Answer-first summary for fast verification
Answer: VaR(15-day) = USD 503 million
## Explanation Given that the daily returns are independently and identically normally distributed with mean zero, and annualized volatilities are equal, the VaR for n-day period should scale with √n (square root of time rule). To find the inconsistent VaR estimate, we calculate the implied 1-day VaR from each option: - **VaR(10-day) = 474 million** - VaR(1-day) = 474 / √10 = 474 / 3.162 ≈ 150 million - **VaR(15-day) = 503 million** - VaR(1-day) = 503 / √15 = 503 / 3.873 ≈ 130 million - **VaR(20-day) = 671 million** - VaR(1-day) = 671 / √20 = 671 / 4.472 ≈ 150 million - **VaR(25-day) = 750 million** - VaR(1-day) = 750 / √25 = 750 / 5 = 150 million The VaR(1-day) calculated from the 15-day VaR (130 million) is inconsistent with the others (150 million), indicating that the VaR(15-day) estimate is incorrect. **Key Concept**: Under the square root of time rule for VaR scaling with independent and identically distributed returns, all n-day VaR estimates should imply the same 1-day VaR when divided by √n.
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A newly hired quantitative analyst at a financial institution has been asked by a portfolio manager to calculate the VaR of a portfolio for 10-, 15-, 20-, and 25-day periods. The portfolio manager notices something wrong with the analyst's calculations. Assuming the annualized volatilities of daily returns for the four periods are equal, and that the daily returns are independently and identically normally distributed with a mean of zero, which of the following VaR estimates for this portfolio is inconsistent with the others?
A
VaR(10-day) = USD 474 million
B
VaR(15-day) = USD 503 million
C
VaR(20-day) = USD 671 million
D
VaR(25-day) = USD 750 million