BUSF_A01.qxd

Chapter 7 • Portfolio theory and its relevance to real investment decisions

which equals:

×

Hence:

=E(ri) −E(rm)

If we set x to be the expression within the square root sign in equation (A7.2), then:

=×

and

σP=x1/2

= x−1/2

or

×

= 2 ασ^2 i+ 2 ασm^2 − 2 αm+2Cov(i,M) − 4 αCov(i,M)

so

If the market is in equilibrium, then M will already contain the appropriate pro-

portion of iand so the portfolio of iand M will contain no excess i. Thus the only point

on iMthat would be expected to occur will be M, that is, the point where α=0.

When α=0, dE(rP)/dσP(above) reduces to:

[E(ri) −E(rm)] × (A7.3)

At M, the chord iMis tangential to (has the same slope as) the capital market line

rfS. The slope of rfSis [E(rm) −rf]/σm. Equating this with (A7.3) above and simplifying

gives:

E(ri) =rf+[E(rm) −rf]Cov(i, M)

σ^2 m

σm

Cov(i, M) −σ^2 m

d

d

Cov

Cov Cov

Er Er Er iM

iM iM

P

P

im

im

imm

() [() ( )] [ ( ) ( ) ( , )]

σ (, ) (, )

ασ α σ α α

ασ ασ σ α

=− ×

+− + −

−−+ −

21 21

22 22 4

22 2 2

22

dx

dα

1

[ασ^22 im ( )+− 121 α σ2 2 ( )+α −αCov( , )]iM

1

2

1

2

dσP

dx

dσP

dx

dx

dα

dσP

dα

dE(rP)

dα

dα

dσP

dE(rP)

dα