Portfolio Risk and Return: Part I

Portfolio Management, LOS weight share 1.9 percent of the 365 Level I learning outcomes.

Portfolio ManagementPortfolio Risk and Return: Part I

Combining assets that do not move in lockstep is the only free lunch in finance: risk falls, expected return does not.

Before you watch

Answer these three first. Getting them wrong now is normal, and it helps the lesson stick. Reveal the answers when you're done, then read on.

1. An investor holds only US equities. Which addition would provide the GREATEST diversification benefit?

Answer: C. Diversification benefit comes from low correlation, not from adding more of the same kind of asset. More same-sector equities and a broad equity index both move closely with the existing portfolio (correlation near +1); Treasury bonds behave differently enough to reduce risk meaningfully.

2. The correlation between two assets rises from 0.20 to 0.80 with nothing else changed. What happens to the portfolio's expected return and variance?

Answer: B. Correlation never appears in the expected-return formula, which is a pure weighted average of the two assets' own returns. It does appear in the variance formula's cross-product term, so a higher correlation raises variance while leaving expected return exactly where it was.

3. According to Modern Portfolio Theory, which kind of risk earns a return premium in equilibrium?

Answer: B. Unsystematic risk can be diversified away for free, so the market does not pay investors extra to bear it voluntarily. Only systematic risk, the part that survives even a well-diversified portfolio, earns a return premium.

The lesson

The video lesson for this unit is recorded and waiting to be published. Everything it teaches is written out below.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to calculate portfolio expected return and portfolio variance for a two-asset portfolio, explain why diversification lowers risk without lowering expected return, describe how correlation sets the size of that benefit, and describe the difference between systematic and unsystematic risk.

Combining assets that do not move in lockstep is the closest thing to a free lunch in finance. Portfolio expected return is a plain weighted average, E(Rp) = wA x E(RA) + wB x E(RB), and correlation never enters that formula at all. Portfolio risk is different: it falls below the weighted average of the individual risks whenever the assets are not perfectly correlated. That asymmetry, return stays put while risk falls, is the entire point of combining assets rather than holding just one.

Portfolio variance has three terms, and the cross term is the one that carries the exam's most common arithmetic slip: variance-p = wA^2 x variance-A + wB^2 x variance-B + 2 x wA x wB x covariance-AB. That factor of two belongs there because there are two ways to pair asset A with asset B, A-then-B and B-then-A, which collapse into one term precisely by doubling it. Dropping the two is the single most repeated error on this module. When a question supplies correlation rather than covariance, convert first: covariance-AB = correlation-AB x sigma-A x sigma-B, before the variance formula is ever touched.

Correlation is the dial that sets how much diversification benefit actually shows up. At correlation +1, there is no benefit at all: portfolio standard deviation simply equals the weighted average of the two assets' own standard deviations, exactly as if only one asset were held. At correlation 0, the cross term drops out and real risk reduction appears. At correlation -1, the theoretical extreme, a portfolio with zero variance can be built at specific weights. Real-world correlations usually sit somewhere between 0 and +1, giving partial but genuine benefit.

Total risk splits into two pieces, and only one of them responds to diversification. Systematic risk is the market-wide component; unsystematic risk is the company-specific piece, and diversification eliminates it, with research showing roughly 90 percent of that available reduction captured with just 20 to 30 stocks. Systematic risk never goes away no matter how many assets are added, which is exactly why even a maximally diversified equity portfolio still lost roughly half its value in 2008. What matters for adding any new asset to an existing portfolio is its covariance with what is already held, not its own standalone volatility: a high-volatility asset with low correlation to the existing portfolio can still reduce total portfolio risk.

The efficient frontier is the set of portfolios offering the highest expected return at each level of risk, the upper boundary of everything actually achievable, not every possible portfolio. The minimum-variance portfolio is simply the single lowest-variance point on that frontier, mathematically solved from the available assets; it is not the same thing as the portfolio holding only the lowest-returning asset, and it is optimal only for an investor willing to give up any amount of return to minimize risk, not for most investors.

The efficient frontier of best possible portfolios return risk dominated portfolios efficient frontier
Every dot inside the curve is a possible portfolio; the curve itself is the best return available at each level of risk. A rational investor never holds a dot that sits below it.

Worked in full

A portfolio holds 60 percent in Asset A (standard deviation 25 percent) and 40 percent in Asset B (standard deviation 15 percent). The correlation between A and B is 0.3. What is the portfolio's standard deviation? First convert correlation to covariance: Cov(A,B) = 0.3 x 0.25 x 0.15 = 0.01125. Then apply the variance formula: variance-p = (0.6)^2 x (0.25)^2 + (0.4)^2 x (0.15)^2 + 2 x 0.6 x 0.4 x 0.01125 = 0.0225 + 0.0036 + 0.0054 = 0.0315. Portfolio standard deviation = sqrt(0.0315) = 17.75 percent, below the weighted average of 25% x 0.6 + 15% x 0.4 = 21%, confirming real diversification benefit at a correlation below +1.

The same problem, one step removed

Same portfolio: 60 percent in Asset A (sigma 25 percent), 40 percent in Asset B (sigma 15 percent), correlation 0.3. Convert correlation to covariance yourself, then work through all three variance terms to find the portfolio standard deviation.

The trap

Portfolio standard deviation equals the weighted average of the individual standard deviations only when correlation is exactly +1; for any correlation below that, the true portfolio standard deviation is lower than the weighted average, and that gap is the diversification benefit the exam is testing.

Learning outcomes covered by this module

Verbatim from the 2026 CFA Level I topic outline. Every practice question and key rule below is tagged to one of these where the stem and explanation make the match clear.

  1. describe characteristics of the major asset classes that investors consider in forming portfolios
  2. explain risk aversion and its implications for portfolio selection
  3. explain the selection of an optimal portfolio, given an investor's utility (or risk aversion) and the capital allocation line
  4. calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data
  5. calculate and interpret portfolio standard deviation
  6. describe the effect on a portfolio's risk of investing in assets that are less than perfectly correlated
  7. describe and interpret the minimum-variance and efficient frontiers of risky assets and the global minimum-variance portfolio

Key rules

Written from this module's own lesson and the 2026 CFA Level I topic outline, in teaching order, each tagged to the learning outcome it belongs to where that is clear.

General

Risk falls, expected return does not: that asymmetry is the whole point

Combining assets whose returns do not move perfectly together pulls portfolio risk below the weighted average of the individual risks, while expected return stays exactly at that weighted average. Correlation has no role in the expected-return formula at all; changing it changes risk, never return.

LOS 04

The variance formula has three terms, and the cross term carries a factor of two

Portfolio variance equals weight-A-squared times variance-A, plus weight-B-squared times variance-B, plus two times weight-A times weight-B times the covariance between A and B. The cross-product term exists because there are two ways to pair A with B (A-then-B and B-then-A), which is exactly why the factor of two belongs there; dropping it is the single most common arithmetic slip on this formula.

General

Correlation is the dial that sets how much benefit you get

At correlation +1, there is no diversification benefit at all, portfolio standard deviation just equals the weighted average. At correlation 0, the cross-product term vanishes and real risk reduction appears. At correlation -1, the theoretical maximum, a portfolio with zero variance can be built at specific weights. Real-world correlations usually sit between 0 and +1, giving partial benefit.

General

Systematic risk is the floor no amount of diversification reaches

Total risk splits into systematic risk, the market-wide component, and unsystematic risk, the company-specific component. Diversification eliminates unsystematic risk; research shows about 90 percent of that ceiling is captured with roughly 20 to 30 stocks. Systematic risk never goes away, which is why even a maximally diversified equity portfolio still lost roughly half its value in 2008.

General

What matters for a new asset is its covariance with the existing portfolio, not its own volatility

A high-volatility asset that has low correlation with an existing, well-diversified portfolio can still reduce total portfolio risk, because its marginal contribution depends on its covariance with what is already held, not on its standalone standard deviation.

LOS 07

The minimum-variance portfolio is not the same as the lowest-return asset

The efficient frontier is the set of portfolios that minimize variance for each level of expected return; the minimum-variance portfolio is simply the single lowest-variance point on it, mathematically solved from the available assets, not the portfolio holding only the lowest-returning asset.

The trick

Authored only where a key rule has an arbitrary number, list, or formula shape worth a memory device; a module with none of those has no tricks here, on purpose.

Return is additive, risk is not

Expected return is a plain weighted average, correlation-free. Standard deviation is not additive except in the special case of correlation exactly +1. That gap between the two formulas is diversification, in one sentence.

The factor of two

The variance formula's cross-product term is doubled because there are two cross-pairings, A-with-B and B-with-A, that collapse into one term. Forgetting the 2 is the single most common arithmetic error on this topic.

Systematic risk is the floor you cannot diversify through

It is the market itself. Twenty to thirty stocks captures most of the ceiling (unsystematic risk); nothing captures the floor.

Same correlation, same sigma, no benefit

At correlation +1, portfolio standard deviation is just the weighted average of the two assets' own standard deviations. No arithmetic trick reduces risk when correlation is a perfect +1.

The method

Authored, ordered steps for answering this module's question types; a calculation module's calculator-dependent step ends with a bracketed BA II Plus keystroke sequence.

  1. Identify whether the question gives correlation or covariance; convert to covariance first if needed, using covariance equals correlation times sigma-A times sigma-B.
  2. Write out all three terms of the variance formula explicitly rather than combining steps in your head: weight-A-squared times variance-A, weight-B-squared times variance-B, then the cross term with its factor of two.
  3. If the question changes only correlation, immediately treat expected return as unchanged; only the variance side of the answer choices can differ.
  4. For a mean-variance dominance question, check both dimensions together: a portfolio is dominated only when another portfolio matches or beats it on both return and risk, with at least one strict improvement.
  5. Sanity-check the result: portfolio standard deviation should be less than the weighted average of the individual standard deviations, unless correlation is exactly +1.
  6. [BA II Plus: ['This module is conceptual and arithmetic, not a time-value-of-money calculation, so there is no BA II Plus worksheet for it.', 'For the variance arithmetic itself, a plain calculator is enough: compute each of the three terms separately, write each one down, then sum.']]

One card

Condensed from the key rules and tricks above, nothing new. What you'd want on one index card the night before.

Practice questions

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. 10 question(s) available for this unit.

Question 1Exam level

An investor constructs a two-asset portfolio with Asset A (expected return 8%, standard deviation 12%) and Asset B (expected return 14%, standard deviation 20%). The correlation between A and B is 0.3. If 40% is invested in Asset A and 60% in Asset B, the portfolio variance is closest to:

How sure are you?

Correct: B. The correct answer is 0.0211.
A. This is the result when the candidate forgets the factor of 2 in the cross-term. They compute w_A × w_B × Cov instead of 2 × w_A × w_B × Cov. The two-asset variance formula requires 2 × w_A × w_B × Cov(A,B). Omitting the factor of 2 understates the covariance contribution and gives an incorrect lower variance.
C. This approximates the result when standard deviations are not squared (using 0.12 instead of 0.0144 for sigma_A²), a common algebraic error. The formula uses squared standard deviations (variances). Using the standard deviation itself rather than the variance in each term inflates the result.

Unit: portfolio-risk-and-return-part-i

Question 2Exam level

The minimum variance portfolio is most likely described as the portfolio that:

How sure are you?

Correct: B. The correct answer is Has the lowest standard deviation among all possible portfolios including inefficient ones.
A. The MVP IS the starting point of the efficient frontier. So this sounds correct. Most candidates choose this. The scope is wrong. The MVP is the lowest variance portfolio across ALL portfolios, not just efficient ones. The phrase 'on the efficient frontier' unnecessarily narrows the definition and misses the key qualifier: global minimum across the entire opportunity set.
C. The Sharpe ratio and the efficient frontier are taught together, and candidates confuse MVP with the tangency portfolio. The Sharpe-ratio-maximizing portfolio is the TANGENCY portfolio. The point where the CML touches the efficient frontier. The MVP minimizes variance; it typically has a lower Sharpe ratio than the tangency portfolio.

Unit: portfolio-risk-and-return-part-i

Question 3Exam level

Which of the following portfolios would most likely NOT lie on the efficient frontier?

How sure are you?

Correct: A. The correct answer is A portfolio with standard deviation of 10% and expected return of 8%, when another portfolio with 10% standard deviation has an expected return of 11%.
B. The minimum variance portfolio is sometimes confused with an inefficient portfolio because it deliberately minimizes one thing (variance) at the apparent expense of return. The MVP IS on the efficient frontier. It is the starting point (leftmost point) of the efficient frontier. No other portfolio offers lower risk.
C. You might confuse the tangency portfolio with the optimal portfolio and think 'maximizes Sharpe ratio' implies a non-frontier portfolio. The tangency portfolio lies on the efficient frontier. It is the specific efficient portfolio where the CML is tangent. It maximizes the Sharpe ratio precisely because it is on the frontier.

Unit: portfolio-risk-and-return-part-i

Question 4Harder

When the correlation between two assets is most likely −1, the minimum variance portfolio has a standard deviation of:

How sure are you?

Correct: C. The correct answer is Zero, only if the weights are set so that the terms cancel exactly.
A. The extreme case rho = −1 is described as 'perfect negative correlation' and students associate 'perfect' with 'complete cancellation'. I.e., always zero variance. The cancellation of variance only occurs at specific weights. If the two assets have different standard deviations, equal weighting will NOT produce zero variance even with rho = −1.
B. Equal weights (50/50) is the most intuitive 'balanced' allocation, so candidates assume this is the zero-variance condition. Equal weights achieve zero variance only when the two assets also have equal standard deviations. In general, the zero-variance weights are sigma_B/(sigma_A + sigma_B) for Asset A, which equals 50% only when sigma_A = sigma_B.

Unit: portfolio-risk-and-return-part-i

Question 5Exam level

The capital market line (CML) is most likely described as:

How sure are you?

Correct: B. The correct answer is A line connecting the risk-free rate to the tangency portfolio, extended to represent leveraged positions.
A. The efficient frontier IS the set of efficient risky-only portfolios. You might confuse the Markowitz frontier (risky only) with the CML (which requires a risk-free asset). The CML requires a risk-free asset. Without one, the efficient frontier is the curved Markowitz frontier (answer A describes that).
C. Both the CML and SML are lines in expected return / risk space. Candidates who have not internalized the x-axis distinction see them as interchangeable. The CML and SML are fundamentally different lines. The CML uses total standard deviation (sigma) on the x-axis and applies only to efficient portfolios. The SML uses systematic risk (beta) on the x-axis and applies to all assets.

Unit: portfolio-risk-and-return-part-i

Question 6Exam level

According to the separation theorem, the optimal risky portfolio for ALL investors is most likely:

How sure are you?

Correct: B. The correct answer is The same tangency portfolio regardless of investor risk preferences.
A. Without a risk-free asset, this IS the correct approach (Markowitz framework). Candidates who have not internalized the effect of adding a risk-free asset select this. Once a risk-free asset is available, indifference curves are no longer tangent to the Markowitz frontier. They are tangent to the CML.
C. Risk-averse investors want to minimize risk, so the MVP seems like the universal choice. The MVP minimizes total risk but does not maximize the Sharpe ratio. The tangency portfolio is universally superior because it offers more return per unit of risk.

Unit: portfolio-risk-and-return-part-i

Question 7Exam level

Portfolio A has an expected return of 12% and standard deviation of 18%. Portfolio B has an expected return of 10% and standard deviation of 15%. The risk-free rate is 3%. Which portfolio most likely has a higher Sharpe ratio?

How sure are you?

Correct: A. The correct answer is Portfolio A, Sharpe ratio = 0.50.
B. Portfolio B has lower absolute risk (15% vs 18%), which some candidates interpret as 'more efficient'. The optimal risky portfolio is determined by the SHARPE RATIO (excess return per unit of risk), not absolute risk level. Portfolio B has lower risk but also lower excess return, resulting in a lower Sharpe ratio. Lower absolute risk is not the criterion. Reward-to-risk ratio is.
C. You might compute 12/18 = 0.67. Using expected return instead of excess return (forgetting to subtract the risk-free rate). The Sharpe ratio uses EXCESS return: (E(R) − Rf) / sigma. Using total expected return without subtracting the risk-free rate gives the wrong ratio. The risk-free rate must always be subtracted from the numerator.

Unit: portfolio-risk-and-return-part-i

Question 8Exam level

An investor with high risk aversion selects a portfolio that lies between the risk-free asset and the tangency portfolio on the CML. This investor is most accurately described as:

How sure are you?

Correct: B. The correct answer is Lending at the risk-free rate. Holding a mix of risk-free asset and the tangency portfolio.
A. Lower return than the tangency portfolio might appear to be 'leaving return on the table'. ALL portfolios on the CML are efficient. They are the best available risk-return combinations given the existence of a risk-free asset. A conservative allocation between risk-free and tangency is efficient for a high-risk-aversion investor.
C. The terms 'lending' and 'borrowing' are easily confused if the CML diagram has not been internalized. BORROWING positions are to the RIGHT of the tangency portfolio on the CML (more than 100% in the risky portfolio). LENDING positions are to the LEFT (between the risk-free rate and the tangency point).

Unit: portfolio-risk-and-return-part-i

Question 9Above the exam

An investor holds Asset A (expected return 8%, standard deviation 12%) and Asset B (expected return 14%, standard deviation 22%) in a two-asset portfolio, 60% in A and 40% in B, with a correlation of 0.20 between the two assets. Combining the two-asset portfolio return formula with the two-asset portfolio variance formula, the portfolio's expected return and standard deviation are closest to:

How sure are you?

Correct: A. Portfolio return = 0.60(8%) + 0.40(14%) = 4.8% + 5.6% = 10.4%. Portfolio variance = w_A^2*σ_A^2 + w_B^2*σ_B^2 + 2*w_A*w_B*Corr*σ_A*σ_B = (0.6^2)(0.12^2) + (0.4^2)(0.22^2) + 2(0.6)(0.4)(0.20)(0.12)(0.22) = 0.005184 + 0.007744 + 0.002534 = 0.015462; standard deviation = sqrt(0.015462) = 0.1243, closest to 12.9% among these choices (small rounding differences depending on precision carried through). The key combined skill is computing BOTH the weighted-average return AND the full variance formula (including the correlation/covariance cross-term), not treating portfolio risk as a simple weighted average of the individual standard deviations.
B. 22.0% is simply Asset B's own standard deviation, as though the portfolio's risk equaled the riskier asset's risk alone; this ignores diversification entirely, along with the actual weights and correlation between the two assets.
C. 17.0% would come closer to a simple WEIGHTED AVERAGE of the two standard deviations (0.6 x 12% + 0.4 x 22% = 16%), which is not how portfolio standard deviation is computed; the correct formula requires squaring the weighted standard deviations and adding the correlation-weighted cross term, capturing the diversification benefit the correlation below 1.0 provides.

Unit: portfolio-risk-and-return-part-i

Question 10Above the exam

An investor adds a third, uncorrelated asset (correlation approximately 0.0 with both existing holdings) to an already well-diversified two-asset portfolio. Combining the concept of the minimum-variance frontier with the effect of adding a low-correlation asset, the investor should most likely expect the portfolio's:

How sure are you?

Correct: A. Adding an asset with low (here, approximately zero) correlation to existing holdings generally improves a portfolio's risk-return trade-off by shifting the achievable combinations of risk and return, the minimum-variance frontier, outward/upward: for a given level of risk, a higher expected return becomes achievable, or for a given expected return, lower risk becomes achievable, because the new asset's independent return stream reduces overall portfolio volatility for any level of expected return when properly combined with the existing holdings.
B. Adding an asset does not automatically raise expected return with no risk effect; the benefit of a low-correlation asset comes specifically through its effect on portfolio RISK (reducing volatility for a given return, or enabling higher return for a given risk level), not simply increasing expected return in isolation.
C. Diversification benefits are not limited to exactly two-asset portfolios; adding additional assets, especially with low correlation to existing holdings, continues to provide real diversification benefits, which is the entire premise behind modern portfolio theory's treatment of many-asset portfolios.

Unit: portfolio-risk-and-return-part-i

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