Answer: Lognormal VaR is less than normal VaR by GBP 17,590

## Explanation To solve this problem, we need to calculate both the normal VaR and lognormal VaR and compare them. **Given:** - Portfolio value (P) = GBP 1,000,000 - Mean return (μ) = 10% - Volatility (σ) = 40% - Confidence level = 95% (z-score = 1.645) - Time horizon = 1 year ### Normal VaR Calculation Normal VaR assumes returns are normally distributed: ``` Normal VaR = P × (μ - z × σ) Normal VaR = 1,000,000 × (0.10 - 1.645 × 0.40) Normal VaR = 1,000,000 × (0.10 - 0.658) Normal VaR = 1,000,000 × (-0.558) Normal VaR = -GBP 558,000 ``` ### Lognormal VaR Calculation Lognormal VaR assumes asset prices follow a lognormal distribution: ``` Lognormal VaR = P × [1 - exp(μ - z × σ)] Lognormal VaR = 1,000,000 × [1 - exp(0.10 - 1.645 × 0.40)] Lognormal VaR = 1,000,000 × [1 - exp(0.10 - 0.658)] Lognormal VaR = 1,000,000 × [1 - exp(-0.558)] Lognormal VaR = 1,000,000 × [1 - 0.5724] Lognormal VaR = 1,000,000 × 0.4276 Lognormal VaR = GBP 427,600 ``` ### Comparison - Normal VaR = GBP 558,000 - Lognormal VaR = GBP 427,600 - Difference = 558,000 - 427,600 = GBP 130,400 Therefore, **lognormal VaR is less than normal VaR by GBP 130,400**. **Key Insight:** The lognormal distribution prevents negative asset prices, which makes it more conservative for calculating VaR. The normal distribution can produce negative asset values, which is unrealistic for most financial assets.

Author: LeetQuiz .