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Efficient Portfolios with no short-sale restriction MGT 4850 Spring 2008 University of Lethbridge Portfolio return • One period return => ET Γ • Matrix of 60 monthly returns for 30 industry portfolios (60x30) • Column vector of equal weights (30x1) • We get 60 period returns of a portfolio of equally weighted industries (column vector) Market risk • Calculate covariance of portfolio returns with market return • Calculate variance of market returns • Beta of the protfolio Copy versus Functions • Transpose of a vector or matrix created with the function changes with change in the origin, e.g. portfolio variance ΓTS Γ will recalculate correctly if we change weights in the original vector of weights. • Another way to avoid this error is to check “validate” when we copy and “paste special” - “transpose” Overview • CAPM and the risk-free asset – CAPM with risk free asset – Black’s (1972) zero beta CAPM • The objective is to learn how to calculate: – Efficient Portfolios – Efficient Frontier Notation • Weights – a column vector Γ (Nx1); it’s transpose ΓT is a row vector (1xN) • Returns - column vector E (Nx1); it’s transpose ET is a row vector (1xN) • Portfolio return ET Γ or ΓT E • 25 stocks portfolio variance ΓTS Γ ΓT(1x25)*S(25x25)* Γ(25x1) • To calculate portfolio variance we need the variance/covariance matrix S. Covariance of two portfolios • Expected return of portfolio X is a column vector Ex (Nx1) • Expected return of portfolio Y is a column vector Ey (Nx1) (note you have the same number of returns, whether the portfolio have the same number of assets or not) • Variance-covariance matrix S (NxN) • Covariance x,y = XTS Y Simultaneous Equations • Solve simultaneously for x and y: x + y=10 x − y=2 • Skip propositions 9.3 – p.164/66 charts intuition: • max slope for the tangent portfolio • finding graphically zero beta portfolio Calculating the efficient frontier • Only four risky assets Find two efficient portfolios • Minimum Variance • Market portfolio • Use proposition two to establish the whole envelope • CML • SML Zero beta CAPM Black (1972) Finding Envelope Portfolios 12% 10% Portfolio mean return x, the tangency portfolio given c 8% 6% c Zero beta portfolio 4% 2% 0% 0% 10% 20% 30% 40% 50% Portfolio standard deviation 60% 70% 80% 90% Notation • • • • R is column vector of expected returns S var/cov matrix c – arbitrary constatnt z – vector that solves the system of linear equations R-c = Sz Solving for z needs inverse matrix of S Simultaneous equations => R-c = Sz • • • • E(r1 )-c= z1σ11 + z2σ12 + z3σ13 + z4σ14 E(r2 )-c= z1σ21 + z2σ22 + z3σ23 + z4σ24 E(r3 )-c= z1σ31 + z2σ32 + z3σ33 + z4σ34 E(r4 )-c= z1σ41 + z2σ42 + z3σ43 + z4σ44 • The vector z assigns proportions to each asset. Find the weights as a proportion of the sum. The Solution is an envelope portfolio • Vector z is: z = S-1 {R-c} • Vector z solves for the weights x x={x1,….. xN} Calculating two envelope portfolios • Choose arbitrary a constant; solve for 0 constant also: • Weight vector is calculated from z by dividing each entry of z by the sum of all entries of the z vector.