Ec 173A—FINANCIAL MARKETS
LECTURE NOTES
Foster, UCSD
12Jul08
TOPIC 5 – MODERN PORTFOLIO THEORY
A. Introduction to Modern Portfolio Theory (MPT)
1.
Background:
a)
In 1952, Harry Markowitz [Nobel, 1990] revolutionized the field of portfolio selection
and management.
1) The old theory of portfolios focused on single assets.
Investors determine a reserva
tion price based on their assessment of an asset's expected return and risk.
If market
price P
0
< the reservation price, they buy the asset at P
0
.
2) Markowitz recognized that wise investors hold portfolios of several assets.
Hence,
the risk and return of any single asset is relevant only to the extent that it affects the
overall risk and return of the portfolio to which it is added.
3) MPT is complicated, but worth the trouble, because the end result is an extremely
simple rule for portfolio management!
b) The assumptions of MPT.
#1 Investors are risk averse and maximize E[U(W)], the expected utility of endof
period wealth, over the same time horizon.
#2 Investors make asset choices based on risk and expected return [σ(r
p
), μ(r
p
)] of port
folios
of assets.
#3 Investors have homogeneous expectations and agree on μ(r) and σ(r) for all assets.
#4 Information is free and simultaneously available to all investors.
#5 There is a riskless asset with yield r
f
, and investors can borrow and lend at this rate.
2.
Portfolio Rates of Return:
a)
Consider a portfolio of one stock and one bond.
NOTATION  for j
∈
{ s, b }
Symbol
Definition
Notes
P
j
Beginning price of stock or bond
n
j
Number of units of asset in portfolio
V
j
Total value of asset in portfolio
V
j
= n
j
P
j
CF
j
Cash flows per unit of asset
r
j
Ex ante
rate of return on asset
r
j
= (CF
j
+
∆
P
j
)/P
j
V
p
Beginning portfolio price (value)
V
p
= V
s
+ V
b
ω
j
Proportion of portfolio value in asset
ω
j
= V
j
/V
p
;
Σω
j
= 1
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MPT
p. 2
b) Portfolio
ex ante
rate of return.
•
b
b
s
s
p
r
r
r
ϖ
+
=
.
Proof:
c)
Portfolio returns and expected returns.
1) We saw above that portfolio return r
p
= ω
s
r
s
+ ω
b
r
b
, a valueweighted linear function
of the individual asset returns.
2) Therefore, r
p
is a linear
function of random
variables r
s
and r
b
, from which we deduce the
portfolio expected return:
.
3.
Portfolio Risk:
a)
Covariance and correlation of
ex ante
rates of return.
1) Asset
ex ante
rates of return are random variables distributed over states of the world
(SW).
This implies that all asset rates of return are jointly distributed.
2) For assets S and B and states of the world SW
j
, j = 1.
..K, with probabilities Pr(j):
b) Portfolio risk will be measured as the standard deviation σ(r
p
) of
ex ante
rate of return r
p
.
1) For our stockbond portfolio, r
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 Spring '09
 Foster
 Capital Asset Pricing Model, CML, Modern portfolio theory

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