Beta, R-squared and what a hedge cannot touch
Regress a stock on the market and its risk splits in two. Beta is correlation rescaled by the ratio of the volatilities; R-squared is the correlation squared; the rest is idiosyncratic. Alpha is whatever is left after beta is paid.
- Compute beta from correlation and the two volatilities, and see why it is not correlation
- Split a stock's variance into systematic and idiosyncratic with R²
- Compute alpha on excess returns
- Recover the correlation a reported beta implies
The intuition
A stock with 30% volatility, a market with 15%, correlation 0.6. Beta is 0.6 × 30 ÷ 15 = 1.20: the stock moves 1.2% for each 1% the market moves, on average. It is not the correlation. A volatile stock can have a high beta with a loose link to the market, and a calm stock a low beta with a tight one. Beta is correlation times the ratio of the volatilities, and the ratio does half the work.
R-squared is the correlation squared, 36% here: the share of the stock's variance the market explains. The systematic part of its volatility is 0.6 × 30% = 18%; the idiosyncratic part is 30% × √(1 − 0.36) = 24%, and 18² + 24² = 30². That 24% is what a perfect market hedge leaves behind. Alpha is the return left after beta is paid its due: the stock's excess return minus beta times the market's.
β = ρ σ_s ÷ σ_m. R² = ρ². Systematic vol = β σ_m = ρ σ_s; idiosyncratic vol = σ_s √(1 − ρ²); σ_s² = systematic² + idiosyncratic². α = (r_s − r_f) − β (r_m − r_f). Implied ρ = β σ_m ÷ σ_s, and above one it means the inputs are inconsistent.
Why it works
- The conventions here: a single-factor regression of the stock's excess returns on the market's, annualized figures. Beta is printed to two decimals and the alpha and implied-correlation answers use the printed beta.
- Beta is a slope. cov(s, m) ÷ σ_m², and since cov = ρ σ_s σ_m, that is ρ σ_s ÷ σ_m. Correlation says how tightly two series move together; beta says how much one moves per unit of the other.
- R² splits the variance. The market explains ρ² of the stock's variance. The two volatilities do not add; the two variances do: β² σ_m² + idiosyncratic² = σ_s².
- Idiosyncratic risk is what the hedge cannot remove. Short β dollars of market per dollar of stock and the systematic part is gone; σ_s √(1 − ρ²) remains. With an R² of 15%, a hedge removes only 15% of the variance.
- Alpha is on excess returns. Subtract the risk-free rate from both, then subtract beta times the market's excess. A high-beta stock should beat a rising market on beta alone; raw outperformance and alpha can disagree.
- Beta drifts with correlation. If correlation falls with the volatilities unchanged, beta falls in proportion and idiosyncratic risk rises. A hedge sized on last year's beta then over-hedges and becomes a net short market bet.
| Beta: 0.60 × 30% ÷ 15% | 1.20 |
| R²: 0.60² | 36% |
| Systematic vol: 1.20 × 15% (= 0.60 × 30%) | 18.0% |
| Idiosyncratic vol: 30% × √(1 − 0.36) | 24.0%, and 18² + 24² = 30² |
| Alpha: (10% − 2%) − 1.20 × (8% − 2%) | 0.80% |
| Correlation a beta of 1.20 implies: 1.20 × 15% ÷ 30% | 0.60 |
If the correlation fell to 0.40 with the same vols, beta would be 0.80 and idiosyncratic vol 27.5%. A hedge still sized at 1.20 would be short 0.40 of market per unit of stock for no reason.
The formulas
Correlation times the ratio of the volatilities.
The market's share of variance, and the two pieces of volatility.
Variances add; volatilities do not.
What is left after beta is paid its due.
The correlation a reported beta implies; above one is a red flag.
Worked example
The ratio of the volatilities first, then times the correlation. The follow-up asks why beta and correlation differ.
See it move
Same stock and market. Change the correlation, the two volatilities and last year's returns.
- Raise the correlation. Beta, R² and the systematic volatility rise; the idiosyncratic volatility falls.
- Raise the stock's volatility. Beta and the idiosyncratic volatility rise; R² does not move, because it is the correlation squared.
- Raise the market's volatility. Beta falls; the split of the stock's own volatility does not change.
- Raise the stock's return. Alpha rises one for one. Raise the market's return instead and alpha falls by beta times the rise.
Run it backwards
A risk system reports a beta and two volatilities. What correlation does that imply?
Rearrange β = ρ σ_s ÷ σ_m: ρ = β σ_m ÷ σ_s.
The follow-up: a result above one says the beta and the volatilities were estimated over different windows. It is a quick sanity check on any risk report.
Traps
Say it in the interview
“A stock has a beta of 1.5 but an R-squared of 15%. What does that tell you?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- β = ρ σ_s ÷ σ_m: correlation times the vol ratio.
- R² = ρ² is the market's share of variance; idiosyncratic vol = σ_s √(1 − ρ²) is what a hedge leaves.
- α = stock excess − β × market excess.
- Implied ρ = β σ_m ÷ σ_s; above one, the inputs disagree.