Testing The Validity Of International Fisher Effect Finance Essay - Free Essay Example

Foreign exchange trading refers to trading one countrys money for that of another country. The need for such trade arises due to tourism, international trade, or investments across boundaries. The foreign-exchange market, as we usually think of it, refers to large commercial banks in financial centers, like New York or London, trading foreign-currency-dominated deposits among each other. Theories aiming to explore and understand interactions in international monetary variables became increasingly more important as deregulations and international integration of financial markets throughout the world continued to evolve and increase faster than it has ever been. One theory linked two important financial variables, exchange rates and interest rates, is the International Fisher Effect (IFE). IFE states that future spot exchange rates can be determined from nominal interest differentials. In turn, real interest rate will be equalized across the world through arbitrage. Differences in observed nominal interest rates will be stemming from differences in expected inflations. These differences in anticipated inflations are embedded in nominal interest rates as described by Fisher equation. The effect of these variables on exchange rates are more likely to occur under flexible exchange rates where currencies are allowed to fluctuates without government interventions b ut rather lift to free market forces to determine the appropriate exchange rates. In this project, some of the most important theories in international finance literature are going to be explored with an attempt to clarify the logic behind them as well as the basic mathematical formulas that describes these theories. After that, a statistical test will be employed using regression analysis on two currencies that are viewed to be the most traded and free of, or more realistically, exhibit minimal government interventions. The test aims to verify the validity of IFE theory and its explanatory power of fluctuations in exchange rates. However, an overview of the market is going to explore the history of foreign exchange market in addition to the major products and players in that particular market. 2. Foreign-Exchange Market: an Overview 2.1 Historical background: Table 1: Historical background Key aspects Major issues Gold Standard From 1876 to 1913 Each currency is convertible into gold at a specified rate Price of each currency relative to the other is determined by gold convertibility rate Suspended when World War I began in 1914 Some attempts were made in the 1920s to go back to the gold standard, but the Great Depression stood for them Bretton Woods Agreement Was signed in 1944 Called for fixed exchange rate between currencies Governments had to prevent their currencies from moving more than 1% By 1971 the dollar was overvalued as demand for dollar was less than supply Governments had difficulty in maintaining exchange rates at their pre-specified levels Smithsonian Agreement Signed in 1971 Devalued the dollar Allowed currencies to fluctuates in either direction by 2.25% Governments had difficulties in maintaining exchange rate despite the wider limits Floating exchange Started in 1973 after floatation of dollar Free market forces dri ve values of currencies 2.2 Market instruments Table 2: Market instruments Definition Issues Spot market Immediate exchange of currencies The most common type of foreign exchange transactions Forward market Buying and selling currencies at a specified price today but future delivery Mostly used by multinational corporations and speculators Used to eliminate uncertainty Currency swaps Buying one currency today and selling another in the future in one transaction It is a spot and forward transaction in one deal Mostly used in interbank trading in order to avoid excessive transaction costs It serves as a borrowing and lending operations combined Options A contract that provides the right to buy or sell a given amount of currency at a specified price in the future The right holder pays a non-refundable premium to the option writer Like forward contracts except forwards are obligation on both parties while options are obligations only on the option writer Mainly used for risk hedging strategies 2.3 Market structure and players Foreign exchange markets involves hundreds of thousands participants at any given day; among those are hundreds of banks facilitating foreign exchange transactions, but the top 20 banks handle about 50% of the market. Deutsche Bank, Citibank and J.P. Morgan Chase are the largest traders in the market. (5) Commercial banks charge fees for conducting foreign exchange transactions. Banks bid (buy) foreign currencies at a price and ask (sell) them at a higher price. The bid/ask spread covers banks costs plus a profit margin. Usually, bid/ask spreads are functions profit margins of banks and of the risk of the currency, liquidity risk for instance. At any given point in time, exchange rate between two currencies should be similar across the various banks otherwise an arbitrage opportunity arises immediately. If a bank experiences a shortage in a particular foreign currency, it can purchase that currency from another bank in what is called interbank market. Usually, the interbank m arket takes place through brokers (10 brokerage firms handle most of the transactions in this market). (5) Central banks play an integral role in the foreign exchange market. Exchange rates affect the competitive position of products of a country in the international market place through changes in the relative price of currencies. This is the main reason for government intervention in the exchange market. Central banks like the Federal Reserve in the U.S. buy and sell currencies to drive values of their currency to level the free market would not establish. For example, buying dollars and selling pounds would lower supply of dollar and increase the pound which will eventually drives the price of the dollar upward and pushes the pound downward (supply and demand game) (6). The course of normal operations of the government might require some foreign currencies which can be another reason for central banks to operate in the foreign exchange market. 3. Theoretical framework: International Fisher Effect In this section, the theoretical frame that describes factors affecting exchange rates are going to be described. The section begins with purchasing power parity (PPP) that relates changes in exchange rates to inflation differential between two countries. The intuition behind the theory and its versions (absolute and relative) are going to be explained in addition to the basic mathematical frame of the theory. After that, the analysis will head to the Fisher equation which states that nominal interest rate in a country is a function of real interest rate plus a premium for anticipated inflation. That theory is not directly related to exchange markets but it builds a necessary bridge to the International Fisher Effect. The section ends with linking the above-mentioned theories, namely PPP and Fisher equation, to form the International Fisher Effect (IFE). IFE relates exchange rate changes to interest rate differentials between two currencies. That theory, if valid, allows fore ign exchange market participants to predict movements in exchange rates using widely available public information which is market interest rates. By the end of this section, we are going to be equipped by enough theory to start analyzing historical data to assess how closely markets move to what Fisher predicted in his theory. 3.1 Purchasing power parity Purchasing power parity (PPP) has two versions: absolute and relative. Absolute PPP states that exchange rate between any two currencies should be equal to the ratio of their price indexes (6). If we let E denotes exchange rate of the foreign currency and Ph and Pf be the price index of the home and foreign country, respectively, the relation can be described mathematically, (1) Equation 1 can be restated differently, (2) Equation 2 is called the Law of One Price (6). That means prices of domestically produced products equal prices of foreign countrys products in the local currency. If that condition was violated, domestic consumers will shift consumption from foreign products to domestic products when prices of foreign products increase relative to domestic products and the vice versa if prices of foreign countrys products decrease. When that happens, exchange rates will adjust to cancel out the price difference between the two countries to retain equilibrium. Re lative PPP stresses the relative change in exchange rate as well as price indexes. It holds that the percentage change in foreign exchange rate equals the difference between percentage change in price indexes of home (domestic) and foreign country (5). Since percentage change in price index is called inflation rate, relative PPP can be stated as percentage change in foreign exchange rate equals inflation differential between the two countries. Mathematically, (3) Where: Percentage change in foreign currencys exchange rate Home countrys inflation Foreign countrys inflation Equation 3 is just an approximation of the mathematical formula of relation PPP. In order to arrive the precise definition of International Fisher Effect, later on, the exact relative PPP needs to be established. First, we assume that price index of home Ph and foreign country Pf are equal. Relative PPP suggests that foreign currency exchange rate will change if inflation differs between any home and foreign country. Now, if we let ef be the percentage change in foreign currency; then, domestic consumer will perceive prices of the foreign countrys goods to be (4) And to maintain relative PPP, the following should be true (5) Solving fore ef (6) Since Ph=Pf, (as we assumed earlier) (7) Equation 7 states, in plain language, that the foreign currency price relative to home currency will appreciate if domestic inflation exceeds foreign inflation, and vice versa, proportionally to inflation differential. The chart bellows shows graphically how inflation differential and relative change in exchange rate should behave if relative PPP holds. The y-axis represents inflation differential while x-axis represents relative change in foreign exchange rate. Arguments of PPP, whether relative or absolute, makes economic sense under very simplified assumptions that does not capture the complexity if the real world. The table bellow shows these assumptions and th eir limitation. Table 3: Limitation of Purchasing Power Parity Assumption Critique Price indexes of different countries is composed of the same basket of goods and services In reality different nations consumes different bundles of goods and services Every product is traded internationally Not necessarily true due to cultural differences like pork in Saudi Arabia and some times due to legal reasons like quotas Products are homogeneous and perfect substitutes for each other In real world, people view different brands differently and companies usually try to differentiate their products from those of competitors No cost is associated with international trade This assumptions ignores tariffs, transportation costs and assumes that information is free 3.2 Fisher equation Economists distinguish between real and nominal interest rates .Nominal interest rate is the rate observed in the market while real interest rate is a concept that measures returns after adjusting for inflation and is assumed to be the same internationally (6). In presence of arbitrage, capital markets will be integrated worldwide. That means real interest rates are determined by global supply and demand of funds. In an internationally integrated capital market, domestic real interest rate is dependent upon events inside as well as outside the country. If real interest rate in a country were higher compared to another, a flow of capital will push supply of funds upward toward equilibrium. Nominal interest rate will tend to incorporate inflation expectations to provide lenders with a real return for the use of their money. This inflation-expectation effect on nominal interest rate is called the Fisher Effect and can be expressed by the Fisher equation. Let i and r denominat es nominal and real interest rate, respectively, then (8) This can be re-arranged as (9) However, this theory, too, has its limitation, namely the equality of real interest rates across the world. The argument behind this idea implicitly assumes that investors view domestic and foreign assets as perfect substitutes. (3) However, many factors prevents free flow of capital across borders including psychological barriers, legal constraints, political risk, exchange rate risk, taxes and transactions costs. If we assumed that markets are perfect and capital is completely mobile then we can safely assume that real interest rates are equal worldwide. Since a relations has been established between interest rate and inflation rate and that we have PPP which links inflation with exchange rate, a relation between interest rate and exchange rate can be established which is known as the International Fisher Effect IFE. 3.3 International Fisher Effect International Fisher Effect is a theory that links PPP with the Fisher effect that were discussed earlier. The rational behind IFE follows the following logical sequence. Since relative changes in exchange rate according to the relative PPP equals inflation differential and since nominal interest rate, too, changes with inflation and since real returns are equal in different countries; then inflation differential exactly equals nominal interest rate differential between any two countries as the chart illustrates graphically. IFE logic, discussed above, can be easily followed using mathematical formulas of relative PPP and the Fisher equation and some algebraic manipulations, Equations 7 (relative PPP) and equation 8 (Fisher equation) discussed earlier are reproduced here (7) (8) Equation 8 can be rearranged to be (10) Substituting equation 10 in equation 7 results in (11) Since (from the fisher effect theory), equation 11 becomes (12) Equations 1 2 is the IFE which states that relative changes in foreign exchange rate is proportional to nominal interest differential, see the chart bellow. 4. Statistical testing and investigation of IFE This is the last section in this paper where IFE is tested based on real world observations rather than logical and conceptual tests. The section begins with a brief discussion of previous tests on PPP, Fisher equation and IFE. Then, a major assumption about the test is established before discussing the statistical model used to test IFE. At the end, the results are shown and discussed in order to find a reasonable interpretation of any deviation from the IFE theory. 4.1 Previous research results Tests of PPP: Much research has been conducted to test whether PPP exists. Most studies found evidence of significant deviations from PPP that persisted even in the long run. Hakkio, however, found that even for exchange rates that deviated from PPP they tend to move toward the value predicted by the theory suggesting that inflation differentials can be used to forecast long-run movements in exchange rates. (5) Tests of International Fisher Effect: Whether the IFE holds in reality depends on the particular time period examined. Although IFE theory may hold during some time frames, there is evidence that it does not consistently hold. Thomson tested the IFE by constructing 216 transactions in order to see whether currencies with higher interest rates would generate excess profit compared to domestic interest rates or the difference will be offset, as IFE predicts, by depreciation of the currency with higher interest rate. The test resulted in 57% profitable transactions. In addition, the average gain exceeded the average loss indicating that the IFE does not hold. (5) 4.2 Statistical test of IFE Since IFE theory was established in the previous section, an immediate question is whether IFE explains changes in foreign exchange market in the real world. In order to do that, a statistical analysis was conducted on historical data of two major currencies against the dollar, namely British pound and Japanese yen. These currencies were chosen for the following reasons. Dollar, pound and yen are the most liquid currencies in the market which eliminates any liquidity risk factor that might affect our analysis. The political system and the stability of governments lower default risk to minimal in order to be able to compare interest rate on government bonds. Finally, availability of data was a main concern, for example, Euro is a relatively new currency that does not have a long historical time series in order to be able to conduct a reliable test. 4.2.1 Data Monthly and quarterly data beginning from 1979 of U.S dollars, British pounds and from 1985 to 2006 of Japanese yen and there respective interest rates were used in the statistical analysis. Data goes back to 1979 for two reasons. First, using monthly or quarterly data for a period from 1979 to 2006 for British pound and American dollar would provide us with a sample of 335 and 111 observations which are reasonable sample sizes (257 and 86, respectively, for Japanese yen). The smaller the sample the less confidence we have in any statistical inference and we might run in period specific results. Second, if an attempt to go further in historical data was made; we might get caught in periods were market forces did not have the power to freely determine the appropriate exchange rates which will obviously backfire the validity of any testing attempt. Data of exchange rate of U.S dollars and British pounds and their respective interest rate data were downloaded from the Federal Reserve and Bank of England websites, respectively. The Japanese Yen exchange rate and Japanese interest rate was download ed from the website of Bank of Japan. In this report, nominal interest rate is defined as the yield on short term government bonds. 4.2.2 Efficient market hypothesis In order to test IFE statistically, we are forced to make one more assumption. The assumption is that foreign exchange markets are efficient, i.e. rational expectation theory applies. Market efficiency implies that a market has many well informed participants who react immediately to any information which will drive prices immediately to their appropriate levels. Due to the fact that an enormous number of well informed individuals, corporations and institutional investors who trade and speculates in more than a trillion dollars a day in the foreign exchange market; it is safe to assume that exchange rates react rationally to changes in the fundamentals that affects prices in a matter of seconds. This assumption is critical to our test. If this assumption was violated then any discrepancy from whatever the ory we are testing might be due to inefficiency and/or irrationality of market participant rather than a deficiency in the theory. 4.2.3 Regression model IFE as discussed earlier can be described mathematically by equation 12 and it is reproduced here, (12) With algebraic manipulation equation 12 becomes (13) If we let represents time then, (14) Equation 14 states that, percentage change in the future spot rate should equal interest rate differential in the current period. Now let ; Equation 14 becomes, (15) Equation 15 is the equation we are going to base our testing of IFE on. However, to make it functional in statistical testing; equation 15 needs to be turned into regression form as the following, (16) Where: is random Equation 16 (the regression model, herein and thereafter) is the model to be used in the statistical test. From equation 15, we know that our parameters should be and If any of these conditions was not met, then IFE doe s not hold. In order to do that 95% confidence level will be used to test these two hypotheses. 4.3 Results A summary of the test results is provided in the table bellow. Pound (monthly data) Pound (Quarterly data) Yen (monthly data) Yen (quarterly data) Period 1979 2006 1979 2006 1985 2006 1985 2006 # observations 335 111 257 86 Does evidence support IFE No No No No Results of data analysis are presented here. The analysis covered two currencies against U.S dollars and covered two pairs which are U.S dollars against 1) British pounds; 2) Japanese Yen. The test was run twice on each pair. The first used monthly data and quarterly in the second. The point from that is to see if changes in the term would have any effect on the result of the test. Complete results of regression analyses are provided in appendix 1. Other pairs could have been tested, too, but that would not have added any value. For example, British pounds against Japanese Yens can be analyzed the same way. However, the exchange rate between them has already been determined from using cross rates of U.S $ / pounds and U.S $ / Yens. That means if Pounds / Yens were tested it merely duplicates an already tested hypothesis. 4.3.1 U.S dollars against British pounds (1979 2006) P-Value 95% confidence interval Notes Period 1979 2006 Sample size 335 Large sample size R2 3.15% 0.001 R2 indicates weak relation between interest and changes in exchange rate ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 0.0048 0.035 (- 0.0093 , 0.0003) Does not include the hypothetical value which is zero ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 -0.2713 0.001 (- 0.4336 , 0.1091) Does not include the hypothetical value which is one From the table above, the data covered a reasonably long period of time providing us with a sufficient sample size. Using the P-value of each parameter ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 and ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 as well as the overall regression model; the model and each estimated parameter are considered statistically significant at 5% significance level. A condition of the accuracy of the model is the randomness of the error term. The chart bellow supports this assumption. Using the regression results shows a very weak relation, as indicated by the R2, between interest differentials and changes in exchange rates which contrast IFE theory. The analysis indicates that interest rate differential explains only 3.5% of the variation in exchange rate leaving 96.5% to be explained by other variables. ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0, which is statistically significant at 5% significance level, did not include zero in the 95% confidence interval which contradicts the hypothetical value established by IFE which is zero. ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 is supposed to be 1 according to IFE. However, the analysis revealed a significant deviation not only in value but also in direction as it carried, at 95% confidence level, a negative value opposed to positive one predicted by IFE. From all of that, it can be stated with 95% confidence that data does not support IFE. Using quarterly data, estimates of parameters differ than monthly data, however, the conclusion is still the same. For more details of the test results of both monthly and quarterly data go to appendix 1. 4.3.2 U.S dollars against Japanese yens (Monthly Data) (1985 2006) P-Value 95% confidence interval Notes Period 1985 2006 Sample size 257 Large sample size R2 2.9% 0.006 R2 indicates weak relation between interest and changes in exchange rate ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 0.01 0.002 (- 0.004 , 0.002) Does not include the hypothetical value which is zero ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 -0.27 0.006 (- 0.46 , 0.077) Does not include the hypothetical value which is one From the table above, the data covered a reasonably long period of time providing us with a sufficient sample size. Using the P-value of each parameter ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 and ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 as well as the overall regression model; the model and each estimated parameter are considered statistically significant at 5% significance level. Another condition of the accuracy of the model is the randomness of the error term. The chart bellow support this assumption. Using the regression results shows a very weak relation, as indicated by the R2, between interest differentials and changes in exchange rates which contrasts IFE theory. The analysis indicates that interest rate differential explains only 2.9% of the variation in exchange rate leaving 97.1% to be explained by other variables. ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0, which is statistically significant at 5% significance level, did not include zer o in the 95% confidence interval which contradicts the hypothetical value established by IFE which is zero. ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 is supposed to be 1 according to IFE. However, the analysis revealed a significant deviation from that not only in value but also in direction as it carried, at 95% confidence level, a negative value opposed to positive one predicted by IFE. From all of that, it can be stated with 95% confidence that data does not support IFE. 4.3.3 U.S dollars against Japanese yens (Quarterly Data) (1985 2006) P-Value 95% confidence interval Notes Period 1985 2006 Sample size 86 Large sample size R2 7,2% 0.01 R2 indicates weak relation between interest and changes in exchange rate ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 0.027 0.005 ( 0.008 , 0.047) Does not include the hypothetical value which is zero ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 0.8 0.012 (0.178 , 1.43) Includes the hypothetical value which is one From the table above, the data covered a reasonably long period of time providing us with a sufficient sample size. Using the P-value of each parameter ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 and ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 as well as the overall regression model; the model and each estimated parameter are considered statistically significant at 5% significance level. Another condition of the accuracy of the model is the randomness of the error term. The chart bellow supports this assumption. Using the regression results shows a weak relation, as indicated by the R2, between interest differentials and changes in exchange rates which contrast IFE theory. The analysis indicates that interest rate differential explains only 7.2% of the variation in exchange rate leaving 92.8% to be explained by other variables. ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0, which is statistically significant at 5% significance level, did not include zero in the 95% confi dence interval which contradicts the hypothetical value established by IFE which is zero. ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 is supposed to be 1 according to IFE. The regression estimate is 0.8 which is quite near that hypothetical value compared to monthly analysis. Moreover, 95% confidence interval includes the hypothetical value which indicates that this data is in line with IFE. Although ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²1 estimate follows IFE, the regression results showed weak relation as indicated by the R2 and showed that ÃÆ'Ã… ½Ãƒâ€šÃ‚ ²0 is not as predicted by the theory. As a result, IFE is rejected using 95% confidence level. 5. Conclusion and implications of results Empirical analysis using linear regression model at 5% significanse level on monthly and quartly data of U.S dollar against the british pound for the period from (1979 2006) and from July (1985-2006) for U.S dollar against Japanese yen did not support IFE. That is in line with previous tests of the theory. The deviation from IFE can be mainly due to weaknesses in the bases it was built on which is PPP and Fisher effect as discussed in earlier sections. Moreover, foreign exchange market fluctuates due to many factors and not limited to interest rate differentials. The factors include, but not limited to, economic growth, fiscal policies, movement of capital due to relative factor costs, market attractiveness (both real and financial markets) in addition to political and legal environments. Moreover, Macroeconomic variables are not the only determinants of exchange rates as market micro-structure has its effects, too. For market participants who try to forecast movement in for eign exchange market by using changes in nominal interest rate, at least in the short run, will be as good as if they flipped a coin to forecast exchange rates. Finally, according to IFE, borrowing from countries with lower interest rates would be as good as borrowing from countries with high interest rate as movement in exchange rates would cancel the advantage of lower interest rates. However, the result of the test supports the idea of finding bargains in countries with low interest rates which contradicts International Fisher Effect theory. The same thing applies for fixed income investments across boundaries. Appendix 1: Full results of the regression models U.S dollar vs. British pound: Monthly data: Regression Statistics Multiple R 0.1774 R Square 0.0315 Adjusted R Square 0.0286 Standard Error 0.0298 Observations 335 ANOVA  df SS MS F Significance F Regression 1 0.0096 0.0096 10.8188 0.0011 Residual 333 0.2954 0.0009 Total 334 0.3050     Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -0.0048 0.0023 -2.1152 0.0352 -0.0093 -0.0003 X Variable 1 -0.2713 0.0825 -3.2892 0.0011 -0.4336 -0.1091 Quarterly data: Regression Statistics Multiple R 0.306794105 R Square 0.094122623 Adjusted R Square 0.085811821 Standard Error 0.01886649 Observations 111 ANOVA  df SS MS F Significance F Regression 1 0.00403119 0.00403119 11.32533624 0.0010 Residual 109 0.038797943 0.000355944 Total 110 0.042829134     Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -0.019254418 0.001790964 -10.7508701 8.17996E-19 -0.0228 -0.0157 X Variable 1 -0.117530134 0.034923976 -3.36531369 0.001056468 -0.1867 -0.0483 U.S dollar vs. Japanese Yen Monthly data: Regression Statistics Multiple R 0.169889 R Square 0.028862 Adjusted R Square 0.025069 Standard Error 0.021709 Observations 258 ANOVA  df SS MS F Significance F Regression 1 0.003586 0.0036 7.6083 0.006229 Residual 256 0.120644 0.0005 Total 257 0.124230     Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.023966 0.0013582 17.645 3.9E-46 0.021291 0.0266 X Variable 1 -0.107714 0.0390505 -2.758 0.00623 -0.184615 -0.030 Quarterly data: Regression Statistics Multiple R 0.2686396 R Square 0.0721672 Adjusted R Square 0.0611216 Standard Error 0.0646435 Observations 86 ANOVA  df SS MS F Significance F Regression 1 0.027302 0.027302 6.533557 0.012386 Residual 84 0.351018 0.004179 Total 85 0.378320     Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 0.027923 0.009696 2.880013 0.005044 0.008643 0.047 X Variable 1 0.803041 0.314169 2.556082 0.012386 0.178282 1.428