All of the examples can be found in Jupyter notebook form here.

# 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 simulataneously. 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 = 1:N
λ = λ_vals[i]
p = minimize( λ*risk - (1-λ)*ret,
sum(w) == 1 )
solve!(p, () -> SCS.Optimizer(verbose = false))
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 = 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(verbose = false))
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 = 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(verbose = false))
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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