I'm having trouble replicating the result from the article Loss Aversion with Multiple Investment Goals written by Enrico G. De Giorgi (2011), the article can be downloaded here.
First consider the choice function proposed by Kahneman and Tversky in prospect theory
$$ u(x) = \begin{cases} \beta^+ x^\alpha, & x > 0 \\ -\beta^-(-x)^\alpha, & x \leq 0 \end{cases} $$
where $\beta^+$ and $\beta^-$ characterize loss aversion.
The main optimization problem is
\begin{cases} \text{max}_{W_T} & E[u(W_T- \overline W)] \\ \text{such that: } & E[\xi_T W_T] \leq \xi_0 W_0 \\ &W_T \geq 0 \\ \end{cases} where $W_0$ is the initial wealth, $W_T$ is the wealth at time horizon T, $\overline W$ is the goal wealth and $\xi$ is the pricing kernel, i.e. stochastic discount factor ($\xi_0 = 1$).
I have two problems:
- Determine the optimal Lagrange multiplier, y, when applying Lagrange relaxation to this optimization problem ($y\geq 0$).
- I don't understand the stochastic discount factor, $\xi_T$.
The Lagrange relaxation mentioned in the $\underline{\text{first problem}}$ (after some algebra) leads to the following equation representing the lagrange multiplier y:
$${ y = \left(\frac{\xi_0 W_0 - \overline W E[\xi_T]}{(\alpha \beta^+)^{\frac{1}{1-\alpha}} E\left[\xi_T^{\frac{\alpha}{\alpha-1}}\right]} \right)^{(\alpha -1)}}$$
and since $y \geq 0$ implies that $\xi_0 W_0$ > $\overline W E[\xi_T]$, which leads to the second problem.
Regarding the $\underline{\text{second problem}}$; if $W_T$ is discounted with discount factor $e^{-r(T-t)}$ and fulfills the budget constraint it is obvious that the terminal wealth $W_T$ can be achieved by simply putting the wealth into a bank account with rate $r$. If this constraint has to be statisfied then one cannot achieve a goal more ambitious than a payoff with rate $r$.
Best regards, Will