FEC 512
Financial Econometrics I
    Behavior of Returns




                           1
What can we say about returns?

    Cannot be perfectly predicted — are random.
    Ancient Greeks:
          Would have t...
Randomness and Probability

    Probability arouse of gambling during the
    Renaissance.
    University of Chicago econo...
Uncertainty in returns


    At time t, Pt+1 and Rt+1 are not only unknown,
    but we do not know their probability
    d...
Leap of Faith


    Future returns similar to past returns
    So distribution of Pt+1 can estimated from
    past data


...
Asset pricing models (e.g. CAPM) use the
    joint distribution of cross-section {R1t, R2t,…
    RNt} of returns on N asse...
Common Model:IID Normal Returns

    R1,R2,...= returns from single asset.
    1. mutually independent
    2. identically ...
Two problems
      The model implies the possibility of
1.
      unlimited losses, but liability is usually
      limited ...
The Lognormal Model

    Assumes: rt = log(1 + Rt)* are IID and normal
    Thus,we assume that
          rt =log(1 + Rt) ~...
Solution to Second Problem

    For second problem:
     1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1)
     log{1 + Rt(k...
Louis Jean-Baptiste Alphonse
  Bachelier
 The lognormal distribution goes back to Louis
Bachelier (1900).
 dissertation at...
In 1827, Brown, a Scottish botanist, observed the
          erratic, unpredictable motion of pollen grains under
         ...
Bachelier stated that
    “The math. expectation of the speculator is
    zero” (this is essentially true of short-term
  ...
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 i...
Matlab and Excel

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




FEC 512            ...
Example 2

    Assume again that 1 + R is lognormal~
    (0,0.12) and i.i.d. Find the probability that a
    simple gross ...
Let’s find a general formula for the kth period
    returns. Assume that
    1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+...
FEC 512   18
Random Walk




FEC 512       19
Random Walk




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5 Random Walks

            30
            25
                                                                       m*t
 ...
10


    8


    6


    4


                                                                                             ...
Geometric Random Walk




    Therefore if the log returns are assumed to be i.i.d
    normals, then the process {Pt:t=1,2...
Geometric Random Walks



      5



      4



      3



                                                               ...
If r1,r2...are i.i.d N(µ,σ2) then the process is
    called a lognormal geometric random walk
    with parameters (µ,σ2).
...
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The effect of drift µ




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A Simulation of Geometric Random Walk


 49,8                                                       1
 44,8
              ...
FEC 512   31
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Example: Daily Prices for Garanti

    Let’s look at return for Garan from 1/3/2002
    to 10/9/2007
    The daily price i...
Sample: 1/03/2002 10/09/2007

                 GARAN          RET         LOGRET

 Mean            3.026870        0.00205...
Example: Monthly Prices for Garanti

                               MONTHLYRET                                  MOTHLYLOGR...
Date: 03/26/09 Time: 12:13
Sample: 1/03/2002 10/09/2007


              MONTHLYRET       MOTHLYLOGRET


Mean           0.0...
FEC 512   38
FEC 512   39
Technical Analysis




FEC 512              40
Efficient Market Hypothesis(EMH)




FEC 512                            41
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FEC 512   43
Three type of Efficiency




FEC 512                    44
Behavioral Finance-a Challange to EMH

    .




FEC 512                                 45
FEC 512   46
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FEC 512.07

  1. 1. FEC 512 Financial Econometrics I Behavior of Returns 1
  2. 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 2
  3. 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 3
  4. 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 4
  5. 5. Leap of Faith Future returns similar to past returns So distribution of Pt+1 can estimated from past data FEC 512 5
  6. 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 6
  7. 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 7
  8. 8. Two problems The model implies the possibility of 1. unlimited losses, but liability is usually limited Rt ≥-1 since you can lose no more than your investment 1 + Rt(k) = (1 + Rt)(1 + Rt-1) ... (1 + Rt-k+1) is 2. not normal Sums of normals are normal but not products But it would be nice to have normality, so math is simple FEC 512 8
  9. 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. y 10 9 This solves the first problem y=e^{x} 8 7 6 5 4 3 (*): log(x) is the natural logarithm of x. 2 1 -3 -2 -1 0 1 2 3 x FEC 512 9
  10. 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 10
  11. 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 11
  12. 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 12
  13. 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 13
  14. 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 14
  15. 15. Matlab and Excel In MATLAB, cdfn(-1.05) = 0.1469 In Excel:NORMDIST(-1.05,0,1,TRUE)=0.1469 FEC 512 15
  16. 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 than 0.9? Solution:log{1 + Rt(2)} = rt + rt-1 [ Rmbr Lec-2: if Z=aX+bY µZ=a µX +b µY σZ2=a2 σX2 +b2 σY2+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 16
  17. 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 17
  18. 18. FEC 512 18
  19. 19. Random Walk FEC 512 19
  20. 20. Random Walk FEC 512 20
  21. 21. 5 Random Walks 30 25 m*t 20 S1 15 S2 S3 10 S4 5 S5 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 -5 S0=0 and µ=0.5, σ=1 Negative Prices can be observed FEC 512 21
  22. 22. 10 8 6 4 S1 2 S2 0 S3 1 23 45 6 78 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 S4 -2 S5 -4 -6 -8 -10 S0=1 and µ=0, σ=1 Similar negative prices can be observed FEC 512 22
  23. 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 23
  24. 24. Geometric Random Walks 5 4 3 P1 2 P2 P4 1 P5 0 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 -1 -2 FEC 512 24
  25. 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 25
  26. 26. FEC 512 26
  27. 27. FEC 512 27
  28. 28. The effect of drift µ FEC 512 28
  29. 29. FEC 512 29
  30. 30. A Simulation of Geometric Random Walk 49,8 1 44,8 0,8 39,8 34,8 0,6 29,8 P (Geom. Random Walk) 24,8 0,4 log returns 19,8 0,2 14,8 9,8 0 4,8 -0,2 -0,2 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 FEC 512 30
  31. 31. FEC 512 31
  32. 32. FEC 512 32
  33. 33. FEC 512 33
  34. 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. GARAN RET LOGRET 12 .20 .20 .16 .15 10 .12 .10 8 .08 .05 .04 6 .00 .00 -.05 4 -.04 -.10 -.08 2 -.15 -.12 0 -.16 -.20 2002 2003 2004 2005 2006 2007 2002 2003 2004 2005 2006 2007 2002 2003 2004 2005 2006 2007 FEC 512 34
  35. 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 35
  36. 36. Example: Monthly Prices for Garanti MONTHLYRET MOTHLYLOGRET 1.2 .8 .6 0.8 .4 0.4 .2 .0 0.0 -.2 -0.4 -.4 -0.8 -.6 2002 2003 2004 2005 2006 2007 2002 2003 2004 2005 2006 2007 FEC 512 36
  37. 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 37
  38. 38. FEC 512 38
  39. 39. FEC 512 39
  40. 40. Technical Analysis FEC 512 40
  41. 41. Efficient Market Hypothesis(EMH) FEC 512 41
  42. 42. FEC 512 42
  43. 43. FEC 512 43
  44. 44. Three type of Efficiency FEC 512 44
  45. 45. Behavioral Finance-a Challange to EMH . FEC 512 45
  46. 46. FEC 512 46

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