Correlation sets direction, vols set size

Regression beta rescales correlation by relative volatility: a stock with correlation 0.6 to the market, stock vol 10 and market vol 2 has beta 0.6 * 10 / 2 = 3. With a zero-intercept linear return model, a 2 percent market return predicts a 6 percent stock return. More generally, the fitted return is alpha + beta times market return. In simple OLS with an intercept and nonzero variances, the share of variance the regression explains is R^2 = 0.6 * 0.6 = 0.36.

Words you need

Variance: The average squared distance from the mean: how spread out results are.
Standard deviation (volatility): The square root of variance, in the same units as the data. In markets it is called volatility.
Covariance: How two quantities move together: positive when they tend to rise together.
Correlation: Covariance scaled to lie between −1 and 1. +1 means they move in lockstep.
Beta: How much a stock moves for each 1% move of the market, from a regression.
R²: The share of a quantity's variance that a regression explains, from 0 to 1.

Why it works

The least-squares slope divides covariance by predictor variance, and covariance factors into correlation times both vols. In simple OLS with an intercept and nonzero variances, squaring the correlation gives the explained-variance share.

beta = Cov(X, Y) / Var(X) = rho sigma_Y / sigma_X, R^2 = rho^2 in simple OLS with intercept

Spot it: An interviewer gives correlation and vols and asks for the hedge slope.

Technique reference and sources

Correlation decides how tightly Y follows X; the vol ratio decides how large the moves are. Beta multiplies the two: direction times size. R^2 then reports how much of Y moves with X at all.

When this method fits

An interviewer gives correlation and vols and asks for the hedge slope.
You need the expected move of a stock from a market move and its beta.
Covariance and predictor variance are given directly as a ratio.
You must convert a correlation into an explained-variance share.

Start from correlation. Note rho = 0.6 between the stock and the market.
Scale by the vol ratio. sigma_Y / sigma_X = 10 / 2 = 5. Then beta = 0.6 * 5 = 3.
Apply beta to moves. With a zero-intercept linear return model, a 2 percent market return predicts 3 * 2 = 6 percent. More generally the fitted return is alpha + beta times market return.
Square for R^2. In simple OLS with an intercept and nonzero variances, explained share is 0.6 * 0.6 = 0.36, so 36 percent of variance is explained.

Beta from covariance: beta = Cov(X, Y) / Var(X). Divide directly when both are given.

Cov 12, Var(X) 4: 12 / 4 = 3
Cov 9, Var(X) 6: 9 / 6 = 1.5

Beta from correlation and vols: beta = rho * sigma_Y / sigma_X. Multiply correlation by the vol ratio.

0.6 with 10 over 2: 0.6 * 5 = 3
0.8 with 10 over 4: 0.8 * 2.5 = 2

R^2 is the square: In simple OLS with an intercept and nonzero variances, R^2 = rho^2, always between 0 and 1. It never exceeds the absolute correlation.

rho 0.8: 0.8 * 0.8 = 0.64
rho 0.6: 0.6 * 0.6 = 0.36

Common traps

Reporting R^2 = 0.8 when rho = 0.8. R^2 squares the correlation: 0.8 * 0.8 = 0.64. The unsquared value overstates the explained share.

Dividing by the wrong variance: beta = Cov / Var(Y). Beta divides by the predictor variance Var(X). Cov 9 over Var(X) 6 gives 1.5.

Dropping the vol ratio: beta = rho always. Beta is rho times sigma_Y / sigma_X. With vols 10 and 2, beta is 5 times the correlation: 3, not 0.6.

Another example: Explained variance from correlation

R^2 = rho * rho
0.8 * 0.8 = 0.64

Source notes

These explanations and questions are written for QuantArc. Links show the underlying mathematics and learning design.

Harvard Stat 110: Probability (Joe Blitzstein)
Free course site with videos and the open companion book Introduction to Probability; the topic spine for counting, Bayes, and linearity. Lesson examples and questions are original.

Grinstead and Snell, Introduction to Probability (free PDF)
Free textbook PDF confirmed live: counting methods, poker-style dealing models, and the random-walk chapter behind gambler's ruin and first-step equations. Lesson examples and questions are original.

IES: Organizing Instruction and Study to Improve Student Learning
Supports worked examples, retrieval, explanations and spaced study. Does not establish QuantArc's exact lesson length, mastery threshold or review intervals.