Market Risk Problem 1 - Calculate VaR using Parametric approach

 Assume that the profit/loss distribution for XYZ is normally distributed with an annual mean of $15 million and a standard deviation of $10 million. Calculate the VaR at the 95% and 99% confidence levels using a parametric approach.

Given Data

• Distribution: Normal.
• Mean ($\mu$): $15 million (annual).
• Standard deviation ($\sigma$): $10 million (annual).
• Confidence levels:
  • 95% ($z = 1.645$).
  • 99% ($z = 2.33$).

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Formula for VaR

The parametric VaR formula is:

$$\text{VaR} = |z| \cdot \sigma - \mu$$

• $|z|$: z-score corresponding to the confidence level.
• $\sigma$: Standard deviation (risk/volatility).
• $\mu$: Mean (expected gain/loss).

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Step-by-Step Calculations

1. VaR at 95% Confidence Level ($z = 1.645$)

$$\text{VaR}_{95} = |1.645| \cdot 10 - 15$$ $$\text{VaR}_{95} = 16.45 - 15 = 1.45 \text{ million}.$$

At 95% confidence, XYZ may lose $1.45 million or more in a year.

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2. VaR at 99% Confidence Level ($z = 2.33$)

$$\text{VaR}_{99} = |2.33| \cdot 10 - 15$$ $$\text{VaR}_{99} = 23.3 - 15 = 8.3 \text{ million}.$$

At 99% confidence, XYZ may lose $8.3 million or more in a year.

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Results

• VaR at 95% confidence: $1.45 million.
• VaR at 99% confidence: $8.3 million.

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Interpretation

• At a 95% confidence level, the company expects that losses will not exceed $1.45 million in a year.
• At a 99% confidence level, the potential worst-case loss increases to $8.3 million.