Transcript
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
top related