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 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 ( μ \mu μ ) : $15 million (annual). Standard deviation ( σ \sigma σ ) : $10 million (annual). Confidence levels : 95% ( z = 1.645 z = 1.645 z = 1.645 ). 99% ( z = 2.33 z = 2.33 z = 2.33 ). Formula for VaR The parametric VaR formula is: VaR = ∣ z ∣ ⋅ σ − μ \text{VaR} = |z| \cdot \sigma - \mu VaR = ∣ z ∣ ⋅ σ − μ ∣ z ∣ |z| ∣ z ∣ : z-score corresponding to the confidence level. σ \sigma σ : Standard deviation (risk/volatility). μ \mu μ : Mean (expected gain/loss). Step-by-Step Calculations 1. VaR at 95% Confidence Level ( z = 1.645 z = 1.645 z = 1.645 ) VaR 95 = ∣ 1.645 ∣ ⋅ 10 − 15 \text{VaR}_{95} = |1.645| \cdot 10 - 15 VaR 95 ​ = ∣1.645∣ ⋅ 10 − 15 VaR 95 = 16.45 − 15 = 1.45  million . \text{VaR}_{95} = 16.45 -...

What is the Parametric Approach? The parametric approach assumes that portfolio returns follow a specific type of mathematical distribution (like a bell-shaped curve for normal distribution). Using some simple formulas, we calculate how much you could lose at a certain confidence level (like 95% or 99%). Steps to Estimate VaR 1. Get the Data You need: Portfolio value (how much your portfolio is worth today). Mean return (average return, usually based on past data). Volatility (how much the returns fluctuate, measured as a standard deviation). Confidence level (how certain you want to be about the worst-case loss, like 95% or 99%). 2. Use a Formula VaR is calculated as: VaR = z ⋅ Volatility ⋅ Portfolio Value \text{VaR} = z \cdot \text{Volatility} \cdot \text{Portfolio Value} VaR = z ⋅ Volatility ⋅ Portfolio Value Here: z z z is a number from statistics that depends on the confidence level: For 95%, z = 1.645 z = 1.645 z = 1.645 . For 99%, z = 2.33 z = 2.33 z = 2.33 . Multipl...

  What is VaR? Value at Risk (VaR) measures the potential loss in value of a portfolio over a defined time period for a given confidence level. It answers the question: "How much could I lose with x % x\% x % confidence over t t t days?" Historical Simulation Approach This is a non-parametric method that uses actual historical returns to estimate potential future losses. It assumes that the past distribution of returns is representative of the future. Steps to Estimate VaR using Historical Simulation Collect Historical Data Gather historical price data for the assets in the portfolio. Calculate daily returns for each asset using: r t = P t − P t − 1 P t − 1 r_t = \frac{P_t - P_{t-1}}{P_{t-1}} r t ​ = P t − 1 ​ P t ​ − P t − 1 ​ ​ where P t P_t P t ​ is the price on day t t t . Simulate Portfolio Returns Compute the daily portfolio return for each historical period. For a portfolio with n n n assets: R t = ∑ i = 1 n w i ⋅ r i , t R_t = \sum_{i=1}^n w_i \cdot r_{i,t} R ...