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Answer: 1-day normal 95% VaR = 3.06% and 1-day lognormal 95% VaR = 4.12%
The question involves calculating the Value at Risk (VaR) for a portfolio using both normal and lognormal distribution assumptions. The manager has provided annualized averages and standard deviations for arithmetic and geometric returns, respectively, along with the current portfolio value and the number of trading days in a year. For the normal distribution (arithmetic returns): 1. The annualized average return is 12%. 2. The annualized standard deviation of returns is 30%. 3. The 95% VaR is calculated using the Z-score for a 95% confidence level, which is 1.645. 4. The daily return is calculated by dividing the annualized average return by the number of trading days (12% / 252). 5. The daily standard deviation is calculated by dividing the annualized standard deviation by the square root of the number of trading days (30% / sqrt(252)). 6. The 1-day 95% VaR is then calculated as -[(0.12/252) - 1.645 * (0.30/sqrt(252))], which equals 3.06%. For the lognormal distribution (geometric returns): 1. The annualized average return is 11%. 2. The annualized standard deviation of returns is 41%. 3. The 95% VaR is calculated using the same Z-score of 1.645. 4. The daily return and daily standard deviation are calculated similarly to the normal distribution. 5. The 1-day 95% VaR for the lognormal distribution is calculated using the exponential function: 1 - exp[(0.11/252) - 0.41 * 1.645 / sqrt(252)], which equals 4.12%. The correct answer is A: 1-day normal 95% VaR = 3.06% and 1-day lognormal 95% VaR = 4.12%. This is because the calculations for both the normal and lognormal VaRs are based on the provided data and the appropriate statistical formulas for each distribution.
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A financial risk manager is evaluating the market risk of a portfolio using two methods: arithmetic returns under the assumption of a normal distribution and geometric returns under the assumption of a lognormal distribution. The following information has been gathered for this assessment:
Considering that daily arithmetic and geometric returns are independently and identically distributed, identify which of the following statements is correct.
Section to focus on: Market Risk Measurement and Management
Reference material: Kevin Dowd, Measuring Market Risk, 2nd Edition (West Sussex, England: John Wiley & Sons, 2005). Chapter 3, Estimating Market Risk Measures: An Introduction and Overview
Learning Objective: Calculate Value at Risk (VaR) using a parametric approach for both normal and lognormal return distributions.
A
1-day normal 95% VaR = 3.06% and 1-day lognormal 95% VaR = 4.12%
B
1-day normal 95% VaR = 3.57% and 1-day lognormal 95% VaR = 4.41%
C
1-day normal 95% VaR = 4.12% and 1-day lognormal 95% VaR = 3.57%
D
1-day normal 95% VaR = 4.46% and 1-day lognormal 95% VaR = 4.49%