Key facts
- Portfolio expected return is a weighted average; portfolio variance also includes covariances.
- Diversification can reduce firm-specific risk, but market exposure remains.
- Beta equals covariance with the market divided by market variance.
- The security market line plots CAPM-required return against beta, not total volatility.
- Sharpe measures excess return per unit of total variability; Treynor measures it per unit of beta.
Portfolio return is a weighted average; portfolio risk also depends on how holdings move together. Diversification can reduce asset-specific risk. The capital asset pricing model (CAPM) links required return to market exposure, measured by beta. Sharpe and Treynor ratios compare excess return with different risk measures. (Naranjo, portfolio risk (secondary source)) (Naranjo, CAPM (secondary source)) (Sharpe, The Sharpe Ratio (1994))
This guide supports study for the CFA® Program Level I, run by CFA Institute. (CFA Institute, Level I exam)Our questions complement CFA Institute's own curriculum; they do not replace it. See the course page, the time value of money guide and the bond duration guide for related foundations.
How do covariance and correlation describe risk?
Covariance measures whether returns tend to deviate from their means together. Positive covariance indicates co-movement; negative covariance indicates opposing deviations. Correlation standardises covariance by dividing it by both standard deviations. It measures linear association on a scale from −1 to +1, provided both standard deviations are positive. (NYU Stern, covariance and correlation (secondary source)) (Naranjo, portfolio risk (secondary source))
Correlation has no units. Covariance depends on the return units: use decimals consistently in the calculations below. Zero correlation removes the covariance term from portfolio variance; it does not remove the individual assets' variance terms. (Naranjo, portfolio risk (secondary source))
A worked portfolio return and risk example
Our original example invests 65% in asset A and 35% in asset B, with no short positions. Assume annual expected returns of 7% and 11.5%, annual standard deviations of 14% and 23%, and correlation of 0.25. These are illustrative inputs, not market estimates; ignore fees and taxes.
For fixed beginning-of-period weights summing to one, expected return is the weighted average of expected asset returns. Two-asset variance includes each weighted variance and a covariance term: (Naranjo, portfolio risk (secondary source))
Expected return is 0.65 × 7% + 0.35 × 11.5% = 8.575%. Covariance is 0.25 × 0.14 × 0.23 = 0.00805. Substituting those inputs gives:
The simple weighted average of volatilities is 17.15%, which overstates this portfolio's volatility. The calculated 13.5735% is even below A's 14%. Mixing in a more volatile asset has reduced risk because its returns are imperfectly correlated with A's.
What can diversification remove?
Hold the weights and individual volatilities fixed. Our formula gives 17.15% volatility at correlation +1, 12.1496% at zero, and 1.05% at −1. Perfect negative correlation would eliminate volatility only if the two weighted volatilities were equal; our 65/35 mix does not achieve that. The formula also shows that, with positive weights and positive asset volatilities, correlation below +1 reduces portfolio volatility relative to the weighted average of individual volatilities. (Naranjo, portfolio risk (secondary source))
Broad diversification can reduce firm-specific risk, but market-wide exposure remains. CAPM rewards that systematic exposure rather than the risk that investors can diversify away. (Naranjo, CAPM (secondary source))
What does the efficient frontier mean?
Plot expected return vertically and standard deviation horizontally. The efficient frontier of risky assets contains portfolios offering the highest expected return for a given risk. It is the upper branch of the minimum-variance frontier, starting at the global minimum-variance portfolio. A lower-return portfolio with the same risk is inefficient. (Naranjo, portfolio risk (secondary source)) (Naranjo, efficient portfolios (secondary source))
For our two assets, minimising the variance formula gives about 79.52% in A and 20.48% in B, with expected return 7.9215% and volatility 13.1281%. Our 65/35 mix accepts more volatility for a higher expected return. Neither allocation is universally best: the choice depends on the investor's willingness to bear risk.
With borrowing and lending at a common risk-free rate, a line from that rate tangent to the risky frontier identifies the portfolio with the highest Sharpe ratio. Under CAPM assumptions, that portfolio is the market; its combinations with the risk-free asset form the capital market line (CML) when the market weight is non-negative. (Naranjo, efficient portfolios (secondary source)) (Naranjo, CAPM (secondary source))
How is beta calculated?
Beta measures an asset's exposure to market returns, not its total volatility. Divide covariance with the market by market variance, or multiply correlation with the market by the ratio of volatilities: (Naranjo, CAPM (secondary source))
In a separate example, suppose annual covariance is 0.018 and market volatility is 12%. Market variance is 0.12² = 0.0144, so beta is 0.018 ÷ 0.0144 = 1.25. Dividing by 0.12 would incorrectly use standard deviation.
The market's beta is one. A beta above one indicates greater market sensitivity; beta need not lie between −1 and +1. Portfolio beta is the weighted average of constituent betas against the same market. This follows from covariance's linearity: 70% at beta 1.25 plus 30% at beta 0.55 gives beta 1.04. (Sharpe, equilibrium and the SML (secondary source)) (Naranjo, portfolio risk (secondary source))
How do CAPM and the security market line work?
CAPM adds a beta-scaled market risk premium to the risk-free rate. The premium is expected market return minus the risk-free rate. (Sharpe, equilibrium and the SML (secondary source))
Assume a 3.2% risk-free rate and 8.8% expected market return. The premium is 5.6 percentage points. For beta 1.25, the required annual return is 3.2% + 1.25 × 5.6% = 10.2%.
The security market line (SML) plots this relationship with beta horizontally and expected return vertically. Its intercept is the risk-free rate and its slope is the market risk premium. It applies to individual securities and portfolios. Unlike the SML, the CML uses standard deviation and describes efficient combinations of the market and risk-free asset. (Sharpe, equilibrium and the SML (secondary source)) (Naranjo, efficient portfolios (secondary source))
A forecast return of 11.2% for our beta-1.25 asset sits one percentage point above its 10.2% requirement. CAPM assumes mean-variance investors with common expectations and frictionless access to assets and risk-free borrowing and lending. Its required return is not a promised outcome. (Naranjo, CAPM (secondary source))
When do Sharpe and Treynor ratios disagree?
Sharpe divides average excess return by its standard deviation. With a constant risk-free rate, that denominator equals portfolio return volatility. Treynor instead divides average excess return by market beta: (Sharpe, The Sharpe Ratio (1994))
Consider separate illustrative historical portfolios with the same annual measurement basis, constant risk-free rate of 3.2%, and betas against the same market. Returns are arithmetic means.
| Portfolio | Mean return | Volatility | Beta | Sharpe | Treynor |
|---|---|---|---|---|---|
| P | 10.7% | 15% | 1.25 | 0.500 | 0.060 |
| Q | 9.2% | 16% | 0.75 | 0.375 | 0.080 |
P's excess return is 7.5 percentage points: 0.075 ÷ 0.15 = 0.500, while 0.075 ÷ 1.25 = 0.060. Q's excess return is 6 percentage points, giving 0.375 and 0.080. Sharpe ranks P higher; Treynor ranks Q higher. Treynor's 0.080 means eight percentage points of excess return per unit of beta, not a dimensionless Sharpe score.
Treynor uses market beta rather than total return volatility. With our constant risk-free rate, Sharpe assesses total return variability. Neither ratio alone captures how a new holding interacts with an existing portfolio. Compare consistent periods and benchmarks; historical ratios do not establish future performance. (Sharpe, The Sharpe Ratio (1994)) (Naranjo, CAPM (secondary source))
Finally, inspect denominators: zero excess-return volatility makes Sharpe undefined; zero beta makes Treynor undefined. Negative beta also defeats the usual simple “higher is better” Treynor ranking. These limits follow directly from the ratios; our comparison uses positive denominators and positive excess returns.
Frequently asked questions
Does diversification require negative correlation?
No. With positive weights and positive asset volatilities, correlation below +1 reduces portfolio volatility relative to the weighted average of individual volatilities.
What is an efficient portfolio?
It offers the highest expected return for its risk. On the risky-asset frontier, efficient portfolios lie on the upper branch from the global minimum-variance portfolio.
Is beta the same as volatility?
No. Volatility measures total return variability. Beta measures market exposure and equals covariance with the market divided by market variance.
How does the SML differ from the CML?
The SML uses beta and applies to individual securities and portfolios under CAPM. The CML uses standard deviation and describes efficient combinations of the market and risk-free asset.
Why can Sharpe and Treynor rank portfolios differently?
They use different denominators: total variability for Sharpe and beta for Treynor. A portfolio can earn more excess return per unit of one risk measure but less per unit of the other.
Related tools and pages
Sources
- Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Portfolios of Risky Assets — expected return, covariance, portfolio variance and the minimum-variance frontier. Accessed 29 October 2026.
- Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on Statistics — covariance and correlation. Accessed 29 October 2026.
- Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Beta Pricing — capital allocation lines, the tangency portfolio and mean-variance efficiency. Accessed 29 October 2026.
- Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): The Capital Asset Pricing Model — assumptions, beta, systematic risk and model limitations. Accessed 29 October 2026.
- William F. Sharpe, Stanford University (academic notes, secondary source): Macro-Investment Analysis: Equilibrium (preliminary) — Expected Return and Beta Values; The Market Portfolio and the Security Market Line. Accessed 29 October 2026.
- William F. Sharpe, The Journal of Portfolio Management (author's Stanford reprint): The Sharpe Ratio — The Ratio, Time Dependence, Correlations and Related Measures (1994). Accessed 29 October 2026.
- CFA Institute: CFA Program Level I exam: format and topic weights. Accessed 26 October 2026.
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