Ralph Vince - Portfolio Mathematics

Classical Portfolio Construction 241

V=

∑N

i= 1

∑N

j= 1

Xi∗Xj∗Ri, j∗Si∗Sj (7.06b)

V=

(

∑N

i= 1

Xi∧ 2 ∗Si∧ 2

)

+ 2 ∗

∑N

i= 1

∑N

j= 1 + 1

Xi∗Xj∗COVi, j (7.06c)

V=

(

∑N

i= 1

Xi∧ 2 ∗Si∧ 2

)

+ 2 ∗

∑N

i= 1

∑N

j= 1 + 1

Xi∗Xj∗Ri, j∗Si∗Sj (7.06d)

where: V=The variance in the expected returns of the portfolio.

N=The number of securities comprising the portfolio.

Xi=The percentage weighting of the ith security.

Si=The standard deviation of expected returns of the ith

security.

COVi, j=The covariance of expected returns between the ith

security and the jth security.

Ri, j=The linear correlation coefficient of expected returns

between the ith security and the jth security.

All four forms of Equation (7.06) are equivalent. The final answer to Equation
(7.06) is always expressed as a positive number.
We can now consider that our goal is to find those values of Xithat, when
summed, equal 1, and result in the lowest value of V for a given value of E.
When confronted with a problem such as trying to maximize (or minimize) a
function, H(X,Y), subject to another condition or constraint, such as G(X,Y),
one approach is to use the method of Lagrange.
To do this, we must form the Lagrangian function, F(X,Y,L):

F(X,Y,L)=H(X,Y)+L∗G(X,Y) (7.07)

Note the form of Equation (7.07). It states that the new function we have
created, F(X,Y,L), is equal to the Lagrangian multiplier, L—a slack variable
whose value is as yet undetermined—multiplied by the constraint function
G(X,Y). This result is added to the original function H(X,Y), whose extreme
we seek to find.
Now, the simultaneous solution to the three equations will yield those
points (X 1 ,Y 1 ) of relative extreme:

FX(X,Y,L)= 0

FY(X,Y,L)= 0

FL(X,Y,L)= 0