FEC 512.07

1

FEC 512

Financial Econometrics I

Behavior of Returns

FEC 512 2

What can we say about returns?

� Cannot be perfectly predicted — are random.

� Ancient Greeks:

� Would have thought of returns as determined by

Gods or Fates (three Goddesses of destiny)

� Did not realize random phenomena exhibit

regularities(Law of large numbers, central limit th.)

� Did not have probability theory despite their

impressive math

FEC 512 3

Randomness and Probability

� Probability arouse of gambling during the

Renaissance.

� University of Chicago economist Frank Knight

(1916) distinguished between

� Measurable uncertainty (i.e.games of

chance):probabilities known

� Unmeasurable uncertainty (i.e.finance):

probabilities unknown

FEC 512 4

Uncertainty in returns

� At time t, Pt+1 and Rt+1 are not only unknown,

but we do not know their probability

distributions.

� Can estimate these distributions: with an

assumption

FEC 512 5

Leap of Faith

� Future returns similar to past returns

� So distribution of Pt+1 can estimated from

past data

FEC 512 6

� Asset pricing models (e.g. CAPM) use the

joint distribution of cross-section {R1t, R2t,…

RNt} of returns on N assets at a single time t.

� Rit is the returns on the ith asset at time t.

� Other models use the time series {Rt, Rt-1,…

R1} of returns on a single asset at a

sequence of times 1,2,…t.

� We will start with a single asset.

FEC 512 7

Common Model:IID Normal Returns

� R1,R2,...= returns from single asset.

1. mutually independent

2. identically distributed

3. normally distributed

� IID = independent and identically distributed

FEC 512 8

Two problems

1. The model implies the possibility of

unlimited losses, but liability is usually

limited Rt ≥-1 since you can lose no more

than your investment

2. 1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1) is

not normal

� Sums of normals are normal but not

products

� But it would be nice to have normality, so

math is simple

FEC 512 9

The Lognormal Model

� Assumes: rt = log(1 + Rt)* are IID and normal

� Thus,we assume that

� rt =log(1 + Rt) ~ N(µ,σ2)

� So that 1 + Rt = exp(normal r.v.) ≥ 0

� So that Rt ≥ -1.

� This solves the first problem

� (*): log(x) is the natural logarithm of x.

-3 -2 -1 0 1 2 3

1

2

3

4

5

6

7

8

9

10

x

y

y=e^{x}

FEC 512 10

Solution to Second Problem

� For second problem:

1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1)

log{1 + Rt(k)} = log{(1 + Rt) ... (1 + Rt-k+1)}

=rt + ... + rt-k+1

� Sums of normals are normal (See Lecture Notes 2)

⇒ the second problem is solved

� Normality of single period returns implies

normality of multiple period returns.

FEC 512 11

Louis Jean-Baptiste Alphonse

Bachelier

� The lognormal distribution goes back to Louis

Bachelier (1900).

� dissertation at Sorbonne called The Theory of

Speculation

� Bachelier was awarded “mention honorable”

Bachelier never found a decent academic

job.

� Bachelier anticipated Einstein’s (1905) theory

of Brownian motion.

FEC 512 12

� In 1827, Brown, a Scottish botanist, observed the

erratic, unpredictable motion of pollen grains under

a microscope.

� Einstein (1905) — movement due to bombardment

by water molecules — Einstein developed a

mathemetical theory giving precise quantitative

predictions.

� Later, Norbert Wiener, an MIT mathematician,

developed a more precise mathematical model of

Brownian motion. This model is now called the

Wiener process.

FEC 512 13

� Bachelier stated that

“The math. expectation of the speculator is

zero” (this is essentially true of short-term

speculation but not of long term investing)

FEC 512 14

Example 1

� A simple gross return (1 + R) is lognormal~ (0,0.12)

– which means that log(1 + R) is N(0,0.12)

What is P(1 + R < 0.9)?

� Solution:

P(1 + R < 0.9) = P{log(1 + R) < log(0.9)}

P{log(1 + R) < -0.105} (log(0.9)= -0.105)

P{ [log(1 + R)-0]/0.1 < [-0.105-0]/0.1}

P{Z<-1.05}=0.1469

FEC 512 15

Matlab and Excel

� In MATLAB, cdfn(-1.05) = 0.1469

� In Excel:NORMDIST(-1.05,0,1,TRUE)=0.1469

FEC 512 16

Example 2

� Assume again that 1 + R is lognormal~(0,0.12) and i.i.d. Find the probability that a simple gross two-period return is less than0.9?

� Solution:log{1 + Rt(2)} = rt + rt-1[ Rmbr Lec-2: if Z=aX+bY µZ=a µX +b µY σZ

2=a2 σX2 +b2 σY

2+2abσXY]

2-period grossreturn is lognormal ~ (0,2(0.1)2)

So this probability is

P(1 + R(2) < 0.9)=P(log[1 + R(2)]<log0.9)= P(Z<-0.745)=0.2281

FEC 512 17

� Let’s find a general formula for the kth period

returns. Assume that

� 1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1)

� log {1 + Ri} ~ N(µ,σ2) for all i.

� The {Ri} are mutually independent.

FEC 512 18

FEC 512 19

Random Walk

FEC 512 20

Random Walk

FEC 512 21

� S0=0 and µ=0.5, σ=1

5 Random Walks

-5

0

5

10

15

20

25

30

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

m*t

S1

S2

S3

S4

S5

Negative Prices can be

observed

FEC 512 22

� S0=1 and µ=0, σ=1

� Similar negative prices can be observed

-10

-8

-6

-4

-2

0

2

4

6

8

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

S1

S2

S3

S4

S5

FEC 512 23

Geometric Random Walk

� Therefore if the log returns are assumed to be i.i.d

normals, then the process {Pt:t=1,2,...} is the

exponential of a random walk.We call it a geometric

random walk or an exponential random walk.

FEC 512 24

Geometric Random Walks

-2

-1

0

1

2

3

4

5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

P1

P2

P4

P5

FEC 512 25

� If r1,r2...are i.i.d N(µ,σ2) then the process is

called a lognormal geometric random walk

with parameters (µ,σ2).

� As the time between steps becomes shorter

and the step sizes shrink in the appropriate

way, a random walk converges to Brownian

motion and a geometric random walk

converges to geometric Brownian motion;

(see Stochastic Processes Lectures.)

FEC 512 26

FEC 512 27

FEC 512 28

The effect of drift µ

FEC 512 29

FEC 512 30

A Simulation of Geometric Random Walk

-0,2

4,8

9,8

14,8

19,8

24,8

29,8

34,8

39,8

44,8

49,8

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

-0,2

0

0,2

0,4

0,6

0,8

1

P (Geom. Random Walk)

log returns

FEC 512 31

FEC 512 32

FEC 512 33

FEC 512 34

Example: Daily Prices for Garanti

� Let’s look at return for Garan from 1/3/2002

to 10/9/2007

� The daily price is taken to be the close price.

0

2

4

6

8

10

12

2002 2003 2004 2005 2006 2007

GARAN

-.16

-.12

-.08

-.04

.00

.04

.08

.12

.16

.20

2002 2003 2004 2005 2006 2007

RET

-.20

-.15

-.10

-.05

.00

.05

.10

.15

.20

2002 2003 2004 2005 2006 2007

LOGRET

FEC 512 35

Sample: 1/03/2002 10/09/2007

GARAN RET LOGRET

Mean 3.026870 0.002054 0.001581

Median 2.240000 0.000000 0.000000

Maximum 10.00000 0.177779 0.163631

Minimum 0.527830 -0.156521 -0.170220

Std. Dev. 2.227524 0.030800 0.030686

Skewness 0.815476 0.274663 0.033390

Kurtosis 2.803732 6.369416 6.243145

Jarque-Bera 162.5863 701.7116 633.5390

Probability 0.000000 0.000000 0.000000

Sum 4376.854 2.968196 2.284883

Sum Sq. Dev. 7169.895 1.369794 1.359728

Observations 1446 1445 1445

FEC 512 36

Example: Monthly Prices for Garanti

-0.8

-0.4

0.0

0.4

0.8

1.2

2002 2003 2004 2005 2006 2007

MONTHLYRET

-.6

-.4

-.2

.0

.2

.4

.6

.8

2002 2003 2004 2005 2006 2007

MOTHLYLOGRET

FEC 512 37

Date: 03/26/09 Time: 12:13

Sample: 1/03/2002 10/09/2007

MONTHLYRET MOTHLYLOGRET

Mean 0.061532 0.046424

Median 0.058146 0.056518

Maximum 1.032785 0.709407

Minimum -0.407409 -0.523250

Std. Dev. 0.174241 0.163564

Skewness 0.675879 -0.153158

Kurtosis 5.621027 4.089647

Jarque-Bera 513.1249 75.58843

Probability 0.000000 0.000000

Sum 87.13000 65.73642

Sum Sq. Dev. 42.95912 37.85555

Observations 1416 1416

FEC 512 38

FEC 512 39

FEC 512 40

Technical Analysis

FEC 512 41

Efficient Market Hypothesis(EMH)

FEC 512 42

FEC 512 43

FEC 512 44

Three type of Efficiency

FEC 512 45

Behavioral Finance-a Challange to EMH

� .

FEC 512 46