FEC 512.01

FEC 512 Financial Econometrics
-About the Course-
• What is Financial Econometrics?
– Science of modeling and forecasting financial time series.
• Who is this course for?
– Students in finance, practitioners in the financial services
sector.
• How is the presentation of the lectures?
– Begins with review of necessary statistics and probability
theory, continues with basics of econometrics reaches up
to the most recent theoretical results
– Uses computer applications, Eviews.
– Course materials are online at online.bilgi.edu.tr
– Students are supposed to choose a data set from the list at
the beginning of the term in order to do assignments
– Attendence is not required.

1

FEC 512 Financial Econometrics
-About the Course-
• Textbooks:
1. (for Statistics part only) Groebner D.F. et al.(2008)
Business Statistics.
2. Ruppert D. (2004), Statistics and Finance, Springer.
3. Brooks, C. “Introductory Econometics for Finance”
4. Stock J.H. and Watson M.W. (2003), Introduction to
Econometrics (first edition), Addison-Wesley.
• Method of Evaluation:
– Assignments (50%)
– Final Examination (50% )

2

Overview
Before getting into applications in financial
econometrics we will first define
• returns on assets
Then we will review
• Probability
– probability density functions, cumulative distribution functions
– expectations, variance, covariance and correlation
• Statistics
– Testing
– Estimation
• We’ll also be studying some new areas of statistics:
– Regression
• interesting connections with portfolio analysis.
– Probit, Logit Analysis
– Time series models
3

I. Preliminary: Asset Return
Calculations

Istanbul Bilgi University
FEC 512 Financial Econometrics-I
Asst. Prof. Dr. Orhan Erdem

FEC 512 Preliminaries and Review Lecture 1-4

Background

How do the prices of stocks and other
financial assets behave?
We will start by defining returns on the prices
of a stock.

Lecture 1-5
FEC 512 Preliminaries and Review

Prices and Returns

Main data of financial econometrics are asset
prices and returns.
Almost all empirical research analyzes
returns to investors rather than prices. Why?
Investors are interested in revenues that are
high r.t. size of the initial invstmnt. Returns
measure this: Changes in prices expressed
as a fraction of the initial price.

Lecture 1-6

Asset Return Calculations

Pt is the price of a stock at time t. Stock pays no
dividends.
Simple return
( Pt − Pt −1 ) = P
Rt = −1
t

P−1 P−1
t t

Simple gross return
Pt
Rt + 1 =
Pt −1

Lecture 1-7

Multi-period returns e.g.
Pt Pt Pt −1
R t (2) = −1 = −1
Pt − 2 Pt −1 Pt − 2
= (1 + R t )(1 + R t −1 ) − 1
In general, k-month gross return is defined as
1 + R t ( k ) = (1 + R t )(1 + R t −1 )....(1 + R t − k + 1 )
Note: For small values of Rt
1 + Rt (k ) ≅ 1 + Rt + ... + Rt − k +1 or
k −1
Rt (k ) ≅ ∑ Rt −i
i =0

Lecture 1-8

Example 1

Suppose that the price of Arçelik stock on January
is 100YTL, and on February is 105YTL, and that you
sell the stock now(on March) at Pt=110YTL. Assume
no dividends,then

Rt=(110-105)/105=0.0476
Rt-1=(105-100)/100=0.05
Rt(2)=(110-100)/100=0.10
Check also that 1+Rt(2)=(1+ Rt)(1+ Rt-1)
1.0476*1.05=1.1

Lecture 1-9

Annualizing Returns

If investment horizon is one year
1+RA =1+R(12) =(1+R1) (1+R2)... (1+R12)

One month inv. with return Rt, (assume Rt=R)
1+RA=(1+R)12
Two month inv. with return Rt(2), (assume
Rt(2)=R(2))
1+RA=(1+R(2))6

Lecture 1-10

Cont. to Example 1

In the first example the one month return was
4.76%. If we assume that we can get this
return for 12 months then the annualized
return is

RA=(1.0476)12-1=1.7472-1=0.7472 or 74.72%

Lecture 1-11

Log-Returns

The log-return is

rt = log( Pt ) − log( Pt −1 ) == log( Pt / Pt −1 ) = ln(1 + Rt )

The log return in the previous example is
rt=ln(0.0476)=0.0465 or 4.65%

The above return measures are very similar
numbers since daily returns are very rarely outside
the range of -10% to 10%.

Lecture 1-12

Log returns are approximately equal to net
returns:
x small ⇒ log(1 + x) ≅ x
Therefore, rt = log(1 + Rt) ≅ Rt
Examples:
* log(1 + 0.05) = 0.0488
* log(1 -0.05) = -0.0513

Lecture 1-13

log(1+x) and x
when x is small

Lecture 1-14

Advantage of Log-Returns

Simplicity of multiperiod returns. Simply the sum:

rt (k ) = log{ + Rt (k )} = log{( + Rt )...(1 + Rt −k +1 )}
1 1
= log( + Rt ) + ... + log( + Rt −k +1 )
1 1
= rt + rt −1 + ...rt −k +1.

Lecture 1-15

Returns are
scale-free, meaning that they do not depend
on monetary units (dollars, cents, etc.)
not unit-less, unit is time; they depend on the
units of t (hour, day, etc.)

Lecture 1-16

Portfolio Return
N
R p = ∑ wi Ri
i =1

where wi is the weight of each asset in the
portfolio.
Example:

Lecture 1-17

About Returns

Returns cannot be perfectly predicted, they
are random.
This randomness implies that a return might
be smaller than its expected value and even
negative, which means that investing involves
RISK.
It took quite some time before it was realized
that risk could be described by probability
theory
Lecture 1-18

II. Review of Probability &
Statistics

FEC 512 Preliminaries and Review Lecture 1-19

Probability and Finance

Because we cannot build purely
deterministic models of the economy, we
need a mathematical representation of
uncertainty in finance (probability, fuzzy
measures etc…)
In economic and finance theory, probability
might have 2 meanings:
As a descriptive concept
1.
As a determinant of the agent decision
2.
making theory.
Lecture 1-20

Probability as a Descriptive Concept

The probability of an event is assumed to be
approx. equal to the rel.freq. of its occurrence
in a large # experiments.
There is one difficulty with this interpretation:
Empirical data have only one realization.
Every estimate is made on a single time-evolving
series.
If stationarity(!) is not assumed, performing
statistical estimation is impossible.

Lecture 1-21

Probability Concepts
Experiment – a process of obtaining
outcomes for uncertain events
Outcome – the possible results of an
observation, such as the price of a security
at t.
However, probability statements are not
made on outcomes but on events, which are
sets of possible outcomes.
The Sample Space is the collection of all
possible outcomes
Lecture 1-22

Examples
Event Example 1: The probability that the price of a security be in a
given range, say (10,12)YTL
Example 2:
Outcome

Sample Space=

The Set of Odd numbers is an Event

Probabilities are defined on events.

Lecture 1-23

Mutually Exclusive Events

If E1 occurs, then E2 cannot occur
E1 and E2 have no common elements

E2 A die cannot be
E1
Odd and Even at
Even the same time.
Odd Numbers
Numbers

Lecture 1-24

Independent and Dependent
Events

Independent: Occurrence of one does not
influence the probability of occurrence of
the other
Dependent: Occurrence of one affects the
probability of the other

Lecture 1-25

Independent vs. Dependent Events
Independent Events
E1 = heads on one flip of fair coin
E2 = heads on second flip of same coin
Result of second flip does not depend on the
result of the first flip.

Dependent Events
E1 = rain forecasted on the news
E2 = take umbrella to work
Probability of the second event is affected by the
occurrence of the first event

Lecture 1-26

Assigning Probability

Classical Probability Assessment
Number of ways Ei can occur
P(Ei) =
Total number of elementary events

Relative Frequency of Occurrence
Number of times Ei occurs
Relative Freq. of Ei =
N

Subjective Probability Assessment
An opinion or judgment by a decision maker about
the likelihood of an event

Lecture 1-27

Rules of Probability

Rules for
Possible Values
and Sum

Individual Values Sum of All Values

k
0 ≤ P(Ei) ≤ 1
∑ P(e ) = 1
i
For any event Ei Rule 1 Rule 2
i=1
where:
k = Number of individual outcomes
in the sample space
ei = ith individual outcome
Lecture 1-28

Addition Rule for
Elementary Events
The probability of an event Ei is equal to
the sum of the probabilities of the
individual outcomes forming Ei.
That is, if:
Ei = {e1, e2, e3}
then:
P(Ei) = P(e1) + P(e2) + P(e3) Rule 3

Lecture 1-29

Complement Rule

The complement of an event E is the
collection of all possible elementary events
not contained in event E. The complement of
event E is represented by E.
E

Complement Rule:
P( E ) = 1 − P(E) E

P(E) + P( E ) = 1
Or,
Lecture 1-30

Addition Rule for Two Events

Addition Rule:
■

P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)

+ =
E1 E2 E1 E2

P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2)
Don’t count common
elements twice!

Lecture 1-31

Addition Rule Example
P( Even or Asal)= P(Even) +P(Asal) - P(Even and Asal)
3/6 + 3/6 - 1/6 = 5/6

2
2,4,6 2,3,5

Lecture 1-32

Addition Rule for
Mutually Exclusive Events

If E1 and E2 are mutually exclusive, then

E1 E2
P(E1 and E2) = 0

So
0 utualvlye
= if m lusi
P(E1 or E2) = P(E1) + P(E2) - P(E1 and E2) c
ex

= P(E1) + P(E2)

Lecture 1-33

Conditional Probability

Conditional probability for any
two events E1 , E2:

P(E1 and E 2 )
P(E1 | E 2 ) =
P(E2 )

P(E2 ) > 0
where

Lecture 1-34

Conditional Probability Example

Of the cars on a used car lot, 70% have air
conditioning (AC) and 40% have a CD
player (CD). 20% of the cars have both.

What is the probability that a car has a CD
player, given that it has AC ?

i.e., we want to find P(CD | AC)

Lecture 1-35

(continued)
Of the cars on a used car lot, 70% have air
conditioning (AC) and 40% have a CD player (CD).
20% of the cars have both.
CD No CD Total
.2 .5 .7
AC
.2 .1
No AC .3
.4 .6 1.0
Total

P(CD and AC) .2
P(CD | AC) = = = .2857
P(AC) .7
Lecture 1-36

(continued)
Given AC, we only consider the top row (70% of the cars). Of
these, 20% have a CD player. 20% of 70% is about 28.57%.

CD No CD Total
.2 .5 .7
AC
.2 .1
No AC .3
.4 .6 1.0
Total

P(CD and AC) .2
P(CD | AC) = = = .2857
P(AC) .7
Lecture 1-37

For Independent Events:

Conditional probability for
independent events E1 , E2:

P(E1 | E 2 ) = P(E1 ) P(E2 ) > 0
where

P(E2 | E1 ) = P(E2 ) P(E1 ) > 0
where

Lecture 1-38

Multiplication Rules

Multiplication rule for two events E1 and E2:

P(E1 and E 2 ) = P(E1 ) P(E2 | E1 )

Note: If E1 and E2 are independent, then P(E2 | E1 ) = P(E2 )
and the multiplication rule simplifies to

P(E1 and E2 ) = P(E1 ) P(E2 )

Lecture 1-39

Bayes’ Theorem

P(Ei )P(B| Ei )
P(Ei | B) =
P(B)
P(Ei )P(B| E i )
=
P(E1 )P(B| E1 ) + P(E2 )P(B| E 2 ) + K + P(Ek )P(B| E k )

where:
Ei = ith event of interest of the k possible events
A = new event that might impact P(Ei)
Events E1 to Ek are mutually exclusive and collectively
exhaustive

Lecture 1-40

More Simply,

P( B | A)P(A)
P(A | B) =
P( B)

Bayes Theorem allows one to recover the
probability of the event A given B from the
probability of the individual events A,B, and
the probability of B given A.

Lecture 1-41

Bayes’ Theorem Example
Suppose that the probability that the price of a
stock will rise on any given day, is 0.5. Thus,
we have the prior probabilities
P(Rise)=0.5 and P(No rise)=0.5.
When it actually rises, the brokers correctly
forecasts the rise 30% of the time. When it
does not rise, they incorrectly forecast rise 6%
of the time. What is the probability that the
prices will rise if the brokers forecasted that it
will rise tomorrow?

Lecture 1-42

Bayes’ Theorem Example (cont.)

Let A: the event that brokers forecast that the price of the stock will
rise.
P(ARise)=30%
P(ANo Rise)= 6%

P( Rise)P(A | Rise) P( Rise)P(A | Rise)
P( Rise | A) = =
P( Rise)P(A | Rise) + P(No Rise)P(A | No Rise)
P( A)
0.30 * 0.5 0.15
= = = 0.83
0.30 * 0.5 + 0.06 * 0.5 0.15 + 0.03

As it can be seen we updated the probability of a rise (0.5) to 0.83
after we heard the brokers’s forecast of rise.

Lecture 1-43

Bayes’ Theorem Example
(continued)

P(Rise) = .5 , P(U) = .5 (prior probabilities)
Conditional probabilities:
P(ARise)=30% P(ANo Rise)= 6%

Revised probabilities
Prior Conditional Joint Revised
Event
Prob.
Prob. Prob. Prob.
Rise .5 .30 .5*.30 = .15 .15/.18 = .833
No Rise .5 .06 .5*.06 = .03 .03/.18 = .166

Sum = .18 Lecture 1-44

Importance of Bayes Law

We update our beliefs in light of new
information.
Revising beliefs after receiving additional info
is smth that humans do poorly without the
help of mathematics.
There is a tendency to put either too little or
too much emphasis on new info
This problem can be mitigated by using
Bayes’ Law.
Lecture 1-45

For Further Study

For Topic I:Ruppert D. (2004), Statistics and
Finance, Springer.
For Topic II:Groebner D.F. et al.(2008)
Business Statistics.

Lecture 1-46

FEC 512.01

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