SK460 Finance Theory, Diderik Lund, 16 January 2001
Administrative
Course used to be 26 lectures (of 90 minutes). Now reduced to 20
lectures. Reason is budgetary, need to save money. Curriculum is
not changed, however. Students have the same amount of time at
disposal, but fewer lectures. Term paper work during weeks 8 and
9 (19 February { 4 March).
Method to speed up lectures: Distribute transparencies via website
before lectures. Students may bring paper copies to lectures. Some
transparencies may be incomplete, in particular, missing diagrams.
Will be drawn during lectures.
Much extra information is given in the lectures, which you should
therefore attend.
Little time to work with solving (exam-like or simpler) problems
during lectures. Instead: Will hand out problems with solutions,
and suggested solutions to a selection of previous exams.
Language: Students may ask questions in Norwegian. Lecturer will
translate before answering in English. Students should also ask for
translations into Norwegian whenever needed.
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Finance Theory: Overview
Main topic: What are the values of various assets?
Both nancial and real assets: Securities (shares of stock, bonds,
options, etc.), investment projects, property
Central feature of theories: Uncertainty about future income
streams connected to the assets, or their values in the future
Equilibrium models: Supply and demand determine values
Lectures 1{8 cover such equilibrium models.
Lectures 11{17 cover another type of models. In some situations
the values of some securities can be derived from the values of other
securities. Method called \absence-of-arbitrage" pricing. Used in
particular to nd the values of options (a kind of securities).
Lecture 9 cover the functioning of capital markets and (briey) the
nancing of corporations.
Lectures 10, 18{20 cover some applications of the theories.
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Relation to other courses
This course does not cover conicts due to asymmetries of information between management, shareholders, and lenders to the corporation (creditors, bondholders). Those topics are covered in SK531
Economics of the Firm.
Lectures 1{3 overlap with SK400 Microeconomic Theory.
Relation to the real world (!)
You will not learn how to make money in the markets
In fact, you will learn why that is very dicult
You will learn basic theory about what determines (and what
does not inuence) security equilibrium prices
You will also learn about the role of these markets in the economy
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Individual choices under uncertainty
(Repeats some material from \andre avdeling" and SK400.)
Basic assumtion: Know probability distribution of outcomes. If
not, \total uncertainty," may form subjective probability distributions and act on basis of those.
To begin with: Uncertainty in one period only. Choices are made
now (often called period zero), with uncertainty about what will
happen next (period one). Each choice alternative gives one probability distribution of outcomes in period one. (Some models also
have consumption in period zero.)
All consequences and the total situation of the decision maker
should be taken into consideration when choices are described. Copeland
and Weston use simple numbers, like:
Choose between (a) keeping $10 and (b) spending it on a
lottery ticket with 1 per cent probability of winning $1000
and 99 per cent of loss.
This is dierent from the problem of choosing between $10010 on
one hand and on the other a 1 per cent probability of $11000 and
99 per cent probability of $10000.
Simplication in nance: Only one good, money. (But theory in
chapters 4 and 5 can deal with vectors of dierent goods.)
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SK460 Finance Theory, Diderik Lund, 16 January 2001
From these simplications: Each individual only interested in probability distribution of wealth at the end of period one.
Consider rst choice between a limited number of dierent alternatives. Examples:
Spending money on education now, receiving uncertain income
from it.
Investing in a farm now, receiving crop with uncertain value.
Buying shares in a corporation now, receiving uncertain value
and dividends.
Buying shares in two corporations now, receiving : : : .
The same amount will be spent in period zero in all alternatives.
Assume that choice is made on the basis of the probability distributions of the stochastic outcomes for wealth one period later,
~ 3; W~ 4:
W~ 1 ; W~ 2; W
Axioms 1 and 2 (C& W, p. 79) say that an individual is able to
compare and choose between such stochastic variables, and that
preferences are transitive. These axioms are known from standard
consumer theory.
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SK460 Finance Theory, Diderik Lund, 16 January 2001
von Neumann and Morgenstern's theory
\Expected utility"
Objects of choice called gambles. Simplication: Each has only
two possible, mutually exclusive outcomes. Notation: G(x; y : )
means:
x
X
XXXXX
XX
1 , XX y
(The G() notation means: After the colon comes the probability
(here: 2 (0; 1)) of the outcome mentioned rst (here: x).)
Axioms 3{5 specic to preferences over gambles. The theory assumes axioms 1{5 hold for the preferences of one individual. Using
the theory, we usually assume it holds for all individuals, but their
preferences may vary within the restrictions given by the theory.
Axiom 3 Strong independence:
Let x; y and z be outcomes of gambles, and assume x y . Then
G(x; z : ) G(y; z : ):
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Axiom 4 Measurability:
If x y z (or x y z ), then there exists a unique such that
y G(x; z : ):
(Not obvious. What about life and death?)
Axiom 5 Ranking:
Assume x y z , and x u z , and y G(x; z : 1), and
u G(x; z : 2). Then
1 > 2 ) y u;
1 = 2 ) y u:
(Actually: None of axioms are obvious.)
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Derivation of theorem of expected utility
Simple version: Assume the set S of outcomes is closed and bounded.
Exists one best outcome, a, and one worst outcome, b, with a b.
With reference to a and b dene a function () such that
for all x 2 S; x G(a; b: (x)):
This probability exists for all x by axiom 4. By axiom 5 it can
be used to rank outcomes, since (x) > (y ) ) x y and
(x) = (y ) ) x y . Thus () is a kind of utility function. Will
prove it can also be used to rank gambles.
Digression: A utility function assigns a real number to any object
of choice, such that a higher number is given to a preferred object,
and equal numbers are given to objects of indierence. If x and y
are money outcomes or otherwise quantities of a (scalar) good, and
there is no satiation, then is an increasing function.
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Consider a gamble G(x1 ; x2: ), which means:
x1
X
XXXXX
XXXX x
1,
2
When x1 G(a; b: (x1 )) and x2 G(a; b: (x2 )), then G(x1 ; x2; )
can be written as
(x1)
(x1) a
+ (1 , )(x2) a
XXXX1X, (x1 )
X
X
XXX b
(1 , (x1 ))
HH
H
H
HHH1 , HHH+ (1 , )(1 , (x2))
(
x
)
2
HHH
HHH
a
HHXX
HHH
(
x
)
XX1X,
2
HHH
XXXXX
HH b
b
So that
G(x1 ; x2: ) G(a; b: (x1) + (1 , )(x2 )):
Thus the \utility" of G(x1 ; x2: ) is (x1) + (1 , )(x2 ).
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Write X~ for the stochastic variable with outcomes x1 with probability and x2 with probability 1 , . The utility expression
(x1) + (1 , )(x2 ) can be interpreted as E [(X~ )], which explains why it is called expected utility.
Notation: Usually the letter U is chosen for the utility function
instead of , and expected utility of X~ is written E [U (X~ )]. (In
SK400: Gravelle and Rees use E [v (X~ )].)
Possible to extend to ordering of gambles of more than two outcomes,
S
X
~
E [U (X )] = sU (xs );
s=1
even to a continuous probability distribution,
Z1
~
E [U (X )] =
U (x)f (x)dx:
,1
Will not look at this more formally.
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Criticism of vN-M expected utility
Some experiments indicate that many people's behavior in some
situation contradicts expected utility maximization.
Exist alternative theories, in particular generalizations (alternative theories in which expected utility appears as one special
case).
Nevertheless much used in theoretical work on decisions under
uncertainty.
Example of when vN-M may not work
Suppose every consumption level below 5 is very bad.
Suppose, e.g., that U (4) = ,10; U (6) = 1; U (8) = 4; U (10) =
5.
Then E [U (4; 10: 0:1)] = 0:1 (,10) + 0:9 5 = 3:5, while
E [U (6; 8: 0:1)] = 0:1 1 + 0:9 4 = 4:6.
But even with the huge drop in U level when consumption
drops below 5, one will prefer the rst of these two alternatives
(the gamble G(4; 10: )) to the other (G(6; 8: )) as soon as drops below 1=12.
If one outcome is so bad that someone will avoided it any cost,
even when its probability is very low, then that person's behavior contradicts the vN-M theory.
In particular, axiom 4 is contradicted.
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SK460 Finance Theory, Diderik Lund, 16 January 2001
Uniqueness of U function?
Given a vN-M preference ordering of one individual, have now
shown we can nd a U function such that
X~ Y~ if and only if E [U (X~ )] > E [U (Y~ )]:
Is U unique? No, depends on a and b, but preferences between X~
and Y~ do not.
Dene an increasing linear transformation of U ,
V (x) c1 U (x) + c0 ;
where c1 > 0 and c0 are constants. This represents the preferences
of the same individual equally well since
E [V (X~ )] = c1E [U (X~ )] + c0
for all X , so that a higher E [U (X~ )] gives a higher E [V (X~ )].
But not possible to do similar replacement of U with any nonlinear transformation of U (as opposed to ordinal utility functions
for usual commodities). For instance, E fln[U (X~ )]g does not necessarily increase when E [U (X~ )] increses. So ln[U ()] cannot be used
to represent the same preferences as U ().
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