portfolio.C

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/// \file
/// \ingroup tutorial_quadp
/// \notebook
/// This macro shows in detail the use of the quadratic programming package quadp .
/// Running this macro :
///
/// ~~~{.cpp}
///     .x portfolio.C+
/// ~~~
///
/// or
///
/// ~~~{.cpp}
/// gSystem->Load("libQuadp");
/// .L portFolio.C+; portfolio()
/// ~~~
///
/// Let's first review what we exactly mean by "quadratic programming" :
///
/// We want to minimize the following objective function :
///
///   \f$ c^T x + ( 1/2 ) x^T Q x \f$  wrt. the vector \f$ x \f$
///
///  \f$ c \f$ is a vector and \f$ Q \f$ a symmetric positive definite matrix
///
/// You might wonder what is so special about this objective which is quadratic in
/// the unknowns, that can not be done by Minuit/Fumili . Well, we have in addition
/// the following boundary conditions on \f$ x \f$:
///
///  \f[
///          A x =  b \\
///  clo \le  C x \le cup \\
///  xlo \le    x \le xup
///  \f]  where A and C are arbitrary matrices and the rest are vectors
///
/// Not all these constraints have to be defined . Our example will only use \f$ xlo \f$,
/// \f$ A \f$ and \f$ b \f$
/// Still, this could be handled by a general non-linear minimizer like Minuit by introducing
/// so-called "slack" variables . However, quadp is tailored to objective functions not more
/// complex than being quadratic . This allows usage of solving techniques which are even
/// stable for problems involving for instance 500 variables, 100 inequality conditions
/// and 50 equality conditions .
///
/// Enough said about quadratic programming, let's return to our example .
/// Suppose, after a long day of doing physics, you have a look at your investments and
/// realize that an early retirement is not possible, given the returns of your stocks .
/// So what now ? ROOT to the rescue ...
///
/// In 1990 Harry Markowitz was awarded the Nobel prize for economics: " his work provided new tools
/// for weighing the risks and rewards of different investments and for valuing corporate stocks and bonds" .
/// In plain English, he developed the tools to balance greed and fear, we want the maximum return
/// with the minimum amount of risk. Our stock portfolio should be at the
/// ["Efficient Frontier"](see http://www.riskglossary.com/articles/efficient_frontier.htm).
/// To quantify better the risk we are willing to take, we define a utility function \f$ U(x) \f$. It describes
/// as a function of our total assets \f$ x \f$, our "satisfaction" . A common choice is \f$ 1-exp(-k*x) \f$ (the reason for
/// the exponent will be clear later)  . The parameter \f$ k \f$ is the risk-aversion factor . For small values of \f$ k \f$
/// the satisfaction is small for small values of \f$ x \f$; by increasing \f$ x \f$ the satisfaction can still be increased
/// significantly . For large values of \f$ k \f$, \f$ U(x) \f$ increases rapidly to 1, there is no increase in satisfaction
/// for additional dollars earned .
///
/// In summary :
///  - small \f$ k \f$ ==> risk-loving investor
///  - large \f$ k \f$ ==> risk-averse investor
///
/// Suppose we have for nrStocks the historical daily returns \f$ r = closing_price(n) - closing_price(n-1) \f$.
/// Define a vector \f$ x \f$ of length of \f$ nrStocks \f$, which contains the fraction of our money invested in
/// each stock . We can calculate the average daily return \f$ z \f$ of our portfolio and its variance using
/// the portfolio covariance Covar :
///
///  \f$ z = r^T x \f$  and \f$ var = x^T Covar x \f$
///
/// Assuming that the daily returns have a Normal distribution, \f$ N(x) \f$, so will \f$ z \f$ with mean \f$ r^T x \f$
/// and variance \f$ x^T Covar x \f$
///
/// The expected value of the utility function is :
///
///  \f[
/// E(u(x)) = Int (1-exp(-k*x) N(x) dx \\
///         = 1-exp(-k (r^T x - 0.5 k x^T Covar x) ) \\
///  \f]
///
/// Its value is maximised by maximising \f$ r^T x -0.5 k x^T Covar x \f$
/// under the condition \f$ sum (x_i) = 1 \f$, meaning we want all our money invested and
/// \f$ x_i \ge 0 \f$, we can not "short" a stock
///
/// For 10 stocks we got the historical daily data for Sep-2000 to Jun-2004:
///
///  - GE   : General Electric Co
///  - SUNW : Sun Microsystems Inc
///  - QCOM : Qualcomm Inc
///  - BRCM : Broadcom Corp
///  - TYC  : Tyco International Ltd
///  - IBM  : International Business Machines Corp
///  - AMAT : Applied Materials Inc
///  - C    : Citigroup Inc
///  - PFE  : Pfizer Inc
///  - HD   : Home Depot Inc
///
/// We calculate the optimal portfolio for 2.0 and 10.0 .
///
/// Food for thought :
/// - We assumed that the stock returns have a Normal distribution . Check this assumption by
///   histogramming the stock returns !
/// - We used for the expected return in the objective function, the flat average over a time
///   period . Investment firms will put significant resources in improving the return prediction .
/// - If you want to trade significant number of shares, several other considerations have
///   to be taken into account :
///    +  If you are going to buy, you will drive the price up (so-called "slippage") .
///       This can be taken into account by adding terms to the objective
///       (Google for "slippage optimization")
///    +  FTC regulations might have to be added to the inequality constraints
/// - Investment firms do not want to be exposed to the "market" as defined by a broad
///   index like the S&P and "hedge" this exposure away . A perfect hedge this can be added
///    as an equality constrain, otherwise add an inequality constrain .
///
/// \macro_image
/// \macro_output
/// \macro_code
///
/// \author Eddy Offermann
 
#include "Riostream.h"
#include "TCanvas.h"
#include "TFile.h"
#include "TMath.h"
#include "TTree.h"
#include "TArrayF.h"
#include "TH1.h"
#include "TF1.h"
#include "TLegend.h"
#include "TSystem.h"
 
#include "TMatrixD.h"
#include "TMatrixDSym.h"
#include "TVectorD.h"
#include "TQpProbDens.h"
#include "TGondzioSolver.h"
 
const Int_t nrStocks = 10;
static const Char_t *stocks[] =
     {"GE","SUNW","QCOM","BRCM","TYC","IBM","AMAT","C","PFE","HD"};
 
class TStockDaily  {
public:
  Int_t fDate;
  Int_t fOpen;     // 100*open_price
  Int_t fHigh;     // 100*high_price
  Int_t fLow;      // 100*low_price
  Int_t fClose;    // 100*close_price
  Int_t fVol;
  Int_t fCloseAdj; // 100*close_price adjusted for splits and dividend
 
  TStockDaily() {
     fDate = fVol = fOpen = fHigh = fLow = fClose = fCloseAdj = 0;
  }
  virtual ~TStockDaily() {}
 
  ClassDef(TStockDaily,1)
};
 
//---------------------------------------------------------------------------
Double_t RiskProfile(Double_t *x, Double_t *par) {
  Double_t riskFactor = par[0];
  return 1-TMath::Exp(-riskFactor*x[0]);
}
 
//---------------------------------------------------------------------------
TArrayF &StockReturn(TFile *f,const TString &name,Int_t sDay,Int_t eDay)
{
  TTree *tDaily = (TTree*)f->Get(name);
  TStockDaily *data = 0;
  tDaily->SetBranchAddress("daily",&data);
  TBranch *b_closeAdj = tDaily->GetBranch("fCloseAdj");
  TBranch *b_date     = tDaily->GetBranch("fDate");
 
  //read only the "adjusted close" branch for all entries
  const Int_t nrEntries = (Int_t)tDaily->GetEntries();
  TArrayF closeAdj(nrEntries);
  for (Int_t i = 0; i < nrEntries; i++) {
    b_date->GetEntry(i);
    b_closeAdj->GetEntry(i);
    if (data->fDate >= sDay && data->fDate <= eDay)
      closeAdj[i] = data->fCloseAdj/100.;
  }
 
  TArrayF *r = new TArrayF(nrEntries-1);
  for (Int_t i = 1; i < nrEntries; i++)
//    (*r)[i-1] = closeAdj[i]-closeAdj[i-1];
    (*r)[i-1] = closeAdj[i]/closeAdj[i-1];
 
  return *r;
}
 
#ifndef __MAKECINT__
//---------------------------------------------------------------------------
TVectorD OptimalInvest(Double_t riskFactor,TVectorD r,TMatrixDSym Covar)
{
// what the quadratic programming package will do:
//
//  minimize    c^T x + ( 1/2 ) x^T Q x
//  subject to                A x  = b
//                    clo <=  C x <= cup
//                    xlo <=    x <= xup
// what we want :
//
//  maximize    c^T x - k ( 1/2 ) x^T Q x
//  subject to        sum_x x_i = 1
//                    0 <= x_i
 
  // We have nrStocks weights to determine,
  // 1 equality- and 0 inequality- equations (the simple square boundary
  // condition (xlo <= x <= xup) does not count)
 
  const Int_t nrVar     = nrStocks;
  const Int_t nrEqual   = 1;
  const Int_t nrInEqual = 0;
 
  // flip the sign of the objective function because we want to maximize
  TVectorD    c = -1.*r;
  TMatrixDSym Q = riskFactor*Covar;
 
  // equality equation
  TMatrixD A(nrEqual,nrVar); A = 1;
  TVectorD b(nrEqual); b = 1;
 
  // inequality equation
  //
  // - although not applicable in the current situation since nrInEqual = 0, one
  //   has to specify not only clo and cup but also an index vector iclo and icup,
  //   whose values are either 0 or 1 . If iclo[j] = 1, the lower boundary condition
  //   is active on x[j], etc. ...
 
  TMatrixD C   (nrInEqual,nrVar);
  TVectorD clo (nrInEqual);
  TVectorD cup (nrInEqual);
  TVectorD iclo(nrInEqual);
  TVectorD icup(nrInEqual);
 
  // simple square boundary condition : 0 <= x_i, so only xlo is relevant .
  // Like for clo and cup above, we have to define an index vector ixlo and ixup .
  // Since each variable has the lower boundary, we can set the whole vector
  // ixlo = 1
 
  TVectorD xlo (nrVar); xlo  = 0;
  TVectorD xup (nrVar); xup  = 0;
  TVectorD ixlo(nrVar); ixlo = 1;
  TVectorD ixup(nrVar); ixup = 0;
 
  // setup the quadratic programming problem . Since a small number of variables are
  // involved and "Q" has everywhere entries, we chose the dense version "TQpProbDens" .
  // In case of a sparse formulation, simply replace all "Dens" by "Sparse" below and
  // use TMatrixDSparse instead of TMatrixDSym and TMatrixD
 
  TQpProbDens *qp = new TQpProbDens(nrVar,nrEqual,nrInEqual);
 
  // stuff all the matrices/vectors defined above in the proper places
 
  TQpDataDens *prob = (TQpDataDens *)qp->MakeData(c,Q,xlo,ixlo,xup,ixup,A,b,C,clo,iclo,cup,icup);
 
  // setup the nrStock variables, vars->fX will contain the final solution
 
  TQpVar      *vars  = qp->MakeVariables(prob);
  TQpResidual *resid = qp->MakeResiduals(prob);
 
  // Now we have to choose the method of solving, either TGondzioSolver or TMehrotraSolver
  // The Gondzio method is more sophisticated and therefore numerically more involved
  // If one want the Mehrotra method, simply replace "Gondzio" by "Mehrotra" .
 
  TGondzioSolver *s = new TGondzioSolver(qp,prob);
  const Int_t status = s->Solve(prob,vars,resid);
 
  const TVectorD weight = vars->fX;
 
  delete qp; delete prob; delete vars; delete resid; delete s;
  if (status != 0) {
    cout << "Could not solve this problem." <<endl;
    return TVectorD(nrStocks);
  }
 
  return weight;
}
#endif
 
 //---------------------------------------------------------------------------
void portfolio()
{
   const Int_t sDay = 20000809;
   const Int_t eDay = 20040602;
 
   const char *fname = "stock.root";
   TFile *f = 0;
   if (!gSystem->AccessPathName(fname)) {
      f = TFile::Open(fname);
   } else if (!gSystem->AccessPathName(Form("%s/quadp/%s", TROOT::GetTutorialDir().Data(), fname))) {
      f = TFile::Open(Form("%s/quadp/%s", TROOT::GetTutorialDir().Data(), fname));
   } else {
      printf("accessing %s file from http://root.cern/files\n",fname);
      f = TFile::Open(Form("http://root.cern/files/%s",fname));
   }
   if (!f) return;
 
   TArrayF *data = new TArrayF[nrStocks];
   for (Int_t i = 0; i < nrStocks; i++) {
      const TString symbol = stocks[i];
      data[i] = StockReturn(f,symbol,sDay,eDay);
   }
 
   const Int_t nrData = data[0].GetSize();
 
   TVectorD r(nrStocks);
   for (Int_t i = 0; i < nrStocks; i++)
      r[i] = data[i].GetSum()/nrData;
 
   TMatrixDSym Covar(nrStocks);
   for (Int_t i = 0; i < nrStocks; i++) {
      for (Int_t j = 0; j <= i; j++) {
         Double_t sum = 0.;
         for (Int_t k = 0; k < nrData; k++) {
            sum += (data[i][k] - r[i]) * (data[j][k] - r[j]);
         }
         Covar(i,j) = Covar(j,i) = sum/nrData;
      }
   }
 
   const TVectorD weight1 = OptimalInvest(2.0,r,Covar);
   const TVectorD weight2 = OptimalInvest(10.,r,Covar);
 
   cout << "stock     daily  daily   w1     w2" <<endl;
   cout << "symb      return sdv              " <<endl;
   for (Int_t i = 0; i < nrStocks; i++)
      printf("%s\t: %.3f  %.3f  %.3f  %.3f\n",stocks[i],r[i],TMath::Sqrt(Covar[i][i]),weight1[i],weight2[i]);
 
   TCanvas *c1 = new TCanvas("c1","Portfolio Optimizations",10,10,800,900);
   c1->Divide(1,2);
 
   // utility function / risk profile
 
   c1->cd(1);
   gPad->SetGridx();
   gPad->SetGridy();
 
   TF1 *f1 = new TF1("f1",RiskProfile,0,2.5,1);
   f1->SetParameter(0,2.0);
   f1->SetLineColor(49);
   f1->Draw("AC");
   f1->GetHistogram()->SetXTitle("dollar");
   f1->GetHistogram()->SetYTitle("utility");
   f1->GetHistogram()->SetMinimum(0.0);
   f1->GetHistogram()->SetMaximum(1.0);
   TF1 *f2 = new TF1("f2",RiskProfile,0,2.5,1);
   f2->SetParameter(0,10.);
   f2->SetLineColor(50);
   f2->Draw("CSAME");
 
   TLegend *legend1 = new TLegend(0.50,0.65,0.70,0.82);
   legend1->AddEntry(f1,"1-exp(-2.0*x)","l");
   legend1->AddEntry(f2,"1-exp(-10.*x)","l");
   legend1->Draw();
 
   // vertical bar chart of portfolio distribution
 
   c1->cd(2);
   TH1F *h1 = new TH1F("h1","Portfolio Distribution",nrStocks,0,0);
   TH1F *h2 = new TH1F("h2","Portfolio Distribution",nrStocks,0,0);
   h1->SetStats(0);
   h1->SetFillColor(49);
   h2->SetFillColor(50);
   h1->SetBarWidth(0.45);
   h1->SetBarOffset(0.1);
   h2->SetBarWidth(0.4);
   h2->SetBarOffset(0.55);
   for (Int_t i = 0; i < nrStocks; i++) {
      h1->Fill(stocks[i],weight1[i]);
      h2->Fill(stocks[i],weight2[i]);
   }
 
   h1->Draw("BAR2 HIST");
   h2->Draw("BAR2SAME HIST");
 
   TLegend *legend2 = new TLegend(0.50,0.65,0.70,0.82);
   legend2->AddEntry(h1,"high risk","f");
   legend2->AddEntry(h2,"low  risk","f");
   legend2->Draw();
}