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# FEC 512.07

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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
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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
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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
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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