Beta & Systematic Risk

The part of risk you cannot diversify away

Finance FundamentalsCAPM & Cost of CapitalFree preview

⏱️ About 15 min

Diversification erased the risk unique to each stock. What is left moves with the whole market - and one number, beta, says how much.

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The big idea: Total risk splits into two parts: idiosyncratic risk, which diversification removes for free, and systematic (market) risk, which it cannot. Beta measures only the systematic part - how strongly an asset moves with the market - and equals the covariance of the asset with the market divided by the market variance.

🎯 By the end, you'll be able to

Separate total risk into systematic and idiosyncratic components.
Define beta as covariance with the market over market variance.
Compute beta from correlation and the ratio of volatilities.

📎 Helpful to know first

The module "Risk & Return" (volatility, correlation, diversification).

Two kinds of risk

The last module showed diversification melting away the risk unique to a single company - a lawsuit, a recall, a failed product. That is idiosyncratic risk, and a broad portfolio drives it toward zero at no cost. What survives is systematic risk: recessions, interest rates, the direction of the whole market. Because you cannot diversify it away, this is the risk investors must be paid to bear.

Beta measures the surviving risk

Beta answers one question: when the market moves 1%, how much does this asset move? A beta of 1 tracks the market; a beta of 2 amplifies it; a beta of 0.5 dampens it. Formally, beta is the covariance of the asset return with the market return, scaled by the market variance.

\[ \beta_{i} = \dfrac{\operatorname{Cov}(R_{i}, R_{m})}{\operatorname{Var}(R_{m})} = \rho_{im}\,\dfrac{\sigma_{i}}{\sigma_{m}} \]

Two ways to the same number

Because covariance equals $\rho_{im}\,\sigma_{i}\sigma_{m}$, dividing by $\sigma_{m}^{2}$ leaves $\rho_{im}\,\sigma_{i}/\sigma_{m}$. So beta rises with the correlation to the market and with how volatile the asset is relative to the market. A wildly volatile asset that is uncorrelated with the market still has a beta near zero - its swings are its own, not the market.

⚠️ Beta is the slope of a line

Plot the asset return against the market return for many periods and fit a straight line. Its slope is beta. The scatter of points around that line is the idiosyncratic risk; the tilt of the line is the systematic risk.

🎮 Beta & Systematic Risk Explorer LIVE

Predict first: Predict first: if you raise the asset volatility but leave its correlation with the market unchanged, does beta rise or stay the same?

Asset-versus-market returns with the best-fit line; the widget reports the covariance, the market variance, and beta (the slope).

📝 Worked example: A stock has a 30% volatility and a correlation of 0.8 with the market, whose volatility is 20%. Find its beta and its covariance with the market.

1. Covariance: $\rho\,\sigma_{i}\sigma_{m} = 0.8(0.30)(0.20) = 0.048$.
2. Market variance: $\sigma_{m}^{2} = 0.20^{2} = 0.040$.
3. Beta: $0.048 / 0.040 = 1.2$.
4. Check via the shortcut: $\rho\,\sigma_{i}/\sigma_{m} = 0.8(30)/20 = 1.2$. Same answer.

✓ Covariance = 0.048; beta = 1.2 (the stock amplifies market moves by 20%)

✏️ Practice: An asset has a 24% volatility and a 0.5 correlation with the market, whose volatility is 16%. What is its beta?

💡 Hint

Use $\beta = \rho\,\sigma_{i}/\sigma_{m} = 0.5(24)/16$.

Answer

Beta = 0.75. The asset moves only three-quarters as much as the market.

Check your understanding

1. The risk that diversification cannot remove is called:

Systematic (market) risk affects all assets together, so holding more names cannot cancel it - unlike idiosyncratic risk, which diversifies away.

2. A stock is very volatile but has zero correlation with the market. Its beta is:

Beta equals correlation times the volatility ratio; with zero correlation the beta is near zero no matter how volatile the asset is on its own.

✅ Key takeaways

Total risk = systematic (undiversifiable) + idiosyncratic (diversifiable).
Beta measures systematic risk: $\beta = \operatorname{Cov}(R_{i},R_{m})/\operatorname{Var}(R_{m}) = \rho_{im}\sigma_{i}/\sigma_{m}$.
Beta is the slope of the asset-versus-market return line.

➡️ Beta measures how much market risk an asset carries; the next lesson turns beta into the return investors should demand for bearing it - the security market line.

Want to test yourself on this? Try the Finance Fundamentals Aptitude test →