# Mean-cVaR model: How can one include transaction cost

$$\min \delta CVaR - (1-\delta) \sum_i^{n} \mu_i x_i \\ \sum x_i = \sum x^{old}_i \\ Losses(s) = \sum x_i - \sum_i^{n} (R(s,i))x_i \\ VaRDev(s) = Losses(s) - VaR \\ CVaR = VaR + \frac{\sum_s^{} p_s VaRDev(s)}{1-\alpha}$$

The Markowitz style Mean-cVaR portfolio is stated above. It Appears in Practical financial optimization by Zenios. $$x^{old}$$ is exciting portfolio and now you want to optimize and create a new portfolio $$x$$. $$s$$ is the different scenarios.

But what if transaction costs existing so you should pay $$c_{variable}$$ % of how much you trade an asset and each trade have a minimum cost as well. How can we add transaction cost to this model?

• I assume that people who knows this model can figure out what the variables stands for. Otherwise please let me know if there anything I need to clarify. – Sanjay Dec 6 '18 at 11:40
• If you know how to include transaction costs in a mean variance optimization, then it's trivial to add it to a mean-cvar one. – John Dec 6 '18 at 14:29
• You could write the weights $x_i= x^{old}_i+u_i-v_i$ where u represents purchases and v represents sales (with both u and v constrained to be >=0). Now the transaction cost can be written as $cu_i+cv_i$. – noob2 Dec 7 '18 at 5:14