To optimize portfolio selection using a specific utility function, we employ the concepts of Modern Portfolio Theory (MPT) by Markowitz. In this framework, we assume rational investors who aim to maximize their expected utility from investment returns.
Let’s consider a single-period investment framework with N assets. The return of each asset is denoted by a random variable Ri, and their expected returns and variances are μi and σi2, respectively, for i = 1, ..., N. The vector of the expected returns is given by μ = [μ1, μ2, ..., μN]T and the covariance matrix of the returns is represented by Σ, with σij = Cov(Ri, Rj).
Assume the investor is considering a certain portfolio denoted by the vector w = [w1, w2, ..., wN]T, where wi represents the proportion of the portfolio invested in asset i. The sum of the weights must be equal to 1, i.e., $\sum_{i=1}^{N} w_i = 1$.
Given the investment universe, the investor wants to find the optimal portfolio allocation, w*, that maximizes their expected utility. To achieve this, the investor adopts a utility function, U( ⋅ ), that represents their preferences over risky assets. Commonly used utility functions are Power utility, Logarithmic utility, and Exponential utility function.
Now, let’s solve the optimization problem with a general utility function, U:
w* = arg maxwU(w)
subject to the constraints:
$$\sum_{i=1}^{N} w_i = 1$$
Every utility function depends on the expected return and risk of the portfolio, denoted as E[Rp] and σp, respectively. Thus, we can rewrite the optimization problem as:
w* = arg maxwU(E[Rp], σp)
For example, let’s consider an investor with the following mean-variance utility function:
$$U(E[R_p], \sigma_p) = E[R_p] - \frac{1}{2} A \sigma_p^2$$
Here, A represents the investor’s risk aversion level. To solve the optimization problem, we apply the following steps:
1. Express expected return and risk for the given portfolio:
E[Rp] = μTw
σp2 = wTΣw
2. Substitute E[Rp] and σp2 in the utility function and simplify:
$$U(\mathbf{w}) = \boldsymbol{\mu}^T\mathbf{w} - \frac{1}{2} A \mathbf{w}^T\mathbf{\Sigma} \mathbf{w}$$
3. Apply optimization techniques like the Lagrange multiplier method to calculate the optimal weights, w*:
$$\mathcal{L}(\mathbf{w}, \lambda) = \boldsymbol{\mu}^T\mathbf{w} - \frac{1}{2} A \mathbf{w}^T\mathbf{\Sigma} \mathbf{w} + \lambda \left(1 - \sum_{i=1}^{N} w_i\right)$$
4. Solve for w* by taking the first-order conditions ($\frac{\partial \mathcal{L}}{\partial \mathbf{w}} = 0$ and $\frac{\partial \mathcal{L}}{\partial \lambda} = 0$) and solving the resulting system.
Once you solve the system, you will obtain the optimal weights, w*, for the given utility function. Adapt this procedure to other utility functions to find the optimal portfolio for various investor preferences.