Portfolio Risk and Return Part I

First Principles Thinking: portfolio standard deviation components and diversification

B is correct. Statement I is correct: the CFA Curriculum derives the two-asset portfolio variance as $\sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1 w_2 \rho_{12} \sigma_1 \sigma_2$, where the third term captures co-movement. Statement II is correct: the CFA Curriculum shows that because $\sigma_f = 0$ for the risk-free asset, the first and third terms vanish, leaving $\sigma_p = (1 - w_1)\sigma_i$. Statement III is incorrect: the CFA Curriculum demonstrates that as N becomes large, the individual variance term becomes negligible and covariance (which depends on correlation) becomes the dominant factor in portfolio risk, not individual asset variance. Therefore, two statements (I and II) are correct.

Option A is incorrect because both the two-asset variance formula and the risk-free asset simplification are explicitly derived in the CFA Curriculum.

Option C is incorrect because Statement III reverses the key insight of diversification — as N grows, it is the average covariance (correlation), not individual variance, that determines portfolio risk.

First Principles Thinking: covariance from correlation and standard deviations

B is correct. $Cov(R_A, R_B) = \rho_{AB} \times \sigma_A \times \sigma_B = 0.60 \times 0.15 \times 0.20 = 0.0180$. The covariance is the product of the correlation coefficient and both standard deviations.

A is incorrect. This results from mistakenly using variances instead of standard deviations: USD 0.60 \times 0.15^2 \times 0.20^2 = 0.60 \times 0.0225 \times 0.04 = 0.00054$, or from a decimal placement error. The formula requires standard deviations, not variances.

C is incorrect. This results from omitting the correlation coefficient entirely: USD 0.15 \times 0.20 = 0.03$. The correlation is a required component of the covariance formula and cannot be ignored.

First Principles Thinking: historical risk-return characteristics of US asset classes

B is correct. According to the CFA Curriculum, small company stocks had an overall annual return of 12.1% versus 10.2% for large company stocks, with risk of 31.7% versus 19.8%, making Statement I correct. Statement II is incorrect because the CFA Curriculum explicitly notes that long-term government bonds had higher total risk (9.9%) than long-term corporate bonds (8.3%) during 1926–2017, which was uncharacteristic. Statement III is correct because Treasury bills did not earn a negative return in any decade shown in the data. Therefore, two statements (I and III) are correct.

Option A is incorrect because it undercounts: both Statements I and III are supported by the historical data in the CFA Curriculum.

Option C is incorrect because Statement II reverses the actual relationship — government bonds had higher, not lower, total risk than corporate bonds over this period.

First Principles Thinking: weighted average portfolio return

A is correct. $R_p = w_A \times R_A + w_B \times R_B = 0.60 \times 8\% + 0.40 \times 14\% = 4.80\% + 5.60\% = 10.40\%$. Portfolio return is always a weighted average of individual asset returns, regardless of correlation.

B is incorrect. This is the simple average of the two returns: $(8\% + 14\%) / 2 = 11\%$. The simple average ignores the different portfolio weights and assumes equal allocation.

C is incorrect. This results from reversing the weights: USD 0.40 \times 8\% + 0.60 \times 14\% = 3.20\% + 8.40\% = 11.60\%$. Applying the wrong weight to the wrong asset is a common error in portfolio return calculations.

First Principles Thinking: diversification benefit on the minimum-variance frontier

B is correct. $\sigma_P^2 = 0.50^2 \times 0.15^2 + 0.50^2 \times 0.15^2 + 2 \times 0.50 \times 0.50 \times 0.40 \times 0.15 \times 0.15$ $= 0.005625 + 0.005625 + 0.0045 = 0.01575$. $\sigma_P = \sqrt{0.01575} = 0.1255 = 12.55\%$. The portfolio risk is below 15% because the correlation is less than +1, demonstrating the diversification benefit.

A is incorrect. This is the portfolio standard deviation if the correlation were 0: $\sqrt{0.005625 + 0.005625} = \sqrt{0.01125} = 10.61\%$. Using zero correlation overstates the diversification benefit when the actual correlation is positive.

C is incorrect. This is the weighted average of standard deviations (or the result when $\rho = +1$): USD 0.50 \times 15\% + 0.50 \times 15\% = 15\%$. This assumes no diversification benefit, which only occurs at perfect positive correlation.

First Principles Thinking: capital allocation line and two-fund separation

B is correct. Statement I is correct: the CFA Curriculum derives the CAL equation and states its slope is $\frac{E(R_i) - R_f}{\sigma_i}$, which represents the additional required return for every increment in risk, sometimes called the market price of risk. Statement II is correct: the CFA Curriculum states the two-fund separation theorem holds that all investors regardless of taste, risk preferences, and initial wealth will hold a combination of the risk-free asset and the optimal portfolio of risky assets. Statement III is incorrect: the CFA Curriculum shows (via the example of Kelly with A=2 and Jane with A=4) that a more risk-averse investor's optimal portfolio lies to the left (lower risk, lower return) on the CAL, not to the right. Therefore, two statements (I and II) are correct.

Option A is incorrect because both the CAL slope definition and the two-fund separation theorem are explicitly stated in the CFA Curriculum.

Option C is incorrect because Statement III reverses the relationship — higher risk aversion leads to an optimal portfolio with less risk (further left on the CAL), not more risk.

First Principles Thinking: Efficient versus Inefficient Portions of the Minimum-Variance Frontier

B is correct. Wait — actually R IS the correct explanation. Let me reconsider. The CFA Curriculum uses the example of Points A and C on the minimum-variance frontier having the same risk, where A (above the global minimum-variance portfolio) has higher return than C (below it). This is exactly why portfolios below are inefficient. Both A and R are true. R directly explains WHY portfolios below the GMVP are inefficient — because there exists a corresponding portfolio above with the same risk but higher return. So A should be the answer.

First Principles Thinking: return statistics and covariance properties

B is correct. Statement I is correct: the CFA Curriculum explicitly states that the geometric mean is never greater than the arithmetic mean. Statement II is incorrect: while the formula $Cov(R_i, R_j) = \rho_{ij} \sigma_i \sigma_j$ is correct, it is the correlation coefficient ($\rho$) that is bounded between −1 and +1, not covariance. The CFA Curriculum states that covariance is unbounded on both sides and difficult to interpret for this reason. Statement III is correct: the CFA Curriculum provides $R_p = w_1 R_1 + (1 - w_1) R_2$ and notes that weights must account for 100% of the investment. Therefore, two statements (I and III) are correct.

Option A is incorrect because both the geometric-arithmetic mean relationship and the weighted average return formula are directly stated in the CFA Curriculum.

Option C is incorrect because Statement II conflates the boundedness of correlation with covariance — covariance is unbounded, which is precisely why correlation is preferred for interpretation.

First Principles Thinking: Risk-Return Trade-off Across Asset Classes

C is correct. According to the CFA Curriculum, US small company stocks earned an overall annualized geometric mean return of 12.1% versus 10.2% for large company stocks over 1926–2017, making Assertion A true. However, small company stocks had the highest risk at 31.7% standard deviation compared to 19.8% for large company stocks. The Reason claims small company stocks had lower risk, which directly contradicts the data. Therefore, R is false.

Option A is incorrect because R is factually false — small company stocks had substantially higher risk, not lower risk, than large company stocks.

Option D is incorrect because Assertion A is true — small company stocks did earn higher returns than large company stocks over the stated period.

First Principles Thinking: efficient frontier properties and the effect of adding asset classes

B is correct. Statement I is correct: the CFA Curriculum states that the curve lying above and to the right of the global minimum-variance portfolio is the Markowitz efficient frontier, containing all portfolios of risky assets that rational, risk-averse investors will choose. Statement II is correct: the CFA Curriculum explicitly notes that the slope of the efficient frontier decreases as one moves from the global minimum-variance portfolio rightward — investors obtain decreasing increases in returns as they assume more risk. Statement III is incorrect: the CFA Curriculum states that adding a new asset class not perfectly correlated with existing assets causes the investment opportunity set to expand outward to the northwest, providing a superior risk-return trade-off, not shift inward. Therefore, two statements (I and II) are correct.

Option A is incorrect because both the definition of the efficient frontier and its decreasing slope property are explicitly described in the CFA Curriculum.

Option C is incorrect because Statement III reverses the direction — new imperfectly correlated asset classes expand the opportunity set outward (northwest), they do not contract it inward.

First Principles Thinking: Perfect Negative Correlation and Risk Elimination

D is correct. Wait — let me re-evaluate. Assertion A states that combining two assets with correlation –1.0 can eliminate portfolio risk. The CFA Curriculum confirms this: "For an extreme case in which ρ12 = –1, the portfolio can be made risk free." The Beachwear/DVDrental example shows a risk-free portfolio earning 10% — the return does NOT equal zero. So Assertion A is true. However, Reason R claims the portfolio return equals zero because gains and losses cancel. This is false — the CFA Curriculum example shows a portfolio of Beachwear and DVDrental with correlation –1.0 earning a stable 10% return, not zero. Risk (variability) is eliminated, but return is preserved. Therefore A is true and R is false, making C the correct answer.

Apologies for the confusion above. C is correct. Assertion A is true: the CFA Curriculum confirms that when two assets have a correlation of –1.0, the portfolio can be made risk-free by choosing appropriate weights. The Beachwear/DVDrental example demonstrates this with a stable 10% return and zero risk. Reason R is false: eliminating risk does not mean the return becomes zero. The portfolio return remains the weighted average of the individual asset returns. Risk elimination means the variability of returns disappears, not the returns themselves.

Option A is incorrect because R is false — portfolio return is preserved even when risk is eliminated through perfect negative correlation.

Option D is incorrect because Assertion A is true as demonstrated in the CFA Curriculum's examples.

First Principles Thinking: two-asset portfolio standard deviation

A is correct. $\sigma_P^2 = w_X^2\sigma_X^2 + w_Y^2\sigma_Y^2 + 2w_Xw_Y\rho_{XY}\sigma_X\sigma_Y$ $= 0.40^2 \times 0.10^2 + 0.60^2 \times 0.20^2 + 2 \times 0.40 \times 0.60 \times 0.30 \times 0.10 \times 0.20$ $= 0.0016 + 0.0144 + 0.00288 = 0.01888$. $\sigma_P = \sqrt{0.01888} = 0.1374 = 13.74\%$.

B is incorrect. This is the portfolio standard deviation if the correlation were 0 instead of 0.30: $\sqrt{0.0016 + 0.0144} = \sqrt{0.0160} = 12.65\%$. Omitting the covariance term understates portfolio risk.

C is incorrect. This is the weighted average of standard deviations: USD 0.40 \times 10\% + 0.60 \times 20\% = 16\%$, which would only be correct if the correlation were +1. Portfolio standard deviation is less than the weighted average when $\rho < 1$.

First Principles Thinking: utility function computation

B is correct. $U = E(r) - \frac{1}{2}A\sigma^2 = 0.12 - 0.5 \times 4 \times 0.25^2 = 0.12 - 0.5 \times 4 \times 0.0625 = 0.12 - 0.125 = -0.005 = -0.50\%$. The high risk aversion and high variance result in a negative utility, indicating this investor would prefer the risk-free asset if it offered more than -0.50%.

A is incorrect. This results from omitting the $\frac{1}{2}$ factor: USD 0.12 - 4 \times 0.0625 = 0.12 - 0.25 = -0.13$. The $\frac{1}{2}$ is an integral part of the utility formula and must not be dropped.

C is incorrect. This equals the expected return alone and ignores the risk penalty entirely, which would only be correct for a risk-neutral investor where $A = 0$.

First Principles Thinking: correlation, diversification benefit, and negative-return assets

B is correct. Statement I is correct: the CFA Curriculum derives that when $\rho_{12} = +1$, $\sigma_p = w_1 \sigma_1 + w_2 \sigma_2$, which is a simple weighted average with no diversification benefit. Statement II is correct: the CFA Curriculum's Example 4 shows that with equal weights (50/50), equal risk (20% each), and $\rho = 0$, portfolio risk drops from 20% to 14%, which is 70% of the individual risk. Statement III is incorrect: the CFA Curriculum explicitly discusses insurance and gold as examples of assets that may have negative expected returns but are valuable because their negative correlation with portfolio assets significantly reduces overall portfolio risk. Therefore, two statements (I and II) are correct.

Option A is incorrect because both the perfect correlation result and the zero-correlation numerical example are directly demonstrated in the CFA Curriculum.

Option C is incorrect because Statement III contradicts the CFA Curriculum's discussion of insurance and put options — assets with negative returns can be beneficial if they significantly reduce portfolio risk through negative correlation.

First Principles Thinking: Shape of Indifference Curves for Risk-Averse Investors

A is correct. According to the CFA Curriculum, indifference curves for risk-averse investors run from the southwest to the northeast (upward sloping) because higher risk must be compensated by higher return. The curves are convex because of diminishing marginal utility of return — as risk increases, the investor needs greater return to compensate at an increasing rate. The Reason accurately explains this convexity: the steepening curve reflects the requirement for increasingly larger return increments per unit of additional risk. Both statements are true and R causally explains A.

Option B is incorrect because R is not merely tangentially true — it directly explains the convex shape of the indifference curves described in A.

Option C is incorrect because R is a true statement about diminishing marginal utility as presented in the CFA Curriculum.

First Principles Thinking: Covariance versus Correlation for Comparing Co-movement

D is correct. The CFA Curriculum states that although covariance is important, it is difficult to interpret because it is unbounded on both sides, whereas the correlation coefficient is bounded between –1 and +1 and provides similar information in a more interpretable form. This makes correlation — not covariance — the more useful measure for comparing co-movement strength across different asset pairs. Therefore, Assertion A is false. However, the Reason is factually true: covariance is indeed unbounded while correlation is bounded between –1 and +1.

Option A is incorrect because A is false — correlation is the more useful comparative measure precisely because of its bounded, standardized nature.

Option C is incorrect because R is a true factual statement about the mathematical properties of covariance and correlation.

First Principles Thinking: Decomposition of Portfolio Risk with Many Assets

B is correct. Assertion A is true: as stated in the CFA Curriculum, when N becomes large, the first term ($\bar{\sigma}^2/N$) becomes smaller and smaller, implying that the contribution of one asset's variance to portfolio variance gradually becomes negligible. Reason R is also true — the formula accurately reflects the portfolio variance decomposition shown in the CFA Curriculum. However, the Assertion specifically states "identical variances and identical pairwise correlations" while the Reason presents the more general formula using average variance and average covariance with equal weights. The formula in R applies to any equally weighted portfolio regardless of whether individual variances and correlations are identical. Therefore, R is true but is the general case, not the specific causal explanation for the identical-variance-and-correlation scenario described in A, where the more specific formula $\sigma_p = \sqrt{\sigma^2/N + ((N-1)/N)\rho\sigma^2}$ applies.

Option A is incorrect because R presents the general average-based formula rather than the specific formula for identical variances and correlations that directly explains Assertion A.

Option D is incorrect because Assertion A is true according to the CFA Curriculum.

First Principles Thinking: risk aversion, utility theory, and indifference curves

B is correct. Statement I is correct: the CFA Curriculum states that indifference curves for risk-averse investors run from southwest to northeast (upward sloping) and are convex because of diminishing marginal utility of return — as risk increases, an investor needs greater return to compensate at an increasing rate. Statement II is correct: the CFA Curriculum explicitly states that the risk aversion coefficient for a risk-neutral investor is 0, meaning changes in risk do not affect utility. Statement III is incorrect: the CFA Curriculum notes that a risk-free asset ($\sigma^2 = 0$) generates the same utility for all individuals, because the risk penalty term vanishes regardless of the value of A. Therefore, two statements (I and II) are correct.

Option A is incorrect because both Statements I and II are directly supported by the CFA Curriculum's discussion of utility theory.

Option C is incorrect because Statement III contradicts the CFA Curriculum — when variance is zero, the utility simply equals E(r), which is identical for all investors regardless of risk aversion.

First Principles Thinking: capital allocation line equation

B is correct. The CAL equation is $E(R_p) = R_f + \frac{E(R_i) - R_f}{\sigma_i} \times \sigma_p$. The slope is $\frac{0.11 - 0.03}{0.20} = 0.40$. At $\sigma_p = 0.10$: $E(R_p) = 0.03 + 0.40 \times 0.10 = 0.03 + 0.04 = 0.07 = 7.00\%$. The investor takes half the risk of the optimal risky portfolio and earns a return halfway between the risk-free rate and the risky portfolio return.

A is incorrect. This would result from using only the risk-free rate plus a small increment, possibly from miscalculating the slope or using the wrong $\sigma_p$ value.

C is incorrect. This is the expected return of the optimal risky portfolio itself, which corresponds to $\sigma_p = 20\%$, not $\sigma_p = 10\%$. This error ignores that lower risk on the CAL also means lower return.

First Principles Thinking: equity risk premium calculation

B is correct. The equity risk premium over bonds equals the real equity return minus the real bond return: USD 6.5\% - 2.0\% = 4.5\%$. This premium compensates investors for the additional risk of holding equities instead of bonds.

A is incorrect. This is the real bond return itself (2.0%), not the premium of equities over bonds. Confusing the bond return with the premium is a common beginner error.

C is incorrect. This is the total real equity return (6.5%), not the premium over bonds. The premium requires subtracting the bond return to isolate the additional compensation for equity risk.

First Principles Thinking: Two-Fund Separation and Optimal Portfolio Identification

A is correct. The CFA Curriculum states that the two-fund separation theorem allows all investors, regardless of taste, risk preferences, and initial wealth, to hold a combination of the risk-free asset and the same optimal risky portfolio. The investment decision (identifying the optimal risky portfolio at the tangent point) is separated from the financing decision (how much to allocate between risk-free and risky). The Reason correctly explains the mechanism: the optimal CAL has the steepest slope (highest reward per unit of risk), and individual preferences expressed through indifference curves only determine where on this line the investor's optimal portfolio lies.

Option B is incorrect because R directly explains why all investors share the same risky portfolio — the tangency point is determined independently of preferences, which then only govern the position along the CAL.

Option D is incorrect because Assertion A is true according to the two-fund separation theorem as stated in the CFA Curriculum.