Financial Risk Manager Part 1

Explanation

Given the MA(2) model: $Y_t = 0.1 + 0.8\epsilon_{t-1} + 0.16\epsilon_{t-2} + \epsilon_t$

Using the lag operator $L$ where $LY_t = Y_{t-1}$:

• $\epsilon_{t-1} = L\epsilon_t$
• $\epsilon_{t-2} = L^2\epsilon_t$
• $\epsilon_t = L^0\epsilon_t = 1\cdot\epsilon_t$

Substituting into the model: $Y_t = 0.1 + 0.8(L\epsilon_t) + 0.16(L^2\epsilon_t) + (1\cdot\epsilon_t)$

Factoring out $\epsilon_t$: $Y_t = 0.1 + \epsilon_t(0.8L + 0.16L^2 + 1)$

This matches option C exactly. The constant term 0.1 remains unchanged since the lag operator doesn't affect constants.

Key points:

• The lag operator $L$ shifts the time index backward
• Constants are unaffected by the lag operator
• The polynomial in $L$ represents the moving average coefficients
• The model can be written as $Y_t = \mu + \theta(L)\epsilon_t$ where $\theta(L)$ is the MA polynomial