U.S./U.K. Exchange Rate: A Statistical Analysis

Transcription

1 U.S./U.K. Exchange Rate: A Statistical Analysis Gabriel Di Capua Fabio Rabinovich Lorenzo Salazar Peter Van Noppen B a b s o n C o l l e g e F i n a n c i a l M a r k e t s a n d I n s t r u m e n t s S e c t i o n 2 P r o f. M i c h a e l G o l d s t e i n 1 2 / 9 /

2 The authors of this paper hereby give permission to Professor Michael Goldstein to distribute this paper by hard copy, to put it on reserve at Horn Library at Babson College, or to post a PDF version of this paper on the internet. I pledge my honor that I have neither received nor provided any unauthorized assistance during the completion of this work. Signature: Date: Signature: Date: Signature: Date: Signature: Date: - 1 -

3 Table of Contents Executive Summary...3 Introduction.. 4 Hypothesis...6 Variables Interest Rates.6 Inflation Unemployment...8 Stock Markets....8 S&P FTSE Statistical Analysis Forecasting Error Analysis Hypothesis (Revisited) 14 Current Events..15 Conclusion...16 Work Referenced..17 Appendix First Regression Model (Exhibit 1.1) First Best Subsets Regression (Exhibit 1.2) Second Regression Model (16 variables) (Exhibit 1.3) Third Regression Model (12 variables) (Exhibit 1.4)..23 Second Best Subsets Regression (12 variables) (Exhibit 1.5)...24 Fourth (Final) Regression Model (6 variables) (Exhibit 1.6).25 Prediction Model for 2006 (Exhibit 1.7) Prediction Model for 2007 (Exhibit 1.8) Prediction Model for 2008 (Exhibit 1.9) Prediction Model for 2009 (Exhibit 1.10) Prediction Model for 2010 (Exhibit 1.11) Prediction Model for 2011 (Exhibit 1.12)..36 Prediction Model for 2012 (Exhibit 1.13)..38 Predicted vs. Actual Exchange Rate (Exhibit 1.14) Uncovered Interest Rate Parity Model (Exhibit 1.15).40 U.S. Unemployment Rate (Exhibit 1.16) Data Plots and Comparison Charts..42 U.S. to U.K. Exchange Rate U.K. Forward Rates for U.S. Dollars U.S. and U.K. 3-Month Treasury Bond Rates Annual Percent Changes in U.S. and U.K. Consumer Price Indices (CPI)..43 U.S. and U.K. Unemployment Rates U.S. and U.K. Home Prices.. 44 U.S. and U.K. Gross Domestic Product (GDP) per Capita

4 Executive Summary During the period of the gold standard the British pound was the main currency used in international trade. After the Bretton-Woods agreement the U.S. dollar became the main currency to which other currencies were pegged. Throughout history these two currencies have played a major role in the world of trade, which is why this paper attempts to discover the underlying factors affecting the exchange rate between the two. Our initial hypothesis was that the exchange rate between the United States dollar and the United Kingdom sterling pound is most strongly affected by interest rates, changes in the Consumer Price Index (CPI), unemployment rates, and stock market opening averages, in each country. We chose ten variables that corresponded to these indicators and ran a regression on data dating from 1980 to 2010, which showed us that the variations in our ten variables only accounted for 17.3% of variations in the U.S./U.K. exchange rate. This was a small figure so we decided to add six more variables to the mix that represented wealth, real estate, and expectations of the future in both countries. We ran more regressions with a total of sixteen variables and through a process of elimination we reached a model that had six variables and helped explain 82.5% of variations in the exchange rate. We used this model to make predictions on the exchange rate from 2006 to 2010, and we ended up with an average percent error margin of 3.17%. We also used the Uncovered Interest Rate Parity Model to make predictions, using the 3-month Treasury bill rates, and the error percent margin for this process was 4.64%. Because of differences in the amount of data used between our six-variable model and the uncovered interest rate parity model it is difficult to compare the accuracy of the two. However, we saw that the annual average of the U.S. 3-month Treasury rate, the S&P 500 opening price, the U.S. unemployment rate, U.K. GDP per capita, U.S. average home prices, and the U.K. 3-month forward rate created a model that fit the data we used, and was 96.83% accurate, on average, in predicting the exchange rate. We believe the error margins in our predictions have a lot to do with the unprecedented low interest rates we are seeing today as a result of the 2008 recession. Because the data we used had no rates that were that low, when we did our predictions we were extrapolating in some cases, which may have led to inaccuracies in the model. Overall, we saw that the U.S./U.K. exchange rate is affected by a wide variety of factors and that our initial hypothesis was flawed in believing that only four economic indicators (interest rates, changes in the Consumer Price Index (CPI), unemployment rates, and stock market opening averages) were enough to explain something as complex as an exchange rate

5 Introduction Exchange rates have long been issues in countries economic and monetary policies. In a simple world exchange rates would only change to keep parity between international price levels. In the complicated world we live in, exchange rates are constantly changing and can deviate from the exchange rate that would create price parity. To control these changes and maintain stability, countries take measures such as having a fixed or floating exchange rate, and many options in between such as a managed, crawling peg or a managed floating rate. 1 From 1870 through 1914 the gold standard was the dominant policy for currency. During this time most currencies were backed by gold either directly or indirectly by being pegged to currencies that were backed by gold, such as the British pound. Because most currencies were pegged to gold, many countries maintained a fixed exchange rate regime through this period. This led to making the preservation of the exchange rate one of the most important concerns in economic policy. 2 The beginning of World War I brought the gold standard era to an abrupt end and the gold standard was not reinstated until after the war. The Great Depression and the financial chaos that came along with it caused shifts in international gold flow, which led many countries, including the United Kingdom, to once again, abandon the gold standard. 3 Toward the end of WWII, the Bretton Woods system was developed. The Bretton Woods Agreement created a system where all foreign currencies were pegged to the U.S. dollar or to currencies that were pegged to the dollar. This brought forth a new period of fixed exchange rates for any country that was involved in the agreement. The Bretton Woods system worked well for many years, but a rising U.S. Balance-of-Payments deficit forced the system to be abandoned in 1971 despite formal and informal restrictions on the convertibility of the dollar into 1 How Stuff Works (#6 on References Page) 2 Investopedia (#5 on References Page) 3 Forex Forum (#7 on References Page) - 4 -

6 gold. An effort was made to re-establish the system, but the effort fully collapsed in Since then, countries have been able to choose what type of foreign exchange policy the country will follow. With improvements in technology and communications our world is heading towards a globalized economy where economies are very interrelated and a shock in one country affects many others, as shown by the global financial system s collapse in As mentioned above, the British pound was the leading world currency during the period of the gold standard. After the Bretton-Woods agreement the dollar became the standard. These two currencies have always been very important in the world of finance and international trade which is why it is important that we get a better understanding of the factors that affect the exchange rate. As we move forward, we hope to build a model that will help deepen our understanding of exchange rates. 4 Ibid - 5 -

7 Hypothesis The purpose of this paper was to figure out the main economic factors that affect the U.S./U.K. exchange rate. We identified ten variables that we believed would provide the best model for predicting the exchange rate based on our preliminary research and prior knowledge from class discussion. For the initial analysis we used annual figures for 10-year U.S. and U.K. government bonds, 3-month U.S. and U.K. Treasury bills, U.S. and U.K. consumer price indices (CPI), U.S. and U.K. unemployment rates, and Standard and Poor s 500 Index average annual opening stock prices and Financial Times and Stock Exchange Index average annual opening stock prices. Based on these variables we developed the following hypothesis to create a foundation for our statistical analysis. We believe that the exchange rate between the United States dollar and the United Kingdom sterling pound is most strongly affected by interest rates, changes in the Consumer Price Index (CPI), unemployment rates, and stock market opening averages, in each country. Variables In this section, we will take a closer look at the ten variables that we selected based upon preliminary research and prior knowledge. Interest Rates We included both a 10-year and 3-month Treasury bill for the U.S. and U.K. because we wanted to account for both long-term and short-term interest rates for the respective countries. From a macroeconomic perspective interest rates are one of the most influential factors in the price of a currency. The higher the interest rate, the more attractive it is to foreign investors. When a country s interest rate increases relative to the rest of the world the investment return increases, creating a capital inflow. To invest in the country with higher interest rates, investors - 6 -

8 need to exchange their money for the local currency, which increases demand for that currency. This increased demand leads to the currency appreciating. The Bank of England has a floating exchange rate with the U.S dollar, which means it does not intervene in the purchase or sale of currency to maintain a fixed rate. A change in exchange rates due to interest rate changes is demonstrated by the uncovered interest rate parity formula: ( ) Where E(S) is the expected rate in the future represented in terms of dollars per pound, S is the current spot rate in terms of dollar per pound, and i us and i uk are the interest rates for the U.S. and the U.K., respectively. For example, if the U.S. has a 10% interest rate and the U.K. has a 15% interest rate, as long as other decisive investment factors are similar in both countries, the value of the pound would increase by approximately 5%. This leads us to believe that interest rates will have a strong effect on the exchange rate. Inflation We used average annual Consumer Price Index figures because the theory of purchasing power parity predicts a change in the exchange rate when there is a change in the purchasing power of a currency. 5 As an indicator of inflation, CPI percent change provides an understanding of a country s economic standing. In accounting for changes in foreign exchange rates, inflation can be used to compare two countries and determine where currency has better purchasing power. A country with high levels of inflation may have a more volatile currency, which would discourage risk-averse investors from putting their money in that country and would leave the currency depreciated. Because of this, inflation is a factor we must include in our model and analysis. 5 Investopedia (#4 on References Page) - 7 -

9 Unemployment We thought that unemployment rates would have significant correlations with the U.S./U.K. exchange rate because of indirect relationships between the variables. If the unemployment rate rises or is worse than expected, one could expect that the country s central bank would keep interest rates low to stimulate the economy; much like the United States is doing now through quantitative easing. 6 Interest rates are an important determinant in exchange rate volatility. If poor economic data is recorded with high unemployment rate, the country would keep interest rates low and thus decrease the demand for the currency and lead to devaluation of the currency. In the long term, the lower exchange rate would probably increase the country s export competitiveness. This would increase demand for the country s exports, increase productivity while decreasing unemployment and strengthen the exchange rate back to equilibrium. Additionally, the unemployment figure is a main indicator of the state of the economy. When lots of people are employed, they are much more likely to have disposable income. With higher levels of disposable income come greater expenditures on luxury goods. 7 For these reasons, we decided to include average annual unemployment rates in our model and analysis. Stock Markets We decided to investigate the possible effects that changes in stock prices have on the dollar-pound exchange rate between the United States and the United Kingdom, since stock trading plays a very important role in today s global economy. When stocks trend upward, there is, generally, an increase in overall wealth and investment among people or companies that own 6 Wall Street Journal (#8 on References Page) 7 Economics Help (#9 on References Page) - 8 -

10 shares of upward trending stocks, as well as subsequent increases in consumption among households and businesses. With this increase in purchasing power comes an increase in domestic and foreign consumption, and thus exchange rates change, in favor of the country, or countries, where overall wealth has increased. 8 In the analysis of the exchange rate between the U.S. dollar and the U.K. pound, we took a closer look at the average annual open prices of two indices over the past few decades: the S&P 500, and the FTSE 100. Standard & Poor s 500 Index (S&P 500) The Standard & Poor s 500 Index (SPX) is a market value-weighted stock index composed of 500 of the United States largest companies that have shares of common stock listed on either the New York Stock Exchange (NYSE) or the NASDAQ. It is one of the most commonly used benchmarks of the entire U.S. stock market because it serves as one of many indicators of the overall health of the nation s economy and, more specifically, of the overall health of the U.S. equity market. 9 When the S&P 500 performs well, it is likely that the entire U.S. stock market is also performing well. If a foreign investor believes that the U.S. economy is doing well, he or she will be more likely to trade their foreign currency for U.S. dollars, therefore affecting the exchange rate in favor of the dollar. Similarly, if the S&P 500 is doing poorly, then foreign investors may be hesitant to keep their money invested in dollars, and may exchange their dollars for another currency where the economy appears to be healthier and less risky. This, in turn, would depreciate the American currency relative to another currency as demand for the dollar falls. Financial Times and Stock Exchange 100 Index (FTSE 100) 8 Exchange Rates and Stock Prices (#10 on References Page) 9 Investopedia (#11 on References Page) - 9 -

11 Similar to the S&P 500, this market capitalization-index of the 100 largest blue chip companies traded on the London Stock Exchange (LSE) is a great indicator of the health of the British economy. 10 As the stocks in this index rise in price, so too does the index; thus, when the index rises, that means that the wealth of the shareholders is increasing as well, which is, generally, a good sign of economic growth. The performance of the FTSE 100 can influence foreign investors into buying or selling British pounds; for example, if the FTSE 100 is doing well, foreign investors may look to exchange their currencies into pounds because they believe that the economy of the United Kingdom is strong and growing. On the other hand, if the FTSE 100 is doing poorly, these same investors may shy away from investing in the British pound, and others who already have their money invested in pounds may choose to exchange their money for a different currency, both of which would negatively affect the value of the British pound. Statistical Analysis In this section we specifically look at the analysis and guide you step by step along the process we employed. We found data for all of our variables mentioned above ranging from 1980 to All of the figures were represented in annual averages to maintain uniformity among the data. We ran our first regression expecting an R-Sq (adj) value of at least 70% for the first time and were disappointed to see that it was only 17.3% (Exhibit 1.1). This tells us that only 17.3% of the variations in the U.S./U.K. exchange rate can be explained by variations in our selected data. The rule of thumb for multiple regressions is that a model is a good fit for the data if the R-Sq (adj) value is greater than 70%, which means our current model is not a good fit for the data and thus cannot be used to predict an accurate exchange rate. We used the same variables to run a Best Subsets Regression in an attempt to see what combination of variables would give us the highest R-Sq (adj) (Exhibit 1.2). The results showed 10 FTSE 100 Index Fact Sheet (#12 on References Page)

12 that with our current variables the best model included the U.K. 3-month Treasury, the U.S. 10 Year Treasury, and the annual percent change in the U.S. CPI. This new 3-variable model left us with an R-Sq (adj) of only 40.8%, which was still not high enough to make it a good fit for the data. We decided that we would leave the ten initial variables in the model and add new ones that may help account for variations in the exchange rate. We decided to think outside the box for new variables and decided to add U.S. and U.K. GDP per capita, U.S. and U.K average home prices represented in dollars, and the 3-month forward rate. After adding these variables we ran a second regression and were surprised to see that the R-Sq (adj) value jumped to 91.9% (Exhibit 1.3). Because the jump was so big we believe that there may be a strong level of collinearity between some of the variables. Collinearity would negatively affect our model because of high similarities among the variables. In this regression we saw that the Variance Inflation Factors (VIF) for some of the variables were in the thousands, which means that our R-Sq (adj) value was extremely inflated because of strong relationships between the predictor variables. To solve this we decided to eliminate four variables that had VIF s above The four variables that we got rid of at this stage were annual averages for: U.K. 3-month Treasury rates, U.S. GDP per capita, U.S. mortgage debt outstanding, and U.K. home prices. Without these four variables we did a second regression and got an R-Sq (adj) value of 73.8% (Exhibit 1.4). With the remaining 12 variables we were comfortable with a 73.8% R-Sq (adj) because it is above the rule of thumb for multiple regression, but we wanted to optimize our model even further and try to obtain an R-Sq (adj) value that was not just marginally higher than the rule of thumb. We did a Best Subsets regression in which we saw that the best model included only six variables and left us with an R-Sq (adj) of 79.8% (Exhibit 1.5). After removing the U.S. 10-year Treasury, the U.K. 10-year Treasury, the percent change in the U.K. CPI, the FTSE 100 annual average of opening price, the percent change in the U.S. CPI, and U.K. Unemployment, we ran a second regression where our new R-Sq (adj) increased to 82.5% (Exhibit 1.6)

13 The new model includes annual averages for U.S. 3-month Treasury, S&P 500 opening prices, U.S. unemployment, U.K. GDP per capita, U.S. home price, and U.K. 3-month forward rate. The VIF values for the remaining six variables are all under 40, which is good considering we started out with values in the thousands (Exhibit 1.6). Therefore, we believe that this regression equation is a good fit for our data and are confident that it will allow us to predict reasonable values for the exchange rate based on the variables in our model. Forecasting To prove that our model was a good fit for the data we decided to use it to predict the exchange rates from 2006 to 2012 using the values for the variables we had previously obtained through our research. 11 To do this we ran predictions on Minitab year by year starting with 2012 and going back to We first attempted to make the predictions by getting a new regression equation using data only from 1980 to 2005, but the R-Sq (adj) value decreased when we did this, so we decided to only eliminate the year we were predicting from the data and all the years after it so as to not alter our model unnecessarily. Exhibits 1.7 to 1.13 show the outcomes of this process. In doing so we found that the percentage of error values between our observed and expected values were: 1.524% for 2006, 1.469% for 2007, 5.144% for 2008, 6.902% for 2009, 4.401% for 2010, 1.222% for 2011, and 1.514% for 2012 (Exhibit 1.14). The average margin of error in our six predictions was 3.168%, leading us to believe that our regression equation is a good fit for our data. We were content with these results because it allowed us to observe the accuracy of our statistical analysis and model. Though not perfect, our model did provide predictions within the 95 percent confidence Intervals provided by Minitab (Exhibits 1.7 to 1.13). We wanted to see how well our model compared to existing models se we also predicted the exchange rate using the uncovered interest rate parity model (Exhibit 1.16). In this model we 11 Data for 2011 and 2012 was gathered from a different source (#3 on References Page); the rest of data was from FRED and Bank of England (#1 and #2 on References Page)

14 used the 3-month rate for Treasury bills in the U.S. and the U.K. because we wanted similar variables to those we had in our model. We used the end of the month spot rate to predict the expected spot rate in 3 months. According to this model the average error margin percentage was 4.64%, which is slightly higher than that of our model. Error Analysis As noted above, and in Exhibit 1.14 there is a higher percentage of error between the predicted and the expected value of the exchange rate in 2008, 2009, and In exhibits for 2009 and 2010 (Exhibits 1.10 and 1.11) we have highlighted the observation of outliers in the predictor variables, including an extreme outlier for We believe that this is because of the rapid drop in interest rates that resulted from the recession. Had we used monthly data instead of yearly averages we believe that the increase in error margins would have been more gradual instead of seeing the jump from 1.469% in 2007 to 5.144% in 2008 and 6.902% in 2009 (Exhibit 1.14). However, our choice of data recording is not as important a fact as the historically low interest rates we are seeing today. The 2008 Recession has created data points that are unprecedented compared to our aggregate data and thus create outliers that make it more difficult to make an accurate model that can yield reasonable predictions. In our regression model the difference between the observed and predicted value of the exchange rate is $ , on average, shown by the standard error of the estimate (Exhibit 1.13). This difference represents 4.248% of , the mean value of our exchange rate data. This shows that our calculated percentage errors of 5.144% and 6.902%, seen in 2008 and 2009, are higher than the 4.248% error margin the model predicted (Exhibit 1.14). The uncovered interest parity model shows an average error margin percent of 4.64%, which we have mentioned is higher than that of our model. However, this does not necessarily mean that this model is less accurate than ours. The uncovered interest rate parity equation only takes into account interest rates from both countries and current spot rates, which means

15 that there are less variables that account for variations in the exchange rate. Also, regarding the use of data for predictions, in our model we used data for 30 years, but it was yearly averages. In the uncovered interest rate parity model we used data for the same range of 30 years, but the data was monthly so it gave us more data points. The more data that is used the more you can approximate the accuracy of a model, which is why we cannot be completely certain that our model is more reliable than the uncovered interest rate parity. Because our data only went back thirty years, there was a limit to how many predictions we could do without compromising the integrity of the model. With the uncovered interest rate parity equation we could make a prediction based on the interest rates 3 months back. Hypothesis (Revisited) Our final model contains only six variables, of which only three are from our original ten variables that made up our hypothesis. To restate our hypothesis: we initially believed that the main factors affecting the exchange rate between the United States dollar and the United Kingdom sterling pound were interest rates, changes in the Consumer Price Index (CPI), unemployment rates, and stock market opening averages, in each country. We chose 10 initial variables to test these four economic indicators, and in the final model only the U.S. 3-month Treasury rate, the S&P 500 annual average of opening price, and U.S. unemployment rate remained. We believe the reason for this is that the initial variables were too closely correlated amongst themselves, and were from a narrow selection that did not take into account other important factors. The other three variables in the final model were U.K. GDP per capita, U.S. average home prices, and the U.K. 3-month forward rate. These three variables expand our model from only including interest rates, stock prices, and unemployment to taking into account wealth, real estate, and expectations of the future. By having added a more eclectic set of variables we were able to, in a sense, diversify our model. We were naive in the formulation of our initial hypothesis in thinking that ten variables from only four economic indicators would

16 generate an accurate model for an exchange rate, which is affected by countless variables and events every day. Current Events Since the Great Recession of central banks around the world have employed various cautionary measures as a way to restore investor confidence and jump-start economies. Now that several years have passed and economies begin to show steady signs of improvement, we want to provide our guidance, based on our statistical models and results, of where we believe the U.S./U.K. exchange rate may be headed. To do so we will first review current events in the U.S. and U.K. Similar to the Federal Reserve s monetary policy, the Bank of England has also maintained interest rates at levels very close to 0. Through its bond-buying program, the Bank of England hopes to stimulate the economy further, but as signs begin to point to strong economic standing, the central bank will begin to taper the bond-buying programs. As it does, interest rates will rise and that will have a tangible effect on the exchange rate between the two currencies. But it is important to note that the Federal Reserve may also cut back on its quantitative easing, thus pushing American interest rates higher. Based on the uncovered interest rate parity equation that we used before, if both countries interest rates increase at the same rate, then the exchange rate should remain unchanged. If, however, interest rates in the U.S. increased at a faster rate compared to the British interest rates, the dollar would probably appreciate relative to the pound. 12 It appears that the quantitative easing has been working quite well, at least in the U.S. it has been; stock markets up are much higher than they were at the beginning of 2013, the housing market is much more healthy than it was for the past few years, and unemployment is at a five-year low (Exhibit 1.16) Wall Street Journal (#13, 14, and 15 on References Page) 13 Wall Street Journal (#16 and 17 on References Page)

17 Conclusion In this paper, we have looked at a number of factors that affect the exchange rate between the United States and Great Britain and narrowed these down to just six variables that have the greatest combined effect on this rate. However, something that we wanted to be able to include in our model, but could not, were the intangible forces that drive exchange rates; things like political and economic events, peoples individual actions, and their own market judgments and sentiments. 14 Certainly, our model cannot account for these variables, which are very unpredictable and difficult (if not impossible) to quantify. Thus, we are quite content with the model that we presented in this paper, especially since it allowed us to make extremely accurate forecasts of exchange rates between the U.S. dollar and the British pound. 14 Starfish FX (#18 on References Page)

18 Works Referenced 1. Bank of England HideNums=-1&ExtraInfo=true&Travel=NIxIRx 2. Federal Reserve Economic Database 3. US Average Home Prices for 2011 and prices 4. Investopedia - Purchasing Power Parity 5. Investopedia History of Forex 6. How Stuff Works How Exchange Rates Work 7. Forex Forum History of Foreign Exchange 8. Wall Street Journal Andrew Huszar: Confessions of a Quantitative Easer 4? 9. Economics Help Unemployment and Exchange Rates The Relationship Between Exchange Rates and Stock Prices: Studied in a Multivariate Model Investopedia S&P 500 Index FTSE 100 Index Fact Sheet Wall Street Journal Booming U.K. Sets Up Policy Tensions Wall Street Journal - Bank of England Keeps Rates on Hold 4? 15. Wall Street Journal Sterling Rises on Bank of England Plans to Cut Home Loan Support

19 2? 16. Wall Street Journal Stocks End Higher; Tech Shares Rally Wall Street Journal U.S. New Home Sales Surged In October Starfish FX Key Factors Affecting Exchange Rates The World Bank United Kingdom The World Bank United States Office for National Statistics U.K Home Prices rty+prices&content-type=reference+table&content-type=dataset 22. Census Bureau U.S. Home Prices The Guardian U.K. CPI Bureau of Labor Statistics U.S. CPI ftp://ftp.bls.gov/pub/special.requests/cpi/cpiai.txt 25. Wall Street Journal U.S. Employers Add 203,000 Jobs; Unemployment Rate Falls to Five-Year Low 6?mod=WSJ_hps_LEFTTopStories

20 Appendix Exhibit 1.1 (First Regression Model) Regression Analysis: UK/US Annual versus UK 3 Month T, US 3 Month T,... The regression equation is UK/US Annual Rate = UK 3 Month Treasury US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate 26 cases used, 5 cases contain missing values Predictor Coef SE Coef T P VIF Constant UK 3 Month Treasury US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate S = R-Sq = 50.4% R-Sq(adj) = 17.3% Analysis of Variance Source DF SS MS F P Regression Residual Error Total MTB >

21 Exhibit 1.2 (First Best Subsets Regression) Best Subsets Regression: UK/US Annual versus UK 3 Month T, US 3 Month T,... Response is UK/US Annual Rate 26 cases used, 5 cases contain missing values S & P U U U K K F 5 S T 0 1 C S 0 C 0 P E P I A I y 1 n e P 0 n P a e 0 A e U U U r r v r K K S, c A g c e v e U z n g o n n 0 e t f t U e M M r A S m o o Y o C n O C p n n e h n p h U l t t a c a e a n o h h r o n O n n e y u g p i g m m T T T p e e n e p e r r r o n g l n e e e n A A o t a a a n P P n y s s s b n r r n m R u u u o u i i u e a Mallows r r r n a c c a n t Vars R-Sq R-Sq(adj) Cp S y y y d l e e l t e X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

22 Exhibit 1.3 (Second Regression with 16 variables) Regression Analysis: UK/US Annual versus UK 3 Month T, US 3 Month T,... The regression equation is UK/US Annual Rate = UK 3 Month Treasury US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate UK Average Home Price (US$) US Average Home Price (US$) US GDP per Capita (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate US Mortgage Debt Outstdg AvgAnn 26 cases used, 5 cases contain missing values Predictor Coef SE Coef T P Constant UK 3 Month Treasury US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate UK Average Home Price (US$) US Average Home Price (US$) US GDP per Capita (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate US Mortgage Debt Outstdg AvgAnn Predictor VIF Constant UK 3 Month Treasury US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate UK Average Home Price (US$) US Average Home Price (US$) US GDP per Capita (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate US Mortgage Debt Outstdg AvgAnn

23 S = R-Sq = 97.1% R-Sq(adj) = 91.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations UK 3 Month UK/US Obs Treasury Annual Rate Fit SE Fit Residual St Resid R R R R R denotes an observation with a large standardized residual. MTB >

24 Exhibit 1.4 (Third Regression Model with 12 Variables) Regression Analysis: UK/US Annual versus US 3 Month T, 10 Year Trea,... The regression equation is UK/US Annual Rate = US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 26 cases used, 5 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury Year Treasury UK 10 year, zero coupon bond UK CPI Percent Change Annual FTSE 100 Avg Ann Open Price S&P 500 AnnAvg of Opening Price US CPI Percent Change Annual US Unemployment UK Unemployment Rate US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 86.4% R-Sq(adj) = 73.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R R denotes an observation with a large standardized residual. MTB >

25 Exhibit 1.5 (Best subsets with 12 Variables) Best Subsets Regression: UK/US Annual versus US 3 Month T, 10 Year Trea,... Response is UK/US Annual Rate 26 cases used, 5 cases contain missing values S & P U U U K K F 5 S U T 0 S 1 C S 0 C 0 P E P A I A I v U U y 1 n e K K e P 0 n P r a e 0 A e U a G 3 U r r v r K g D S, c A g c e P M e v e U o 3 1 z n g o n n H p n 0 e t f t U e o e t M r A S m m r h o Y o C n O C p e n e h n p h U l C F t a c a e a n o P a o h r o n O n n e y r p r u g p i g m m i i w T T p e e n e p e c t a r r o n g l n e a r e e n A A o t d a a n P P n y ( ( s s b n r r n m R U U R u u o u i i u e a S S a Mallows r r n a c c a n t $ $ t Vars R-Sq R-Sq(adj) Cp S y y d l e e l t e ) ) e X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X MTB >

26 Exhibit 1.6 (Final Regression with 6 Variables) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 27 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 86.5% R-Sq(adj) = 82.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R R denotes an observation with a large standardized residual. MTB >

27 Exhibit 1.7 (Prediction for 2006) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 22 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 82.0% R-Sq(adj) = 74.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.6523, ) (1.5867, )X X denotes a point that is an outlier in the predictors

28 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

29 Exhibit 1.8 (Prediction for 2007) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 23 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 83.8% R-Sq(adj) = 77.7% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.8664, ) (1.8053, )X X denotes a point that is an outlier in the predictors

30 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

31 Exhibit 1.9 (Prediction for 2008) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 24 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 87.2% R-Sq(adj) = 82.7% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.8474, ) (1.7685, )

32 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

33 Exhibit 1.10 (Prediction for 2009) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 25 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 87.2% R-Sq(adj) = 82.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.2599, ) (1.2095, )XX XX denotes a point that is an extreme outlier in the predictors

34 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

35 Exhibit 1.11 (Prediction for 2010) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 26 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 86.7% R-Sq(adj) = 82.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.3328, ) (1.2699, )X X denotes a point that is an outlier in the predictors

36 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

37 Exhibit 1.12 (Prediction for 2011) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 27 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 86.5% R-Sq(adj) = 82.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.5061, ) (1.4361, )

38 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

39 Exhibit 1.13 (Prediction for 2012) Regression Analysis: UK/US Annual versus US 3 Month T, S&P 500 AnnA,... The regression equation is UK/US Annual Rate = US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate 27 cases used, 4 cases contain missing values Predictor Coef SE Coef T P VIF Constant US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate S = R-Sq = 86.5% R-Sq(adj) = 82.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Source DF Seq SS US 3 Month Treasury S&P 500 AnnAvg of Opening Price US Unemployment US Average Home Price (US$) UK GDP per Capita (US$) UK 3 Month Forward Rate Unusual Observations UK/US US 3 Month Annual Obs Treasury Rate Fit SE Fit Residual St Resid R R R R denotes an observation with a large standardized residual. Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI (1.4770, ) (1.3925, )

40 Values of Predictors for New Observations S&P 500 UK GDP AnnAvg of US Average per US 3 Month Opening US Home Price Capita UK 3 Month New Obs Treasury Price Unemployment (US$) (US$) Forward Rate MTB >

41 Exhibit 1.14 Predicted Exchange Rate vs. Actual Exchange Rate Observation Predicted Exchange Rate (US$/UK) Actual Exchange Rate (US$/UK) Percent Error 1.524% 1.469% 5.144% 6.902% 4.401% 1.222% 1.514% Exhibit 1.15 Uncovered Interest Rate Parity Model (1980 to 2013; only showing 1980, 1981, 2012, 2013) Date End of Month Discount Rate, 3 month Treasury bills, Sterling End of Month Discount Rate, 3 month Treasury Bills, US Dollar Expected exchange rate, US$/UK Spot exchange rate, US$/UK Percentage of Error 31-Jan % % $ Feb % % $ Mar % % $ Apr % % $2.20 $ % 31-May % 8.170% $2.24 $ % 30-Jun % 7.980% $2.15 $ % 31-Jul % 8.670% $2.16 $ % 31-Aug % % $2.19 $ % 30-Sep % % $2.20 $ % 31-Oct % % $2.22 $ % 30-Nov % % $2.31 $ % 31-Dec % % $2.33 $ % 31-Jan % % $2.41 $ % 28-Feb % % $2.40 $ % 31-Mar % % $2.43 $ % 30-Apr % % $2.44 $ % 31-May % % $2.27 $ % 30-Jun % % $2.28 $ % 31-Jul % % $2.22 $ % 31-Aug % % $2.16 $ % 30-Sep % % $1.99 $ % 31-Oct % % $1.88 $ % 30-Nov % % $1.90 $ %

42 31-Dec % % $1.81 $ % 31-Jan % 0.060% $1.61 $ % 29-Feb % 0.080% $1.57 $ % 31-Mar % 0.070% $1.54 $ % 30-Apr % 0.090% $1.57 $ % 31-May % 0.070% $1.59 $ % 30-Jun % 0.090% $1.59 $ % 31-Jul % 0.110% $1.62 $ % 31-Aug % 0.090% $1.54 $ % 30-Sep % 0.100% $1.56 $ % 31-Oct % 0.110% $1.56 $ % 30-Nov % 0.080% $1.59 $ % 31-Dec % 0.050% $1.61 $ % 31-Jan % 0.070% $1.61 $ % 28-Feb % 0.110% $1.60 $ % 31-Mar % 0.070% $1.61 $ % 30-Apr % 0.050% $1.58 $ % 31-May % 0.040% $1.51 $ % 30-Jun % 0.040% $1.51 $ % 31-Jul % 0.040% $1.55 $ % 31-Aug % 0.020% $1.51 $ % 30-Sep % 0.020% $1.51 $ % 31-Oct % 0.040% $1.51 $ % Exhibit 1.16 U.S. Unemployment Rate Average 4.64%