1CongNghe.Com: Code for Markowitz Efficient Frontier Calculation using Black’s Shortcut

Thursday, August 12, 2010

Code for Markowitz Efficient Frontier Calculation using Black’s Shortcut



#include


// This pseudo-code class entails the risk and expected return values for efficient frontier. In real code, accessors and mutators should be provided
class EfficientFrontier()
{
	float[] return;	// series of expected returns on efficient portfolios
	float[] risk;	// series of standard deviation of returns of efficient portfolios
	float gmvpReturn;	// expected return of Global Minimum Variance Portfolio
	float gmvpRisk;	// standard deviation of returns of Global Minimum Variance Portfolio
	float tangentReturn;		// expected return of the efficient portfolio tangent to Capital Market Line
	float tangentRisk;	// standard deviation of returns of the efficient portfolio tangent to Capital Market Line
	float shortConstraintMaxReturn;	// maximum expected return that an efficient portfolio can reach
	float shortConstraintMaxRisk;	// standard deviation of returns of the efficient portfolio with maxmimum return
}

// This pseudo-code function produces position of Markowitz Efficient Frontier
EfficientFrontier getEfficientFrontier(float[][] price)
{
	const float c1 = 0.1;	// first substitute of risk-free rate for Black's shortcut
	const float c2 = 0.2;	// second substitute of risk-free rate for Black's shortcut
	float[] annualReturn = annualReturn(price);
	float[][] annualVarcovar = annualVarcovarMatrix(price);
	annualVarcovar = InverseMatrix(annualVarcovar);

	for (i = 0 ; i < NUMBER_OF_VARIABLES ; i++)
	{
		for (j = 0 ; j < NUMBER_OF_VARIABLES ; j++)
		{
			z1[i] += annualVarcovar[i][j] * (annualReturn[j] - c1);
		}
		sumz1 += z1[i];
		x1[i] = z1[i] / sumz1;
	}

	// repeat for z2 & x2;

	float expectedReturn1, expectedReturn2;

	for (i = 0 ; i < NUMBER_OF_VARIABLES ; i++)	// calculate expected return
	{
		expectedReturn1 += annualReturn[i] * x[i];
	}

	// repeat for expectedReturn2;

	variance1 = z1[] * annualVarcovar * z1[];	// this is a shorthand for tri-matrices multiplication. Actual code should extend the calculation of matrix multiplication

	// repeat for variance2;

	covariance = z1[] * annualVarcovar * z2[];

	return = discreteReturn(price);
	risk = stdDev(price);

	for (i = 0 ; i < NUMBER_OF_OBSERVATIONS ; i++)
	{
		if (maxReturn < return[i])	maxReturn = return[i];
		if (maxRisk < risk[i])	maxRisk = risk[i];
		// the maximum return and standard deviation here are not exactly value due to limitation of the method
	}

	do
	{
		w2 = 1 - w1;	// rebalance the two asset portfolio
		EfficientFrontier.return[i] = w1 * expectedReturn1;	// for demonstration only, a mutator should be used
		EfficientFrontier.risk[i] = (w1^2 * variance1 + w2^2 * variance2 + 2 * w1 * w2 * covariance)^(1/2);	// for demonstration only, a mutator should be used
		w1 += INCREMENT;
		i++;
	}
	while(EfficientFrontier.return[i] <= maxReturn);	// for demonstration only, an accessor should be used
}