Summary Lectures Portfolio Theory (for MSc. Finance)

Portfolio Theory Lectures
Summary and notes of the course’s lectures

Joris Wellen – University of Groningen




Overview
Lecture 1 – Single Period Utility Analysis........................................................................................................ 2
1.1 Utility Functions..............................................................................................................................................2
1.2 Absolute and Relative Risk Aversion..............................................................................................................4
1.3 Certainty Equivalent.......................................................................................................................................5
1.4 Optimal Portfolios..........................................................................................................................................7
1.5 Matrix Algebra and Portfolio Analysis...........................................................................................................9

Lecture 2 – Mean Variance Analysis............................................................................................................. 11
2.1 Efficient Portfolios........................................................................................................................................11
2.2 Advanced Mathematical Portfolio Analysis.................................................................................................13

Lecture 3 – Index Models and the Return Generating Process.......................................................................15

Lecture 4 – Asset Pricing Models.................................................................................................................. 25
4.1 Capital Asset Pricing Model.........................................................................................................................25
4.2 Arbitrage Pricing Theory..............................................................................................................................28

Lecture 5 – Empirical Tests of the CAPM....................................................................................................... 34
5.1 Dual Hypothesis Testing Problem................................................................................................................34
5.2 Time Series Tests..........................................................................................................................................34
5.3 Fama & French Models................................................................................................................................37

Lecture 6 – Dynamic Asset Allocation........................................................................................................... 42
6.1 Introduction..................................................................................................................................................42
6.2 Dynamic Asset Allocation (DAA)..................................................................................................................44

Lecture 7 – Performance Measurement and Performance Attribution..........................................................54
7.1 Evaluation of Portfolio Performance............................................................................................................54
7.2 Risk-Adjusted Performance Measures.........................................................................................................57
7.3 Performance Attribution..............................................................................................................................64

, Portfolio Theory Lectures – University of Groningen – Joris Wellen



Lecture 1 – Single Period Utility Analysis
Book: Chapter 4

1.1 Utility Functions




A utility analysis is the standard neo-classical approach for decision-
making under risk. The assumption is that the investor is rational (see the
six basic axioms of utility analysis). There are also alternative
approaches:
1. Prospect Theory (Kahneman & Tiversky)  violates the six axioms.
2. Decision-making when we are not able to specify the uncertainty in
terms of possible outcomes and a probability distribution. We do
not know the possible states or their probabilities. This is also
known as ambiguity = Knightian uncertainty.

Six axioms of utility theory
1. People have preferences.
2. People’s preferences are transitive.
3. Investment opportunities with equal expected utility are equally
desirable.
4. Utility can be used with risky decisions.
5. If we have two ranked risky investment alternatives, adding a third
one that is unrelated (irrelevant) to the first two will not affect our
ranking.
Suppose that I have two alternatives A and B, with U(A) > U(B). If I
add a third investment alternative E (for instance a lottery ticket),
which is unrelated (not correlated with) to either A or B, then
U(A+B) > U(B+E).
6. People make risky decisions by maximizing their expected utility.

Optimizing the expected utility of wealth, one uses the following
equation:


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, Portfolio Theory Lectures – University of Groningen – Joris Wellen

n
E [ U ] =∑ pi U (W i )
i=1


Where n are the possible outcomes (scenarios), pi is the probability of the
scenario, and W i is the wealth acquired in that particular scenario.
One can see this as a scoring system. The shape of the utility function
determines risk aversion.


Example: soccer scores
 In the past, this was used in
counting soccer scores in a




tournament.
The first team plays a risky
strategy involving an offensive
playing style. The second team
plays a conservative strategy
involving a defensive playing
style.

The public probably dislike too many ties. To encourage risk taking, the
system was changed into one that assigns 3 points for a win, 1 point for a
tie and zero for a lost match.

 Results under the new system.
Conclusions:
 The scoring
system can be
adjusted to
accommodate to
specific risk
preferences
 I
n


utility theory, the risk preferences are
implemented by choosing a utility
function
 The shape of the utility function determines the risk
aversion (which should match the underlying preferences
of the individual.

Ultimately, we are interested in the utility of wealth. However, for

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, Portfolio Theory Lectures – University of Groningen – Joris Wellen

reasons of convenience, we transform wealth into one-period returns
without affecting the preference order.

W T =W 0 ( 1+r ) =W 0 +W 0 r

We prefer returns over wealth for many reasons, including comparability
between individuals with different wealth, statistical properties in time
series analysis (stationarity).

1.2 Absolute and Relative Risk Aversion
Consider three investors, with the following utility functions
A U ( r )=100 r −50 r 2
B U ( r )=100 r
C U ( r )=100 r +50 r 2
Since all the three assets have the
same expected return, preference
orderings must arise from
differences in risk attitude.



Below one can see the final preferences for each alternative




Utility function A chooses the least risky portfolio (risk aversion).
Utility function C chooses the riskiest portfolio (risk loving).

Properties of utility functions:
1. Non-satiation: economic subjects prefer more over less
∂U
U ' [W ]= >0
∂W
2. Risk preference
U”(W) < 0  risk-aversion
U”(W) = 0  risk-neutrality
U”(W) > 0  risk-loving

Diminishing marginal utility leads to risk avoiding behavior because a
risky investment has a lower expected utility than a risk-free investment
with the same expected return. Another way to frame this is in terms of a
fair game. A fair game is a lottery with an expected value of zero.

An investor with initial wealth of 1,000 considers a lottery with a 50%
chance of winning 300 and a 50% chance of losing 300. Participating in
the lottery does not require an investment.

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