Instrument Analysis

Linear regression analysis is a statistical method used to examine the relationship between two variables: the returns of an instrument (e.g., a stock or fund) and the returns of a benchmark (e.g., a market index).

The analysis fits a straight line to the data using the formula:

Y = α + βX + ε

• Y: Return of the instrument
• X: Return of the benchmark
• α (Alpha): Intercept — the excess return not explained by the benchmark
• β (Beta): Slope — sensitivity of the instrument to benchmark movements
• ε: Error term

In this context:

• Alpha > 0: Instrument outperformed the benchmark risk-adjusted.
• Beta = 1: Instrument moves like the benchmark.
• Beta > 1: Instrument is more volatile.
• Beta < 1: Instrument is less volatile.

Rolling monthly volatility measures the variability of monthly returns over a moving window of time.

It is calculated as the standard deviation of the monthly returns within the specified rolling window (e.g., 6 months).

Mathematically, for a window size n, the rolling volatility at time t is:

\[ \sigma_t = \sqrt{ \frac{1}{n-1} \sum_{i=t-n+1}^t (r_i – \mu)^2 } \]

• \(\sigma_t\): Rolling volatility at time t
• \(r_i\): Monthly return at time i
• \(\mu\): Mean monthly return over the window
• \(n\): Window size (e.g., 6 months)

Rolling volatility helps understand how the risk (variability of returns) changes over time.

Value at Risk (VaR) is a statistical measure used to estimate the maximum potential loss of a financial asset or a portfolio over a specific time horizon and at a given confidence level.

VaR can be applied to:

• Portfolios: to assess total downside risk across multiple instruments.
• Individual financial instruments: such as a stock, bond, or ETF, to measure its specific risk exposure.

In our case, the time horizon is set to one month. Therefore, the VaR we display corresponds to the maximum expected loss over a monthly period at a given confidence level.

For example:

• VaR 95% means that with 95% confidence, the asset or portfolio will not lose more than the calculated amount over the next month.
• VaR 99% represents a more conservative estimate, allowing only a 1% probability of exceeding the loss threshold in a month.

The most common (parametric) VaR formula under the assumption of normally distributed returns is:

\( \text{VaR}_\alpha = z_\alpha \cdot \sigma \cdot \sqrt{t} \)

• \( z_\alpha \): z-score for the selected confidence level (e.g., 1.645 for 95%, 2.33 for 99%)
• \( \sigma \): standard deviation of returns (volatility)
• \( t \): time period — in our case, \( t = 1 \) month

VaR is typically reported as a negative number to indicate potential loss. For instance, a monthly VaR 95% of -€1,500 means there’s a 95% chance the asset or portfolio will not lose more than €1,500 in the coming month.

Important: VaR does not provide any information about losses that might occur beyond the selected confidence level — it does not model tail risk (extreme losses).