In plain English
One asset's risk is its standard deviation. Two assets' combined risk depends on whether they move together. If they rise and fall in step, holding both changes nothing; if they move independently, or in opposite directions, the combination is calmer than either one alone. That is diversification, and it is a statistical fact rather than a strategy.
The advanced view
Portfolio variance is a quadratic form w′Σw in the weight vector and covariance matrix. As holdings grow, the average-variance term falls with 1/n while the average-covariance term does not, so portfolio variance converges to average covariance — the undiversifiable, systematic component. Beta is the slope of the regression of asset excess returns on market excess returns, equal to Cov(i,m)/Var(m), and it prices exactly that component.
Covariance measures whether two variables deviate from their means in the same direction. Its sign is informative and its size is not, because it carries the units of both variables. Divide by the two standard deviations and you get correlation, bounded between −1 and +1, which is comparable across any pair. Correlation of 1 means an identical straight line, 0 means no linear relationship, −1 means a perfect mirror.
The two-asset variance formula is worth being able to write from memory, because it makes the diversification benefit visible: the third term carries the correlation, so lower correlation directly lowers portfolio risk. With equal weights and equal volatilities, correlation 1 leaves risk unchanged, correlation 0 cuts it by about 29%, and correlation −1 can eliminate it entirely. Nothing about expected return changed — only the variance did.
Diversification has a floor. Adding assets removes the idiosyncratic part of risk but not the shared part, so portfolio variance approaches the average covariance between holdings and stops there. That residue is systematic risk, and because it cannot be diversified away it is the only risk the market pays you to bear — which is the entire logic of CAPM, where beta measures exposure to it and alpha is what is left over.
Beta is a regression slope, so everything you know about regression applies: it is estimated with error, it depends on the window and frequency you chose, and R² tells you how much of the asset's variance the market explains. A beta of 1.4 with an R² of 0.15 is a weak claim about a mostly idiosyncratic stock. And correlations are not stable — they rise in crises, exactly when diversification was supposed to help.
Co-movement formulas
Cov(x,y) = Σ(xᵢ − x̄)(yᵢ − ȳ) / (n − 1) ρ = Cov(x,y) / (σₓ σᵥ) −1 ≤ ρ ≤ 1 Two-asset variance: σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ Beta β = Cov(i, m) / Var(m) = ρ_im × (σᵢ / σ_m) Portfolio beta = Σ wᵢ βᵢ CAPM E(Rᵢ) = R_f + βᵢ (E(R_m) − R_f) Large-n limit: σ²_p → average covariance
Essential vocabulary
- Covariance
- Co-movement in raw units. Sign meaningful, magnitude scale-dependent.
- Correlation ρ
- Standardised covariance in [−1, 1]. Measures linear association only.
- Systematic risk
- The shared component that diversification cannot remove. Priced, and measured by beta.
- Idiosyncratic risk
- Asset-specific variation. Diversifiable, therefore unpriced in CAPM.
- Beta
- Sensitivity to market moves; the slope from regressing asset excess returns on market excess returns.
- R²
- Share of variance explained. For a single-stock market regression, the share of risk that is systematic.
- Correlation breakdown
- The empirical tendency of correlations toward 1 in stressed markets.
Common pitfalls
- ×Reading correlation as causation, or as evidence of any non-linear relationship — ρ only sees straight lines.
- ×Comparing covariances across pairs. Only correlations are comparable.
- ×Believing a historical correlation matrix holds in a crisis. It does not, and that is when it matters.
- ×Quoting beta without the estimation window, frequency and R². All three change the number.
- ×Thinking diversification can reach zero risk. It converges to average covariance, not to zero.
- ×Averaging betas of subsidiaries without weighting by value.
Worked example — two assets and a beta
Step 1 of 11
- 1Asset A: σ₁ = 20%, Asset B: σ₂ = 30%, weights 50/50, ρ =
Where this shows up
Beta from here becomes the cost of equity in WACC, which becomes the discount rate in a DCF. A sloppy correlation estimate at this step propagates all the way into a valuation you will have to defend.
Deeper
Deeper: correlation is not stability
Correlation is a linear, in-sample, average measure. It misses non-linear relationships, it is unstable across regimes, and it rises towards one in crises exactly when the diversification is needed. Estimated betas are noisy too, which is why practitioners shrink them towards one (Blume: 0.67β + 0.33).
Beta measures only co-movement with the market. A stock with 40% volatility and beta 0.8 is risky but not systematically risky; CAPM will not pay you for its idiosyncratic part, though a concentrated owner still feels it.
Must know cold
- ✓ρ = cov(x, y) ÷ (σx σy), always between −1 and 1.
- ✓β = ρ × σi ÷ σm.
- ✓R² of a single-factor regression = ρ².
- ✓Correlation ≠ causation, and correlation ≠ stability of the relationship.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
Stock σ 36%, market σ 18%, ρ 0.5. Compute β and the share of variance that is systematic.
Exercise 2
Two 50/50 portfolios: ρ = 0.9 in normal times, 0.99 in a crisis. Both assets σ 25%. Quantify the lost diversification.
Regress excess stock returns on excess market returns and beta is the slope, alpha the intercept, and the residual scatter the idiosyncratic risk that diversification removes. R² here is the share of the stock's variance that is market risk.