Explanation:
To solve this problem, we need to calculate both the normal VaR and lognormal VaR and compare them.
Given:
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
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 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
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
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.
Ultimate access to all questions.
The annual mean and volatility of a portfolio are 10% and 40%, respectively. The current value of the portfolio is GBP 1,000,000. How does the 1-year 95% VaR that is calculated using a normal distribution assumption (normal VaR) compare with the 1-year 95% VaR that is calculated using the lognormal distribution assumption (lognormal VaR)?
A
Lognormal VaR is greater than normal VaR by GBP 130,400
B
Lognormal VaR is greater than normal VaR by GBP 17,590
C
Lognormal VaR is less than normal VaR by GBP 130,400
D
Lognormal VaR is less than normal VaR by GBP 17,590
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