org.akutan.blacklitterman
Class BlackLitterman

java.lang.Object
  org.akutan.blacklitterman.BlackLitterman

Direct Known Subclasses:

FusaiMeucci, HeLitterman, IdzorekBLExample, KrishnanMains

public class BlackLitterman
extends java.lang.Object

This class encapsulates the top level Black-Litterman functionality in terms of backing the risk aversion out of the market portfolio, and then on to calculating the expected return with views.

Since:

24 April 2006

Field Summary

protected boolean

modifyVariance Flag indicating if posterior variance should also be calculcated, this flag must be enabled to match He and Litterman, 1999.

Constructor Summary

BlackLitterman()
Constructs

Method Summary

BLResults	applyBL(java.lang.String[] assets, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D pi, BLViews views, double tau, double delta, cern.colt.matrix.DoubleMatrix1D w) Called to apply the Black-Litterman model to the inputs in order to derive a posterior view of the asset returns and covariances.
BLResults	applyBL(java.lang.String[] assets, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D pi, BLViews views, double tau, double delta, cern.colt.matrix.DoubleMatrix1D w, cern.colt.matrix.DoubleMatrix1D refW) Called to apply the Black-Litterman model to the inputs in order to derive a posterior view of the asset returns and covariances.
double	computeDelta(double risk_premium, double sigma_m) Called to compute the risk aversion of the market portfolio.
protected cern.colt.matrix.DoubleMatrix1D	computeEr(cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D pi, BLViews views, double tau) Called to apply the Black-Litterman transform to the inputs to compute a new excess return vector assuming 100% certainty in the estimates.
protected cern.colt.matrix.DoubleMatrix1D	computeImpliedWeights(double delta, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D mu) Given the risk aversion coefficient, the covariance matrix and the excess returns we can calculate the implied weights.
double	computeInterval(double confidence, double width) This function is called to compute the variance of a distribution given a confidence level and a width.
double	computeKLIC(BLResults results) Computes the Kullback-Leibler information criteria as a measure of the distance between the prior and the posterior distributions.
protected cern.colt.matrix.DoubleMatrix1D	computeLambda(BLViews views, double tau, cern.colt.matrix.DoubleMatrix1D wEq, cern.colt.matrix.DoubleMatrix1D wStar) Computes the He-Litterman Lambda value when posterior variance varies
protected cern.colt.matrix.DoubleMatrix1D	computeLambdaN(BLViews views, double tau, double delta, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D pi) Computes the He-Litterman Lambda value when posterior variance does not vary.
cern.colt.matrix.DoubleMatrix2D	computePartialTevSensitities(double delta, BLResults results) Computes the partial sensitivites of the tracking error
double	computeTev(BLResults results) Computes the standard tracking error of the active weights.
cern.colt.matrix.DoubleMatrix1D	computeTevSensitities(double lambda, BLResults results) Computes the sensitivites of the tracking error from the Braga-Natale paper.
protected cern.colt.matrix.DoubleMatrix2D	computeV(cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D pi, BLViews views, double tau) Computes the posterior variance using a robust formula which works for any value of the view variance.
double	getViewVariance(double confidence, double width) Given the confidence specified as a decimal (area under the cdf) and the total width of the region specified in units, the variance is returned in units squared.
boolean	isModifyVariance() State of the modifyVariance flag
protected double	maxValue(cern.colt.matrix.DoubleMatrix2D m) Little helper function that runs through the matrix and finds the maximum absolute value for any entry.
double	mehalanobisDistance(double tau, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D mu, cern.colt.matrix.DoubleMatrix1D x) Called to compute the Mehalanobis distance (multidimensional Z-score) for this return vector given the mean return and variance matrix.
double	overallProbability(double tau, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D mu, cern.colt.matrix.DoubleMatrix1D x) Called to compute a probability for the updated returns given that the mehalanobis distance will be distributed as a Chi squared with n degrees of freedom.
cern.colt.matrix.DoubleMatrix1D	reverseOptimize(double delta, double rf, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D w) Computes market implied excess returns given the following parameters
cern.colt.matrix.DoubleMatrix1D	sensitivities(BLViews views, double tau, cern.colt.matrix.DoubleMatrix2D V, cern.colt.matrix.DoubleMatrix1D mu, cern.colt.matrix.DoubleMatrix1D x) Called to compute the sensitivities of the Mahalanobis Distance to the various views.
void	setModifyVariance(boolean modifyVariance) Sets state of the modifyVariance flag

Methods inherited from class java.lang.Object

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

Field Detail

modifyVariance

protected boolean modifyVariance

Flag indicating if posterior variance should also be calculcated, this flag must be enabled to match He and Litterman, 1999.

Constructor Detail

BlackLitterman

public BlackLitterman()

Constructs

Method Detail

isModifyVariance

public boolean isModifyVariance()

State of the modifyVariance flag

Returns:

true if the modifyVariance flag is true

setModifyVariance

public void setModifyVariance(boolean modifyVariance)

Sets state of the modifyVariance flag

Parameters:

modifyVariance - New value for modifyVariance flag

computeTev

public double computeTev(BLResults results)

Computes the standard tracking error of the active weights. From the paper TEV Sensitivity to Views in Black-Litterman Model, by Braga and Natale.

Parameters:

results - Container of Black-Litterman model results

Returns:

The TEV metric for the results specified.

computeTevSensitities

public cern.colt.matrix.DoubleMatrix1D computeTevSensitities(double lambda,
                                                             BLResults results)

Computes the sensitivites of the tracking error from the Braga-Natale paper.

Parameters:

lambda - The risk aversion parameter

results - Container of Black-Litterman model results.

Returns:

Vector of sensitivites by asset

computeKLIC

public double computeKLIC(BLResults results)

Computes the Kullback-Leibler information criteria as a measure of the distance between the prior and the posterior distributions. Note that in order for this function to evaluate, all the weights must be > 0.

Parameters:

results - Container of Black-Litterman model results.

Returns:

The KLIC metric

computePartialTevSensitities

public cern.colt.matrix.DoubleMatrix2D computePartialTevSensitities(double delta,
                                                                    BLResults results)

Computes the partial sensitivites of the tracking error

Parameters:

delta - Risk aversion coefficient

results - The BL results

Returns:

The matrix of TEV partial sensitivies (by asset by TODO)

computeInterval

public double computeInterval(double confidence,
                              double width)

This function is called to compute the variance of a distribution given a confidence level and a width.

Parameters:

confidence - Expressed in percent

width - of return confidence internal Expressed in percent

Returns:

The standard deviation required for a normal distribution

mehalanobisDistance

public double mehalanobisDistance(double tau,
                                  cern.colt.matrix.DoubleMatrix2D V,
                                  cern.colt.matrix.DoubleMatrix1D mu,
                                  cern.colt.matrix.DoubleMatrix1D x)

Called to compute the Mehalanobis distance (multidimensional Z-score) for this return vector given the mean return and variance matrix.

Parameters:

tau - Parameter of uncertainty of the prior estimates of the returns

V - Covariance matrix of the returns

mu - Vector of prior estimates of the returns

x - Vector of posterior estimates of the returns (to be measured).

Returns:

Mehalanobis Distance of the posterior estimates from the prior estimates

overallProbability

public double overallProbability(double tau,
                                 cern.colt.matrix.DoubleMatrix2D V,
                                 cern.colt.matrix.DoubleMatrix1D mu,
                                 cern.colt.matrix.DoubleMatrix1D x)

Called to compute a probability for the updated returns given that the mehalanobis distance will be distributed as a Chi squared with n degrees of freedom. This is from the paper by Fusai and Meucci.

Parameters:

tau - Parameter of uncertainty of the prior estimates of the returns

V - Covariance matrix for the assets

mu - Vector of prior estimates of the returns

x - Vector of posterior estimates of the returns (to be measured).

Returns:

Overall probability of that the posterior estimates are consistent with the covariance matrix and prior estimates.

sensitivities

public cern.colt.matrix.DoubleMatrix1D sensitivities(BLViews views,
                                                     double tau,
                                                     cern.colt.matrix.DoubleMatrix2D V,
                                                     cern.colt.matrix.DoubleMatrix1D mu,
                                                     cern.colt.matrix.DoubleMatrix1D x)

Called to compute the sensitivities of the Mahalanobis Distance to the various views. This is from the paper by Fuasi and Meucci.

Parameters:

views - Container of the views

tau - Parameter of uncertainty of the prior estimates of the returns

V - Covariance matrix for the assets

mu - Vector of prior estimates of the returns

x - Vector of posterior estimates of the returns (to be measured).

Returns:

Vector of sensitivites of the Mehalanobis Distance to the individual views

getViewVariance

public double getViewVariance(double confidence,
                              double width)

Given the confidence specified as a decimal (area under the cdf) and the total width of the region specified in units, the variance is returned in units squared.

Parameters:

confidence -

width -

Returns:

The variance of the view given the confidence interval

computeDelta

public double computeDelta(double risk_premium,
                           double sigma_m)

Called to compute the risk aversion of the market portfolio.

Parameters:

risk_premium - Expected risk premium of market portfolio over the risk free rate (percent).

sigma_m - Standard deviation of the market portfolio (percent).

Returns:

lamdba (risk aversion coefficient for Black-Litterman)

reverseOptimize

public cern.colt.matrix.DoubleMatrix1D reverseOptimize(double delta,
                                                       double rf,
                                                       cern.colt.matrix.DoubleMatrix2D V,
                                                       cern.colt.matrix.DoubleMatrix1D w)

Computes market implied excess returns given the following parameters

Parameters:

delta - Market portfolio implied risk aversion

rf - Long run risk free rate

V - Long run covariance matrix for the individual assets

w - Current equilibrium weights of the assets in the market portfolio

Returns:

Vector of implied excess returns from unconstrained reverse optimization

computeImpliedWeights

protected cern.colt.matrix.DoubleMatrix1D computeImpliedWeights(double delta,
                                                                cern.colt.matrix.DoubleMatrix2D V,
                                                                cern.colt.matrix.DoubleMatrix1D mu)

Given the risk aversion coefficient, the covariance matrix and the excess returns we can calculate the implied weights.

Parameters:

delta - The risk aversion coefficient

V - The matrix of asset return covariances

mu - The vector of mean expected asset returns

Returns:

Vector of implied weights

applyBL

public BLResults applyBL(java.lang.String[] assets,
                         cern.colt.matrix.DoubleMatrix2D V,
                         cern.colt.matrix.DoubleMatrix1D pi,
                         BLViews views,
                         double tau,
                         double delta,
                         cern.colt.matrix.DoubleMatrix1D w)

Called to apply the Black-Litterman model to the inputs in order to derive a posterior view of the asset returns and covariances. Also computes the implied weights for the unconstrained minimum variance portfolio.

Parameters:

assets - Array of asset names

V - Matrix of asset return covariances

pi - Vector of expected asset mean returns

views - Structure of investors views

tau - Confidence in the prior

delta - Risk aversion coefficient

w - Equilibrium asset weights

Returns:

A structure containing the results of applying the investors views to the prior.

applyBL

public BLResults applyBL(java.lang.String[] assets,
                         cern.colt.matrix.DoubleMatrix2D V,
                         cern.colt.matrix.DoubleMatrix1D pi,
                         BLViews views,
                         double tau,
                         double delta,
                         cern.colt.matrix.DoubleMatrix1D w,
                         cern.colt.matrix.DoubleMatrix1D refW)

Called to apply the Black-Litterman model to the inputs in order to derive a posterior view of the asset returns and covariances. Also computes the implied weights for the unconstrained minimum variance portfolio.

Parameters:

assets - Array of asset names

V - Matrix of asset return covariances

pi - Vector of expected asset mean returns

views - Structure of investors views

tau - Confidence in the prior

delta - Risk aversion coefficient

w - Equilibrium asset weights

Returns:

A structure containing the results of applying the investors views to the prior.

computeLambda

protected cern.colt.matrix.DoubleMatrix1D computeLambda(BLViews views,
                                                        double tau,
                                                        cern.colt.matrix.DoubleMatrix1D wEq,
                                                        cern.colt.matrix.DoubleMatrix1D wStar)

Computes the He-Litterman Lambda value when posterior variance varies

Parameters:

views - Structure of investors views

tau - Confidence in the prior/equilibrium returns

wEq - Vector of equilibrium asset weights

wStar - Vector of asset weights after views have been applied.

Returns:

Vector of lambda values (variance of views)

computeLambdaN

protected cern.colt.matrix.DoubleMatrix1D computeLambdaN(BLViews views,
                                                         double tau,
                                                         double delta,
                                                         cern.colt.matrix.DoubleMatrix2D V,
                                                         cern.colt.matrix.DoubleMatrix1D pi)

Computes the He-Litterman Lambda value when posterior variance does not vary.

Parameters:

views - Structure of investors views

tau - Confidence in prior/equilibrium returns

delta - Risk aversion coefficient

V - Matrix of covariances of asset returns

pi - Vector of expected mean asset returns

Returns:

Vector of Lambda values, variance of the view for each view

maxValue

protected double maxValue(cern.colt.matrix.DoubleMatrix2D m)

Little helper function that runs through the matrix and finds the maximum absolute value for any entry.

Parameters:

m - Input matrix

Returns:

maximum value in any cell in the matrix

computeV

protected cern.colt.matrix.DoubleMatrix2D computeV(cern.colt.matrix.DoubleMatrix2D V,
                                                   cern.colt.matrix.DoubleMatrix1D pi,
                                                   BLViews views,
                                                   double tau)

Computes the posterior variance using a robust formula which works for any value of the view variance.

Parameters:

V - Covariance matrix of expected returns

pi - Market equilibrium returns

views - List of user views

tau - Confidence parameter for the prior covariance

Returns:

New covariance matrix adjusted by the views

computeEr

protected cern.colt.matrix.DoubleMatrix1D computeEr(cern.colt.matrix.DoubleMatrix2D V,
                                                    cern.colt.matrix.DoubleMatrix1D pi,
                                                    BLViews views,
                                                    double tau)

Called to apply the Black-Litterman transform to the inputs to compute a new excess return vector assuming 100% certainty in the estimates.

Parameters:

V - Matrix of covariances of asset returns

pi - Vector of asset mean expected returns

views - Structure of investors views

tau - Confidence in the prior/equilibrium returns

Returns:

Vector of asset returns after views have been applied.