely to reflect luck than skill.