- Portfolio Management 117
the minimum variance portfolio) satisfy the condition
γwC=m−μu (5.17)for some real numbersγ>0andμ.
Proof
Letwbe the weights of a portfolio, other than the minimum variance portfolio,
belonging to the efficient frontier. The portfolio has expected returnμV =
mwTand standard deviationσV=
√
wCwT.Ontheσ, μplane we draw the
tangent line to the efficient frontier through the point representing the portfolio.
This line will intersect the vertical axis at some point with coordinateμ,the
gradient of the line being
mwT−μ
√
wCwT. This gradient is maximal among all lines
passing through the point on the vertical axis with coordinateμand intersecting
the set of attainable portfolios. The maximum is to be taken over all weightsw
subject to the constraintuwT= 1. We put
F(w,λ)=mw√ T−μ
wCwT−λuwT,whereλis a Lagrange multiplier. A necessary condition for a constrained max-
imum is that the partial derivatives ofFwith respect to the weights should be
zero. This gives
m−λσVu=μV−μ
σ^2 V wC.Multiplying bywTon the right and using the constraint, we find thatλ=σμV.
Forγ=μVσV− 2 μthis gives the asserted condition. Because the tangent line has
positive slope, we haveμV>μ,thatis,γ>0.
Remark 5.5
An interpretation ofγandμfollows clearly from the proof:γσVis the gradient
of the tangent line to the efficient frontier at the point representing the given
portfolio,μbeing the intercept of this tangent line on theσ, μplane.
Exercise 5.15
In a market consisting of the three securities in Exercise 5.12, consider
the portfolio on the efficient frontier with expected returnμV = 21%.
Compute the values ofγandμsuch that the weightswin this portfolio
satisfyγwC=m−μu.