Portfolio Theory
Tuomo Haapalainen
Introduction to Market Analysis
1
The eggs
2
Volatility in Investment Returns

1.
2.
Two types of risk cause the volatility in investment
returns
Market risk: reflects the degree to which investment
values vary when the level of prices in the
underlying market changes
Unsystematic risk: factors specific to the asset can
generate volatility, for instance, a change of
management within a company. This type of
volatility is unique to the asset
3
Diversification





By diversifying you are able to take away some of the
stock-specific risk, but not all
Some standard deviation remains, which is the
market risk part
You only get compensation for bearing the market risk
This compensation is the so-called market risk
premium (return for risky investment minus return for
riskless investment)
Riskless return is compensation for waiting (time
value of money) and risk premium for insecurity of the
return
4
Diversification
Standard
Deviation
(Risk)
Specific Risk
Market Risk
1
20
5
Expected return of portfolio

E(Rp) is the weighted average of expected
returns of individual assets in a portfolio, and
w’s are weights
6
Portfolio risk

General form of standard deviation:

Covariance of two assets:
7
About covariances



Diversification utility of a stock portfolio’s stockspecific risk depends on the covariances between
stocks, not the variances
Portfolio’s standard deviation is the weighted
avarage of individual stocks’ standard deviations
only if the correlations are +1 (i.e., diversification
does not reduce portfolio risk)
Equivalently, maximal utility is reached with
correlation -1
8
Case: Example of a portfolio
with three stocks
9
Case: Example of a portfolio
with three stocks
10
Case: Variance-Covariance
matrix
11
Efficient frontier




Combinations of securities form an area that
includes all possible combinations
This is the efficient opportunity set, which is
convex
Investors are interested in the lowest
possible variances of portfolios at any given
return expectations
These are optimal portfolios that together
form the efficient frontier
12
Efficient frontier (between A and B)
E(R)
A
B
Opportunity
set
C
Minimum
Variance Portfolio
Portfolio
Standard
Deviation
13
Efficient Frontier


Portfolios with the best combinations of
weights lie on the efficient frontier, where a
target expected return is achieved with
minimum risk
The efficient frontier contains a unique
minimum variance, and overall there will be a
minimum variance portfolio
14
Solving the efficient frontier
15
Capital allocation line

Assume there is also a risk-free security
16
Capital allocation line

Invest (1-wi) into risk-free asset:
17
Capital allocation line


Rf is a constant and the slope is now the
relationship between expected return
and standard deviation
The slope is usually called the Sharpe
ratio, which is a common performance
measure
18
Tangency portfolio



The point where the capital allocation
line touches the efficient frontier
Tangent portfolio offers the best possible
risk premium per risk unit
Thus, tangency portfolio maximizes the
Sharpe ratio
19
Example
20
Required return
21