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11 · Beta and factor exposure

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Statistics and Regression, lesson 11

Beta and factor exposure

Regression in finance: how much of a return the market explains.

Most stocks rise and fall with the market, and a regression measures by how much. That slope, beta, sizes a hedge and splits a stock's risk into the market's part and its own. You need the least-squares line from Linear regression and r from Correlation.

By the end, you can:

  1. Compute a stock's beta and alpha from its returns and the market's.
  2. Size a market hedge, and work out the risk it leaves.
  3. Read a factor regression, and see how adding a factor can erase an alpha.

Part 1. Beta is a slope

Compute a stock's beta and alpha from its returns and the market's.

In Linear regression you fitted GULL = 0.2 + 0.8 × market to five weeks. In finance that slope has a name, and a second way to compute it, from a covariance and a variance.

Example 1.1

GULL's beta from five weeks

Over five weeks the market returns +1%, −1%, +3%, 0% and +2%, and GULL returns 0%, −1%, +2%, +1% and +3%. Find GULL's beta and alpha, and the share of its variance the market explains.

  1. Take distances from the averagesmarket: 0, −2, 2, −1, 1; GULL: −1, −2, 1, 0, 2

    Both average 1%.

  2. Covariance(0 + 4 + 2 + 0 + 2) / 4 = 2

    Multiply each week's two distances, add them up, and divide by n − 1.

  3. Betaβ = 2 / 2.5 = 0.8

    Divide by the market's variance, (0 + 4 + 4 + 1 + 1) / 4 = 2.5.

  4. Alphaα = 1 − 0.8 × 1 = 0.2

    The line passes through both averages, so alpha is what's left of GULL's average.

  5. R²r = 2 / √(2.5 × 2.5) = 0.8, so R² = 0.64

    GULL's variance is also 2.5.

Answerβ = 0.8 and α = 0.2% a week. The market explains 64% of GULL's variance.

Covariance over variance is the least-squares slope again: the n − 1 cancels, leaving Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 8 / 10. Write the covariance as the correlation times both volatilities and beta becomes the correlation times a ratio of volatilities: 0.8 × 1.58 / 1.58.

−2−22244market, %GULL, %1%0.8%α = 0.2β = 0.8
Figure 1The five weeks and their least-squares line. For each 1% the market moves, the line moves GULL 0.8%: that's beta. Where the market is flat, the line sits at 0.2%: that's alpha.

Beta and alpha

Regress a stock's return on the market's: stock = α + β × market + e. Beta, β, is the slope: how far the stock moves on average for each 1% the market moves. Alpha, α, is the intercept: the return left once the market's part is taken out. e is the stock's own move. ρ, read "rho", is the correlation.

β = cov(stock, market) / var(market) = ρ × σ(stock) / σ(market)

Beta counts only the part of a move that comes with the market. A stock with a low correlation can have a small beta and still swing more than the market does, because most of its moves are its own. Part 2 measures that part.

Your turn

A stock's correlation with the market is 0.5. Its volatility is 40% a year and the market's is 16%. What's its beta?

β = ρ × σ(stock) / σ(market) = 0.5 × 40 / 16 = 1.25. 2.5 forgets the correlation, 0.2 turns the ratio of volatilities upside down, and 0.625 uses R² = 0.25 in place of ρ.

Part 2. Hedge out the market

Size a market hedge, and work out the risk it leaves.

Keep going with Pro

You've done the first part. Pro opens the rest of this lesson, and every other one.

Still to come
  1. Hedge out the market
  2. More than one factor
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Next lessonPrincipal componentsThe few hidden drivers behind many series that move together. Start lesson 12

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