Lecture 1 – Portfolio theory
Portfolio theory is designed by Harry Markowitz
Utility analysis
Standard neo-classical approach for facilitating decision making under risk.
The assumption is that the investor is rational > to make sure that rationality is
enforced with the utility model: six axioms
1. People have preferences
2. Peoples preferences are transitive: A over B, B over C -> A over C
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
6. People make risky decisions by maximizing their expected utility.
Which portfolio to take? A or B?
The difference between A and B depends on your preferences towards risk.
These preferences are modelled as an utility function (e.g. iso-utility curve).
Maximizing expected utility
We like to optimize the expected utility of wealth
--
Where n are the possible outcomes (scenarios), pi is the probability of the
scenario, and wi is the wealth acquired in that particular scenario
The shape of the utility function determines risk aversion.
1
,Example: soccer scores
In the past, the following was used in counting soccer scores in a tournament.
Risk attitudes
The first team plays a risky strategy involving an offensive playing style
while the second team plays a conservative strategy involving a
defensive playing style
Public didn’t like too many tie’s so decided to change the wins to 3
points instead of 2: people like risk.
Intermediate conclusion:
The weighting scheme can be adjusted to accommodate risk preference
In utility theory, the risk preferences are implemented by choosing a
utility function
Risk aversion/preference can be illustrated via 2 ways:
o Shape of function (exponential vs. quadratic)
o Parameters used (2x^3 or 3x^2)
2
,Utility of wealth and returns
Ultimately, we are interested in the utility of wealth (as opposed to the
utility of returns because in the end, the wealth (money amount able to
spend), determines the purchasing power and so utility of consumption).
However, for reasons of convenience, we transform wealth into one-
period returns without affecting the preference order.
--
Initial wealth level where r means the return
We prefer returns over wealth for many reasons, including comparability
between individuals with different wealth, statistical properties in time
series analysis (stationary)
Wealth = explosive process, you can admit when you plot the numbers
Returns as % = approximately with constant deviation
Example expected utility of returns
Since all three assets have the same expected return, preference orderings
must arise from differences in risk attitude.
3
, Properties of utility functions
1. Non-satiation: economic subjects prefer more over less. This can be seen
through the first derivative: positive
a. First derivative >0 ---
2. Risk preference: second derivative
a. Second derivative <0: risk aversion (concave) = negative but
doesn’t mean that you don’t want to take risk, but want to be
awarded for it
b. Second derivative = 0: risk -neutrality
c. Second derivative >0: risk-loving / preference (convex)
Another way of looking at risk aversion is the fair game concept
A fair game is essentially a lottery with an expected value of zero. The
investment equals the outcome, so the expected value is zero.
Example of a fair gamble
An investor with initial wealth of 1000 considers a lottery with 50% chance of
winning 300 and 50% chance of losing 300. Participating in the lottery does not
require an investment.
If the expected utility after playing the game exceeds the expected utility
before playing the game, then the investor should take the lottery.
--
The investor would reject this lottery
4
Portfolio theory is designed by Harry Markowitz
Utility analysis
Standard neo-classical approach for facilitating decision making under risk.
The assumption is that the investor is rational > to make sure that rationality is
enforced with the utility model: six axioms
1. People have preferences
2. Peoples preferences are transitive: A over B, B over C -> A over C
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
6. People make risky decisions by maximizing their expected utility.
Which portfolio to take? A or B?
The difference between A and B depends on your preferences towards risk.
These preferences are modelled as an utility function (e.g. iso-utility curve).
Maximizing expected utility
We like to optimize the expected utility of wealth
--
Where n are the possible outcomes (scenarios), pi is the probability of the
scenario, and wi is the wealth acquired in that particular scenario
The shape of the utility function determines risk aversion.
1
,Example: soccer scores
In the past, the following was used in counting soccer scores in a tournament.
Risk attitudes
The first team plays a risky strategy involving an offensive playing style
while the second team plays a conservative strategy involving a
defensive playing style
Public didn’t like too many tie’s so decided to change the wins to 3
points instead of 2: people like risk.
Intermediate conclusion:
The weighting scheme can be adjusted to accommodate risk preference
In utility theory, the risk preferences are implemented by choosing a
utility function
Risk aversion/preference can be illustrated via 2 ways:
o Shape of function (exponential vs. quadratic)
o Parameters used (2x^3 or 3x^2)
2
,Utility of wealth and returns
Ultimately, we are interested in the utility of wealth (as opposed to the
utility of returns because in the end, the wealth (money amount able to
spend), determines the purchasing power and so utility of consumption).
However, for reasons of convenience, we transform wealth into one-
period returns without affecting the preference order.
--
Initial wealth level where r means the return
We prefer returns over wealth for many reasons, including comparability
between individuals with different wealth, statistical properties in time
series analysis (stationary)
Wealth = explosive process, you can admit when you plot the numbers
Returns as % = approximately with constant deviation
Example expected utility of returns
Since all three assets have the same expected return, preference orderings
must arise from differences in risk attitude.
3
, Properties of utility functions
1. Non-satiation: economic subjects prefer more over less. This can be seen
through the first derivative: positive
a. First derivative >0 ---
2. Risk preference: second derivative
a. Second derivative <0: risk aversion (concave) = negative but
doesn’t mean that you don’t want to take risk, but want to be
awarded for it
b. Second derivative = 0: risk -neutrality
c. Second derivative >0: risk-loving / preference (convex)
Another way of looking at risk aversion is the fair game concept
A fair game is essentially a lottery with an expected value of zero. The
investment equals the outcome, so the expected value is zero.
Example of a fair gamble
An investor with initial wealth of 1000 considers a lottery with 50% chance of
winning 300 and 50% chance of losing 300. Participating in the lottery does not
require an investment.
If the expected utility after playing the game exceeds the expected utility
before playing the game, then the investor should take the lottery.
--
The investor would reject this lottery
4
