Least Squares Criterion on Philippine Stocks Exchange

8/13/2019 Least Squares Criterion on Philippine Stocks Exchange

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Using Least Squares Criterion to Forecast the Profit of an

Investor using Dollar/Pesos Cost Averaging

Paul Samuel S. Lacap

Institute of Mathematical Sciences and Physics, University of the Philippines-Los Banos

Journal Info

Journal history:

Made: 28 September 2013

Submitted: 4 October 2013

Keywords:

Stock Market

Shares

Profits

Curve-Fit

Errors

Least Squares Criterion

Least Square Fit

Power Fit

ABSTRACT

In the financial world, stock market has been a

bridge that connects famous companies and the

ordinary citizen to help each other in business. A

company can get the funds it needs by selling

shares, and the investors also profits from it

waiting for the right time to sell the shares that

he/she bought from the company. However,

investing in the stock market comes with a risk.

Instead of gaining money, the investor might

lose money because of miscalculations. Due to

this reason, various strategies have been

implemented if not to remove, but minimize the

probability of losing money in the stock market.

An example of such method is called Dollar

Cost Averaging, (called Pesos Cost Averaging in

the Philippines, the name depends on the

currency of the country) with this method, the

probability of losing in the stock market islowered, due to some factors like inflation rate,

and its property that it makes an investor invest

less when the prices of shares are high and

invest more when the prices of shares are low.

Note: The Dollar in Dollar Cost Averaging is not really a part of the name of the investment strategy.

It just specifies what currency is being used when the investment strategy is applied; it is because in this

paper the words Dollar Cost Averaging and Pesos Cost Averaging are frequently interchanged. There is

only a change in the currency but it is still the same strategy.


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Acknowledgements

I would like to thank Ptr. Adonis Reyes for giving me the idea on Pesos Cost Averaging and stock

market, without him, it would have been impossible for me to think of this paper. I would also like to

thank my parents for always encouraging me to do my best regardless of the result. I would like to thank

my friends for always cheering me up no matter how stressful my life is, I would like to thank Sir

Jonathan Mamplata for extending the deadline thus making this paper possible to be finished completely,

and lastly to an everloving God that guides me and supports me in every way.


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1. Introduction

Dollar/Pesos Cost Averaging is a technique used

by investors in the stock market to lower the risk

of losing money. By using this strategy, the

investor will refrain from investing a large

portion of his money in a single investment but

rather will invest partial amounts of money in a

fixed interval through a long period of time

regardless of the share price. It is assumed that

by using this method, it will result in the

investor in buying fewer shares when the prices

are high and buying more shares when the prices

are low as time goes by. Because of this, the

investor will profit in the long run regardless of

how volatile the changes of prices in the stock

market. Based on the properties of Dollar/PesosCost Averaging, this study will aim to

approximate the future profit of an investor if

he/she wants to apply Dollar/Pesos Cost

averaging to buy stocks of a certain company.

By forecasting the profit, the investor will have

an idea on how much he/she will earn at the end

of his/her investment plan and make decisions

based on the forecast. He/She can also compare

his/her approximated future profit of a certain

company to the approximated future profit of

other companies suppose he/she invest their

instead.

2. Review of Related Literature

Curve fitting has been widely used in

forecasting the cost of the goods and

commodities that have volatile price changes.

An example is the paper of Stephen Alberth

entitled, Forecasting Technology costs via the

Learning Curve Myth or Magic?, wherein

he did an in depth analysis on the use of learning

curve, weighted least squares with exponentially

increasing weights, and Ordinary Least Square

to forecast technology cost. It can be seen in his

research that it is possible to forecast prices

using curve fitting.

Meanwhile, In the University of

Melbourne, Carlton 3010, Australia, a research

was done by Md. Rafiul Hassan, a computer

science, and Baikunth Nat, a software engineer,

that forecast the stock prices for interrelated

markets by using the Hidden Markov Model.This is the graph they obtained when Hidden

Markov Model was used:

Graph 1: Actual Prices vs Related Prices

As it can be seen, although there are errors, it is

acceptable just by looking at the graph and the

predicted price has a similar trend to the trend ofthe actual price.

Aside from the two researches, stock market has

been a center of research on prediction because

of its ability to give the investor profits. By

forecasting, the investor will have a guide on

investing and not invest randomly on different

companies. As it can be seen, although the

prices of the stock markets are volatile in nature,

it is possible to make a model to forecast the

shares of the prices in the future. Using thisobtained knowledge and alteration of some data

values: Instead of forecasting price, the profits

are the one forecasted. It is possible to make a

simple least squares model with acceptable

errors that can forecast an investors profit in the

long run.


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3. Theoretical Framework

To further illustrate and to make it easier to explain mathematically what dollar/pesos cost averaging is,

an example is given below:

An example taken from: http://pesonijuan.blogspot.com/2013/01/peso-cost-averaging-in-stocks-

investing.html (No copyright infringement intended for educational use only)

BPI Common Stock Example: Stock Code: BPI

Definition Variable where each corresponding value is stored

Duration: 2 years T

Board lot* (minimum number of shares one canpurchase): 10

B

Number of times invested in an interval R

Time equivalent of an interval (normally onemonth)

I

Amount of I it takes to make 1 year C

The last period of investment i Fixed Amount Money that is Planned to beinvested per period

W (This is a constant to be given, at this example itis 5000)

Money Planned to be invested for each interval:P5000.00 initially

Share Price Dates: January 2011 to December 2012 *Board lot means that a person can only buy shares in multiple of the value of the board lot

If on January 2011, the price per share is P51.50, the P5,000 pesos in an investorscash position can buy

only 90 shares based on the board lot. With the price per share of P51.50, the total cost of 90 shares is

only P4,635.00. The remainder of P365 from the P5,000.00 ( which is just equal to P5,000 P4,635) will

stay in his/her trading account and will be added to his/her next deposit of P5,000 to be used to buy sharesagain next month. So generally the following Equations are made (equations are own interpretation, it is

not included in the website):

wherein this must be a multiple of B

( )


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Thus, using those mathematical equations, a table is obtained below

Investment data at the end of December 2012, will be:

Total Shares Purchased: 1,860 shares Total Amount Invested: P119,341.00

Amount Left in Cash Position: P659.00 Average Price per Share: P66.07 per share

At the end of December 2012, the total present worth of investment will be:

Present worth of Investment = 1,860 shares x P95.00 per share

Total Profits at Present time = P176,700P119,341.00

Profits at Present time = P57,359.00

The Returns on Investment within two (2) years of building the investment thru PCA will be:
http://1.bp.blogspot.com/-w1piY5rqNQc/UPq5l8lJU3I/AAAAAAAAAVo/00NhY9UhEhU/s1600/BPI+Peso+Cost+Averaging+Table.PNG


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, note PCA means Pesos Cost Averaging

% ROI after 2 years = P57,359.00 / P119,341.00 x 100%

% ROI after 2 years = 48.06% (Percentage of how much you profit)

As it can be seen, even though the prices of the shares keeps going up and down, the investor still

manages to profit a large amount of money at the end of the investment period when Dollar/Pesos Cost

averaging is applied. Technically, this paper will focus on approximating the Overall Profits

(Profits+Money Left) by using curve fitting. The same process will be used for obtaining the needed

values to create satisfactory model for prediction.

About Curve Fitting and Regression Analysis

Curve fitting or Regression Analysis is a

widely used tool when it comes to forecasting if

given a set of data. One of the most common

Regression Analysis techniques is Linear

Regression, and this study will use one of the

popular forms of linear regression called Least-

Squares fitting.

Least Squares Fitting

One of the reasons that Least-Squares fit

is popular is because its mathematically

constructed in a way that it minimizes the

squared summation of errors.

In Equation form: ( ()) Using the criterion above, equations

have been formed to produce many models that

can be used to curve fit and forecast. The two

models that this journal will use (which will be

explained why only those two are chosen later)

is the linear-least squares line and Power fit ofdegree 2.

Linear-Least Squares Line

This technique uses the equation

() wherein:

to find the best-fitting line to a given set of

points with the property of minimizing thesquared summation of errors.

Power Fit of Degree 2

In Power Fit, the model is no longer

limited to a line, but different degrees of

polynomials can now be used to find the best

fitting line given a set of data points. The

technique uses the equation () , whereM is a known constant. Just like the previous

technique the equation of the Power fit mustsatisfy an equation (given below) to obtain the

value for:


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4. Discussion

Without losing realism, assumptions are made to

Dollar/Pesos cost averaging to create such

conditions that using curve fitting will only

requires approximately at most a 2nd degree

polynomial. The assumptions are as of follows:

Assumption 1: An investor will invest in a well-

known successful company like PLDT, SM,

Nike etc.

Assumption 2: The investor will invest two

times a month, he will invest at the 15th day and

at the 30th day (the normal days when salary is

given). When no data of the stock market is to

be obtained on a specific month, the nearest date

after the supposed to be date recorded will be

used, for example: No records on Stock Market

on July 30 is to be found, data on August 1 will

be used instead.

Assumption 3: Extreme Deflation will not occur

(occurs very rarely in real life and almost all

countries as much as possible avoid this event)

Effects of the Assumptions

Via Assumption 1, we are sure to some extentthat the company to be invested in will not go

bankrupt and leave the stock market.

Assumption 2 will be the investment plan for

pesos cost averaging, and through Assumption

3, we can assume an increasing relationship for

the real data values to be collected, thus making

it possible to create a polynomial/model using

least squares criterion. By using the

assumptions, it is now easy to create a model by

using curve fitting.

Application

Least squares criterion will now be used in the

records of data obtained from Sun Life TSE

(Toronto Stocks Exchange)

Investor will invest in: Sun Life TSE

Currency: Dollars

Fixed Investment Amount: $100

Investment Period: January 3, 2012 to December

31, 2013

How many times to invest in an interval: Once

in every interval

Interval of Investment: Every 15th and 30th day

of the month

Condition of the Price of the share: Average:

(Max Price of the day+ Min Price of the day)/2

Assumed Board lot*: Multiple of 2

*Failed to find the data for the board lot of Sun

Life

Using the Mathematical Equations formed

earlier in the previous example, the following

table is formed below


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Table 2.1 Records of Investments

i Date Price of a share

Money

()Shares Bought

Amount Invested

Money left

3 1/3/2012 19.36 100 4 77.44 22.56

16 1/16/2012 20.15 122.56 6 120.9 1.66

30 1/30/2012 19.9 101.66 4 79.6 22.06

46 2/15/2012 19.9 122.06 6 119.4 2.66

61 3/1/2012 22.04 102.66 4 88.16 14.5

75 3/15/2012 22.7 114.5 4 90.8 23.7

90 3/30/2012 22.04 123.7 4 88.16 35.54

107 4/16/2012 22.7 135.54 4 90.8 44.74

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

320 11/15/2012 25.34 118.92 4 101.36 17.56

335 11/30/2012 27.09 117.56 4 108.36 9.2

352 12/17/2012 27.11 109.2 4 108.44 0.76

366 12/31/2012 26.34 100.76 2 52.68 48.08

381 1/15/2013 27.61 148.08 4 110.44 37.64

396 1/30/2013 29.5 137.64 4 118 19.64

412 2/15/2013 29.05 119.64 4 116.2 3.44

426 3/1/2013 28.5 103.44 2 57 46.44

440 3/15/2013 28.68 146.44 4 114.72 31.72

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.534 6/17/2013 30.35 117.34 2 60.7 56.64

549 7/2/2013 31.21 156.64 4 124.84 31.8

562 7/15/2013 33.12 131.8 2 66.24 65.56

577 7/30/2013 33.26 165.56 4 133.04 32.52

593 8/15/2013 33.39 132.52 2 66.78 65.74

608 8/30/2013 32.13 165.74 4 128.52 37.22

625 9/16/2013 33.65 137.22 4 134.6 2.62


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Table 2.2 Records of Profits

i Date Total shares Share

Value

Amount Invested

Profits

Profits+Money

Left

3 1/3/2012 4 77.44 77.44 0 22.56

16 1/16/2012 10 201.5 198.34 3.16 4.8230 1/30/2012 14 278.6 277.94 0.66 22.72

46 2/15/2012 20 398 397.34 0.66 3.32

61 3/1/2012 24 528.96 485.5 43.46 57.96

75 3/15/2012 28 635.6 576.3 59.3 83

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

320 11/15/2012 98 2483.32 2182.44 300.88 318.44

335 11/30/2012 102 2763.18 2290.8 472.38 481.58

352 12/17/2012 106 2873.66 2399.24 474.42 475.18

366 12/31/2012 108 2844.72 2451.92 392.8 440.88

381 1/15/2013 112 3092.32 2562.36 529.96 567.6

396 1/30/2013 116 3422 2680.36 741.64 761.28

412 2/15/2013 120 3486 2796.56 689.44 692.88

.

.

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.

.

534 6/17/2013 146 4431.1 3543.36 887.74 944.38

549 7/2/2013 150 4681.5 3668.2 1013.3 1045.1

562 7/15/2013 152 5034.24 3734.44 1299.8 1365.36

577 7/30/2013 156 5188.56 3867.48 1321.08 1353.6

593 8/15/2013 158 5275.62 3934.26 1341.36 1407.1

608 8/30/2013 162 5205.06 4062.78 1142.28 1179.5

625 9/16/2013 166 5585.9 4197.38 1388.52 1391.14


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Analysis of the Table

Although the overall profits are volatile,

it increases in the long run. Starting from an

overall profit of $22.56, it goes up and down and

finally increases to the value of $1391.14 on the

last given data 9/16/2013. This relationship will

be used to obtain a least-square model that will

approximate the overall profit in the long run

(future).

Least Squares Model

Initial Conditions

1. There is no limit on how much the profit will

increase.

2. There is no limit on how long the investor

wants to invest; he can keep investing as long as

he wants.

Due to the given conditions, it is not advisable to

use a polynomial of degree greater than or equal

to three, because a graph of a cubic polynomial

normally shows a limit on how much the y

values will increase (it has an asymptote and an

upper bound). Furthermore, a high degree

polynomial will tend to extremely wiggle, butthe assumption of dollar/pesos cost averaging is

that the values may be volatile, but it increases

in the long run (it only creates very small

wiggles with an increasing value of y in the

long run). Thus, the recommended degree of the

least-squares polynomial is of degree less than

or equal to 2.

Modelling

With the use of SciLab andimplementation of a Statistical Regression

Model the following Models were obtained:

Table 3.1 Key Values for Least Squares

Criterion

Method: Linear Least Squares

Value of A Value of B

Model - 236.01841+2.2029371x

Method: Power Fit of Degree 2

Value of A Value of B n/a

Model

Table 3.2 Value of the Overall Profits

based on the two models

Linear Least Squares

Time/Day Y-Value

1 -233.8154729

2 -231.6125358

.

...

304 433.6744684

305 435.8774055

.

...

720 1350.096302


Time/Day Y-Value

1 0.0035088

2 0.0140352

.

...

304 324.2692608

305 326.40612..

.

.

720 1818.96192


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Procedures of Graphing

x-axis-values- will be the time/day counter

wherein 1 January 2012 is day 1, 2 January 2012

is day 2 31 December 2013 is day 720

y-axis-values- will be the profit obtained when

an investor sold all of his shares on a specific

date and add it with the money left/unspent and

subtracts it from the total amount of money he

invested from the beginning up to that specific

date.

In formula: y= present worth of investment +

money left - total amount of money invested

starting from moment he started to use pesos

cost averaging.

Graph of Data Values (SciLab)

Thus using SciLab to plot the values of Table

2.2, the following graph is obtained:

Graph 1: Graph of the Exact data Values

Graph of Models with Data Values (SciLab)

Then by using the models/polynomials obtained

using least squares criterion, the following graph

is obtained:

Graph 2: Graph of the Exact Data Values

with models

Black Line: Least-Squares Line

Red Line: Power Fit of Degree 2

Blue Dots: Exact Data Values

Since the unique property of the least squares

criterion is that it minimizes the squared

summation of errors, the model that will be

chosen between the two approximations will be

the one with the least squared summation of

errors.

Squared Summation of Errors:

Linear Least Squares: 1,296,068.873

Power Fit of Degree 2: 545,400.5557

Therefore, the Power Fit of Degree 2 will be

used to forecast the profit at any given time. (In

this journal, forecasts the investors profit at the

end of the investment period: 31 December

2013). Thus, to obtain the future profit in 31December 2013, the number of days is

calculated from 1 January 2012 to 31 December

2013 which is 720 days. Then the x value to be

substituted to Power Fit is 720 as shown in the

next page.


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()Thus, the investors expected profit in 31

December 2013, assuming that he/she did notstop Dollar/Pesos Cost averaging is

approximately $1818.96192

5. Conclusion

Least-Squares Criterion is a curve fitting

method that minimizes the squared summation

of errors. However, it is not recommended when

the approximation requires a polynomial of

degree greater than 6. Fortunately Dollar/Pesos

Averaging exhibits such nature that the investors

profit will increase in the long run. It can be

clearly seen in Graph 1 that although the data

points are going up and down, the profits are

increasing in the long run starting from an

overall profit of $22.56 it increases to a value

greater than $1000 at the year 2013. Thus, such

nature enables the use of a low degree

polynomial for least squares criterion which

shows, as seen on Graph 2, acceptable results.

To answer the original problem of this study and

to meet the objective, the obtained model will be used to forecast since itsatisfies the assumptions made and is much

more accurate than linear-least squares. By using

the model, it is now possible to forecast the

investors profit at any date starting from 1

January 2013. Thus, by using the obtained

model, the approximate profit of the investor

when using Dollar/Pesos Cost Averaging at the

end of the planned investment period namely 31

December 2013 is $1818.96


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Appendix

History of the Prices of the Shares Obtained from the Official Sun Life Webpage at

http://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CA

Appendix Table 1: Records of the Share Prices Used in this Journal

i Date Price of a share

3 1/3/2012 19.36

16 1/16/2012 20.15

30 1/30/2012 19.9

46 2/15/2012 19.9

61 3/1/2012 22.04

75 3/15/2012 22.7

90 3/30/2012 22.04

107 4/16/2012 22.7121 4/30/2012 24.18

136 5/15/2012 22.62

151 5/30/2012 20.83

167 6/15/2012 22.03

185 7/3/2012 22.56

198 7/16/2012 21.75

212 7/30/2012 21.53

228 8/15/2012 22.66

243 8/30/2012 22.95

261 9/17/2012 24.59275 10/1/2012 23.03

290 10/15/2012 23.8

304 10/30/2012 24.86

320 11/15/2012 25.34

335 11/30/2012 27.09

352 12/17/2012 27.11

366 12/31/2012 26.34

381 1/15/2013 27.61

396 1/30/2013 29.5

412 2/15/2013 29.05

426 3/1/2013 28.5

440 3/15/2013 28.68457 4/1/2013 27.59

471 4/15/2013 27.27

486 4/30/2013 28.03

501 5/15/2013 29.59

516 5/30/2013 30.91

534 6/17/2013 30.35

549 7/2/2013 31.21

562 7/15/2013 33.12

577 7/30/2013 33.26

593 8/15/2013 33.39608 8/30/2013 32.13

625 9/16/2013 33.65
http://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CAhttp://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CAhttp://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CA


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Appendix Table 2: Records of Values of X and Y of the Model

Linear Least Squares

Time Y-Value

1 -233.8154729

2 -231.61253583 -229.4095987

4 -227.2066616

5 -225.0037245

6 -222.8007874

7 -220.5978503

8 -218.3949132

9 -216.1919761

10 -213.989039

.

.

.

.

.

.

499 863.2472029

500 865.45014

501 867.6530771

502 869.8560142

503 872.0589513

504 874.2618884

505 876.4648255

506 878.6677626

507 880.8706997

508 883.0736368

509 885.2765739

510 887.479511

.

.

.

713 1334.675742

714 1336.878679

715 1339.081617

716 1341.284554

717 1343.487491

718 1345.690428

719 1347.893365

720 1350.096302


Time Y-Value

1 0.0035088

2 0.01403523 0.0315792

4 0.0561408

5 0.08772

6 0.1263168

7 0.1719312

8 0.2245632

9 0.2842128

10 0.35088

.

.

.

.

.

.

499 873.6947088

500 877.2

501 880.7123088

502 884.2316352

503 887.7579792

504 891.2913408

505 894.83172

506 898.3791168

507 901.9335312

508 905.4949632

509 909.0634128

510 912.63888

.

.

.

.

.

.

713 1783.765147

714 1788.772205

715 1793.78628

716 1798.807373

717 1803.835483

718 1808.870611

719 1813.912757

720 1818.96192


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Appendix Table 3: Table of Errors

Error Measurement: Linear Least Squares

Time Y-Value Real Values Errors

3 -229.4095987 22.56 63488.67867

16 -200.7714164 4.82 42267.830530 -169.930297 22.72 37114.13693

46 -134.6833034 3.32 19044.91175

61 -101.6392469 57.96 25471.91961

75 -70.7981275 83 23653.86402

90 -37.754071 76.36 13022.0212

107 -0.3041403 106.68 11445.60628

121 30.5369791 163.24 17610.09176

136 63.5810356 65.12 2.368411424

151 96.6250921 -0.54 9441.055123

167 131.8720857 71.34 3664.133399

185 171.5249535 64.6 11432.94568

198 200.1631358 29 29296.81906

212 231.0042552 28.8 40886.56082

228 266.2512488 115 22876.94026

243 299.2953053 98.18 40447.36603

261 338.9481731 227.74 12367.25776

275 369.7892925 107.7 68690.79724

290 402.833349 178.72 50226.7932

304 433.6744684 274.68 25279.24098

320 468.921462 318.44 22644.67041

335 501.9655185 481.58 415.5693645

352 539.4154492 475.18 4126.192934

366 570.2565686 440.88 16738.2965

381 603.3006251 567.6 1274.534633

396 636.3446816 761.28 15608.83378

412 671.5916752 692.88 453.1927728

426 702.4327946 669.88 1059.684436

440 733.273914 677.12 3153.262058457 770.7238447 529.42 58227.54547

471 801.5649641 478.74 104215.9574

486 834.6090206 568.46 70835.30117

501 867.6530771 824.56 1857.013294

516 900.6971336 985.72 7228.887811

534 940.3500014 944.38 16.24088872


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549 973.3940579 1045.1 5141.742132

562 1002.03224 1365.36 132007.061

577 1035.076297 1353.6 101457.3496

593 1070.32329 1407.1 113418.5522

608 1103.367347 1179.5 5796.180883

625 1140.817278 1391.14 62661.4654

Summation of Errors 1296068.873

Error Measurement: Power Fit of Degree 2

Time Y-Value Real Values Errors

3 0.0315792 22.56 507.5297437

16 0.8982528 4.82 15.3801011

30 3.15792 22.72 382.6749739

46 7.4246208 3.32 16.84791191

61 13.0562448 57.96 2016.347231

75 19.737 83 4002.207169

90 28.42128 76.36 2298.120875

107 40.1722512 106.68 4423.28065

121 51.3723408 163.24 12514.37317

136 64.8987648 65.12 0.048945014

151 80.0041488 -0.54 6487.359906

167 97.8569232 71.34 703.147216

185 120.08868 64.6 3078.993608

198 137.5589952 29 11785.05544

212 157.6995072 28.8 16615.08296

228 182.4014592 115 4542.956702

243 207.1911312 98.18 11883.42673

261 239.0229648 227.74 127.3052947

275 265.353 107.7 24854.46841

290 295.09008 178.72 13541.99552

304 324.2692608 274.68 2459.094787

320 359.30112 318.44 1669.631128

335 393.77508 481.58 7709.703976352 434.7543552 475.18 1634.232757

366 470.0248128 440.88 849.4201131

381 509.3409168 567.6 3394.120775

396 550.2359808 761.28 44539.57804

412 595.5977472 692.88 9463.83671

426 636.7629888 669.88 1096.736431


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440 679.30368 677.12 4.768458342

457 732.8093712 529.42 41367.23632

471 778.3957008 478.74 89793.53902

486 828.7645248 568.46 67758.44563

501 880.7123088 824.56 3153.081784

516 934.2390528 985.72 2650.287925

534 1000.555373 944.38 3155.672509

549 1057.555829 1045.1 155.1476711

562 1108.233427 1365.36 66114.07444

577 1168.181275 1353.6 34380.10351

593 1233.866011 1407.1 30010.01488

608 1297.077043 1179.5 13824.36109

625 1370.625 1391.14 420.865225

Summation of Errors 545400.5557


18/20

Appendix: Program Algorithm: Least-Squares Modelling

cleardisp("Least Squares Line")

N=42

//value for x(i)Xi(1)=3Xi(2)=16Xi(3)=30Xi(4)=46Xi(5)=61Xi(6)=75Xi(7)=90Xi(8)=107Xi(9)=121Xi(10)=136Xi(11)=151Xi(12)=167

Xi(13)=185Xi(14)=198Xi(15)=212Xi(16)=228Xi(17)=243Xi(18)=261Xi(19)=275Xi(20)=290Xi(21)=304Xi(22)=320Xi(23)=335Xi(24)=352Xi(25)=366Xi(26)=381Xi(27)=396Xi(28)=412Xi(29)=426Xi(30)=440Xi(31)=457Xi(32)=471Xi(33)=486Xi(34)=501Xi(35)=516Xi(36)=534Xi(37)=549

Xi(38)=562Xi(39)=577Xi(40)=593Xi(41)=608Xi(42)=625

disp("thank you")disp("----------")

//Value for y(i)Yi(1)=22.56Yi(2)=4.82Yi(3)=22.72

Yi(4)=3.32Yi(5)=57.96Yi(6)=83Yi(7)=76.36Yi(8)=106.68Yi(9)=163.24Yi(10)=65.12Yi(11)=-0.54Yi(12)=71.34Yi(13)=64.6Yi(14)=29Yi(15)=28.8Yi(16)=115

Yi(17)=98.18Yi(18)=227.74Yi(19)=107.7Yi(20)=178.72Yi(21)=274.68Yi(22)=318.44Yi(23)=481.58Yi(24)=475.18Yi(25)=440.88Yi(26)=567.6Yi(27)=761.28Yi(28)=692.88Yi(29)=669.88Yi(30)=677.12Yi(31)=529.42Yi(32)=478.74Yi(33)=568.46Yi(34)=824.56Yi(35)=985.72Yi(36)=944.38Yi(37)=1045.1Yi(38)=1365.36Yi(39)=1353.6Yi(40)=1407.1Yi(41)=1179.5

Yi(42)=1391.14

//Get summation XkSXi=sum(Xi)

//Get summation of X^2kX2i=Xi^2SX2i=sum(X2i)

//summation of X(i)Y(i)fori=1:NXY(i)=(Xi(i)*Yi(i))

end

SXY=sum(XY)

//summation of YkSYi=sum(Yi)

//Get AAL=(N*SXY-SXi*SYi)*1/(N*SX2i-SXi^2)

//Get BBL=(SX2i*SYi-

SXi*SXY)*1/(N*SX2i-SXi^2)

//Polynomialsx=poly(0,"x")y=poly(0,"y")

y=AL*x+BL

//Power fitM=input("Please inputexponent variable M: ")fori=1:NPXMi(i)=Xi(i)^M*Yi(i)PX2Mi(i)=Xi(i)^(2*M)endSPXMi=sum(PXMi)SPX2Mi=sum(PX2Mi)

AP=SPXMi/SPX2Mi

xp=poly(0,'xp')yp=poly(0,'yp')

yp=AP*xp^M

HX=[1:0.1:720]HY=horner(y,HX)

plot(HX,HY)

HYP=horner(yp,HX)plot(HX,HYP)


19/20

Appendix Program: Plotting of Data Values

cleardisp("Least Squares Line")

//value for x(i)

Xi(1)=3Xi(2)=16Xi(3)=30Xi(4)=46Xi(5)=61Xi(6)=75Xi(7)=90Xi(8)=107Xi(9)=121Xi(10)=136Xi(11)=151Xi(12)=167Xi(13)=185

Xi(14)=198Xi(15)=212Xi(16)=228Xi(17)=243Xi(18)=261Xi(19)=275Xi(20)=290Xi(21)=304Xi(22)=320Xi(23)=335Xi(24)=352Xi(25)=366

Xi(26)=381Xi(27)=396Xi(28)=412

Xi(29)=426Xi(30)=440Xi(31)=457Xi(32)=471

Xi(33)=486Xi(34)=501Xi(35)=516Xi(36)=534Xi(37)=549Xi(38)=562Xi(39)=577Xi(40)=593Xi(41)=608Xi(42)=625

disp("thank you")

disp("----------")//Value for y(i)Yi(1)=22.56Yi(2)=4.82Yi(3)=22.72Yi(4)=3.32Yi(5)=57.96Yi(6)=83Yi(7)=76.36Yi(8)=106.68Yi(9)=163.24Yi(10)=65.12

Yi(11)=-0.54Yi(12)=71.34Yi(13)=64.6

Yi(14)=29Yi(15)=28.8Yi(16)=115Yi(17)=98.18

Yi(18)=227.74Yi(19)=107.7Yi(20)=178.72Yi(21)=274.68Yi(22)=318.44Yi(23)=481.58Yi(24)=475.18Yi(25)=440.88Yi(26)=567.6Yi(27)=761.28Yi(28)=692.88Yi(29)=669.88Yi(30)=677.12

Yi(31)=529.42Yi(32)=478.74Yi(33)=568.46Yi(34)=824.56Yi(35)=985.72Yi(36)=944.38Yi(37)=1045.1Yi(38)=1365.36Yi(39)=1353.6Yi(40)=1407.1Yi(41)=1179.5Yi(42)=1391.14

plot(Xi,Yi,".")


20/20

References

Stock Forecast Methods: Stock Trading and Investing. Retrieved from

http://stockforecast.wordpress.com/

Uminga, Russel Gabriel (2013),Invest in Stocks using Peso Cost Averaging Strategy. Retrieved from:

http://pesonijuan.blogspot.com/2013/01/peso-cost-averaging-in-stocks-investing.html

Matthews, John H. and Fink, Kurtis D. (1999) Numerical Methods Using MATLAB Third Edition

Ryabenkii, Victor S and Tsynkov, Semyon V. (2007) A THEORETICAL INTRODUCTION TO

NUMERICAL ANALYSIS. United States of America. Taylor & Francis Group

Return on Investment ROI retrieved from:http://www.investopedia.com/terms/r/returnoninvestment.asp 4

October 2013

Weisstein, Eric W. "Least Squares Fitting." From MathWorld--A Wolfram Web Resource.

http://mathworld.wolfram.com/LeastSquaresFitting.html

Sun Life Financial. Share History. Retrieved from

http://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CA

Hassan, Md. Rafiul and Nath Baikunth (2005). Stock Market Using Hidden Markov Model: A New

Approach. The University of Melbourne, Carlton 3010, Australia

Stephen, Albert. Forecasting Technology costs via Learning Curve-Myth or Magic?. Judge Business

School, University of Cambridge

More info: On Pesos Cost Averaging

http://www.youtube.com/watch?v=XYU4F1Gn5tY

Headings Format: (Taken from Purdue Owl)

Formatted, unnumbered:

Level 1 Heading: bold, flush left

Level 2 Heading: italics, flush left

Level 3 Heading: centered, bold

Level 4 Heading: centered, italics

Level 5 Heading: underlined, flush left

The Purdue OWL. Purdue U Writing Lab, 2010. Web. September 28, 2013
http://www.investopedia.com/terms/r/returnoninvestment.asp%204%20October%202013http://www.investopedia.com/terms/r/returnoninvestment.asp%204%20October%202013http://www.investopedia.com/terms/r/returnoninvestment.asp%204%20October%202013http://www.investopedia.com/terms/r/returnoninvestment.asp%204%20October%202013http://mathworld.wolfram.com/LeastSquaresFitting.htmlhttp://mathworld.wolfram.com/LeastSquaresFitting.htmlhttp://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CAhttp://www.youtube.com/watch?v=XYU4F1Gn5tYhttp://www.youtube.com/watch?v=XYU4F1Gn5tYhttp://www.sunlife.com/Global/Investors/Share+performance/Share+history?vgnLocale=en_CAhttp://mathworld.wolfram.com/LeastSquaresFitting.htmlhttp://www.investopedia.com/terms/r/returnoninvestment.asp%204%20October%202013http://www.investopedia.com/terms/r/returnoninvestment.asp%204%20October%202013

Least Squares Criterion on Philippine Stocks Exchange

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