Financial Risk Manager Part 1

Explanation

For an AR(1) process of the form:

$y_t = c + \phi y_{t-1} + u_t$

where $u_t$ has variance $\sigma_u^2$, the unconditional variance is given by:

$\text{Var}(y) = \frac{\sigma_u^2}{1 - \phi^2}$

In this case:

• $\phi = 0.3$ (autoregressive coefficient)
• $\sigma_u^2 = 1$ (given disturbances have unit variance)

Substituting the values:

$\text{Var}(y) = \frac{1}{1 - (0.3)^2} = \frac{1}{1 - 0.09} = \frac{1}{0.91} \approx 1.0989$

Key points:

• The constant term (0.2) does not affect the variance calculation
• The formula only applies when $|\phi| < 1$ for stationarity
• The denominator $1 - \phi^2$ comes from the geometric series expansion of the AR(1) process