Physics307L F08:People/Barron/labsum~finance

Practical Finance and Data Analysis Lab Summary

Here is the lab manual page.

Here are my lab notes.

Partner: Justin Muehlmeyer

Introduction

This lab was basically undefined to start with, with simple goals of gathering knowledge on finance in general, learning where to find data, and using data analysis methods to glean information from the data. We came up with several grand ideas, but finally condensed our goals- the entire process can be seen in the lab notes.

Approach

In the end, we generated fixed-length time windows with random start dates over the period of price data acquired for the Dow Jones Industrial Average, then tried to find an average growth rate using both "snapshots" and least-squares linear fits. We defined snapshots as slopes between the start and end points in each time window, reflecting what a person would receive if he actually invested over exactly each window. Least-squares fits take into account fluctuations in the price between the start and end points, so that someone investing on a shorter interval would probably see results more akin to the least-squares fit of his specific time window. We compared the two methods over several different time windows, then analyzed the least-squares data to see if it fit any sort of distribution.

Final Results

Least-squares:

Least-squares Slope Data [$/week] ± σravg	Sample Window
10 Year Window 1) 2.4689 ± 0.0007 11) 2.6043 ± 0.0007 2) 3.1063 ± 0.0011 12) 3.2287 ± 0.0011 3) 2.5977 ± 0.0008 13) 2.1363 ± 0.0009 4) 2.4893 ± 0.0007 14) 2.8116 ± 0.0008 5) 2.8451 ± 0.0007 15) 2.3922 ± 0.0005 6) 2.1571 ± 0.0007 16) 2.7420 ± 0.0009 7) 3.0676 ± 0.0009 17) 3.1118 ± 0.0008 8) 3.4700 ± 0.0009 18) 2.4796 ± 0.0009 9) 2.1050 ± 0.0006 19) 2.3554 ± 0.0009 10) 2.2250 ± 0.0008 20) 2.0223 ± 0.0009
20 Year Window 1) 2.9810 ± 0.0003 11) 2.5314 ± 0.0003 2) 2.8751 ± 0.0004 12) 2.5233 ± 0.0003 3) 2.1529 ± 0.0003 13) 2.0260 ± 0.0003 4) 2.5026 ± 0.0002 14) 2.8666 ± 0.0003 5) 2.3646± 0.0002 15) 2.8161 ± 0.0003 6) 3.5018 ± 0.0004 16) 2.0481 ± 0.0003 7) 2.3696 ± 0.0002 17) 2.8723 ± 0.0003 8) 2.7557 ± 0.0003 18) 2.6072 ± 0.0003 9) 2.0702 ± 0.0003 19) 3.0362 ± 0.0004 10) 2.6607 ± 0.0003 20) 1.8809 ± 0.0003
30 Year Window 1) 1.6386 ± 0.0002 11) 1.7714 ± 0.0002 2) 1.6168 ± 0.0002 12) 1.9832 ± 0.0002 3) 1.9087 ± 0.0002 13) 1.7668 ± 0.0002 4) 1.4801 ± 0.0002 14) 1.2027 ± 0.0002 5) 1.8070 ± 0.0002 15) 2.0399± 0.0002 6) 2.3613 ± 0.0002 16) 1.9900 ± 0.0002 7) 1.9316 ± 0.0002 17) 2.1310 ± 0.0002 8) 2.3433 ± 0.0002 18) 2.3271 ± 0.0002 9) 2.1834 ± 0.0002 19) 2.1681 ± 0.0002 10) 1.6845 ± 0.0002 20) 1.9814 ± 0.0002
40 Year Window 1) 1.2491 ± 0.0002 11) 1.5066 ± 0.0002 2) 1.6391 ± 0.0002 12) 1.6332 ± 0.0002 3) 1.4006 ± 0.0002 13) 1.7675 ± 0.0002 4) 1.6526 ± 0.0001 14) 1.7078 ± 0.0002 5) 1.8456 ± 0.0002 15) 1.4713 ± 0.0002 6) 1.4773 ± 0.0002 16) 1.6195 ± 0.0002 7) 1.5594 ± 0.0001 17) 1.6892 ± 0.0002 8) 1.5667 ± 0.0002 18) 1.5924 ± 0.0002 9) 1.6973 ± 0.0001 19) 1.7595 ± 0.0002 10) 1.3066 ± 0.0002 20) 1.3776 ± 0.0002
50 Year Window 1) 1.5937 ± 0.0001 11) 1.4359 ± 0.0002 2) 1.3238 ± 0.0002 12) 1.4196 ± 0.0002 3) 1.5460 ± 0.0001 13) 1.5010 ± 0.0001 4) 1.4301 ± 0.0002 14) 1.6835 ± 0.0002 5) 1.4891 ± 0.0002 15) 1.4429 ± 0.0001 6) 1.3474 ± 0.0001 16) 1.4743 ± 0.0001 7) 1.6312 ± 0.0002 17) 1.4178 ± 0.0001 8) 1.4696 ± 0.0002 18) 1.5034 ± 0.0002 9) 1.5794 ± 0.0001 19) 1.4420 ± 0.0001 10) 1.3559 ± 0.0001 20) 1.3633 ± 0.0001

Window	Average Least-squares Slope after 100 iterations and 20 Trials ($/week)
10 Year	2.782 ± .0002
20 Year	2.442 ± .0001
30 Year	1.916, error < .0001
40 Year	1.576, error < .0001
50 year	1.472, error < .0001

Snapshot:

Snap Shot Window	Average Slope after 100 iterations and 20 Trials ($/week)
10 Year	2.997
20 Year	2.287
30 Year	2.002
40 Year	1.892
50 year	1.999

The snapshot and least-squares methods are roughly comparable, although not within error bars generated with least-squares fitting. I believe more iterations of the snapshot method would be necessary to make up for its lack of sensitivity of trends. This would in a sense be a "least-squares" analysis taking a much longer route.

Distribution

The least-squares data returned histograms with leftward-leaning "shoulders," which make sense given the preponderance of smaller slope in the total time period of the DJIA price:

With this in mind, fitting with known distributions seemed frivolous, but did provide elementary insight into the distribution fitting process in general. The log-likelihood rating given fits intrigues me in particular, but online resources are cryptic at best. Open science has not conquered this one just yet!

Examples of fits I tried on 500 iterations of average rates:

500 Iteration Probability Distribution	Fit Parameters
10-year ravg Distribution Best: Log Normal	10 year: Distribution: Normal Log likelihood: -336.618 Domain: -Inf < y < Inf Mean: 2.79947 Variance: 0.225507 Parameter Estimate Std. Err. mu 2.79947 0.0212371 sigma 0.474876 0.0150395 Estimated covariance of parameter estimates: mu sigma mu 0.000451013 -1.07538e-18 sigma -1.07538e-18 0.000226185 Distribution: Lognormal Log likelihood: -332.103 Domain: 0 < y < Inf Mean: 2.79986 Variance: 0.231355 Parameter Estimate Std. Err. mu 1.01503 0.00762697 sigma 0.170544 0.00540119 Estimated covariance of parameter estimates: mu sigma mu 5.81707e-05 -3.97263e-19 sigma -3.97263e-19 2.91729e-05 Distribution: Nakagami Log likelihood: -332.751 Domain: 0 < y < Inf Mean: 2.79988 Variance: 0.222772 Parameter Estimate Std. Err. mu 8.91723 0.553746 omega 8.0621 0.120739 Estimated covariance of parameter estimates: mu omega mu 0.306634 1.83823e-09 omega 1.83823e-09 0.014578
20-year ravg Distribution Best: Nakagami	20 year: Distribution: Normal Log likelihood: -189.022 Domain: -Inf < y < Inf Mean: 2.41135 Variance: 0.124957 Parameter Estimate Std. Err. mu 2.41135 0.0158086 sigma 0.353492 0.0111952 Estimated covariance of parameter estimates: mu sigma mu 0.000249913 2.0193e-19 sigma 2.0193e-19 0.000125333 Distribution: Lognormal Log likelihood: -190.842 Domain: 0 < y < Inf Mean: 2.41174 Variance: 0.13012 Parameter Estimate Std. Err. mu 0.869288 0.00665194 sigma 0.148742 0.0047107 Estimated covariance of parameter estimates: mu sigma mu 4.42483e-05 2.60163e-21 sigma 2.60163e-21 2.21907e-05 Distribution: Nakagami Log likelihood: -187.868 Domain: 0 < y < Inf Mean: 2.41137 Variance: 0.124614 Parameter Estimate Std. Err. mu 11.7865 0.735135 omega 5.93933 0.0773678 Estimated covariance of parameter estimates: mu omega mu 0.540424 2.95263e-09 omega 2.95263e-09 0.00598578
30-year ravg Distribution Best: Nakagami	30 year: Distribution: Normal Log likelihood: 6.36398 Domain: -Inf < y < Inf Mean: 1.90213 Variance: 0.0571925 Parameter Estimate Std. Err. mu 1.90213 0.0106951 sigma 0.239149 0.00757394 Estimated covariance of parameter estimates: mu sigma mu 0.000114385 1.29821e-18 sigma 1.29821e-18 5.73646e-05 Distribution: Lognormal Log likelihood: 7.37713 Domain: 0 < y < Inf Mean: 1.90226 Variance: 0.058345 Parameter Estimate Std. Err. mu 0.635044 0.00565599 sigma 0.126472 0.0040054 Estimated covariance of parameter estimates: mu sigma mu 3.19902e-05 -1.3048e-19 sigma -1.3048e-19 1.60432e-05 Distribution: Nakagami Log likelihood: 7.99048 Domain: 0 < y < Inf Mean: 1.90219 Variance: 0.0568658 Parameter Estimate Std. Err. mu 16.0294 1.00341 omega 3.67519 0.0410522 Estimated covariance of parameter estimates: mu omega mu 1.00684 2.23054e-09 omega 2.23054e-09 0.00168528
40-year ravg Distribution Best: Nakagami	40 year: Distribution: Normal Log likelihood: 191.058 Domain: -Inf < y < Inf Mean: 1.60399 Variance: 0.0273207 Parameter Estimate Std. Err. mu 1.60399 0.00739199 sigma 0.16529 0.00523478 Estimated covariance of parameter estimates: mu sigma mu 5.46414e-05 -3.82513e-19 sigma -3.82513e-19 2.74029e-05 Distribution: Lognormal Log likelihood: 191.951 Domain: 0 < y < Inf Mean: 1.60404 Variance: 0.0276637 Parameter Estimate Std. Err. mu 0.46718 0.00462479 sigma 0.103413 0.00327514 Estimated covariance of parameter estimates: mu sigma mu 2.13887e-05 -1.04169e-19 sigma -1.04169e-19 1.07265e-05 Distribution: Nakagami Log likelihood: 192.237 Domain: 0 < y < Inf Mean: 1.60402 Variance: 0.0271876 Parameter Estimate Std. Err. mu 23.7815 1.49365 omega 2.60005 0.0238439 Estimated covariance of parameter estimates: mu omega mu 2.23098 2.58228e-09 omega 2.58228e-09 0.000568533
50-year ravg Distribution Best: Nakagami	50 year: Distribution: Normal Log likelihood: 316.326 Domain: -Inf < y < Inf Mean: 1.50232 Variance: 0.0165531 Parameter Estimate Std. Err. mu 1.50232 0.0057538 sigma 0.128659 0.00407467 Estimated covariance of parameter estimates: mu sigma mu 3.31062e-05 -1.89468e-19 sigma -1.89468e-19 1.66029e-05 Distribution: Lognormal Log likelihood: 317.216 Domain: 0 < y < Inf Mean: 1.50234 Variance: 0.0166775 Parameter Estimate Std. Err. mu 0.403344 0.00383718 sigma 0.0858019 0.00271737 Estimated covariance of parameter estimates: mu sigma mu 1.47239e-05 -4.44414e-20 sigma -4.44414e-20 7.38411e-06 Distribution: Nakagami Log likelihood: 317.246 Domain: 0 < y < Inf Mean: 1.50233 Variance: 0.0164799 Parameter Estimate Std. Err. mu 34.3622 2.1628 omega 2.27348 0.0173446 Estimated covariance of parameter estimates: mu omega mu 4.67769 3.29831e-09 omega 3.29831e-09 0.000300836

Thoughts

This lab basically turned into a large exercise on data manipulation and information-hunting. I also was able to combine concepts from lecture on linear least-squares fitting with more general error propagation.

The distribution analysis is gratifying in that I see a reflection in the processed data corresponding to trends in the raw data. Something to try regarding the distribution fit would be to generate a fake DJIA from the best-fit distribution function and compare with the real data. I imagine this would involve something like executing our lab process in reverse, generating huge numbers of linear windows with random start dates and distribution-derived slopes. One could then "smooth" all these disjointed, overlapping linear sections into a fake DJIA.