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Answer: The 1-day normal 95% VaR is equal to 2.74% and the 1-day lognormal 95% VaR is equal to 2.92%.
## Explanation To calculate the 1-day VaR using both normal and lognormal distributions: ### Normal VaR (Arithmetic Returns) - Daily mean return = 16% / 252 = 0.0006349 - Daily standard deviation = 27% / √252 = 0.27 / 15.8745 = 0.01701 - 95% VaR = -[μ - zσ] = -[0.0006349 - 1.645 × 0.01701] = -[0.0006349 - 0.02798] = -[-0.02735] = 2.735% ### Lognormal VaR (Geometric Returns) - Daily mean return = 13% / 252 = 0.0005159 - Daily standard deviation = 29% / √252 = 0.29 / 15.8745 = 0.01827 - 95% VaR = 1 - exp[μ - zσ] = 1 - exp[0.0005159 - 1.645 × 0.01827] = 1 - exp[0.0005159 - 0.03006] = 1 - exp[-0.02954] = 1 - 0.9709 = 2.91% The calculations match option C: **Normal VaR = 2.74%** and **Lognormal VaR = 2.92%**. **Key Points:** - Normal VaR uses arithmetic returns and assumes normal distribution - Lognormal VaR uses geometric returns and assumes lognormal distribution - The formulas differ: Normal VaR = -[μ - zσ], Lognormal VaR = 1 - exp[μ - zσ] - Annual parameters must be converted to daily using 252 trading days
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A risk manager is estimating the market risk of a portfolio using both the arithmetic returns with normal distribution assumptions and the geometric returns with lognormal distribution assumptions. The manager gathers the following data on the portfolio:
Assuming both daily arithmetic returns and daily geometric returns are serially independent, which of the following statements is correct?
A
The 1-day normal 95% VaR is equal to 1.63% and the 1-day lognormal 95% VaR is equal to 1.76%.
B
The 1-day normal 95% VaR is equal to 2.69% and the 1-day lognormal 95% VaR is equal to 2.88%.
C
The 1-day normal 95% VaR is equal to 2.74% and the 1-day lognormal 95% VaR is equal to 2.92%.
D
The 1-day normal 95% VaR is equal to 3.26% and the 1-day lognormal 95% VaR is equal to 3.48%.
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