FRM Quantitative Analysis 1 — Questions and Answers
Question 1: 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%
Approximately 95% of observations fall within ±2 standard deviations in a normal distribution, following the empirical 68-95-99.7 rule.
Question 2: Which statistical measure describes the asymmetry of a probability distribution?
- Kurtosis
- Variance
- Skewness (Correct answer)
- Standard deviation
Correct answer: Skewness
Skewness measures the asymmetry of a distribution, with positive skewness indicating a longer right tail and negative skewness indicating a longer left tail.
Question 3: A leptokurtic distribution compared to a normal distribution has:
- Excess kurtosis less than 0
- Excess kurtosis equal to 0
- Excess kurtosis greater than 0 (Correct answer)
- Negative skewness
Correct answer: Excess kurtosis greater than 0
A leptokurtic distribution has excess kurtosis greater than 0 (kurtosis > 3), indicating heavier tails and a higher peak than a normal distribution, which is critical for risk modeling.
Question 4: The covariance between two assets is 0.006, and their standard deviations are 0.10 and 0.15. What is their correlation coefficient?
- 0.20
- 0.30
- 0.40 (Correct answer)
- 0.60
Correct answer: 0.40
Correlation = Covariance / (σ1 × σ2) = 0.006 / (0.10 × 0.15) = 0.006 / 0.015 = 0.40.
Question 5: Which statement best describes the Central Limit Theorem (CLT)?
- All financial returns are normally distributed
- The distribution of sample means approaches normality as sample size increases regardless of the underlying distribution (Correct answer)
- The mean of a population always equals the mean of its samples
- Large samples always have lower variance than small samples
Correct answer: The distribution of sample means approaches normality as sample size increases regardless of the underlying distribution
The CLT states that the sampling distribution of the mean approaches a normal distribution as sample size increases, regardless of the population's underlying distribution.
Question 6: In hypothesis testing, a Type I error occurs when:
- We fail to reject a false null hypothesis
- We reject a true null hypothesis (Correct answer)
- We accept a false alternative hypothesis
- The p-value exceeds the significance level
Correct answer: We reject a true null hypothesis
A Type I error (false positive) occurs when we reject a null hypothesis that is actually true, with its probability equal to the significance level α.
Question 7: For a two-asset equal-weighted portfolio where each asset has variance 0.04 and the assets have a correlation of 0.50, the portfolio variance is:
- 0.02
- 0.03 (Correct answer)
- 0.04
- 0.06
Correct answer: 0.03
Portfolio variance = (0.5)²(0.04) + (0.5)²(0.04) + 2(0.5)(0.5)(0.2)(0.2)(0.5) = 0.01 + 0.01 + 0.01 = 0.03.
FRM Quantitative Analysis 1
In a normal distribution, approximately what percentage of observations fall within two standard deviations of the mean?