optimization - Maximum Certainty Equivalent Portfolio with Transaction Costs

Out of curiosity I tried to compute the portfolio weights of a maximum certainty equivalent allocation, however, by incorporating (quadratic) transaction costs. However, my result is not as intuitive as I thought =( I would be happy for each and every hint to solve this problem:

Let the parameters of the return distribution be $\Sigma$ and $\mu$. The current allocation vector is $\omega_c$. The risk aversion factor of the investor is defined as $\gamma$. When shifting his wealth to allocation $\alpha$, the investor pays a fee of the form $T = c(\alpha - \omega_c)'(\alpha - \omega_c)$ with some parameter $c$, therefore transaction costs increase quadratically by factor $c$. Therefore, at time point $t+1$ the investor expects the portfolio returns to be

$$\mu_\text{PF} = \alpha'\mu - T(\alpha,\omega_c)$$ and the corresponding variance of the portfolio $$\sigma^2_\text{PF} = \alpha'\Sigma\alpha.$$ In one line, the allocation is chosen as the solution to the maximization problem $$\alpha^* = \arg \max _{\sum \alpha = 1} \alpha'\mu - T(\alpha,\omega_c) - \frac{\gamma}{2}\alpha'\Sigma\alpha.$$ Equivalently, we have: $$\alpha^* = \omega_c + \arg \max _{\sum \Delta = 0} (\omega_c+\Delta)'\mu - c\Delta'\Delta - \frac{\gamma}{2}(\omega_c+\Delta)'\Sigma(\omega_c+\Delta).$$ $$\Delta^* = \arg \max _{\sum \Delta = 0} \underbrace{\omega_c '\mu - \frac{\gamma}{2}\omega_c'\Sigma\omega_c}_{CE(\omega_c)} +\Delta'\mu - c\Delta'\Delta - \frac{\gamma}{2}\Delta'\Sigma\Delta - \gamma \Delta'\Sigma\omega_c.$$ $$\Delta^* = \arg \max _{\sum \Delta = 0} \Delta'\mu - \Delta'\underbrace{(c I + \frac{\gamma}{2}\Sigma)}_{:=A}\Delta - \gamma \Delta'\Sigma\omega_c.$$

The first-order conditions take the form:

$$\mu - 2A\Delta - \gamma \Sigma\omega_c -\lambda\iota= 0$$ $$\iota ' \Delta = 0$$ It follows that $$A^{-1} (\mu-\gamma \Sigma \omega_c - \lambda \iota) = 2\Delta$$ Evaluating $\iota'\Delta = 0$ with $\Delta$ as above results in $$\lambda = \frac{1}{\iota' A^{-1}\iota}\iota'A^{-1}[\mu - \gamma \Sigma \omega_c]$$ Plug-in gives $$\Delta = A^{-1} (I - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}\iota) (\mu-\gamma \Sigma \omega_c ) = 0\iota.$$ In other words, no matter how sub-optimal the current allocation and irrespective of the sice of $c$, there will never be any rebalancing. I do not believe this result but I also do not see the mistake in my computations. Anyone an idea, where did I miss something/ did something wrong?

Answer

Seems like a small mistake in the last equation. It should read

$\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\right]$,

which is not equivalent to your result.