Portfolio Optimization - Markowitz Efficient Frontier

In this problem, we will find the unconstrained portfolio allocation where we introduce the weighting parameter $\lambda \;(0 \leq \lambda \leq$ 1) and minimize $\lambda * \text{risk} - (1-\lambda)* \text{expected return}$. By varying the values of $\lambda$, we trace out the efficient frontier.

Suppose that we know the mean returns $\mu \in \mathbf{R}^n$ of each asset and the covariance $\Sigma \in \mathbf{R}^{n \times n}$ between the assets. Our objective is to find a portfolio allocation that minimizes the risk (which we measure as the variance $w^T \Sigma w$) and maximizes the expected return ($w^T \mu$) of the portfolio of the simultaneously. We require $w \in \mathbf{R}^n$ and $\sum_i w_i = 1$.

This problem can be written as

\[\begin{array}{ll} \text{minimize} & \lambda*w^T \Sigma w - (1-\lambda)*w^T \mu \\ \text{subject to} & \sum_i w_i = 1 \end{array}\]

where $w \in \mathbf{R}^n$ is the vector containing weights allocated to each asset.

using Convex, SCS    #We are using SCS solver. Install using Pkg.add("SCS")

# generate problem data
μ = [11.5; 9.5; 6] / 100          #expected returns
Σ = [
    166 34 58              #covariance matrix
    34 64 4
    58 4 100
] / 100^2

n = length(μ)                   #number of assets

3

If you want to try the optimization with more assets, uncomment and run the next cell. It creates a vector or average returns and a variance-covariance matrix that have scales similar to the numbers above.

using Random Random.seed!(123)

n = 15 #number of assets, CHANGE IT?

μ = (6 .+ (11.5-6)*rand(n))/100 #mean A = randn(n,n) Σ = (A * A' + diagm(0=>rand(n)))/500; #covariance matrix

First we solve without any bounds on $w$

N = 101
λ_vals = range(0.01, stop = 0.99, length = N)

w = Variable(n)
ret = dot(w, μ)
risk = quadform(w, Σ)

MeanVarA = zeros(N, 2)
for i in 1:N
    λ = λ_vals[i]
    p = minimize(λ * risk - (1 - λ) * ret, sum(w) == 1)
    solve!(p, SCS.Optimizer; silent_solver = true)
    MeanVarA[i, :] = [evaluate(ret), evaluate(risk)]
end

Now we solve with the bounds $0\le w_i \le 1$

w_lower = 0                     #bounds on w
w_upper = 1

MeanVarB = zeros(N, 2)   #repeat, but with 0<w[i]<1
for i in 1:N
    λ = λ_vals[i]
    p = minimize(
        λ * risk - (1 - λ) * ret,
        sum(w) == 1,
        w_lower <= w,     #w[i] is bounded
        w <= w_upper,
    )
    solve!(p, SCS.Optimizer; silent_solver = true)
    MeanVarB[i, :] = [evaluate(ret), evaluate(risk)]
end

using Plots
plot(
    sqrt.([MeanVarA[:, 2] MeanVarB[:, 2]]),
    [MeanVarA[:, 1] MeanVarB[:, 1]],
    xlim = (0, 0.25),
    ylim = (0, 0.15),
    title = "Markowitz Efficient Frontier",
    xlabel = "Standard deviation",
    ylabel = "Expected return",
    label = ["no bounds on w" "with 0<w<1"],
)
scatter!(sqrt.(diag(Σ)), μ, color = :red, label = "assets")

We now instead impose a restriction on $\sum_i |w_i| - 1$, allowing for varying degrees of leverage.

Lmax = 0.5

MeanVarC = zeros(N, 2)   #repeat, but with restriction on Sum(|w[i]|)
for i in 1:N
    λ = λ_vals[i]
    p = minimize(
        λ * risk - (1 - λ) * ret,
        sum(w) == 1,
        (norm(w, 1) - 1) <= Lmax,
    )
    solve!(p, SCS.Optimizer; silent_solver = true)
    MeanVarC[i, :] = [evaluate(ret), evaluate(risk)]
end

plot(
    sqrt.([MeanVarA[:, 2] MeanVarB[:, 2] MeanVarC[:, 2]]),
    [MeanVarA[:, 1] MeanVarB[:, 1] MeanVarC[:, 1]],
    xlim = (0, 0.25),
    ylim = (0, 0.15),
    title = "Markowitz Efficient Frontier",
    xlabel = "Standard deviation",
    ylabel = "Expected return",
    label = ["no bounds on w" "with 0<w<1" "restriction on sum(|w|)"],
)
scatter!(sqrt.(diag(Σ)), μ, color = :red, label = "assets")

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