Portfolio Optimization

Sec. 4.2 Optimization Derivation 69

Cross multiplying gives

t

m

α

0

+ t

2

m

α

1

− t

m

r

m

= β

0

+ β

2

t

2

m

.

Simplifying and solving for t

m

now g ives

t

m

=

β

0

α

0

− r

m

,

which is the identical result obtained in Theorem 4.1 (b).

We can perform a similar analysis beginning with Theorem 4.2 (b).

Set t

m

= t

1

and x

m

= x

1

in Theorem 4.2 (b). Because x

m

is optimal

for (2.3), µ

m

= α

0

+ t

m

α

1

and σ

2

m

= β

0

+ t

2

m

β

2

. Then Theorem 4.2(b)

implies

r

m

= µ

′

x

m

−

x

′

m

Σx

m

t

m

(4.14)

= α

0

+ t

m

α

1

−

β

0

+ t

2

m

β

2

t

m

.

Multiplying through by t

m

, using α

1

= β

2

and simplifying gives

t

m

=



β

0

α

0

− r

m



and µ

m

= α

0

+



β

0

α

0

− r

m



α

1

, (4.15)

in agreement with Theorem 4.1(b).

There are other somewhat simpler methods to derive the results of

(4.15) (see ...