tech.nmfin.portfoliooptimization.markowitz

Class MarkowitzByQP

java.lang.Object

tech.nmfin.portfoliooptimization.markowitz.MarkowitzByQP

Direct Known Subclasses:

MarkowitzByCLM

public class MarkowitzByQP
extends Object

Modern portfolio theory (MPT) is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets.

For a given "risk aversion", q, the mathematical formulation is the following Markowitz’s mean-variance formulation: \[ \begin{aligned} w_{\mathrm{eff}} &=\operatorname*{arg\,max}_{w} \left\{\mathbb{E}[w^{\mathsf T}r]-q\,\operatorname{Var}(w^{\mathsf T}r)\right\}\\ &=\operatorname*{arg\,min}_{w} \left\{q\,\operatorname{Var}(w^{\mathsf T}r)-\mathbb{E}[w^{\mathsf T}r]\right\}\\ &=\operatorname*{arg\,min}_{w} \left\{q\,w^{\mathsf T}\Sigma w-w^{\mathsf T}r\right\}. \end{aligned} \] s.t., \(w^{\mathsf T}\mathbf{1}=1\), and any additional constraints on \(w\). q is the slope of a portfolio on the efficient frontier.

Input data and solve lifecycle

1. Align asset order across expected returns, covariance rows/columns, and weight bounds. Use fractional returns, not percentages. Expected returns, covariances and the benchmark must refer to the same horizon.
2. Supply a symmetric, well-conditioned covariance estimate. The implementation may adjust a Hessian that is not numerically positive definite; do not treat that adjustment as a replacement for estimating a valid covariance matrix.
3. Construct the optimizer, then call setRiskAversionCoefficient(double) with a positive q before reading getOptimalWeights() or fw(). The weights are null before a successful solve.
4. Check the budget and all bounds using tolerances, then independently compute return and variance with PortfolioUtils.

The unity constraint is added automatically. q penalizes variance, not standard deviation. Holding expected returns fixed while scaling covariance changes the effective risk penalty. A benchmark changes Sharpe-ratio calculations, not the fixed-q objective itself. Treat inputs as immutable while optimizing.

Each risk-aversion setter solves again and replaces the stored weights. Retain the returned weight vector for each frontier point before the next solve. If a solve throws, do not assume a previously stored portfolio solves the newly requested q. Inconsistent bounds or other constraints can make the QP infeasible. This class neither forecasts returns nor backtests rebalancing, turnover, fees, slippage or cash flows.

A fully invested portfolio with explicit bounds

Use estimates for a common horizon, request the solve explicitly, then check feasibility and reconcile the objective sign.

 import dev.nm.algebra.linear.matrix.doubles.matrixtype.dense.DenseMatrix;
 import dev.nm.algebra.linear.vector.doubles.Vector;
 import dev.nm.algebra.linear.vector.doubles.dense.DenseVector;
 import tech.nmfin.portfoliooptimization.markowitz.MarkowitzByQP;
 import tech.nmfin.portfoliooptimization.PortfolioUtils;
 DenseVector expectedReturns = new DenseVector(0.08, 0.12);
 DenseMatrix covariance = new DenseMatrix(new double[][]{{0.04, 0}, {0, 0.09}});
 MarkowitzByQP optimizer = new MarkowitzByQP(expectedReturns, covariance,
     new DenseVector(0.2, 0.2), new DenseVector(0.8, 0.8));
 assert optimizer.getOptimalWeights() == null;
 optimizer.setRiskAversionCoefficient(1.0);
 Vector weights = optimizer.getOptimalWeights();
 assert Math.abs(weights.get(1) + weights.get(2) - 1) < 1e-7;
 for (int i = 1; i <= weights.size(); ++i) {
     assert weights.get(i) >= 0.2 - 1e-7 && weights.get(i) <= 0.8 + 1e-7;
 }
 double expectedReturn = PortfolioUtils.getPortfolioReturns(weights, expectedReturns);
 double variance = PortfolioUtils.getPortfolioVariance(weights, covariance);
 assert Math.abs(optimizer.fw() - (variance - expectedReturn)) < 1e-10;
 

See Also:

• Wikipedia: Modern portfolio theory
• "Tze Leung Lai, Haipeng Xing, Zehao Chen. "Eq. 1.2, Mean-variance portfolio optimization when means and covariances are unknown," Annals of Applied Statistics 2011, Vol. 5, No. 2A, 798-823"

Constructor Summary

Constructors

Constructor and Description
MarkowitzByQP(Vector mu, Matrix sigma) Constructs a Markowitz portfolio from expected future returns and future covariance, assuming no short selling constraint and zero benchmark rate.
MarkowitzByQP(Vector mu, Matrix sigma, QPConstraint constraints) Constructs a Markowitz portfolio from expected future returns and future covariance, assuming zero benchmark rate for Sharpe ratio calculation.
MarkowitzByQP(Vector mu, Matrix sigma, QPConstraint constraints, double benchmarkRate) Constructs a Markowitz portfolio from expected future returns and future covariance.
MarkowitzByQP(Vector mu, Matrix sigma, Vector lower, Vector upper) Constructs a Markowitz portfolio from expected future returns and future covariance, with lower and upper limits on asset weights, assuming zero benchmark rate.
MarkowitzByQP(Vector mu, Matrix sigma, Vector lower, Vector upper, double benchmarkRate) Constructs a Markowitz portfolio from expected future returns and future covariance for a given benchmark rate, with lower and upper limits on asset weights.

Method Summary

Modifier and Type	Method and Description
double	fw() Evaluates the minimized objective \(q\operatorname{Var}(w^{\mathsf T}r)-\mathbb{E}[w^{\mathsf T}r]\) at the stored weights.
double	getOptimalRiskAversionCoefficient() Gets the optimal risk aversion coefficient w.r.t.
double	getOptimalRiskAversionCoefficient(double lower, double initial, double upper) Gets the optimal risk aversion coefficient w.r.t.
protected Vector	getOptimalW(double q) Solves w_eff = argmin {q * (w' Σ w) - w'r}.
Vector	getOptimalWeights() Gets the Markowitz optimal portfolio weights, for a given risk aversion coefficient.
double	getRiskAversionCoefficientForTargetReturn(double r, double lower, double upper, int maxIterations)
double	getRiskAversionCoefficientForTargetVariance(double var, double lower, double upper, int maxIterations)
void	setRiskAversionCoefficient(double q) Sets the risk aversion coefficient, effectively moving along the efficient frontier.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

MarkowitzByQP

public MarkowitzByQP(Vector mu,
                     Matrix sigma,
                     QPConstraint constraints)

Constructs a Markowitz portfolio from expected future returns and future covariance, assuming zero benchmark rate for Sharpe ratio calculation.

Parameters:

mu - expected future returns of assets

sigma - future covariance of assets

constraints - the additional constraints on w_eff

MarkowitzByQP

public MarkowitzByQP(Vector mu,
                     Matrix sigma,
                     QPConstraint constraints,
                     double benchmarkRate)

Constructs a Markowitz portfolio from expected future returns and future covariance.

Parameters:

mu - expected future returns of assets

sigma - future covariance of assets

constraints - the additional constraints on w_eff

benchmarkRate - the benchmark rate used to compute Sharpe ratio

MarkowitzByQP

public MarkowitzByQP(Vector mu,
                     Matrix sigma)

Constructs a Markowitz portfolio from expected future returns and future covariance, assuming no short selling constraint and zero benchmark rate.

Parameters:

mu - expected future returns of assets

sigma - future covariance of assets

MarkowitzByQP

public MarkowitzByQP(Vector mu,
                     Matrix sigma,
                     Vector lower,
                     Vector upper)

Constructs a Markowitz portfolio from expected future returns and future covariance, with lower and upper limits on asset weights, assuming zero benchmark rate.

Parameters:

mu - expected future returns of assets

sigma - future covariance of assets

lower - the lower limits of asset weights

upper - the upper limits of asset weights

MarkowitzByQP

public MarkowitzByQP(Vector mu,
                     Matrix sigma,
                     Vector lower,
                     Vector upper,
                     double benchmarkRate)

Constructs a Markowitz portfolio from expected future returns and future covariance for a given benchmark rate, with lower and upper limits on asset weights.

Parameters:

mu - expected future returns of assets

sigma - future covariance of assets

lower - the lower limits of asset weights

upper - the upper limits of asset weights

benchmarkRate - the benchmark rate used to compute Sharpe ratio

Method Detail

setRiskAversionCoefficient

public void setRiskAversionCoefficient(double q)
                                throws QPInfeasible,
                                       Exception

Sets the risk aversion coefficient, effectively moving along the efficient frontier. Recompute the optimal weights.

Parameters:

q - the risk aversion coefficient

Throws:

QPInfeasible

Exception

getOptimalW

protected Vector getOptimalW(double q)
                      throws Exception

Solves w_eff = argmin {q * (w' Σ w) - w'r}.

Parameters:

q - the risk-aversion coefficient

Returns:

the optimal weights w_eff

Throws:

QPInfeasible - when the underlying quadratic programming problem is infeasible

Exception

getOptimalWeights

public Vector getOptimalWeights()

Gets the Markowitz optimal portfolio weights, for a given risk aversion coefficient.

Returns:

the Markowitz optimal portfolio weights

fw

public double fw()

Evaluates the minimized objective \(q\operatorname{Var}(w^{\mathsf T}r)-\mathbb{E}[w^{\mathsf T}r]\) at the stored weights.

Returns:

risk penalty minus expected return; the negative of mean-variance utility

getOptimalRiskAversionCoefficient

public double getOptimalRiskAversionCoefficient(double lower,
                                                double initial,
                                                double upper)
                                         throws Exception

Gets the optimal risk aversion coefficient w.r.t. Sharpe ratio.

Parameters:

lower - lower bound for the search, e.g., 0

initial - an initial guess of where the optimal risk aversion coefficient may be

upper - upper bound for the search

Returns:

the optimal risk aversion coefficient

Throws:

Exception

getOptimalRiskAversionCoefficient

public double getOptimalRiskAversionCoefficient()

Gets the optimal risk aversion coefficient w.r.t. Sharpe ratio.

Returns:

the optimal risk aversion coefficient

getRiskAversionCoefficientForTargetReturn

public double getRiskAversionCoefficientForTargetReturn(double r,
                                                        double lower,
                                                        double upper,
                                                        int maxIterations)
                                                 throws NoRootFoundException

Throws:

NoRootFoundException

getRiskAversionCoefficientForTargetVariance

public double getRiskAversionCoefficientForTargetVariance(double var,
                                                          double lower,
                                                          double upper,
                                                          int maxIterations)
                                                   throws NoRootFoundException

Throws:

NoRootFoundException