FORECASTING EXCHANGE RATES

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Forecasting Exchange Rates

Ashley Rose Granvik

Degree Thesis International Business

2010

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BACHELOR’S DEGREE THESIS Arcada Polytechnic

Degree Programme: International Business Identification number: 8253

Author: Ashley Rose Granvik

Title: Forecasting Exchange Rates

Supervisor (Arcada): Karis Badal Durbo Commissioned by:

Abstract:

The work of this thesis primarily revolves around the concept of forecasting the daily exchange rates of the European Euro valued in United States Dollars. Forecasting is a relatively important issue in business operations, however it is also one of the most problematic. With the uncertainty of the future, forecasts are difficult to assess. The aim of this thesis is to successfully forecast the future exchange rates of the European Euro in terms of United States Dollars for the month of May 2010, and determine whether the forecasting models properly work when applied to exchange rates, why or why not, and their measure of accuracy. The methodology of this thesis revolves extensively around quantitative research, largely including a probability and forecasting approach.

Throughout a period of three months, actual exchange rate values were collected and recorded to form a data set. The exchange rate data was then used in the application of a variety of mathematical forecasting models to forecast the daily exchange rates for a future, one-month period. Upon measuring the accuracy of the forecasts, the forecasted exchange rates contained very little error. Therefore, the forecasts are considered to be successful, and the hypothesis that exchange rates could be determined with the aid of a mathematical forecasting model is accepted. Though it is very difficult to consistently estimate exchange rates successfully, the work of this thesis shows there is always a greater probability of benefiting from a forecast.

Keywords:

Economics, Exchange Rates, Forecasting, Mathematics, Quantitative

Number of pages: 88

Language: English

Date of acceptance: 25/11/2010

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1 INTRODUCTION ... 11

1.1 Description and Motivation for Topic ... 11

1.2 Aim of Research ... 11

1.3 Hypotheses... 11

1.4 Description of Methods... 12

1.5 Limitations ... 12

1.6 Technical frame of reference... 12

2 DESCRIPTIVE MEASURES ... 14

2.1 Numerical Descriptive Methods... 14

2.2 Measures of Central Tendency ... 14

2.2.1 Mean ...14

2.2.2 Median ...15

2.2.3 Mode ...15

2.3 Measures of Variability ... 15

2.3.1 Range ...16

2.3.2 Variance...16

2.3.3 Standard Deviation ...17

2.4 Simple Index Numbers ... 18

2.5 Example Calculations of Numerical Descriptive Methods ... 19

2.5.1 Data ...19

2.5.2 Mean ...19

2.5.3 Median ...20

2.5.4 Mode ...20

2.5.5 Range ...20

2.5.6 Population Variance...20

2.5.9 Population Standard Deviation ...21

2.5.8 Sample Variance...21

2.5.10 Sample Standard Deviation ...22

2.5.8 Simple Index Numbers...22

3 BACKGROUND OF FORECASTING ... 24

3.1 Forecasting Approaches... 24

3.1.1 Quantitative Forecasting ...24

3.1.2 Econometric Forecasting Models...25

3.1.3 Time-series Forecasting Models...25

3.2.1 Types of Forecasts ...26

3.2.2 Time Horizons...27

3.3 Forecasting System... 27

3.4 Importance of Forecasting... 28

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4 FORECASTING MODELS ... 29

4.1 Naive Approach ... 29

4.2 Simple Moving Averages... 29

4.3 Exponential Smoothing... 30

4.3.1 Smoothing Constant ...30

4.4 Exponential Smoothing with Trend Adjustment... 31

4.5 Linear Regression ... 32

4.6.1 Assumptions Underlying Linear Regression ...33

4.6.2 The Standard Error of Estimate ...33

4.6.3 Correlation Analysis ...34

5 MEASURING ERROR... 36

5.1 Measures of Forecast Accuracy ... 36

5.2 Mean Absolute Deviation... 36

5.3 Mean Squared Error ... 36

5.4 Root Mean Square Error ... 37

5.5 Mean Absolute Percentage Error ... 37

6 EXCHANGE RATES ... 39

6.1 Background of Foreign Exchange Rates... 39

6.2 Forecasting Exchange Rates ... 39

6.2.1 Influencing Factors...39

6.2.2 Economical Forecasting Models ...40

6.2.3 Time Horizons...42

6.3 Limitations ... 42

7 DATA... 44

8 CALCULATIONS FOR FORECASTING THE EXCHANGE RATES ... 46

8.1 Naive Approach ... 46

8.2 Moving Average... 47

8.3 Exponential Smoothing... 48

8.4 Exponential Smoothing With Trend Adjustment ... 49

8.5 Linear Regression ... 51

9 CALCULATIONS FOR MEASURING ERROR... 55

9.1 Mean Absolute Deviation... 55

9.2 Mean Squared Error ... 59

9.3 Root Mean Squared Error ... 63

9.4 Mean Absolute Percentage Error ... 67

10 RESULTS... 72

11 CONCLUSION ... 74

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BIBLIOGRAPHY ... 76

APPENDICES ... 77

Appendix 1... 77

Appendix 2... 77

Appendix 3... 78

Appendix 4... 79

Appendix 5... 79

Appendix 6... 80

Appendix 7... 81

Appendix 8... 81

Appendix 9... 82

Appendix 10... 83

Appendix 11... 83

Appendix 12... 84

Appendix 13... 85

Appendix 14... 85

Appendix 15... 86

Appendix 16... 87

Appendix 17... 87

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TABLES

Table 1. ... 19

Table 2. ... 44

Table 3. ... 46

Table 4. ... 47

Table 5. ... 49

Table 6. ... 51

Table 7. ... 54

Table 8. ... 56

Table 9. ... 56

Table 10. ... 57

Table 11. ... 58

Table 12. ... 58

Table 13. ... 60

Table 14. ... 60

Table 15. ... 61

Table 16. ... 62

Table 17. ... 62

Table 18. ... 64

Table 19. ... 65

Table 20. ... 65

Table 21. ... 66

Table 22. ... 67

Table 23. ... 68

Table 24. ... 69

Table 25. ... 70

Table 26. ... 70

Table 27. ... 71

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FORMULAS

Formula 1. ... 14

Formula 2. ... 16

Formula 3. ... 16

Formula 4. ... 17

Formula 5. ... 18

Formula 6. ... 18

Formula 7. ... 18

Formula 8. ... 29

Formula 9. ... 29

Formula 10. ... 30

Formula 11. ... 31

Formula 12. ... 31

Formula 13. ... 31

Formula 14. ... 32

Formula 15. ... 32

Formula 16. ... 32

Formula 17. ... 33

Formula 18. ... 34

Formula 19. ... 36

Formula 20. ... 37

Formula 21. ... 37

Formula 22. ... 38

Formula 23. ... 40

Formula 24. ... 41

Formula 25. ... 42

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ABBREVIATIONS

• EUR/EURO: European Euro

• FX: Foreign Exchange

• MAD: Mean Absolute Deviation

• MAPE: Mean Absolute Percent Error

• MSE: Mean Squared Error

• PPP: Purchasing Power Parity

• RMSE: Root Mean Squared Error

• U.S: United States

• USD: United States Dollar

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DEFINITIONS

• Actual Data – Any number of values that have been historically recorded.

• Data – A collection of organized information from which conclusions may be drawn.

• Equation – Two mathematical models expressed as being equal to each other.

• Expected Value – The average value that an experiment is expected to produce if it is repeated a large amount of times.

• Forecasting – The science of predicting future events, which may involve taking historical data and projecting them into the future with the application of a mathematical model.

• Formula – A systematical process of a mathematical model.

• Model – A system of numbers and/or variables that represent a real-life situation.

• Probability – The likelihood or chance that an event will occur.

• Sample – A limited quantity of something that is intended to represent the whole.

• Sample Size – The number of observations that is to be included within a defined sample.

• Trend – A general direction of movement

.

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1 INTRODUCTION

1.1 Description and Motivation for Topic

The work of this thesis primarily revolves around the concept of forecasting the daily exchange rates of the European Euro (EUR) valued in United States Dollars (USD).

Throughout a period of three months, actual exchange rate values were collected and recorded to form a data set. The exchange rate data was then used in the application of a variety of forecasting models to forecast the daily exchange rates for a future, one- month period. As the month approached, the actual exchange rates were collected and recorded. Lastly, the actual exchange rates were compared with the forecasted

exchange rates to form an analysis that measured the forecasting accuracy of the various models.

The thesis has been largely motivated by an operations management course and a statistics course completed by the author. With a curiosity to perform a forecast upon completion of the operations management course, the thesis topic was developed instantaneously.

1.2 Aim of Research

Forecasting is a relatively important issue in business operations, however it also happens to be one of the most problematic of issues. Because the future is always uncertain, forecasts are difficult to assess; yet they are rarely ever avoidable in effective business planning. The aim of this thesis is to successfully forecast the future exchange rates of the EUR in terms of USD for the month of May 2010.

Furthermore, the author expects to determine whether the forecasting models properly work when applied to exchange rates, why or why not, and their measures of

accuracy.

1.3 Hypotheses

1. Future exchange rates can be determined with the aid of a mathematical forecasting model.

2. Future exchange rates cannot be determined with the aid of a mathematical forecasting model.

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1.4 Description of Methods

The methodology of this thesis revolves extensively around quantitative research, and largely includes a probability and forecasting approach. The forecasting system that is implemented involves the determining of an appropriate sample size, a collection of the necessary data, the application of relevant forecasting models, and an analysis of the results. Additionally, a variety of mathematical models that are commonly used in forecasting are applied to the sample of collected exchange rate data. These models include a naive approach, a moving average, exponential smoothing, and linear regression.

1.5 Limitations

It is evident that forecasting exchange rates is no simple task. Therefore, a variety of limitations are present within the work of this thesis. Foremost, there are

approximately one hundred eighty-two currencies in existence throughout the world.

The foreign exchange rate data that is used in this thesis only includes two currencies, which happen to be one of the most common pairs in currency trading. Due to the lack of randomness, a biased selection of variables exist; making it unknown whether or not the same results would be achieved when using a different, unbiased, pair of currencies. Furthermore, there are numerous forecasting models available, however, this thesis only includes models that are familiar to the author. Therefore, it is uncertain whether these models are actually efficient in forecasting exchange rates.

Additionally, it is imperative to bear in mind that the forecasting models used in this thesis do not support the probability of natural disasters or political/economical factors occurring. Lastly, since exchange rates do not typically follow an identifiable trend, it is difficult to conclude whether or not the same results would be achieved at a different period of time.

1.6 Technical frame of reference

A majority of the theories that this thesis is based upon have been referenced from four significant works of literature. Lind, Marchal, and Mason’s “Statistical Techniques in Business & Economics” (2002) was a fundamental element in interpreting the theories of correlation, linear regression, indexes, and dispersion.

Mendenhall, Reinmuth, and Beaver’s “Statistics for Management and Economics”

(1993) was essential in supporting the theories of the measures of central tendency, as

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well as the measures of forecast accuracy. Heizer and Render’s “Operations Management” (2004) was the core foundation for explaining the theory of forecasting. Lastly, Henderson’s “Currency Strategy” (2002) and Rosenberg’s

“Exchange-Rate Determination” (2003) provided the framework for the context of exchange rates. Additional text was referenced from relevant internet sites that contained theories pertaining to forecasting and exchange rates.

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2 DESCRIPTIVE MEASURES

2.1 Numerical Descriptive Methods

“When presented with a set of quantitative data, most people have difficulty making any sense out of it” (Mendenhall, Reinmuth, & Beaver, 1993, p. 16). An efficient way of illustrating the significances of a data set is to present the data’s numerical

descriptive methods. Descriptive methods signify the first approach to organizing, summarizing, and presenting a set of data informatively. Descriptive methods are particularly useful in quickly gathering an initial description of data because they provide a generalization of the center and spread of the set of data. Numerical

descriptive measures can be divided into two categories: measures of central tendency and measures of variability. The measures of central tendency identify the center of the data, while the measures of variability describe how closely the data values are located to the data’s center.

(Mendenhall, Reinmuth, & Beaver, 1993),(Lind, Marchal, & Mason, 2002)

2.2 Measures of Central Tendency

A measure of central tendency is a descriptive method that is used to locate the center of distribution of a set of data. Measures of central tendency summarize the data by determining a single numerical value to represent the data set. There are numerous measures of central tendency, and presented below are a few of the most common.

(Mendenhall, Reinmuth, & Beaver, 1993), (Lind, Marchal, & Mason, 2002) 2.2.1 Mean

The arithmetic mean, also referred to as simply the mean, is a very useful and quite common measure of central tendency. The mean is used in mathematics to measure the average of a set of data. The arithmetic mean of a data set can be expressed as the following formula:

x = x_i

i=1 n

∑

n (1)

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Where

€

x = the symbol used to denote the mean n = the number of measurements in the sample, and

€

x_i

i=1 n

∑

= the summation of the data values.

The mean is a delicate measure of central tendency, however, as it is easily influenced by extreme data values. Therefore, it is important to note that the arithmetic mean may not always be accurately representative of the data set.

(Mendenhall, Reinmuth, & Beaver, 1993) 2.2.2 Median

The median is a second example of a measure of central tendency. When the data is arranged in order of ascending value, the median is the value that divides the data into two equal parts. The median can easily be identified with an odd-quantity set of data, however when there is an even-quantity set of data the median must be found by calculating the average of the two middle values.

(Mendenhall, Reinmuth, & Beaver, 1993) 2.2.3 Mode

A third example that can also be used as a measure of central tendency is the mode.

The mode is found by identifying the data value that has the greatest frequency of occurrence. Though the mode is not used as often as the mean or median, it can be useful in cases where the frequency is considered to be a significant part of the evaluation.

(Mendenhall, Reinmuth, & Beaver, 1993)

2.3 Measures of Variability

Once the center of distribution for a data set has been identified, the next part of the process is to provide a measure of the variability. Variation is a very important property of data, as its measure is necessary in visualizing the distribution of frequencies.

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Dispersion is a term used in statistics to describe the variation within a data set. While the mean or median identify the average value for a set of data, the measure of

dispersion determines the distance from the data’s average to the minimum and maximum values. Dispersion allows the forecaster to determine whether the average is, or is not, representative of the data according to how close the data values are situated to the measure of central tendency. A large measure of dispersion indicates a wide variation in data values, while a small measure of dispersion indicates little variation. Therefore, a small dispersion measure would indicate that the mean or median is representative of the data.

(Mendenhall, Reinmuth, & Beaver, 1993), (Lind, Marchal, & Mason, 2002) 2.3.1 Range

The simplest way to measure the dispersion of a data set is by using the range. The range can be determined by calculating the difference between the highest and lowest values within a data set. The following exemplifies this as an equation:

Range = highest value – lowest value (2)

(Lind, Marchal, & Mason, 2002), (Mendenhall, Reinmuth, & Beaver, 1993) 2.3.2 Variance

The variance can be defined as the arithmetic mean of the squared deviations from the mean of a set of data, in which the deviation is the difference between the mean of a data set and its data values. The variance is a measure that is used to describe a data set’s variation from the mean. By squaring the deviations, it becomes impossible for the variance to be negative in value. Additionally, the variance will only be zero if the values of the data set are equivalent.

Population Variance

The formula for the population variance is:

€

σ² =

x_i−µ

( )

²

i=1 n

∑

n (3)

Where σ² = the symbol used to denote the population variance n = the number of items in the sample

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€

i=1 n

∑

= the summation of n items, and

€

x_i−µ

( )

² = the squared value of the mean subtracted from an item in the sample.

Sample Variance

The formula for the sample variance is:

€

σ_s² =

x_i−x

( )

²

i=1 n

∑

n−1 (4)

Where

€

σ_s² = the symbol used to denote the sample variance n = the number of items in the sample

€

i=1 n

∑

€

x_i−x

( )

Similar to the other measures of variability and the measures of central tendency, the variance is a descriptive method that can be used to compare the variability in two or more sets of data. It is often difficult, however, to interpret the significance of the variance in one set of data due to its squared value.

(Lind, Marchal, & Mason, 2002) 2.3.3 Standard Deviation

A simpler way to interpret the variance is with the measure of the standard deviation.

The standard deviation is defined as the square root of the variance. By un-squaring the variance, the value is transformed into the same unit of measurement as the set of data. In the previous section we mentioned it is impossible for the variance to be negative in value. This is essential for the standard deviation formula to work, as the square root does not work with negative numbers.

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Population Standard Deviation

The formula for the population standard deviation is:

€

σ =

x_i−µ

( )

²

i=1 n

∑

n (5)

Where

€

σ = the symbol used to denote the population standard deviation n = the number of items in the sample

€

i=1 n

∑

€

x_i−µ

( )

Sample Standard Deviation

The formula for the sample standard deviation is:

€

σ_s =

x_i −x

( )

²

i=1 n

∑

n−1 (6)

Where

€

σ_s = the symbol used to denote the sample standard deviation n = the number of items in the sample

€

i=1 n

∑

€

xi−x

( )

(Lind, Marchal, & Mason, 2002)

2.4 Simple Index Numbers

A simple index number, also known as just an index number, refers to a number for a defined period of time that expresses a change in value relative to a base period. The change in value of the indexed number is for a single variable and can be represented by the following equation:

P=

( )

p^p₀^t ^∗100 ⁽⁷⁾

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Where

€

p_t = the amount at any given period, and

€

p₀ = the base-period amount.

The use of index numbers conveniently expresses the change within a diverse set of data by showing the ratios of one time period to another. Additionally, by converting the data into an index, it is easier to observe any visible trends within the data set.

2.5 Example Calculations of Numerical Descriptive Methods

In addition to individually explaining each of the basic numerical descriptive methods, a simple example of each method is solved below to demonstrate the application of these formulas.

2.5.1 Data

The following table provides a fictive set of data to be used in solving the examples.

Table 1. Sample data used for solving the numerical descriptive methods.

Sample Data

X 11 25 38 25 6

2.5.2 Mean

x = x_i

i=1 5

∑

5

x = (x5 + x4 + x3 + x2 + x1) / 5 x = (6 + 25 + 38 + 25 + 11) / 5

!

x = x_i

i=1 n

"

n

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€

x = 105 / 5

€

x = 21

2.5.3 Median

Since there is no equation for finding the median, the median is found by simply identifying the number in the sample that is located directly in the middle. In this case the median is 38.

2.5.4 Mode

Since there is no equation for finding the mode, the mode is found by simply

identifying the number in the sample that has the greatest frequency of occurrence. In this case the mode is 25.

2.5.5 Range

Range = highest value – lowest value

Before the range can be calculated, the data must first be arranged in order from smallest to largest.

Data(smallest to largest) = 38, 25, 25, 11, 6

The formula for finding the range is then calculated as follows:

Range = x5 – x1

Range = 38 – 6 Range = 32

2.5.6 Population Variance

€

σ² =

x_i− x_i

i=1 5

∑

⁵

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣

⎢ ⎤

⎦

⎥

i=1 5

∑

2

5

σ² =

x_i− x₅+x₄ +x₃+x₂+x₁ 5

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

2

i=1 5

∑

5

!

"² =

x_i#µ

( )

²

i=1 n

$

n

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€

σ² =

x_i − 6+25+38+25+11

(

5

)

[ ]

²

i=1 5

∑

5

€

σ² =

x_i − 105

(

5

)

[ ]

²

i=1 5

∑

5

€

σ² =

x_i−21

( )

²

i=1 5

∑

5

€

σ² = [(x5 – 21)² + (x4 – 21)² + (x3 – 21)² + (x2 – 21)² + (x1 – 21)²] / 5

€

σ² = [(6 – 21)² + (25 – 21)² + (38 – 21)² + (25 – 21)² + (11 – 21)²] / 5

€

σ² = [(-15)² + (4)² + (17)² + (4)² + (-10)²] / 5

€

σ² = (225 + 16 + 289 + 16 + 100) / 5

€

σ² = 646 / 5

€

σ² = 129.2

2.5.9 Population Standard Deviation

€

σ= σ²

€

σ= 129.2

€

σ = 11.367

2.5.8 Sample Variance

€

σ_s² =

x_i−x

( )

²

i=1 n

∑

n−1

σ_s² =

x_i− x_i

i=1 5

∑

⁵

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

i=1 5

∑

2

5−1

!

" =

x_i#µ

( )

²

i=1 n

$

n

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€

σ_s² =

x_i− x₅+x₄ +x₃+x₂+x₁ 5

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

2

i=1 5

∑

4

€

σ_s² =

x_i − 6+25+38+25+11

(

5

)

[ ]

²

i=1 5

∑

4

€

σ_s² =

x_i − 105

(

5

)

[ ]

²

i=1 5

∑

4

€

σ_s² =

x_i−21

( )

²

i=1 5

∑

4

€

σs

2 = [(x5 – 21)² + (x4 – 21)² + (x3 – 21)² + (x2 – 21)² + (x1 – 21)²] / 4

€

σ_s² = [(6 – 21)² + (25 – 21)² + (38 – 21)² + (25 – 21)² + (11 – 21)²] / 4

€

σ_s² = [(-15)² + (4)² + (17)² + (4)² + (-10)²] / 4

€

σs

2 = (225 + 16 + 289 + 16 + 100) / 4

€

σ_s² = 646 / 4

€

σ_s² = 161.5

2.5.10 Sample Standard Deviation

€

σs =

x_i −x

( )

²

i=1 n

∑

n−1

€

σ_s = σ_s²

€

σ_s= 161.5

€

σ_s = 12.708

2.5.8 Simple Index Numbers

P1 = (P1 / P1) * 100

!

P=

( )

p^p₀^t ^"100

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P1 = (11 / 11) * 100 P1 = 1 * 100

P1 = 100

P2 = (P2 / P1) * 100 P2 = (25 / 11) * 100 P2 = 2.272 * 100 P2 = 227.2

P3 = (P3 / P1) * 100 P3 = (38 / 11) * 100 P3 = 3.454 * 100 P3 = 345.4

P4 = (P4 / P1) * 100 P4 = (25 / 11) * 100 P4 = 2.272 * 100 P4 = 227.2

P5 = (P5 / P1) * 100 P5 = (6 / 11) * 100 P5 = 0.545 * 100 P5 = 54.5

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3 BACKGROUND OF FORECASTING

Forecasting is best described as the art and science of predicting a future event. It is the estimating of a future outcome of a random process. One of the main assumptions of forecasting is that the results achieved from the random process will provide accurate and reliable evidence of the future. However, limits do exist in what can be expected from a forecast. Though a forecast can provide a great amount of insight, it is rarely ever perfect. There are numerous unpredictable circumstances that can affect the actual outcome of a forecast. Nevertheless, businesses cannot afford to avoid the process of forecasting, as effective planning in both the short and long run often depend greatly on a forecast.

(Heizer & Render, 2004), (Mendenhall, Reinmuth, & Beaver, 1993)

3.1 Forecasting Approaches

There are two distinct approaches to making a forecast. One of the approaches, known as quantitative forecasting, is an arithmetical method and involves the implementation of a mathematical model to a set of data. It typically relies on historical data and does not include emotional or intuitional factors, like qualitative forecasting. Qualitative models are intended for situations where essential historical data is not available to the forecaster.

(Heizer & Render, 2004), (Armstrong, 2001), (Mendenhall, Reinmuth, & Beaver, 1993)

3.1.1 Quantitative Forecasting

Quantitative forecasting largely involves the use of historical data, which is projected into the future with the use of one ore more mathematical models. It may, however, in some cases also be a subjective or intuitive prediction based on qualitative

forecasting, which focuses on subjective inputs that have been obtained from various sources like historical analogies or personal judgment. In such cases, quantitative forecasting becomes a combination of these two approaches and uses a mathematical model that is adjusted by the forecaster’s intuition.

Two subcategories of quantitative forecasting are time-series models and econometric models. The main difference between these two types of models is that while

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econometric models include the use of supporting variables, time series models do not.

(Heizer & Render, 2004)

3.1.2 Econometric Forecasting Models

An econometric model can be defined as one or more equations that demonstrate the relationship amongst a variety of variables. The variables used in econometric forecasting are often a combination of a time series variable and any number of economic variables. An econometric model consists of one dependent variable and any number of independent variables. Econometric models are probabilistic models, meaning that they are calculated by estimating the probability that the dependent variable will reoccur based on the impact of the independent variables. While econometric models forecast by relating independent variables to a dependent variable, time series models forecast based on a dependent variable’s history.

(Mendenhall, Reinmuth, & Beaver, 1993) 3.1.3 Time-series Forecasting Models

“Time-series models predict on the assumption that the future is a function of the past” (Heizer & Render, 2004, p. 107). Since time is often an important factor in decision-making, time-series forecasts were developed to utilize time in the

foundation of the forecast. Therefore, time-series forecasts are based on a sequence of historical data points that are measured over any length of time, including daily, monthly or yearly. When a variable is measured over time, it allows for the visibility of possible trends. Once a trend has been identified, the most recent data is projected into the future according to the movement of the identified trend.

The analyses of time series forecasts are largely influential, and often essential, to the formulating of future estimations, predictions, and many other major decisions that pertain to long-term planning. A few examples of the time-series models that have been implemented in the forecasting of this thesis include: naive approach, moving average, exponential smoothing, and trend projection.

(Mendenhall, Reinmuth, & Beaver, 1993), (Lind, Marchal, & Mason, 2002), (Heizer

& Render, 2004), (Shim, 2000)

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3.2 Choosing a Forecasting Model

Though there are many forecasting methods, there is no such thing as one superior method for use in all forecasting scenarios. “What works best in one firm under one set of conditions may be a complete disaster in another organization; or even in a different department of the same firm” (Heizer & Render, 2004, p. 104). Additionally, though it is relatively simple to apply many of the widely used methods, using a more complex forecasting model may improve the accuracy of the forecast. Nevertheless, these more complex methods often involve a meticulous data collection procedure and/or the application of a computer program, which can become quite costly.

One of the most important parts of forecasting is being able to properly match a forecasting model to the data that is being forecasted. It is the responsibility of the forecaster to select a forecasting model that makes best use of the data that is

available and fits within the forecaster’s time and budget. Choosing a model is often intuitive and becomes recognizable to the forecaster through literary knowledge and experience from trial and error.

During the process of selecting an appropriate forecasting model, the forecaster must review an assortment of criteria as part of the decision-making. The criterion includes the various types of forecasts and the time horizons of the forecasting period.

(Heizer & Render, 2004) 3.2.1 Types of Forecasts

The type of forecast depends on how the forecaster intends to use the information that is received through the forecasting of the data. There are three main types of forecasts that are commonly used in the planning of the future:

1. Economic forecasts: These types of forecasts address the business cycle.

2. Technological forecasts: These are concerned with the rate of technological progress.

3. Demand forecasts: Project the demand for a company’s products or services.

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3.2.2 Time Horizons

Another consideration when selecting a forecasting model is the time horizon for the forecasting period. Once the use of the forecast has been determined, the forecast is categorized by its time horizon. “Some models are more accurate for short-term time horizons and others are more reliable for long-term horizons” (Mendenhall,

Reinmuth, & Beaver, 1993, p. 667). There are three kinds of time horizons:

1. Short-ranged forecasts: May have a time span of up to one year but are normally less than three months.

2. Medium-ranged forecasts: Generally span from three months to three years.

3. Long-ranged forecasts: Any length of time lasting three years or longer.

With the constant change in demand patterns, long-term forecasts are normally used as a planning model for product lines and capital investment decisions, while short- term forecasts usually involve forecasting sales, price changes, and customer demand.

3.3 Forecasting System

The forecasting system is an exemplary guide for successfully completing a forecast.

There are seven steps that form the forecasting system, and together they present a systematic way of initiating, designing, and implementing a forecasting project. The steps of the forecasting system have been applied throughout the forecasting within this thesis. The seven steps that form the forecasting system are:

1. Determine the use of the forecast.

2. Select the items to be forecasted.

3. Determine the time horizon of the forecast.

4. Select the forecasting model(s).

5. Gather the data needed to make the forecast.

6. Make the forecast.

7. Validate and implement the results.

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3.4 Importance of Forecasting

Forecasting plays an important role in the success of many companies around the world. Inventory must be ordered without confidence of what sales will be, new equipment must be purchased with uncertainty of the final products’ demand, and investments are commonly made without knowing what profits will be. Thus, it is important for forecasters to consistently make estimates of what will happen in the future. The projection of a good estimate is the core purpose of forecasting, as it is essential to efficient service and manufacturing operations. “A forecast is the only estimate of demand until actual demand becomes known” (Heizer & Render, 2004, p.

105). Therefore, effective planning in both the short and long run depends on forecasts, making it the drive behind many critical decisions.

Due to the uncertainty of what the future may hold, it is very unlikely for a forecast to be completely accurate. Oftentimes, the further a forecast is projected into the future, the more apprehensive its outcome becomes. A majority of the results produced by a forecast are in some way influenced by one or more unexpected factors. A forecast, long or short-term, is rarely able to offset the effects of each and every possible influencing factor. Still, forecasting is often inevitable due to the probability of benefitting from the forecast, compared to not forecasting at all.

It is important to consider forecasting as a practice that can consistently be perfected.

As the forecaster gains experience, they become more capable of adjusting their methods according to changes within the forecast’s environment. It is possible for even the worst forecasters to occasionally produce good forecasts, while even the best forecasters sometimes miss completely.

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4 FORECASTING MODELS 4.1 Naive Approach

The naive approach is the simplest way to make a forecast because there are no actual calculations involved in its determination. The approach is to assume that the demand in the next period will be the same as the demand in the most recent period.

€

F_t =A_t−1 (8)

Where

€

F_t = the new forecast, and

€

A_t−1 = the previous period’s actual value.

(Heizer & Render, 2004), (Shim, 2000)

4.2 Simple Moving Averages

A simple moving average, also referred to as moving average, is an approach that bases its forecast on an average of a fixed number of previous cycles within a data set.

As time moves forward, the earliest data is excluded from the average to include the most recent cycle’s actual data, creating a continuous inflow of current data and smoothes fluctuations. Therefore, by following this process, any irregular variations that may be present in the set of data are eliminated. This allows for any visible long- term trends within the set of data to be presented. The formula for a simple moving average is:

€

F_t = A_i

i=1 n

∑

n (9)

Where

€

F_t = the new forecast

n = the number of previous periods in the sample, and

€

A_i

i=1 n

∑

= the summation of the actual values for the previous n periods.

Moving averages are useful when the forecaster can assume that the data will move steadily with time. When the data exhibits extreme, random movement a moving

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average forecasting model is likely to provide inaccurate results.

(Lind, Marchal, & Mason, 2002), (Heizer & Render, 2004), (Shim, 2000), (Mendenhall, Reinmuth, & Beaver, 1993)

4.3 Exponential Smoothing

An exponential smoothing model can be described as a combination of a naive approach and a moving-average model, with an added smoothing constant.

Exponential smoothing is used to reveal possible trends and seasonal and cyclic effects within a set of data. Similar to the other time-series models, exponential smoothing assumes that past trends and cycles will continue into the future. Though the exponential smoothing formula forecasts by extending these patterns, it involves very little record keeping and only uses one period of previous data.

Naturally, a pattern does not always occur exactly the same as it did previously. Thus, a smoothing constant is included to reduce the random fluctuations that may exist within a pattern. The formula for exponential smoothing is as follows:

€

F_t =F_t−1+α

(

A_t−1−F_t−1

)

⁽¹⁰⁾

Where

€

F_t−1 = the previous period’s forecast

α = the smoothing constant (0 ≤ α ≤ 1), and

€

A_t−1 = the previous period’s actual value.

(Mendenhall, Reinmuth, & Beaver, 1993), (Heizer & Render, 2004), (Shim, 2000) 4.3.1 Smoothing Constant

The symbol α, read as alpha, is used to represent a smoothing constant. A smoothing constant is a number that is used as a weight in forecasting to make adjustments to the previous forecast’s error. The smoothing constant may range between 0 and 1 and is chosen by the forecaster. Deciding on the value of a smoothing constant for a set of data is no simple task. No formula exists for finding the best value. Thus, the

smoothing constant is often found through trial and error. A high smoothing constant is used to give more weight to recent data while a low smoothing constant gives more weight to past data. When the smoothing constant is 1 the forecast relies solely on the

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most recent data period and assumes that the new forecast is equal to the previous period’s actual value, making it identical to a naive approach.

A second smoothing constant, β, is used in addition to α when forecasting with trend adjustment. Pronounced as beta, this second smoothing constant is used to weight a measure of trend before it is added to an exponentially smoothed forecast to develop a forecast that accounts for trend. Similar to α, the value of β is chosen by the

forecaster and must be a number that ranges between 0 and 1. Choosing a proper value for both smoothing constants is crucial and can often make the difference between an accurate forecast and an inaccurate forecast.

(Mendenhall, Reinmuth, & Beaver, 1993), (Heizer & Render, 2004)

4.4 Exponential Smoothing with Trend Adjustment

Exponential smoothing with trend adjustment is a modified exponential smoothing model that is able to respond to trends. In a simple exponential smoothing model, a lag between the forecast and actual amount is often present. A trend-adjusted exponential smoothing model is applied to reduce any lag and improve the forecast.

The formula for exponential smoothing with trend adjustment is:

€

FIT_t=F_t+T_t (11)

Where FIT = the abbreviation for forecasting with trend F = the exponentially smoothed forecast, and

T = the exponentially smoothed trend.

The exponentially smoothed forecast and trend can be found by the following formulas:

€

F_t =α

(

A_t−1

)

⁺(¹⁻^α)

(

^Ft−1+T_t−1

)

⁽¹²⁾

€

T_t =β

(

F_t−F_t−1

)

⁺

(

¹⁻^β

) ( )

^Tt−1 ⁽¹³⁾

Where

€

α = the smoothing constant for the average (0 ≤ α ≤ 1) A_t−1 = the previous period’s actual value

F_t−1 = the previous period’s forecast

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€

T_t−1 = the previous period’s trend

€

T_t = the new trend, and

β = the smoothing constant for the trend (0 ≤ β ≤ 1).

4.5 Linear Regression

Linear regression can be described as a forecasting model that is used to express a direct relationship between two or more variables. Unlike the previous mentioned forecasting models, linear regression is an econometrics model; meaning that the dependent variable is predicted based on a selected value of one or more independent variables. Furthermore, the linear regression model uses the same formula that is used to graph a line. By arranging the set of data in a linear form, the general movement of the data is displayed. This allows better accuracy in measuring the trend, which allows for a more accurate forecast. Once the slope and the y-intercept of the line have been determined, the line is extended to forecast for a future period of time. The formula for linear regression is written as follows:

€

y =ˆ a+bx (14)

Where

€

y ˆ = the predicted value of the y variable for a selected value of x a = the y-intercept

b = the slope of the line, and

x = any value of the independent variable.

The y-intercept and the slope of the linear regression model can be calculated using the following formulas:

€

b=

x_iy_i

i=1 n

∑

⁻

^{( )}

ⁿ

^{( )}

^x

^{( )}

^y

x_i²

i=1 n

∑

⁻

^{( )}

ⁿ

^{( )}

^x²

(15)

€

a=y −bx (16)

Where a = the y-intercept y = the average of the y values

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b = the slope of the regression line

€

x = the average of the x values

x = a value of the independent variable y = a value of the dependent variable, and n = the number of items in the sample.

(Heizer & Render, 2004),(Swift, 1997), (Lind, Marchal, & Mason, 2002) 4.6.1 Assumptions Underlying Linear Regression

In the application of a linear regression model, there are certain assumptions that must be met in order for the model to properly forecast. For each value of the dependent variable, there must be a value for the independent variable(s). This means that the amount of observations must be the same for all variables. Additionally, the values for each independent variable must not be biased towards other values of the same

variable. In other words, the values for an independent variable cannot be dependent on one another. This assumption is particularly important when the set of data is collected over a period of time, as the forecast error for a time period can often be correlated with another period of time.

4.6.2 The Standard Error of Estimate

Since it is uncommon for a set of data to lie exactly on a linear regression line, there is some degree of error present in the estimating of the dependent variable. The standard error of estimate is used to measure this error and describe how precise or inaccurate the forecast is, according to the dispersion of the observed values from the linear regression line. The standard error of estimate is noted as

€

S_y,x, and can be determined with the following equation:

€

S_y,x=

∑

y²−a

∑

y⁻^b

∑

^xy

n−2 (17)

Where

€

S_y,x = the symbol used to denote the standard error of estimate for x and y y = the values of the dependent variable

a = the y-intercept

b = the slope of the regression line

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x = the values of the independent variable, and n = the number of items in the sample.

(Lind, Marchal, & Mason, 2002) 4.6.3 Correlation Analysis

Correlation analysis can be described as the precise study of a relationship between two variables, in which a group of techniques are applied to measure the strength of the association between variables. The first technique that is applied to the data is commonly a scatter diagram. The dependent variable, the variable that is being predicted, is scaled on the vertical axis as the independent variable, the variable that provides the basis for estimation, is scaled on the horizontal axis. After evaluation, if a general relationship is identifiable, the coefficient of correlation is applied to determine the strength and direction of the relationship.

The coefficient of correlation is a formula used to establish a numerical measure of a relationship’s strength. Referred to as r, the coefficient can assume any value ranging between -1 and +1. An exact measure of either a positive or negative 1 signifies a perfect correlation, meaning that the dependent variable directly relates to the independent variable. The stronger the relationship, the more linear the scatter plot will be. A positive or negative coefficient will indicate the direction of the

relationship’s movement, with a negative coefficient representing a negative direction and a positive coefficient representing a positive direction. Conversely, if there is absolutely no relationship the coefficient will be 0. The coefficient of correlation formula can be indentified as follows:

€

r=

n

(

x_i∗y_i

)

i=1 n

∑

⁻ ^xⁱ

i=1 n

∑

^∗ ^yⁱ

i=1 n

⎛

∑

⎝ ⎜ ⎞

⎠ ⎟

n x²− x

i=1 n

⎛

∑

⎝ ⎜ ⎞

⎠ ⎟

2

i=1 n

⎡

∑

⎣

⎢

⎤

⎦

⎥

⎥ n y² − y

i=1 n

⎛

∑

⎝ ⎜ ⎞

⎠ ⎟

2

i=1 n

⎡

∑

⎣

⎢

⎤

⎦

⎥

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Where r = the symbol used to denote the correlation (–1 ≤ r ≤ 1) n = the number of paired samples

x_i

i=1 n

∑

^∗^yⁱ = the summation of the products of the x values and y values

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€

x_i

i=1 n

∑

= the summation of the x variables

€

y_i

i=1 n

∑

= the summation of the y variables

€

x²

i=1 n

∑

= the summation of the x variables squared

€

x

i=1 n

⎛

∑

⎝ ⎜ ⎞

⎠ ⎟

2

= the squared summation of the x variables

€

y²

i=1 n

∑

= the summation of the y variables squared, and

€

y

i=1 n

⎛

∑

⎝ ⎜

⎞

⎠ ⎟

2

= the squared summation of the y variables.

If there is a strong relationship between two variables, we are easily tempted to assume that an increase or decrease in one variable will cause a change in the other variable. This, however, is not what the measure of the correlation interprets. A strong coefficient of correlation can only conclude that a relationship between the two

variables exists.

(Lind, Marchal, & Mason, 2002), (Heizer & Render, 2004)

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5 MEASURING ERROR

5.1 Measures of Forecast Accuracy

As previously mentioned, it is quite rare for a forecast to be entirely perfect.

Measuring the amount of error that exists within a forecast is very important in the interpreting of its results. Measures of forecast accuracy are used to summarize the overall precision of the application of a forecasting model. The closer a forecast is to its actual value, the more accurate the forecasting model is. Therefore, through examination of a forecast’s error, the quality of the model applied can be evaluated.

5.2 Mean Absolute Deviation

The mean absolute deviation is a relatively simple formula that can be used to measure the overall error of a forecasting model. Also known as the MAD, the mean absolute deviation is computed by the following formula:

€

MAD=

A_i−F_i

i=1 n

∑

n (19)

Where MAD = the abbreviation for Mean Absolute Deviation n = the number of periods

€

i=1 n

∑

= the summation of n periods, and

€

A_i−F_i = the absolute value of the forecast error for n periods.

5.3 Mean Squared Error

The mean squared error, also referred to as the MSE, is another way to measure the overall accuracy of a forecasting model by a forecast’s error. The main difference between the MSE and the MAD formula is that the MSE penalizes extreme errors more heavily than the formula of the MAD, causing the extreme deviations to be accentuated.

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The formula for the MSE is:

€

MSE =

A_i−F_i

( )

²

i=1 n

∑

n (20)

Where MSE = the abbreviation for Mean Squared Error n = the number of periods

€

i=1 n

∑

€

A_i−F_i

( )

² = the squared absolute value of the forecast error for n periods.

5.4 Root Mean Square Error

The root mean squared error, or RMSE, is simply the square root of the MSE. The formula for the RMSE is as follows:

€

RMSE =

A_i −F_i

( )

²

i=1 n

∑

n (21)

Where RMSE = the abbreviation for Root Mean Squared Error n = the number of periods

€

i=1 n

∑

€

A_i−F_i

( )

² = the squared absolute value of the forecast error for n periods.

5.5 Mean Absolute Percentage Error

One disadvantage in using the MAD and MSE in measuring forecast accuracy is that their values are dependent on the magnitude of the item that is forecasted. When a forecasted item is measured in large terms, the MAD and MSE values tend to be very large. By using the mean absolute percentage error, this problem can be avoided.

Conversely, the MAPE is not as effective when a forecasted item is measured in extremely small terms.

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The mean absolute deviation, also known as the MAPE, can be calculated by the following formula:

€

MAPE =

(

100

)

∗ A_i−F_i A_i

i=1 n

⎛

∑

⎝ ⎜ ⎞

⎠ ⎟

n (22)

Where MAPE = the abbreviation for Mean Absolute Percentage Error n = the number of periods

€

i=1 n

∑

= the summation of n periods,

€

A_i−F_i = the absolute value of the forecast errors for n periods, and

€

A_i = the actual values for n periods.

Since the MAPE is measured as a percentage it is a very simple to interpret its results.

The application of the MAPE is particularly useful in comparing the performance of a model for a variety of time-series.

(Mendenhall, Reinmuth, & Beaver, 1993), (Heizer & Render, 2004)

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6 EXCHANGE RATES

6.1 Background of Foreign Exchange Rates

Around the world, numerous forms of money are in existence. Commonly referred to as a currency, each form of money serves a universal purpose – to be exchanged in return for goods and/or services. Ultimately, however, currencies serve as much more than simply a medium of exchange. Often varying by country, a currency is generally representative of a country’s market value. Therefore, by converting two or more currencies into a single form, it is possible to evaluate a country’s worth. This process of measuring two or more currencies in the form of a single country’s currency is known as a foreign exchange rate.

6.2 Forecasting Exchange Rates

Accurately estimating exchange rates is often critical in the financial world.

Unfortunately, it is very difficult to consistently estimate exchange rates successfully.

A variety of reasons exist as to why forecasts for exchange rates tend to deviate from their actual values. The forecaster’s expectations for a currency’s future may move in the opposite direction of the exchange rate’s actual value. Conversely, the forecaster’s expectations may be correct, however, unexpected factors may influence the

exchange rate’s movement to suddenly change. Lastly, the chosen forecasting model may not be as efficient in forecasting a currency as compared to an alternative forecasting model.

(Rosenberg, 2003)

6.2.1 Influencing Factors

The value of a currency is constantly affected and influenced by numerous factors, with many of them often occurring simultaneously. Since the amount of impact that a factor may have on a currency can vary drastically, the forecasting of exchange rates is often complicated. Due to their effect on the value of an exchange rate, however, the influencing factors can be difficult to avoid. Some of the most common

influencing factors include:

• Gross domestic product

• Interest rates

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• Government budget deficits/surpluses

• Monetary policies

• Inflation

• Current-account balance

• Political stability

• Natural disasters (Rosenberg, 2003)

6.2.2 Economical Forecasting Models

A variety of exchange rate forecasting models that predict on the assumption that foreign exchange rates revert to an ideal equilibrium level exists in economics. The models provide a framework for forecasting exchange rates by alerting the forecaster with important changes in the economy and, in turn, how those may affect the

exchange rates over a medium or long-term time horizon. The models, however, do not inform the forecaster of when such events are likely to occur. Therefore, the models are quite limited in their ability to forecast exchange rates. A few of the most common economical forecasting models include purchasing power parity, the

monetary approach, the interest rate approach, and the balance of payments approach.

(Henderson, 2002)

Purchasing Price Parity

Purchasing power parity, often abbreviated as PPP, is one of the most familiar exchange rate models in currency analysis and is commonly referred to as the law of one price. The main concept of the PPP model is that by having no barriers in free trading, the price of a good will be valued the same throughout the world. As a result, the exchange rate will move towards a long-term equilibrium level. In order for the purchasing power parity model to work properly, there must not be any trading barriers or transaction costs. The purchasing power parity is calculated by the following formula:

€

E = P_t

P₀ (23)

FORECASTING EXCHANGE RATES