Risk Assessment Example: Quantifying Portfolio Risk Using GARCH Models
Question
You manage a portfolio and want to estimate its risk using daily returns data. The goal is to quantify the conditional volatility (risk) of the portfolio using a GARCH(1,1) model.
Given daily returns (\(r_t\)): \[ r_t = [0.001, -0.002, 0.0015, -0.003, 0.0025, \dots] \]
Solution
Step 1: Model Definition
A GARCH(1,1) model specifies:
\[ r_t = \mu + \epsilon_t, \quad \epsilon_t \sim N(0, \sigma_t^2) \] \[ \sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \]Where:
- \(\sigma_t^2\): Conditional variance (volatility) at time \(t\).
- \(\omega\): Constant term (long-run variance).
- \(\alpha\): Weight on past shocks (\(\epsilon_{t-1}^2\)).
- \(\beta\): Weight on past volatility (\(\sigma_{t-1}^2\)).
Step 2: Estimate Model Parameters
Use software like Python or R to fit the GARCH(1,1) model to the returns data:
from arch import arch_model
model = arch_model(returns, vol='Garch', p=1, q=1)
result = model.fit()
print(result.summary())
The output provides estimates for \(\omega\), \(\alpha\), and \(\beta\).
Step 3: Calculate Conditional Volatility
Use the estimated parameters to compute \(\sigma_t^2\) iteratively:
\[ \sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \]Example Calculation
Assume:
- \(\omega = 0.0001\)
- \(\alpha = 0.05\)
- \(\beta = 0.90\)
- Initial \(\sigma_0^2 = 0.0002\)
- \(\epsilon_0 = 0.001\)
For \(t = 1\):
\[ \sigma_1^2 = 0.0001 + 0.05 \times (0.001)^2 + 0.90 \times 0.0002 = 0.000281 \]Volatility (\(\sigma_1\)):
\[ \sigma_1 = \sqrt{\sigma_1^2} = \sqrt{0.000281} \approx 0.0168 \, (1.68\%) \]Step 4: Interpret Results
The conditional volatility (\(\sigma_t\)) represents the portfolio's risk at time \(t\). Higher volatility indicates greater uncertainty in returns.
Practical Use
- Risk Metrics: Use \(\sigma_t\) to compute Value-at-Risk (VaR) or Expected Shortfall.
- Risk Management: Adjust portfolio weights to mitigate periods of high volatility.
