3.2 Portfolio with Risky Assets Only
19
and its transpose
w
T
= =r
T
C
−1
λ 1 + e
T C
−1
λ 2 ,
(3.13)
provided that the covariance matrix is invertible. This is the case if there are no
risk-free assets in the portfolio, a point we relax in the next section. But for now we
insert the expression for the weights of the portfolio vector w into the condition for
the desired return ρ and obtain
ρ = w
T
r = =r
T
C
−1
rλ 1 + e
T C
−1
rλ 2 .
(3.14)
The normalizing condition can be written as 1 = w
T
· e, which yields
1 = w
T e = =r
T
C
−1 eλ 1 + e
T C
−1 eλ 2 .
(3.15)
These two equations form a set of linear equations that can be cast into a matrixvalued equation
ρ
1
=
r
T
C
−1
r e
T C
−1
r
r
T
C
−1 e e
T C
−1 e
λ 1
λ 2
,
(3.16)
which is easily solved for the Lagrange multipliers λ 1 and λ 2 by inverting the 2 × 2
matrix appearing in (3.16)
λ 1
λ 2
=
1
e
T C
−1 e −e
T C
−1
r
−−r
T
C
−1 e r
T
C
−1
r
ρ
1
,
(3.17)
where is the determinant of the 2 × 2 matrix. Inserting λ 1 and λ 2 into (3.12),
we obtain the optimal weight vector w for the portfolio as a function of the desired
portfolio return ρ
w = λ 1 C
−1
r + λ 2 C
−1 e ,
(3.18)
which contains only known quantities such as the average return of shares r and
their variance as well as the desired portfolio return ρ specifying the investor’s
eagerness to make a profit. The resulting volatility σ of the portfolio, given by w, is
σ
2
= w
T Cw.
We can now determine the relation between desired return ρ and the resulting
volatility by plotting ρ versus σ, which is done in Fig. 3.2 for a sample portfolio with
three assets, approximately having returns r 1 ≈ 1%, r 2 ≈ 2%, and r 3 ≈ 3%
with approximate volatilities of 1, 2, and 3%, respectively. The plot was generated with the MATLAB script from Appendix B.1 and uses sequences of correlated
random numbers with returns and volatilities shown as the three black asterisks in
Fig. 3.2. We also show the relation between the desired portfolio return ρ and the
portfolio volatility σ as the solid blue line for our three-stock portfolio. The portfolio with a minimum volatility given by the left-most point in the knee of the blue
hyperbola. The coordinates of this point can be calculated from the requirement that
19
and its transpose
w
T
= =r
T
C
−1
λ 1 + e
T C
−1
λ 2 ,
(3.13)
provided that the covariance matrix is invertible. This is the case if there are no
risk-free assets in the portfolio, a point we relax in the next section. But for now we
insert the expression for the weights of the portfolio vector w into the condition for
the desired return ρ and obtain
ρ = w
T
r = =r
T
C
−1
rλ 1 + e
T C
−1
rλ 2 .
(3.14)
The normalizing condition can be written as 1 = w
T
· e, which yields
1 = w
T e = =r
T
C
−1 eλ 1 + e
T C
−1 eλ 2 .
(3.15)
These two equations form a set of linear equations that can be cast into a matrixvalued equation
ρ
1
=
r
T
C
−1
r e
T C
−1
r
r
T
C
−1 e e
T C
−1 e
λ 1
λ 2
,
(3.16)
which is easily solved for the Lagrange multipliers λ 1 and λ 2 by inverting the 2 × 2
matrix appearing in (3.16)
λ 1
λ 2
=
1
e
T C
−1 e −e
T C
−1
r
−−r
T
C
−1 e r
T
C
−1
r
ρ
1
,
(3.17)
where is the determinant of the 2 × 2 matrix. Inserting λ 1 and λ 2 into (3.12),
we obtain the optimal weight vector w for the portfolio as a function of the desired
portfolio return ρ
w = λ 1 C
−1
r + λ 2 C
−1 e ,
(3.18)
which contains only known quantities such as the average return of shares r and
their variance as well as the desired portfolio return ρ specifying the investor’s
eagerness to make a profit. The resulting volatility σ of the portfolio, given by w, is
σ
2
= w
T Cw.
We can now determine the relation between desired return ρ and the resulting
volatility by plotting ρ versus σ, which is done in Fig. 3.2 for a sample portfolio with
three assets, approximately having returns r 1 ≈ 1%, r 2 ≈ 2%, and r 3 ≈ 3%
with approximate volatilities of 1, 2, and 3%, respectively. The plot was generated with the MATLAB script from Appendix B.1 and uses sequences of correlated
random numbers with returns and volatilities shown as the three black asterisks in
Fig. 3.2. We also show the relation between the desired portfolio return ρ and the
portfolio volatility σ as the solid blue line for our three-stock portfolio. The portfolio with a minimum volatility given by the left-most point in the knee of the blue
hyperbola. The coordinates of this point can be calculated from the requirement that