CAIA Quantitative Methods and Statistics 1 — Questions and Answers
Question 1: What does the Sharpe ratio measure?
- Excess return per unit of total risk (Correct answer)
- Excess return per unit of systematic risk
- Total return per unit of downside risk
- Risk-adjusted return relative to a benchmark
Correct answer: Excess return per unit of total risk
The Sharpe ratio is calculated as (Rp - Rf) / σp, representing excess return over the risk-free rate per unit of total portfolio risk (standard deviation).
Question 2: Which statistical measure best captures 'fat tail' risk in alternative investment return distributions?
- Standard deviation
- Skewness
- Kurtosis (Correct answer)
- Variance
Correct answer: Kurtosis
Kurtosis measures the degree of fat tails (excess probability of extreme outcomes) relative to a normal distribution; excess kurtosis above 3 signals leptokurtic (fat-tailed) distributions common in alternatives.
Question 3: In a normal distribution, approximately what percentage of observations fall within two standard deviations of the mean?
- 68%
- 90%
- 95% (Correct answer)
- 99%
Correct answer: 95%
The empirical rule states that approximately 68% of observations fall within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean.
Question 4: Why is the geometric mean return typically lower than the arithmetic mean return for a volatile investment?
- Geometric mean ignores compounding effects
- Geometric mean accounts for compounding and is reduced by variance drag (Correct answer)
- Arithmetic mean penalizes negative returns more heavily
- They are equal unless leverage is applied
Correct answer: Geometric mean accounts for compounding and is reduced by variance drag
The geometric mean approximates the arithmetic mean minus half the variance (g ≈ r - σ²/2), so higher volatility creates a larger variance drag, widening the gap between the two measures.
Question 5: What does negative skewness in the distribution of hedge fund returns indicate?
- The distribution has a long right tail with more extreme positive outcomes
- The distribution has a long left tail with more extreme negative outcomes (Correct answer)
- Returns are normally distributed around the mean
- Volatility is lower than measured by standard deviation
Correct answer: The distribution has a long left tail with more extreme negative outcomes
Negative skewness means the left tail is longer, indicating a higher probability of large negative returns (crash risk) relative to a normal distribution — a common feature of strategies that sell optionality.
Question 6: Value at Risk (VaR) at the 95% confidence level over a one-day horizon is best defined as:
- The maximum loss at the 95% confidence level over one day (Correct answer)
- The average loss when losses exceed the 95th percentile threshold
- The probability that losses exceed a given threshold over one day
- The portfolio standard deviation scaled to a one-day period
Correct answer: The maximum loss at the 95% confidence level over one day
VaR at 95% confidence means there is a 5% probability that losses will exceed the stated VaR amount over the given time horizon; it defines the loss threshold, not the expected loss beyond it.
Question 7: Why is autocorrelation in returns particularly important when evaluating alternative investment funds?
- Autocorrelation measures systematic risk exposure and helps compute beta accurately
- Positive autocorrelation can signal smoothed or stale pricing in illiquid assets, understating true volatility (Correct answer)
- Autocorrelation corrects for survivorship bias in historical databases
- Negative autocorrelation indicates market timing skill
Correct answer: Positive autocorrelation can signal smoothed or stale pricing in illiquid assets, understating true volatility
Illiquid assets (e.g., private equity, real estate) are often marked to stale prices, artificially inducing positive autocorrelation and causing standard deviation and correlation estimates to understate true risk.
CAIA Quantitative Methods and Statistics 1
What does the Sharpe ratio measure?