Lecture 7: Portfolio Theory
RSM 332
Capital Market Theory
Rotman School of Management
University of Toronto
Mike Simutin
October 26/27, 2011
Portfolio Theory
RSM 332, 1/21
Why Is This Important?
We as investors want to achieve the highest possible expected
return for a given level of risk
Modern Portfolio Theory (MPT) allows us to find the optimal
way to allocate our money to achieve this goal
Whether you are managing your own or someone else’s money,
MPT tells you how to build the ‘best’ portfolio
MPT and its offsprings are still widely studied and used
MPT is a stepping stone to the very popular asset pricing
models such as the CAPM
Portfolio Theory
RSM 332, 2/21
Intuition Behind MPT
Investing is a tradeoff between risk and return
Rather than selecting assets on their own merits, we should
consider how returns of every asset correlate with returns of
every other asset in the portfolio
We can combine assets to achieve the highest possible
expected return for a given level of risk
Or alternatively, we can combine the assets to achieve the
lowest possible level of risk for a given level of expected return
MPT can be thought of as a form of diversification:
minimizing risk without hurting expected return
Portfolio Theory
RSM 332, 3/21
Modern Portfolio Theory
We will consider three cases
Two risky assets
N
risky assets
N
risky assets and a riskfree asset
Caveat
: MPT relies on a number of assumptions, including:
Returns follow a normal distribution
Investors only care about mean and variance of returns
Correlations between returns of different assets are predictable
Markets are eﬃcient
Investors are rational
There are no transaction costs
Assets are infinitely divisible
Portfolio Theory
RSM 332, 4/21
TwoStock Case
Let’s start with just two risky assets (stocks),
A
and
B
Define expected return and variance of each stock
i
=
A
,
B
as
E
(
R
i
) and
Var
(
R
i
) =
σ
2
i
Define covariance and correlation between the returns of stocks
A
and
B
as
σ
AB
and
ρ
AB
, respectively
If we invest a fraction
w
A
of our money in stock
A
and the
remaining fraction
w
B
= 1
−
w
A
in
B
, expected return and variance
of such portfolio
P
will be
E
(
R
P
)
=
w
A
E
(
R
A
) +
w
B
E
(
R
B
)
σ
2
P
=
w
2
A
σ
2
A
+
w
2
B
σ
2
B
+ 2
w
A
w
B
σ
AB
=
w
2
A
σ
2
A
+
w
2
B
σ
2
B
+ 2
w
A
w
B
σ
A
σ
B
ρ
AB
We can vary weights
w
A
and
w
B
and plot the expected return and
standard deviation of returns of the resulting portfolios
Risk of the portfolios we obtain this way will importantly depend on
correlation
ρ
AB
Portfolio Theory
RSM 332, 5/21
TwoStock Case
Expected Return, Percent
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
Standard Deviation, Percent
0
1
2
3
4
5
6
7
8
9
10
11
Stock A:
E(R)=1%,
=5%
Stock B:
E(R)=1.5%,
=10%
Portfolio Theory
RSM 332, 6/21
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 Fall '08
 RAYMONDKAN
 Capital Asset Pricing Model, Portfolio Theory, Modern portfolio theory, wA

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