In this paper, the authors make a simple model with:
(1) A global bank, who is risk-neutral but has a Value-at-Risk constraint:
$$\max_{x_t^B} E_t[x_t^B\prime R_{t+1}]$$ s.t. $$\alpha (Var(x_t^B\prime R_{t+1}))^{\frac{1}{2}} <= 1$$ where $R_{t+1}$ is a (n x 1) vector of returns, $x_t^B$ is a (n x 1) vector of weights $\alpha$ is a parameter, and $Var$ is the variance operator.
I tried setting up a lagrangean which should yield:
$$ \mathcal{L} = E_t[x_t^B\prime R_{t+1}] - \lambda_t (1 - \alpha (Var(x_t^B\prime R_{t+1}))^{\frac{1}{2}}) $$
The first order conditions w.r.t. $x_t^B$ are yielding me:
$$ E_t(R_{t+1}) - \lambda_t \alpha Var(x_t^B\prime R_{t+1})^{-\frac{1}{2}} Var(R_{t+1}) x_t^B = 0 $$
In comparison with the solution given on the paper it seems that I have an extra term $ Var(x_t^B\prime R_{t+1})^{-\frac{1}{2}}$.
Can anyone help me on this? Thanks.