Two kinds of risk
A single share carries two very different exposures. The first is the risk that something goes wrong with that specific company: a plant fails, a product flops, a key customer leaves. The second is the risk that the whole market moves: interest rates, recessions, commodity prices. The first is diversifiable. The second is not.
Diversifying across many holdings cancels the company-specific surprises against each other, because they are largely independent. What remains is the market-wide movement that every asset shares. That is why an investor is not paid for total risk, only for the risk that cannot be removed by holding a portfolio.
Key points
- Idiosyncratic or company-specific risk can be diversified away.
- Systematic or market risk cannot, so the market prices it.
- Total risk = systematic risk + idiosyncratic risk.
Diversification and correlation
σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2
Portfolio risk is not the average of the parts; correlation decides how much of it survives.
- w
- Weight of each asset
- σ
- Volatility of each asset
- ρ
- Correlation between the two return series
The portfolio volatility formula is the heart of the matter. If two assets each have a volatility of 20% and are held in equal weights, a naive average would say the portfolio risk is 20%. It is not, unless the two move in perfect lockstep (ρ = 1).
At a correlation of 0.3 the portfolio volatility falls to about 16.1%. At a correlation of 0 the fall is deeper, to about 14.1%. When correlation is high the assets fall together and diversification has almost nothing to work with; when correlation is low or negative the portfolio is calmer than any single holding.
Key points
- Correlation, not the number of holdings, decides how much risk diversification removes.
- Only perfect correlation leaves portfolio risk equal to the weighted average.
- Adding a negatively correlated asset can cut risk sharply.
Beta: the systematic part, measured
β = Cov(Ra, Rm) ÷ Var(Rm) = ρ × (σa ÷ σm)
How much a holding moves when the market moves by one unit.
- Cov(Ra, Rm)
- Covariance of the asset with the market
- Var(Rm)
- Variance of the market return
Beta packages systematic risk into one number. A beta of 1 means the holding moves with the market. A beta of 2 means it amplifies each market move by two, and a beta of 0.5 means it is half as sensitive. Because idiosyncratic risk is diversified away inside a portfolio, beta is the risk that actually drives the required return.
The two expressions above are the same statement. The covariance form is what you compute from returns; the correlation form shows that beta rises with correlation and with relative volatility. It is the bridge from a return series to the CAPM required return.
Key points
- Beta measures systematic risk, not total risk.
- CAPM turns beta into a required return: Rf + β × (Rm − Rf).
- A portfolio beta is the weighted average of its holdings' betas.
Separating skill from exposure
Sharpe = (E(R) − Rf) ÷ σ
Excess return per unit of total risk. Suits a whole portfolio.
Treynor = (E(R) − Rf) ÷ β
Excess return per unit of systematic risk. Suits one sleeve of a diversified book.
α = E(R) − [Rf + β × (Rm − Rf)]
The return left over once the beta has been paid for.
Two portfolios can earn the same return and deserve very different conclusions. Sharpe divides excess return by total volatility and answers whether the ride was worth it. Treynor divides by beta instead, which suits a holding that sits inside an already diversified book where idiosyncratic risk is irrelevant.
Jensen's alpha asks the sharpest question: after paying the return that CAPM says this beta deserves, is anything left over? Positive alpha suggests value added; negative alpha means the exposure did worse than a passive combination with the same beta would have done. One period is never proof of skill, but alpha is the right scoreboard.
Key points
- Sharpe uses total risk; Treynor uses systematic risk.
- Alpha is the excess return after adjusting for beta.
- All three are noisy over short windows; consistency matters more than one year.
A worked example
Take five periods of returns. The asset returned 2%, 6%, 4%, −2%, and 0%; the market returned 1%, 3%, 2%, −1%, and 0%; the risk-free rate is 1%.
The asset's average return is 2% and its sample volatility is 3.16%. The market's average is 1% and its sample volatility is 1.58%. The covariance of the two series is 5 and the market variance is 2.5, so beta is 5 ÷ 2.5 = 2.0. The correlation is 5 ÷ (3.16 × 1.58) = 1.0, so this asset moved perfectly with the market: it behaved like a levered index position.
Sharpe is (2% − 1%) ÷ 3.16% = 0.32. Jensen's alpha is 2% − [1% + 2.0 × (1% − 1%)], which is 2% − 1% = +1.0%. The whole excess return came from carrying twice the market's beta, not from a selection skill that beat the beta-adjusted benchmark.
Interactive experiment
Beta, Sharpe, and Jensen's alpha
One asset, one market. Measure how much systematic risk the asset carries (beta), how well it is paid for total risk (Sharpe), and whether it beat what CAPM says it should earn (alpha).
Sharpe = (E(R) - Rf) / SD . B = Cov(Ra, Rm) / Var(Rm)
Your inputs
Return available with no risk.
Result
- Beta
- 2.00×
- Sharpe ratio
- 0.32×
- Jensen's alpha
- 1.0%
- Correlation with the market
- 1.00×
- Asset mean return
- 2.0%
- Asset volatility
- 3.2%
- Market mean return
- 1.0%
- Treynor ratio
- 0.00×
Beta of 2.00 means the asset moves 2.00x with the market. Sharpe is 0.32 and the result shows positive alpha: it beat its CAPM benchmark.
03 / Apply
The study fund's scoreboard
A student investment fund earned 2% over a window when the market returned 1% and cash returned 1%. Its beta to the market was 2.0 and its volatility was 3.16%. Members disagree about whether the year was a success.
The assumptions
- The fund is a concentrated, high-beta book rather than a diversified index.
- Fund and market returns are measured over the same five periods.
- Risk-free cash yielded 1% over the window.
What to take away
01
A Sharpe ratio of about 0.32 says the excess return was modest for the volatility taken.
02
Beta of 2.0 means the return was amplified market exposure, not stock selection.
03
Alpha is the tie-breaker: after paying for beta, how much was actually added?
Your turn
0 of 3 answered
01Question
Two assets are held in equal weights. Each has a volatility of 20% and their correlation is 0.3. What is the portfolio volatility, in percent, to one decimal place?
02Question
An asset's covariance with the market is 0.045 and the market's variance is 0.02. What is its beta?
03Question
Which component of risk does the market reward?
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OpenStax · Principles of Finance (free textbook) ↗Educational sample content. Worked examples use hypothetical figures. Dates refer to the note's latest content update.