1. Theory of Capital Markets
Security Markets VI
Miloslav S Vosvrda
Theory of Capital Markets

2. The explanation of security prices
One of the principal applications of security market
theory is the explanation of security prices. The static
Capital Asset Pricing Model or CAPM, begin with a
set Y of random variables with finite variance on
some probability space. Each y in Y corresponds to
the random payoff of some security.

3. span(Y) The space of linear combination of elements of Y,
meaning x is in L if and only if
for some scalars and some
in Y. The elements of L are portfolios.
Some portfolio in L denoted 1 is riskless. The utility
functional of each agent i is assumed to be strictly
variance averse, meaning that whenever
and

4. The market portfolio The total endowment M of portfolios is the market
portfolio, and is assumed to have non-zero variance.
Suppose is a competitive equilibrium
for this economy. The equilibrium price functional
p is represented by a unique portfolio in L via the
formula:

5. For the equilibrium choice of agent i, consider
the least squares regression of on :
where the residual term e has zero expectation and
zero covariance with ; that is
Since both 1 and are available portfolios, agent
could have chosen the portfolio

6. We have
implying that is budget feasible for agent i.
Since and
string variance aversion implies that .
Since is optimal for agent i, it follows that e=0.
For any portfolio , relation
implies that

7. where k=
and K= ,

8. Defining the return on any portfolio x with non-
zero market value to be and
denoting the expected return by
and
where
which is known as the beta of portfolio x.

9. The Capital Asset Pricing Model
is relation
By words:
The expected return on any portfolio in excess of
the riskless rate of return is the beta of that
portfolio multiplied by the excess expected return
of the market portfolio.

10. the riskless expected rate of return.
Consider the linear regression of
on , where aby is any portfolio with
non-zero variance. The solution is
For the particular case of y=M, we have
By taking expectations shows that
This is a special property distinguishing
the market portfolio.
A portfolio whose return is uncorrelated with the
market return has
the riskless expected rate of return.