Portfolio Optimization

Find the optimal portfolio weights that balance expected return and risk based on simulated asset data

In the Markowitz portfolio model, the goal is to balance expected return against risk. Each asset has its own average performance and volatility, but the overall portfolio behavior also depends on how the assets move together. When assets are imperfectly correlated, losses in some can be offset by gains in others, which allows diversification to reduce total risk.

Simulate daily percentage returns for six assets from a normal distribution and smooth each series with a moving average to reduce noise and obtain clearer return patterns:

In[1]:=

SeedRandom[123];

In[2]:=

n=6;returns=RandomVariate[NormalDistribution[20,50],{6000,n}];timeSeries=MovingAverage[#,300]&/@Transpose[returns];

Plot a few of the smoothed return series to visualize how the simulated assets fluctuate around their mean and differ in volatility:

In[3]:=

ListLinePlot[timeSeries〚1;;3〛,PlotLegends{"Asset 1","Asset 2","Asset 3"},FrameTrue,FrameLabel{"Time","Smoothed Return"},PlotRangeAll]

Out[3]=

Compute the average return of each asset and the covariance matrix, which captures how the assets move together:

In[4]:=

μ=Mean/@timeSeries;

In[5]:=

Σ=Covariance[Transpose@timeSeries];

Define functions for portfolio return and portfolio risk (variance) based on asset weights:

In[6]:=

portfolioReturn[w_]:=μ.w;

In[7]:=

portfolioRisk[w_]:=w.Σ.w;

Introduce an auxiliary variable  that limits the portfolio's total variance:

In[8]:=

riskConstraint[s_,w_]:=portfolioRisk[w]≤s;

Define the objective of maximizing expected return while penalizing risk through a risk-aversion parameter α:

In[9]:=

objectiveFunction[α_?NumericQ,s_,w_]:=portfolioReturn[w]-αs;

Model trading cost with a convex penalty that grows faster than linearly in the position size. For each weight

𝑤

𝑖

introduce an auxiliary variable

𝑡

𝑖

so that

1.5



𝑤

𝑖



is represented through simple inequality constraints:

In[10]:=

tVars=Array[t,n];

In[11]:=

marketImpactConstraints[w_]:=Thread[Abs[w]^1.5≤tVars];

Include a budget condition where both invested capital and the scaled trading cost must add up to one:

In[12]:=

budgetConstraint[w_]:=Total[w]+0.1Total[tVars]1;

Define the optimization variables and combine the risk bound, budget condition, market-impact constraints, and non-negativity requirements:

In[13]:=

wVars=Array[w,n];

In[14]:=

allConstraints[α_]:={riskConstraint[s,wVars],budgetConstraint[wVars],marketImpactConstraints[wVars],wVars≥0,s≥0,tVars≥0};

Set up the optimization that returns the optimal portfolio for a given value of α:

In[15]:=

solution[α_?NumericQ]:=NMaximize[{objectiveFunction[α,s,wVars],allConstraints[α]},Flatten[{wVars,s,tVars}]]

Compute the optimal risk–return pairs across a range of risk-aversion parameter values. Each solution gives an optimal portfolio, from which we extract its risk and expected return:

In[16]:=

αValues=Subdivide[0.1,10,100];

In[17]:=

frontier=Table[res=solution[α];{portfolioRisk[wVars]/.res〚2〛,portfolioReturn[wVars]/.res〚2〛},{α,αValues}];

In[18]:=

Show[{ListLinePlot[frontier,FrameTrue,FrameLabel{"Risk","Return"},PlotRangeAll,ImageSize400],ListPlot[frontier,PlotStyleLighter[Red,0.55]]}]

Out[18]=

Compute the weights for a specified collection of risk-aversion parameter values:

In[19]:=

riskAversionParameter=N@{1/1000,1/100,1/2,1,5,10,15,20};res=Table[{α,Quiet@NMaximize[{objectiveFunction[α,s,wVars],riskConstraint[s,wVars],budgetConstraint[wVars],marketImpactConstraints[wVars]},Flatten[{wVars∈Vectors[6,Reals],s,tVars∈Vectors[6,Reals]}]]〚2〛},{α,riskAversionParameter}];

A diversified portfolio can be obtained for low risk aversion, but when the risk aversion is high, the market impact cost dominates, due to purchasing less diversified stock:

In[20]:=

LegendedBarChart[Append[#,1-Total[#]]&/@res〚All,-1,1;;6,-1〛,ChartLayout"Stacked",ChartStyleColorData[3,"ColorList"]〚2;;8〛,ChartLabels{riskAversionParameter,None},AxesFalse,FrameTrue,FrameStyleDirective[Black,12],FrameLabel{"Risk Aversion Parameter","Fraction of budget"},ImageSize300],

SwatchLegend[

]



Out[20]=

Publisher Information

Contributed by: Wolfram Staff

Wolfram Language Example Repository

Portfolio Optimization

See Also

Publisher Information

Portfolio Optimization

See Also

Related Symbols

Publisher Information