Market Risk : Estimate VaR using a historical simulation approach

 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 $x\%$ confidence over $t$ days?"

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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.

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Steps to Estimate VaR using Historical Simulation

1. Collect Historical Data

   • Gather historical price data for the assets in the portfolio.
   • Calculate daily returns for each asset using: $$r_t = \frac{P_t - P_{t-1}}{P_{t-1}}$$ where $P_t$ is the price on day $t$.

2. Simulate Portfolio Returns

   • Compute the daily portfolio return for each historical period. For a portfolio with $n$ assets: $$R_t = \sum_{i=1}^n w_i \cdot r_{i,t}$$ where $w_i$ is the portfolio weight of asset $i$, and $r_{i,t}$ is its return on day $t$.

3. Rank Historical Returns

   • Sort the historical portfolio returns $R_t$ in ascending order.

4. Determine the Confidence Level

   • Choose the confidence level $\alpha$ (e.g., 95% or 99%).
   • Identify the $q$-th percentile of the sorted returns where: $$q = (1 - \alpha) \times N$$ Here, $N$ is the total number of historical returns.

5. Estimate VaR

   • The $q$-th percentile value in the sorted returns represents the VaR. This is the maximum loss at the given confidence level over the specified time horizon.

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Example

Scenario:

• You have a portfolio with a single asset.
• Historical daily returns over the last 100 days are sorted in ascending order.
• Confidence level: 95%.

Steps:

1. With 95% confidence, $q = (1 - 0.95) \times 100 = 5$.
2. The 5th percentile corresponds to the 5th value in the sorted returns.
3. If the 5th lowest return is $-3.2\%$, then: $$\text{VaR} = -3.2\% \times \text{Portfolio Value}.$$

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Advantages of Historical Simulation

1. No assumption about the distribution of returns.
2. Directly uses historical data, which captures real-world characteristics like skewness or kurtosis.

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Limitations

1. Relies on the assumption that the future resembles the past.
2. Sensitive to the choice of historical window length.
3. Cannot model scenarios outside historical data.