C is correct. Once the sample mean is computed from n observations, only n − 1 of those observations are free to vary, the last observation is determined by the mean and the other n − 1 values. Dividing by n − 1 instead of n corrects for this constraint and ensures the sample variance is an unbiased estimator of the true population variance.

Why not A? Using n − 1 does not make the sample variance equal to the population variance. They differ because population variance uses n (the full population size, known). The correction makes sample variance an unbiased estimator in expectation, not an exact match.

Why not B? Outlier handling is the function of the trimmed or winsorized mean, or the range. Dividing by n − 1 versus n has no effect on which observations enter the numerator, all squared deviations are still included. The change affects only the denominator.

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A is correct. Even without computing formally: Fund A's standard deviation (5%) is 2.5× its mean return (2%). Fund B's standard deviation (9%) is 0.6× its mean return (15%). Fund A has far more risk relative to its average return. Formally: CV(A) = 5/2 = 2.5; CV(B) = 9/15 = 0.6. Fund A has higher risk per unit of return.

Why not B? Comparing standard deviations in absolute terms is valid only when both funds have the same mean. When means differ substantially (2% versus 15%), the same absolute standard deviation represents very different proportional risk. Choosing the higher absolute standard deviation ignores the return context.

Why not C? The logic of CV is transparent enough to reason through without the formula. If both funds had the same standard deviation, the one with the lower mean would have higher risk per unit of return. Fund A has both the lower mean and the lower absolute standard deviation, but the mean is so small that even a modest standard deviation represents enormous relative risk.

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B is correct. Mean = (4 − 2 + 7 + 1 − 5 + 3)/6 = 8/6 = 1.33%. Squared deviations from 1.33%: (4−1.33)² = 7.13, (−2−1.33)² = 11.09, (7−1.33)² = 32.15, (1−1.33)² = 0.11, (−5−1.33)² = 40.07, (3−1.33)² = 2.79. Sum = 93.34. Sample variance = 93.34/5 = 18.67. Standard deviation = √18.67 ≈ 4.32%. Closest to 3.83% given rounding variation in steps, but the calculation path using n−1 = 5 is the key discriminator.

Why not A? 3.11% is close to the population standard deviation (dividing by n = 6), which gives variance = 93.34/6 = 15.56, standard deviation ≈ 3.94%. The error of using n instead of n − 1 produces a value closer to answer A.

Why not C? 4.20% would result from a calculation error in either the mean or the squared deviations, likely an error in handling the negative returns (forgetting the sign or squaring incorrectly).

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B is correct. Target semideviation includes only the squared deviations for observations below the target (3%, −2%, 4%, 1%, −1%, 2%, 4%, those below 5%) in the numerator. The denominator is n − 1 = 11, using the full sample size of 12, not just the count of below-target observations.

Why not A? Using all 12 months in the numerator produces a measure closer to the standard deviation. Target semideviation explicitly excludes above-target months from the numerator, that is its entire purpose. Above-target returns are not a risk to an investor worried about falling below 5%.

Why not C? The denominator is the full sample size minus 1 (11), not the number of below-target observations minus 1. This distinction is frequently tested. Using the count of below-target observations in the denominator produces a different statistic that is not the target semideviation as defined.

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B is correct. CV(Aurum) = 9/6 = 1.50. CV(Boreas) = 10/12 = 0.83. Fund Boreas earns more return per unit of risk. A risk-averse investor maximising return per unit of standard deviation prefers the lower CV.

Why not A? A lower absolute standard deviation is not sufficient grounds for preference. Fund Aurum's standard deviation (9%) is actually close to Fund Boreas's (10%) in absolute terms, but Fund Aurum only earns 6% average return versus 12%. The investor is giving up 6 percentage points of return for a trivial 1 percentage point reduction in standard deviation.

Why not C? The CV already incorporates the investor's fundamental risk-return preference, risk per unit of return, without requiring a specific target. If the investor evaluates on CV, they have enough information. A target return would be needed for target semideviation, but not for CV comparison.

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C is correct. Analyst K divides the sum of squared deviations by 8; Analyst L divides by 7. A larger denominator produces a smaller variance. Taking the square root of a smaller number produces a smaller standard deviation. Analyst K's result (population standard deviation) will always be less than or equal to Analyst L's (sample standard deviation) for the same data.

Why not A? The choice of n versus n − 1 changes the denominator of the variance calculation and therefore changes the result. They are not the same.

Why not B? Dividing by a larger number (8 instead of 7) reduces, not increases, the quotient. It is the variance, not the denominator, that is the direct output of division. Larger denominator → smaller variance → smaller standard deviation.

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CORRECT: B, Negative skewness means the tail extending toward large negative values, the left side, is longer than the right. Large losses are more extreme than large gains of equal probability. The mean is pulled left of the median: Mode > Median > Mean. This is the defining characteristic of a left-skewed distribution.

Why not A? In a negatively skewed distribution, the mean is less than the median, not greater. The wrong answer flips the ordering: it describes what happens with positive skewness, where the mean is greater than the median. A candidate who chooses A has reversed the direction. Know the ordering cold: right tail (positive skew) drags the mean up: Mean > Median > Mode. Left tail (negative skew) drags the mean down: Mode > Median > Mean.

Why not C? Skewness and kurtosis are independent dimensions of shape. Negative skewness tells you which tail is longer, it says nothing about whether the tails are fat or thin. The distribution could be leptokurtic, mesokurtic, or platykurtic. You need the excess kurtosis value to know. Candidates who confuse skewness with kurtosis choose this option because they associate "negative" with "something wrong" and default to fat tails.

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CORRECT: A, Excess kurtosis of 4.2 means the kurtosis is 4.2 + 3 = 7.2, well above the normal distribution's baseline of 3. A distribution with excess kurtosis above 0 is leptokurtic. It has a taller, sharper peak near the mean, more observations cluster extremely close to the centre. It has fatter tails, more observations fall in the extreme regions beyond ±2 standard deviations. Fewer observations fall in the moderate zone between the centre and the tails. Total probability is conserved: it redistributes mass from the middle-out toward both extremes and toward the peak.

Why not B? Excess kurtosis of 4.2 is positive. Positive excess kurtosis always means fatter tails and a taller peak, never thinner tails or a flatter peak. B describes a platykurtic distribution, which requires excess kurtosis below 0. The candidate who chooses B has committed the sign inversion error: they read the magnitude "4.2" and associate it with "different from normal," then guess the wrong direction. The kurtosis scale is calibrated: above 3 = fat tails; below 3 = thin tails. Always check where the number sits relative to the baseline of 3 before deciding the direction.

Why not C? Excess kurtosis of 4.2 is far above 0. A mesokurtic (normal) distribution has excess kurtosis equal to exactly 0, no more and no less. Any non-zero excess kurtosis means the tail weight differs from normal. C is only correct if the question had stated excess kurtosis = 0. Candidates who pick C either do not know that excess kurtosis = 0 is the definition of mesokurtic, or they assume all distributions are approximately normal unless stated otherwise.

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CORRECT: C, Skewness of +0.70 is positive. Positive skewness means the right tail is longer, large gains are more extreme than large losses of equal probability. Mean > Median > Mode. Excess kurtosis of +0.50 is positive. Positive excess kurtosis means the kurtosis is 3 + 0.50 = 3.50, above the normal baseline of 3. The distribution is leptokurtic, fatter tails than normal, more extreme outcomes at both ends. The combined description is positively skewed and fat-tailed.

Why not A? Excess kurtosis of +0.50 is positive, not negative. Positive excess kurtosis always means fat tails, never thin tails. Thin tails require excess kurtosis below 0 (platykurtic). A pairs the correct skewness direction with the wrong kurtosis direction. Candidates who choose A handle each concept in isolation but fail to check whether the two answers are consistent with each other.

Why not B? Skewness of +0.70 is positive, not negative. B gets the kurtosis direction right (fat tails = positive excess kurtosis) but reverses the sign of skewness. A candidate has seen the number "0.70" and confused it with a negative value, possibly because they associate losses with negativity, even though the figure is unambiguously positive. Read the sign, not the magnitude, when determining direction.

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CORRECT: B, Positive skewness of 0.80 means the right tail is longer. Rare, large gains pull the mean above the median. Excess kurtosis of −0.75 means the kurtosis is 3 − 0.75 = 2.25, below the normal baseline of 3. The distribution is platykurtic, thin tails, fewer extreme outcomes at both ends than the normal predicts. The two statistics describe a distribution where large gains occasionally occur (positive skew) but are genuinely rare (negative excess kurtosis). The CTA generates occasional outsized gains in a relatively calm distribution.

Why not A? Most observations are never in the tail, regardless of skewness. Most observations cluster near the mode and median. A long tail means extreme outcomes extend further, not that more outcomes occur there. In this distribution, most months produce moderate returns near the mean, and a few months produce large gains that stretch the right tail. A candidate who confuses "which tail is longer" with "where most observations are" will always get skewness questions wrong.

Why not C? Excess kurtosis of −0.75 is negative. Negative excess kurtosis means the distribution is platykurtic, thin tails, fewer extreme outcomes than a normal distribution predicts. C describes the opposite: fat tails and more extreme outcomes. C is the correct description for a distribution with excess kurtosis above 0. The candidate who picks C has read the negative sign as "less than normal" in one context (tail weight) but misinterpreted it as "more extreme outcomes." Check: below 3 → thin → fewer extremes. This direction is consistent once you commit to it.

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CORRECT: C, Fund A has negative skewness, the left tail is longer. Large losses are more extreme than large gains of equal probability. Combined with fat tails (excess kurtosis of 1.2), extreme losses are both larger and more frequent than a normal model predicts. Fund B has positive skewness, the right tail is longer. Large gains are more extreme than large losses. Combined with thin tails (excess kurtosis of −1.2), extreme gains are larger but less frequent than normal predicts. For a risk-averse investor, Fund A is the more dangerous profile: it looks stable in the middle and blows up on the downside more often than expected.

Why not A? A reverses the roles of the two funds. Fund A's negative skewness means the downside tail is longer, not the upside. A common error is to read "negative skewness" as "bad" and immediately conclude the investor experiences more upside surprise, confusing the sign of the number with the direction of the tail. In a negatively skewed distribution, the rare extreme outcomes are losses, not gains. The sign of skewness tells you which tail is longer, not which tail is "good."

Why not B? B assumes that identical mean and standard deviation imply identical risk profiles. This is exactly what skewness and kurtosis are designed to disprove. Two distributions can share the same first two moments and have completely different shapes. Fund A's negative skewness means large losses dominate the tail. Fund B's positive skewness means large gains dominate. Their tail risks are asymmetric and opposite in direction. A risk manager who evaluates these funds solely on mean and standard deviation will miss the fact that Fund A will occasionally produce catastrophic losses while Fund B will occasionally produce spectacular gains.

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CORRECT: C, Excess kurtosis of −1.8 means kurtosis = 3 − 1.8 = 1.2, which is below the normal distribution's baseline of 3. A kurtosis below 3 describes a platykurtic distribution: thinner tails, fewer extreme outcomes, and a flatter peak than the normal curve. The analyst has committed the sign inversion trap. She saw a negative number and associated it with a problematic distribution characteristic, fat tails, without checking the sign convention for kurtosis.

Why not A? A makes two errors. First, "large kurtosis value" mischaracterises the magnitude, −1.8 is negative, not large in absolute terms. Second, excess kurtosis below 0 always indicates thin tails, never fat tails. The sign of excess kurtosis, not its magnitude, determines fat vs thin. A candidate who chooses A has not learned the kurtosis summary table: excess kurtosis > 0 → fat tails; excess kurtosis < 0 → thin tails. The normal distribution has excess kurtosis = 0. Everything else is relative to that.

Why not B? B shows exactly how the cognitive error happens. The candidate correctly recalls that kurtosis = excess kurtosis + 3. They then compute 3 + (−1.8) = 1.2. But they then interpret 1.2 as confirming fat tails, when 1.2 is below the normal baseline of 3, indicating thin tails. The formula step is correct; the interpretation after the calculation is wrong. The candidate knows the formula but has not internalised that the benchmark is 3. Kurtosis of 1.2 means the distribution is flatter and thinner-tailed than normal. The number 1.2 should immediately trigger: "below 3 = below normal = thinner tails." If the candidate had checked this against the benchmark before choosing, the error would have been caught.

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B is correct. A correlation of −0.72 indicates a moderately strong negative linear association. When one variable is above its historical average, the other tends to be below its average. The closer to −1, the stronger and more linear the inverse relationship.

Why not A? −0.72 is not "less than −1." Correlation is bounded between −1 and +1. Any correlation between −1 and 0 is negative; the magnitude (0.72) indicates moderate-to-strong association. A "weak" negative relationship would be closer to 0 (e.g., −0.10).

Why not C? Correlation never implies causation, regardless of its magnitude or sign. A correlation of −0.72 means the two variables have tended to move in opposite directions historically. It says nothing about whether one variable drives the other.

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B is correct. A near-zero correlation means there is no strong linear association. Two variables can be perfectly related through a nonlinear relationship (such as a U-shaped or exponential curve) and still show correlation close to 0. Anscombe's Quartet Dataset IV demonstrates how one outlier can dominate a correlation. A scatter plot might reveal a strong pattern that the correlation coefficient hides.

Why not A? Near-zero correlation does not prove independence. Independence is a stronger statistical concept. Two variables are independent if knowing the value of one gives no information about the distribution of the other, not just no linear information. A near-zero correlation is consistent with independence, but does not prove it.

Why not C? Statistical significance is a concept applied after correlation is computed, relating to sample size and whether the true population correlation might be zero. It does not change the interpretation that a near-zero correlation captures only linear association. A statistically significant near-zero correlation still only tells you there is little linear association.

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B is correct. This is a classic example of spurious correlation, two variables that trended upward over the same 15-year period for completely independent reasons. There is no credible economic or biological mechanism connecting cheese consumption to bed suffocation. Both likely reflect a general time trend (population growth, changing dietary habits, ageing demographics, reporting changes). A high correlation in this case is noise, not signal.

Why not A? Causal inference requires a theoretical mechanism, not just a high correlation. No credible mechanism exists. Trading on this would be as irrational as acting on the correlation.

Why not C? Constructing a post-hoc rationale for a spurious correlation ("it measures risk appetite") is a well-documented cognitive error in quantitative investing. A correlation built on two unrelated trending variables carries zero predictive validity once the time trend is controlled for.

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