Challenges of portfolio optimization

Intermediate Portfolio Analysis in R

Ross Bennett

Instructor

Challenges

Many solvers are not specific to portfolio optimization
Understanding the capabilities and limits of solvers to select the appropriate solver for the problem or formulate the problem to fit the solver
Difficult to switch between solvers
Closed-Form solver (e.g., quadratic programming)
Global solver (e.g., differential evolution optimization)

Quadratic utility

Maximize: $\quad \displaystyle \omega^T * \mu - \lambda * \omega^T * \Sigma * \omega $
Subject to:

$$\omega_{i} >= 0$$

$$\sum_{i=1}^{n} \omega_i = 1$$

$\omega$ is the weight vector
$\mu$ is the expected return vector
$\lambda$ is the risk aversion parameter
$\Sigma$ is the variance - covariance matrix

Quadratic programming solver

Use the R package quadprog to solve the quadratic utility optimization problem
solve.QP() solves quadratic programming problems of the form:

$$min(-d^Tb+\frac{1}{2}b^TDb)$$

Subject to the constraint:

$$A^Tb>=b_0$$

library(quadprog)
data(edhec)
dat <- edhec[,1:4]

# Create the constraint matrix
Amat <- cbind(1, diag(ncol(dat)), -diag(ncol(dat)))
# Create the constraint vector
bvec <- c(1, rep(0, ncol(dat)), -rep(1, ncol(dat)))
 # Create the objective matrix
Dmat <- 10 * cov(dat)
# Create the objective vector
dvec <- colMeans(dat)

# Specify number of equality constraints
meq <- 1

# Solve the optimization problem
opt <- solve.QP(Dmat, dvec, Amat, bvec, meq)

Let's practice!

Intermediate Portfolio Analysis in R