4  Description and justification of the Cost

he cost function in this investment model combines the expected return of the portfolio and the penalty for risk, capturing the trade-off between risk and return. The objective is to maximize a CRRA function over his wealth over the investment horizon, \(t = 0, 1..., T\). \[ U_t = E_t(W_T^{1-\gamma}) \]

If \(\gamma\) > 1 the investor tries to avoid risk and he is intrested about the variance of returns as well as expected return; with higher values of the parameter associated with greater risk aversion. If \(\gamma\) = 1, the investor is risk neutral and behaves so as to maximise the log of terminal wealth. If γ < 1, the investor is willing to risk in search to maximaze his wealth. In this project we are intrested in \(\gamma \geq 1\).

The objective is identical to a mean-variance objective. So we can write it as \[ U_t = W_t^{1-\gamma} E_t\left( \left( \prod_{i=t+1}^T w_{i-1}e^{r_i} \right)^{1-\gamma} \right) \]

If we denote Cov(rt) = Σ and the diagonal elements of the covariance matrix by the n-vector σ2 = diag(Σ) then we can aproximate to

\[ U_t \approx W_t^{1-\gamma}E_t \left( \left( \prod_{i=t+1}^T e^{w_{i-1}r_i +\frac{1}{2}w_{i-1}(\sigma^2 - \Sigma w_{i-1})} \right)^{1-\gamma} \right) \]

\[ = W_t^{1-\gamma} E_t \left( e^{(1-\gamma)\sum_{i=t+1}^Tw_{i-1}(\sigma^2 - \Sigma w_{i-1})} \right) \]

We have now aproximated out objetive to as the expected value of a exponential of a sum of period cost functions.

\[ C_i := w_{i-1}r_i + \frac{1}{2}w_{i-1}(\sigma^2 -\Sigma w_{i-1}) \]