Short sell Portfolio optimalization Portfolio Theory Lecture 4 Luděk Benada Department of Finance - 402, benada.esf@gmail.com March 14, 2016 □ S1 Luděk Benada MPF APOT Q Short sell Q Portfolio optimalization Luděk Benada MPF APOT Short sell Portfolio optimalization The assumption for the technique: Pt+n < Pt /-\ ia/iI 11 —rr\r\ +hon +■ h a CQrunf iQC mi ict h r . I Luděk Benada MPF APOT • The assumption for the technique: Pt+n < Pt o Investor has to borrow securities, which will sell and use the sources for a portfolio construction Cöf 11 y 1 —i— i Til "il "li li T— ' a Luděk Benada MPF APOT • The assumption for the technique: Pt+n < Pt o Investor has to borrow securities, which will sell and use the sources for a portfolio construction • When the portfolio will be realized, then the securities must be returned I I Y11\/ —i— i Til "il "li li i— / Luděk Benada MPF APOT • The assumption for the technique: Pt+n < Pt o Investor has to borrow securities, which will sell and use the sources for a portfolio construction • When the portfolio will be realized, then the securities must be returned o At that point the investor must purchase on the market the security —i— I Til "il "li II t— ' Luděk Benada MPF APOT • The assumption for the technique: Pt+n < Pt o Investor has to borrow securities, which will sell and use the sources for a portfolio construction • When the portfolio will be realized, then the securities must be returned o At that point the investor must purchase on the market the security • The owner of the security has rights on all CF's Luděk Benada MPF APOT • The assumption for the technique: Pt+n < Pt o Investor has to borrow securities, which will sell and use the sources for a portfolio construction • When the portfolio will be realized, then the securities must be returned o At that point the investor must purchase on the market the security • The owner of the security has rights on all CF's • An example... Luděk Benada MPF APOT Short sell Portfolio optimalization of SS on the EPF • The Short Sell expands the feasible set of portfolios Luděk Benada MPF APOT Short sell Portfolio optimalization 1 ie im pact of SS on tf ^e EPF • The Short Sell expands the feasible set of portfolios 9 ...expands the Efficient Portfolio Frontier ~he SS could be used even the r■ ^> 0 Luděk Benada MPF APOT Short sell Portfolio optimalization The impact of SS on the EPF • The Short Sell expands the feasible set of portfolios 9 ...expands the Efficient Portfolio Frontier • The SS could be used even the T\ > 0 Luděk Benada MPF APOT Short sell Portfolio optimalization The impact of SS on the EPF • The Short Sell expands the feasible set of portfolios 9 ...expands the Efficient Portfolio Frontier • The SS could be used even the T\ > 0 • ... but there will be a lost of the diversification offect. Luděk Benada MPF APOT Short sell Portfolio optimalization The impact of SS on the EPF • The Short Sell expands the feasible set of portfolios 9 ...expands the Efficient Portfolio Frontier • The SS could be used even the T\ > 0 • ... but there will be a lost of the diversification offect. • The graphical representation ... Luděk Benada MPF APOT Short sell Portfolio optimalization • The purpos is to find an extrem of given function Luděk Benada MPF APOT The purpos is to find an extrem of given function There are two feasible approaches: 9 .. .to maximize the expected returne of the oortf □ S1 Luděk Benada MPF APOT Short sell Portfolio optimalization • The purpos is to find an extrem of given function • There are two feasible approaches: • .. .to maximize the expected returne of the portfolio Luděk Benada MPF APOT Short sell Portfolio optimalization • The purpos is to find an extrem of given function • There are two feasible approaches: • .. .to maximize the expected returne of the portfolio • .. .to minimize the risk of a change in expected returne Luděk Benada MPF APOT Two ways of problem solving: +■ /^v ti n/H n r\c/^vl i ifc rv» i n Luděk Benada MPF APOT • Two ways of problem solving: • ... to find an absolute minimum risk Luděk Benada MPF APOT • Two ways of problem solving: • ... to find an absolute minimum risk • ... to find a minimum risk for a given return Luděk Benada MPF APOT • Solving the optimalization problem I I • • • J p Luděk Benada MPF APOT □ S1 Short sell Portfolio optimalization • Solving the optimalization problem • The weights are found by solving the objectie function, assuming the restriction... \^ V_- L. 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I \J <7r ■->■ mm = 1 Luděk Benada MPF APOT Short sell Portfolio optimalization • Solving the optimalization problem • The weights are found by solving the objectie function, assuming the restriction... • Objective function: o£ —> min Luděk Benada MPF APOT Short sell Portfolio optimalization • Solving the optimalization problem • The weights are found by solving the objectie function, assuming the restriction... • Objective function: o£ —> min o Restrictive conditions: L/^Li wi — 1 Luděk Benada MPF APOT Luděk Benada MPF APOT Luděk Benada MPF APOT Short sell Portfolio optimalization • f(w)^min • Lagrange function: L(w) = L(W,X] = X;fi{w) • For finding the solution will be used the Lagrange multipliers: dL(Y) r\ \j -,vv J = U, V; O Y j Luděk Benada MPF APOT Short sell Portfolio optimalization • f(W)^min • Lagrange function: L(W 9 For finding the solution will dL(Y)_0 y. /.(^a) = i/^1a//;-(i^: be used the Lagrange multipliers: Luděk Benada MPF APOT Short sell Portfolio optimalization Multiplie L(^ = a|(Mž5 + Ai(EÍllW/-l) —>► Luděk Benada MPF APOT Short sell Portfolio optimalization Multiplie Deriving with respect to all variables we obtain n+1 equiations —>► ž — i Luděk Benada MPF APOT Short sell Portfolio optimalization Multiplie L(Y) =