Deriving Portfolio Weights Under a Variance Constraint

# Maximization with risk-neutral investors and VaR constraints


# Maximization with risk-neutral investors and VaR constraints












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.

## Answer by Kiwiakos (score 1)

https://quant.stackexchange.com/a/25077

Regarding your question, you appear to treat variance as a linear operator, which it is not. But hard to tell as your parantheses don't match.

## Answer by Quantuple (score 1)

https://quant.stackexchange.com/a/25078

Completely agree with @Kiwiakos' remark, this is not a 'VaR' (Value-At-Risk) constraint but rather a 'Var' (Variance) constraint.

Here's how I would do it, I'll drop the superscript $B$ to keep notations uncluttered.

First, square the original constraint to obtain an inequality involving variance instead of standard deviation. Next build the Lagrangian

$$ \mathcal{L}(x_t^\prime) = E_t[ x_t^\prime R_{t+1} ] - \lambda \left( 1 - \alpha^2 \text{Var}(x_t^\prime R_{t+1}) \right) $$

Assuming no further constraints on the portfolio weights, KKT stationary condition writes:

$$ \frac{\partial \mathcal{L}}{\partial x_t^\prime}(x_t^\prime) = \frac{\partial}{\partial x_t^\prime} \left( E_t[ x_t^\prime R_{t+1} ] - \lambda \left( 1 - \alpha^2 \left( E_t[\left(x_t^\prime R_{t+1}\right)^2] - E_t[x_t^\prime R_{t+1}]^2 \right) \right) \right) = 0$$

By linearity of the expectation operator, the RHS equally writes $$ \frac{\partial \mathcal{L}}{\partial x_t^\prime}(x_t^\prime) = E_t[ R_{t+1} ] + \lambda \alpha^2 \left( E_t[2(x_t^\prime R_{t+1})R_{t+1}] - 2 E_t[x_t^\prime R_{t+1}] E_t[R_{t+1}] \right) $$

Finally, because the portfolio weights $x_t$ are known at time $t$ \begin{align} \frac{\partial \mathcal{L}}{\partial x_t^\prime}(x_t^\prime) &= E_t[ R_{t+1} ] + 2 x_t^\prime \lambda \alpha^2 \left( E_t[R_{t+1}^2] - E_t[R_{t+1}]^2 \right) \\ &= E_t[ R_{t+1} ] + \tilde{\lambda} x_t^\prime \text{Var}(R_{t+1}) \end{align}

Is that the result given in you referenced paper? I can't quite confirm since it appears to be behind a paywall.