FEC 512.01

1

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.

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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% )

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Overview

Before getting into applications in financialeconometrics 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

FEC 512 Preliminaries and Review Lecture 1-4

I. Preliminary: Asset Return

Calculations

Istanbul Bilgi University

FEC 512 Financial Econometrics-I

Asst. Prof. Dr. Orhan Erdem


Background

� How do the prices of stocks and other

financial assets behave?

� We will start by defining returns on the prices

of a stock.


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.


Asset Return Calculations

Pt is the price of a stock at time t. Stock pays no

dividends.

� Simple return

� Simple gross return

1

1 tt

t

PR

P −

+ =

( )11 1

1t t t

t

t t

P P PR

P P

−

− −

−= = −


� Multi-period returns e.g.

� In general, k-month gross return is defined as

� Note: For small values of Rt

1

2 1 2

1

(2) 1 1

(1 )(1 ) 1

t t tt

t t t

t t

P P PR

P P P

R R

−

− − −

−

= − = −

= + + −

1 11 ( ) (1 )(1 )....(1 )t t t t kR k R R R− − ++ = + + +

∑−

=−

+−

≅

+++≅+1

0

1

)(

or ...1)(1

k

i

itt

kttt

RkR

RRkR


Example 1

� Suppose that the price of Arçelik stock on January is 100YTL, and on February is 105YTL, and that yousell the stock now(on March) at Pt=110YTL. Assumeno 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


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


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%


Log-Returns

� The log-return is

� 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%.

)1ln()/log()log()log( 11 tttttt RPPPPr +===−= −−


� 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


log(1+x) and x

when x is small


Advantage of Log-Returns

� Simplicity of multiperiod returns. Simply the sum:

....

)1log(...)1log(

)}1)...(1log{()}(1log{)(

11

1

1

+−−

+−

+−

++=

++++=

++=+=

kttt

ktt

ktttt

rrr

RR

RRkRkr


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.)


Portfolio Return

� where wi is the weight of each asset in the

portfolio.

Example:

1

N

p i i

i

R w R=

=∑


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


II. Review of Probability &

Statistics


Probability and Finance

� Because we cannot build purelydeterministic models of the economy, weneed a mathematical representation of uncertainty in finance (probability, fuzzymeasures etc…)

� In economic and finance theory, probabilitymight have 2 meanings:

1. As a descriptive concept

2. As a determinant of the agent decisionmaking theory.


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.


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


Sample Space=

Outcome

The Set of Odd numbers is an Event

Event Example 1: The probability that the price of a security be in a

given range, say (10,12)YTL

Example 2:

Probabilities are defined on events.

Examples


Mutually Exclusive Events

� If E1 occurs, then E2 cannot occur

� E1 and E2 have no common elements

Odd

Numbers

Even

Numbers

A die cannot be

Odd and Even at

the same time.

E1

E2


� Independent: Occurrence of one does not

influence the probability of occurrence of

the other

� Dependent: Occurrence of one affects the

probability of the other

Independent and Dependent

Events


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


Assigning Probability

� Classical Probability Assessment

� Relative Frequency of Occurrence

� Subjective Probability Assessment

P(Ei) =Number of ways Ei can occur

Total number of elementary events

Relative Freq. of Ei =Number of times Ei occurs

N

An opinion or judgment by a decision maker about

the likelihood of an event


Rules of Probability

Rules for

Possible Values

and Sum

Individual Values Sum of All Values

0 ≤ P(Ei) ≤ 1

For any event Ei

1)P(ek

1i

i =∑=

where:

k = Number of individual outcomes

in the sample space

ei = ith individual outcome

Rule 1 Rule 2


Addition Rule forElementary Events

Rule 3

� 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)


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.

� Complement Rule:

P(E)1)EP( −= E

E

1)EP(P(E) =+Or,


Addition Rule for Two Events

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

E1 E2


Don’t count common

elements twice!

■ Addition Rule:

E1 E2+ =


Addition Rule Example

� P( Even or Asal)= P(Even) +P(Asal) - P(Even and Asal)

3/6 + 3/6 - 1/6 = 5/6

2,4,6 2,3,5 2


Addition Rule for Mutually Exclusive Events

� If E1 and E2 are mutually exclusive, then

P(E1 and E2) = 0

So


= P(E1) + P(E2)

= 0

E1 E2

if mutu

ally

exclusiv

e


Conditional Probability

� Conditional probability for any

two events E1 , E2:

)P(E

)EandP(E)E|P(E

2

2121 =

0)P(Ewhere 2 >


� 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)

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.



No CDCD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

� 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.

.2857.7

.2

P(AC)

AC)andP(CDAC)|P(CD ===

(continued)



No CDCD Total

AC .2 .5 .7

No AC .2 .1 .3

Total .4 .6 1.0

� 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%.

.2857.7

.2

P(AC)

AC)andP(CDAC)|P(CD ===

(continued)


For Independent Events:

� Conditional probability for

independent events E1 , E2:

)P(E)E|P(E 121 = 0)P(Ewhere 2 >

)P(E)E|P(E 212 = 0)P(Ewhere 1 >


Multiplication Rules

� Multiplication rule for two events E1 and E2:

)E|P(E)P(E)EandP(E 12121 =

)P(E)E|P(E 212 =Note: If E1 and E2 are independent, thenand the multiplication rule simplifies to

)P(E)P(E)EandP(E 2121 =


Bayes’ Theorem

� 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

)E|)P(BP(E)E|)P(BP(E)E|)P(BP(E

)E|)P(BP(E

P(B)

)E|)P(BP(EB)|P(E

kk2211

ii

iii

+++=

=

K


More Simply,

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.

)(

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

BP

BP=


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?


Bayes’ Theorem Example (cont.)

Let A: the event that brokers forecast that the price of the stock will

rise.

P(ARise)=30%

P(ANo Rise)= 6%

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.

83.003.015.0

15.0

5.0*06.05.0*30.0

5.0*30.0

Rise) No|ise)P(A P(NoRise)|ise)P(AP(

Rise)|ise)P(AP(

)(

Rise)|ise)P(AP(A)|(

=+

=+

=

+==

RR

R

AP

RRiseP


� P(Rise) = .5 , P(U) = .5 (prior probabilities)

� Conditional probabilities:

P(ARise)=30% P(ANo Rise)= 6%

� Revised probabilities

Bayes’ Theorem Example

.03/.18 = .166.5*.06 = .03.06.5No Rise

.15/.18 = .833.5*.30 = .15.30.5Rise

Revised

Prob.

Joint

Prob.

Conditional

Prob.

Prior

Prob.Event

Sum = .18

(continued)


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.


For Further Study

� For Topic I:Ruppert D. (2004), Statistics and

Finance, Springer.

� For Topic II:Groebner D.F. et al.(2008)

Business Statistics.

FEC 512.01

Economy & Finance

mutually exclusive events

mutually exclusive

month inv

sample space

ewhere 2

bayes law

addition rule

fec 512 preliminaries