Portfolio Optimization

196 Sharpe Rat io s Under Constraints and Kinks Chap. 9

= α

0j

+ t

∗

α

1j

−

β

0j

t

∗

− t

∗

β

2j

= α

0j

−

β

0j

t

∗

. (9.15)

Since β

0j

≥ 0 and t

∗

> 0 it follows from (9.15) that r

m

≤ α

0j

, as

required. 

Note that in the proo f of Theorem 9.5, if β

2j

> 0 then r

m

< α

0j

We next consider the problem of maximizing the implied risk free

return, r

m

, when the Sharpe ra t io , R, is given. This was studied in

Section 4.2 and its generalization to linear inequality constraints is

max{µ

′

x − R(x

′

Σx)

1

2

| Ax ≤ b }. (9.16)

The gradient of the o bj ective function for this problem is

µ −

"

R

(x

′

Σx)

1

2

#

Σx.

After rearrangement, the optimality conditions (see Exercise 5 .1 1) f or

(9.16) are

Ax ≤ b, ( 9.17) ...