CFA Quantitative Methods & Statistics 2 — Questions and Answers
Question 1: A portfolio has an arithmetic mean return of 12% and a geometric mean return of 11.3%. Which statement best explains the difference?
- The geometric mean is always higher than the arithmetic mean for positive returns
- Volatility causes the geometric mean to be lower than the arithmetic mean
- The arithmetic mean ignores compounding effects and thus overstates long-run growth (Correct answer)
- Both means should be identical when returns are normally distributed
Correct answer: The arithmetic mean ignores compounding effects and thus overstates long-run growth
The arithmetic mean overstates long-run growth because it does not account for the compounding drag caused by return volatility.
Question 2: When testing H₀: μ = 50 against H₁: μ ≠ 50 at the 5% significance level, a two-tailed z-test yields z = 1.85. What is the correct conclusion?
- Reject H₀ because 1.85 > 1.645
- Fail to reject H₀ because 1.85 < 1.96 (Correct answer)
- Reject H₀ because the p-value is less than 0.025
- Fail to reject H₀ because 1.85 < 2.576
Correct answer: Fail to reject H₀ because 1.85 < 1.96
For a two-tailed test at 5% significance, the critical z-value is ±1.96, and since 1.85 < 1.96 we fail to reject H₀.
Question 3: An analyst calculates a 95% confidence interval for a population mean as [8.2%, 11.8%]. Which interpretation is correct?
- There is a 95% probability the true mean lies in this specific interval
- If the sampling procedure were repeated many times, 95% of such intervals would contain the true mean (Correct answer)
- The population mean equals 10% with 95% certainty
- 95% of individual observations fall between 8.2% and 11.8%
Correct answer: If the sampling procedure were repeated many times, 95% of such intervals would contain the true mean
A confidence interval is a frequentist statement: 95% of identically constructed intervals from repeated samples would capture the true population mean.
Question 4: Which probability distribution is most appropriate for modeling the number of credit defaults occurring in a bond portfolio over one year, assuming defaults are independent and have a constant probability?
- Normal distribution
- Binomial distribution (Correct answer)
- Uniform distribution
- Lognormal distribution
Correct answer: Binomial distribution
The binomial distribution models the number of successes (defaults) in a fixed number of independent Bernoulli trials with constant probability.
Question 5: A covariance matrix shows Cov(X,Y) = 0. What can be concluded about X and Y?
- X and Y are independent
- X and Y have no linear relationship (Correct answer)
- X and Y have no relationship of any kind
- X and Y have a correlation of 1
Correct answer: X and Y have no linear relationship
Zero covariance indicates no linear relationship, but X and Y could still have a nonlinear dependence.
Question 6: The continuously compounded equivalent of a 10% effective annual rate is closest to:
- 10.00%
- 9.53% (Correct answer)
- 10.52%
- 9.09%
Correct answer: 9.53%
The continuously compounded rate is ln(1.10) ≈ 9.53%, derived from the relationship e^r = 1 + EAR.
Question 7: Roy's Safety-First criterion selects the portfolio that:
- Maximizes the Sharpe ratio relative to the risk-free rate
- Minimizes the probability of falling below a specified threshold return (Correct answer)
- Maximizes expected return for a given level of standard deviation
- Minimizes variance regardless of expected return
Correct answer: Minimizes the probability of falling below a specified threshold return
Roy's Safety-First criterion chooses the portfolio with the highest ratio (E[Rp] − RL)/σp, which minimizes the probability of return falling below the threshold RL.
CFA Quantitative Methods & Statistics 2
A portfolio has an arithmetic mean return of 12% and a geometric mean return of 11.3%.
Which statement best explains the difference?