Given the log returns of a portfolio, calculate the 1-day 99% VaR.?

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 r_t is the log return at day t, P_t is the portfolio value at day t, and P_t − 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:

VaR_1day^99% = 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:

VaR_1day^99% = 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.