To calculate the 1-day 99% VaR, you would first need to have the log returns of the portfolio. Once you have the log returns, follow these steps:
1. Calculate the mean (µ) and standard deviation (σ) of the log returns.
2. Determine the corresponding z-score for the 99% confidence level.
3. Compute the Value at Risk (VaR) using the mean, standard deviation, and z-score.
Let’s go through these steps in detail:
**Step 1**: Calculate the mean (µ) and standard deviation (σ) of the log returns.
Assume you have the log returns as a time series: $r_t = \ln(\frac{P_t}{P_{t-1}})$, where rt is the log return at day t, Pt is the portfolio value at day t, and Pt − 1 is the portfolio value at day t-1.
Calculate the mean (µ) and standard deviation (σ) of the log returns:
$$\mu = \frac{\sum_{t=1}^n r_t}{n}$$
$$\sigma = \sqrt{\frac{\sum_{t=1}^n (r_t - \mu)^2}{n-1}}$$
**Step 2**: Determine the corresponding z-score for the 99% confidence level.
For quantile calculation, you will be using the inverse of the standard normal cumulative distribution function. In this case, you want the z-score for the 99% confidence level, which corresponds to the 1% tail. The z-score for this quantile is approximately 2.33.
**Step 3**: Compute the Value at Risk (VaR) using the mean, standard deviation and z-score.
The 1-day 99% VaR can be calculated as:
VaR1day99% = Portfolio Value × (μ − 2.33 × σ)
To demonstrate with an example, let’s assume the following log returns of a portfolio:
Log Returns = [-0.02, 0.01, -0.005, -0.01, 0.008, 0.015, -0.03, 0.025, 0.01, 0.005]
First, calculate the mean (µ):
$$\begin{aligned}
\mu &= \frac{1}{10} \left(-0.02 + 0.01 - 0.005 - 0.01 + 0.008 + 0.015\right.\\
&\qquad\left.- 0.03 + 0.025 + 0.01 + 0.005\right) \\
&= -0.0015\end{aligned}$$
Next, calculate the standard deviation (σ):
$$\sigma = \sqrt{\frac{((-0.02 - (-0.0015))^2 + \cdots + (0.005 - (-0.0015))^2)}{9}} \approx 0.0172$$
Assume the current portfolio value is $100,000.
Calculate the 1-day 99% VaR:
VaR1day99% = 100, 000 × ( − 0.0015 − 2.33 × 0.0172) ≈ 4, 004.96
Thus, the 1-day 99% VaR for the portfolio is approximately $4,004.96. This means that there is a 1% chance that the portfolio will lose at least $4,004.96 in value over a single day.