BUSF_A01.qxd

Appendix: Derivation of CAPM

APPENDIX Derivation of CAPM

In Figure 7.9:

Figure 7.9
The relationship
of an individual
security iwith the
market portfolio M
and the capital
market line

The line iM is a series of various combinations of individual security i and the market

portfolio M.

l EE is the efficient frontierof risky assets;

l rfis the risk-free rate;

l rfS is the efficient frontier of possibilities once the risk-free asset is introduced; and

l iMis the risk return profile of various portfolios containing proportions of an indi-

vidual inefficientasset iand the marketportfolio M.

Let αbe the proportion of iin the portfolio (P) of iand M. Then:

E(rP) =αE(ri) +(1 −α)E(rm) (A7.1)

and

σp= (A7.2)

The slope of iM at any point is given by:

dE(rP)

dσP

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