In plain English
Do not put everything in one place. If two investments do not move in lockstep, holding both makes the ride smoother without giving up much return.
The advanced view
Portfolio variance is a quadratic form in the covariance matrix, so risk is dominated by covariances rather than individual variances once you hold more than a handful of assets. That leaves systematic risk as the only compensated risk, giving CAPM's beta, and the tangency portfolio plus the risk-free asset as the efficient set (two-fund separation).
Markowitz showed that risk is a portfolio property, not an asset property. Combining assets with correlations below 1 reduces variance without giving up expected return — the only free lunch in finance. The efficient frontier is the set of portfolios with maximum return for each level of risk; adding a risk-free asset produces the capital market line and the tangency portfolio.
σp² = w1²σ1² + w2²σ2² + 2w1w2ρσ1σ2 Sharpe = (Rp − Rf) / σp CAPM: E(Ri) = Rf + βi × (Rm − Rf) βi = Cov(Ri, Rm) / Var(Rm)
CAPM says only systematic risk is compensated because idiosyncratic risk is diversifiable. Empirically it explains less than it should, which is why factor models dominate practice: Fama–French three-factor adds size (SMB) and value (HML), Carhart adds momentum, and the five-factor version adds profitability and investment. These are what practitioners use for attribution and risk decomposition.
Essential vocabulary
- Systematic risk
- Market-wide risk that cannot be diversified away. Measured by beta.
- Idiosyncratic risk
- Company-specific risk — a recall, a CEO departure. Diversifiable.
- Alpha (α)
- Return above what the factor model predicts. Most funds have negative alpha after fees.
- Information ratio
- Alpha divided by tracking error. Active skill per unit of active risk.
Intuition
Diversification is free risk reduction, and covariance is where it comes from. Adding an asset that moves differently lowers portfolio volatility even if the asset is individually risky. The market only pays you for the risk you cannot diversify away, which is exactly what beta measures.
Common pitfalls
- ×Averaging standard deviations instead of combining variances and covariance.
- ×Assuming correlations are stable — they rise in crises, exactly when diversification is needed.
- ×Confusing total volatility with systematic risk when setting a cost of equity.
Worked example — two-asset portfolio
Step 1 of 5
- 1σ_A = 20%, σ_B = 30%, ρ =
Why it works
Diversification works because variances add slower than expected returns. With n equally weighted assets of variance σ² and average correlation ρ, portfolio variance tends to ρσ² as n grows: the idiosyncratic part is averaged away, the common part is not. That residual is exactly what beta measures and what the market pays you for.
How it is used — two-asset risk in your head
Step 1 of 4
- 150/50 split, both σ =
Deeper
Deeper: from the efficient frontier to CAPM
Portfolio variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂. Because the cross term uses correlation, combining assets with ρ < 1 lowers risk without lowering expected return. That free lunch traces out the efficient frontier; add a risk-free asset and the best mix is the tangency portfolio, giving the capital market line.
CAPM follows if everyone holds that same tangency portfolio: only non-diversifiable risk is priced, so E(r) = rf + β(E(rm) − rf), with β = cov(i, m) ÷ var(m). Idiosyncratic risk is uncompensated because it can be diversified away for free. Extensions (Fama–French size, value, momentum, quality) exist because empirical returns are not fully explained by β.
Must know cold
- ✓σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
- ✓β = cov(i, m) ÷ var(m) = ρ × σi ÷ σm.
- ✓Sharpe = (rp − rf) ÷ σp; the tangency portfolio maximises it.
- ✓Only systematic risk is compensated; diversifiable risk is not.
Exercises
Try each one on paper before revealing the worked solution.
Exercise 1
Two assets, σ 20% and 30%, equal weights, ρ = 0.2. What is the portfolio volatility, and what if ρ = 1?
Exercise 2
A stock has σ 30%, market σ 18%, ρ 0.6. Compute beta and the CAPM return at rf 3%, ERP 5%.
Every dot is a feasible portfolio; the curve is the set with the highest return for each level of risk. Adding a risk-free asset gives the straight capital market line, whose tangency point is the market portfolio — the geometric argument that produces CAPM.