Institute of Mathematics, Academic Sinica

Final Project Report

Using Dimension Reduction Techniques

to Model Dynamics in Foreign Exchange Markets

Author:

黃大維

國立清華大學 計量財務金融學系

蔡秉軒

國立交通大學 應用數學系

陳範恩

國立臺灣大學 電機工程學系

August 14, 2015

Data Description and Motivation

Most of theories about foreign exchange markets focus on relationships among exchange rates,

interest rates, and price levels of two countries. However, it is reasonable to assume there are some

common factors and linkages among FX rates of different countries, and thus I would like to present a

factor analysis of exchange rates. Then, I would like to test the arbitrage pricing model, a famous

model to analyze returns in finance using factor analysis, for the international foreign exchange market.

I download daily returns of foreign exchanges of 16 countries from 2014/05/21 to 2015/05/22. The

data source is the TEJ profile database. In the data set, there are 16 columns (countries) and 240 rows

(days). A sample of the first 5 rows and 5 columns is presented in table 1. The goal is to present a

factor analysis of the data set.

Table 1: Sample of the Data Set

日期 ARS 阿根廷披索 BRL 巴西雷阿爾 CAD 加幣 MXN 墨西哥披索 INR 印度盧比

2014/5/21 0 0 0 0.001548 0

2014/5/22 0 0.004515 0 -0.00387 -0.00445

2014/5/23 0 0 0 -0.00155 0.00137

2014/5/26 0 0 0 0.000777 0.003757

2014/5/27 0 0.008969 0 -0.00078 0.00544

Explorative Data Analysis

Before starting to do the factor analysis, I conduct some explorative data analysis first. To learn

more about daily returns, a heat map is shown in figure 1. Here I exclude the record of CHF at

2015/01/15 on the heat map to get a clear figure. (Note that the unusual return was caused by the

significant change of SBC’s policy.) I also exclude the Rusiian Ruble since it is extremely volatile due

to the sanctions from the Western countries. From the plot, one can find an interesting thing. Since

some of the countries such as Taiwan and Thailand have a relatively fixed exchange policy, they seemed

to be less volatile comparing with other countries.

To check whether a factor analysis is suitable for the data set, I also present the heat map of

correlation matrix in figure. One can find that all the variables are positive correlated, and the four

currencies AUD/CAD/GBP/EUR have correlations larger than 0.4, so it is not bad to conduct a factor

analysis. Note that all currencies are positive correlated with each other since exchange rates are all

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Figure 1: Heat Map for Daily Return Data

based on US dollar. To further check the adequacy of factor analysis, I also calculate the KMO

measure of sampling adequacy (MSA) and conduct the Bartlett’s test of sphericity, and since the

MSA= 0.89 and the p-value is less than 2.2 × 10−6 (i.e. the correlation matrix is not a unit matrix), it

is reasonable to do a factor analysis.

Figure 2: Heat Map of Correlation Matrix

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Theoretical Aspect of the Arbitrage Pricing Theory

Our target is estimating the expected return of a currency. Formulated by Ross (1976), the

arbitrage pricing theory (APT) assumes the rate of return on any security is a linear function of m

factors, that is,

Ri − E(Ri) = li1F1 + li2F2 + · · · + limFm + εi,

where Fj’s and εi’s satisfy the assumption of the orthogonal factor model, and the expected return

E(Ri)’s are all constants. The matrix form is R− E(R) = LF + ε, where Rp×1 is the vector of asset

returns. Note that to analyze the return data, we have i = 1, 2, · · · , 16.

For a portfolio with weight vector wp×1, the portfolio return is

Rportfolio = wR = wE(R) + wLF + wε,

where we assume wε ≈ 0. Then, the portfolio return can be decomposed as a constant return wE(R)

plus a risk premium term wLF. If we select the weight w such that wLF = 0, or equivalently

wL = 0,then Rportfolio = wE(R). In theoretical aspect of security market equilibrium, wE(R) = 0

must holds in this case by the no arbitrage condition.

Thus, in market equilibrium, there exists a weight w such that wL = 0 and wE(R) = 0. The

above result guarantee us to write down the arbitrage pricing model E(R) = λ01p×1 + LΛ, where

Λ = [λ1 λ2 · · · λp]′. Here, most scholars assume λ0 to be the risk-free interest rate Rf , and I set it

as zero for since the foreign exchange trade is actually a zero-sum game. Here, each λi is called the

”market risk premium of factor i”, and the coefficient lij is called the sensitivity of security i for the

risk factor i. To test the APT, we still need to estimate E(R). One can rewrite the factor model

R− E(R) = LF + ε as E(R) = R− LF− ε. Hence, we can approximate the cross-sectional expected

return by E(R) ≈ R− LF.

Empirical Test for the Arbitrage Pricing Theory

Now, I would like to summarize the steps to test the arbitrage pricing theory empirically. Assume

the model R16×249 − E(R) = L16×mFm×249 + ε16×249.

1. Decide the number m of factors. Use principle factor solution and maximum likelihood

estimation to get the estimate L̂ of the coefficients matrix L. Apply the Varimax criteria to get a

rotated factor loading matrix. Comparing the result of different methods.

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2. Use the regression method to get the factor score F̂, and then derive the proxy of expected return

R∗ = R− L̂F̂. Note that each element R∗it is the estimated expected return of security i at time t.

3. To get the market risk premium Λ, we run the Fama-MacBeth cross-sectional regression

Rt = Λ0t + L̂Λt + εt,

where Λ0t is the vector of intercepts, and Rt is the tth column of R. Note that the risk-free rate

is assumed to be zero in foreign exchange rate market.

4. To test the arbitrage pricing theory, we need to test

(1) whether Λ0 = 0 (no abnormal returns),

(2) whether the residuals are white noises, and

(3) whether the fitted value R̂t from the regression is close to real value Rt.

Besides testing the arbitrage pricing theory, there are two things we can notice. First, we are

interested in the explanation of factors since the meaning represents the linkage of exchange rates of

different currencies. The second thing is that whether Λt is time-invariant.

Results of the Analysis

Step 1: Construct the Market Factor Model

The first step is to construct the foreign exchange market factor model R− E(R) = LF + ε. From

the Hotelling’s T 2 test, the p-value is 0.5896, which suggests that we can directly do the factor analysis

without eliminating the sample mean.

Parallel analysis (PA) is a Monte-Carlo based simulation method that compares the observed

eigenvalues with those obtained from uncorrelated normal variables. A factor or component is retained

if the associated eigenvalue is bigger than the 95th of the distribution of eigenvalues derived from the

random data. The parallel analysis scree plot is shown below, and it suggests that the number of

factors m = 3, so we extract three factors from the return data.

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Figure 3: Parallel Analysis Scree Plot

The result of principle factor solution (using correlation matrix) is shown in table 2. From the

loadings with varimax factor rotation, one can see a very clear clustering phenomena. The first group

includes currencies of well-developed country such as JPY, SGD, NZD, AUD, EUR, GBP, and CHF.

The second group including MXN, ZAR, BRL, INR, CAD, RUB, and THB. They are all developing

countries with strong economic growth, but their currencies are easily attacked by international

speculative funds. The third group contains only two countries, Taiwan and Korea. These two countries

have very similar foreign exchange policy, and their major country to export is United States. Note

that the proportion of variance explained by PA1, PA2, and PA3 are 18%, 18%, and 9%, respectively.

Table 2: Factor Loadings from Principle Factor Solution

CurrencyPFS (No Rotation) PFS (After Rotation)

Specific VariancePA1 PA2 PA3 PA1 PA2 PA3

JPY.日圓 0.5879 0.4837 0.0075 0.6798 0.0073 0.3428 0.42

SGD.新加坡元 0.6629 0.2706 0.0985 0.6436 0.2249 0.2398 0.48

NZD.紐西蘭幣 0.7104 0.1318 0.2062 0.6414 0.3695 0.1292 0.44

AUD.澳幣 0.7349 -0.0191 0.2574 0.5909 0.5043 0.0567 0.39

EUR.歐元 0.5117 0.1392 0.1870 0.5060 0.2351 0.0698 0.68

GBP.英鎊 0.3777 0.1435 0.1332 0.3956 0.1421 0.0658 0.82

CHF.瑞士法郎 0.2931 0.2928 0.0302 0.3818 -0.0339 0.1601 0.83

MXN.墨西哥披索 0.6997 -0.4070 -0.0615 0.1904 0.7546 0.2308 0.34

ZAR.南非蘭特 0.7609 -0.3139 -0.0088 0.3104 0.7266 0.2309 0.32

BRL.巴西雷阿爾 0.5262 -0.3721 -0.1309 0.0651 0.6117 0.2326 0.57

INR.印度盧比 0.5580 -0.2772 -0.2035 0.1094 0.5541 0.3325 0.57

CAD.加幣 0.5733 -0.1453 0.1855 0.3762 0.4920 0.0260 0.62

RUB.俄羅斯盧布 0.2650 -0.3342 0.1584 0.0477 0.4370 -0.1175 0.79

THB.泰銖 0.5480 -0.0099 -0.0689 0.3239 0.3534 0.2745 0.69

KRW.韓元 0.6683 0.3732 -0.3557 0.5008 0.1163 0.6694 0.29

TWD.台幣 0.5876 0.0804 -0.3418 0.2791 0.2894 0.5540 0.53

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Table 3 presents factor loadings and specific variances estimated from the maximum likelihood

estimation (using correlation matrix). Note that the group behaviour are very similar with the result

from principle factor solution. The specific variance of each country is also close to the result from

principle factor solution. Note that the proportion of variance explained by FA1, FA2, and FA3 are

19%, 19%, and 8%, respectively.

Table 3: Factor Loadings from Maximum Likelihood Estimation

CurrencyMLE (No Rotation) MLE (After Rotation)

Specific VarianceML1 ML2 ML3 ML1 ML2 ML3

MXN.墨西哥披索 0.7055 0.4345 -0.1419 0.7990 0.1670 0.2010 0.29

ZAR.南非蘭特 0.7615 0.3417 -0.0180 0.7521 0.3212 0.1676 0.30

BRL.巴西雷阿爾 0.5284 0.3374 -0.1931 0.6165 0.0673 0.2139 0.57

INR.印度盧比 0.5576 0.2473 -0.1620 0.5639 0.1472 0.2420 0.60

CAD.加幣 0.5600 0.1586 0.0939 0.4718 0.3441 0.0804 0.65

RUB.俄羅斯盧布 0.2358 0.3466 0.1245 0.4016 0.0644 -0.1607 0.81

JPY.日圓 0.5960 -0.4423 0.1483 0.0350 0.6826 0.3252 0.43

NZD.紐西蘭幣 0.7001 -0.0563 0.3062 0.3779 0.6633 0.0667 0.41

SGD.新加坡元 0.6579 -0.1983 0.2010 0.2541 0.6395 0.1975 0.49

AUD.澳幣 0.7144 0.0689 0.2922 0.4832 0.6052 0.0266 0.40

EUR.歐元 0.5026 -0.0607 0.1985 0.2582 0.4729 0.0739 0.70

CHF.瑞士法郎 0.2824 -0.2704 0.1537 -0.0381 0.4021 0.1156 0.82

GBP.英鎊 0.3753 -0.1060 0.1406 0.1477 0.3771 0.0889 0.83

THB.泰銖 0.5433 -0.0120 0.0300 0.3387 0.3751 0.2020 0.70

KRW.韓元 0.7048 -0.4583 -0.2748 0.1370 0.5065 0.7121 0.22

TWD.台幣 0.6004 -0.1334 -0.2822 0.3161 0.2817 0.5278 0.54

Step 2: Derived the Expected Returns

The second step is to derive the estimated expect return R∗. Here I compare R∗ with standardized

returns since I use the correlation matrix to do the factor analysis. The time series plots of estimated

expected returns and real returns of 16 currencies are shown below. The black lines are the real

standardized returns, the red line are expected returns estimated from principle factor solution, and the

red line are expected returns estimated from the MLE. Note that the maximum in sample root mean

square error among 16 variables is 1.788859. This result suggest that under the assumption of factor

model, the foreign exchange market is not in equilibrium.

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Figure 4: Estimated and Real Returns for the 16 Currencies

Step 3: Fama-MacBeth Regression

The Fama-MacBeth regression is an empirical method to test the arbitrage pricing theory and

predict the future return. I run the cross-sectional regression Rit = λ̂0t + λ̂1tli1 + λ̂2tli2 + λ̂3tli3 + εit

with 16 countries from t = 1 to t = 249. Note that each specific risk premium λ̂i is estimated from the

previous factor analysis.

The first question is that whether or not the abnormal return λ̂0t is zero. The plots of p-values of

sample t test for λ0 = 0 are shown below. For the principle factor solution, there are 45 days having

significant abnormal returns, and there are 44 out of 249 days for the maximum likelihood estimation.

In general, we cannot reject the arbitrage pricing model.

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Figure 5: P-values for testing λ0t = 0

Next, I test the white noise assumption. The p-values of portmanteau test are shown in table 4.

Since the test result from PFS and the result from MLE are very similar, I only present the PFA case.

Approximately half of residuals are serial correlated, which suggest that there are some improvements

we can do for modeling returns. However, there are still half of currencies having uncorrelated

residuals, so the evidence is not strong enough to reject the arbitrage pricing theory.

Table 4: P-values of Portmanteau Test (Box-Ljun Test for Serial Correlation)

Lag 1 Lag 5 Lag 10 Lag 15 Lag 20

BRL 巴西雷阿爾 0.274 0.630 0.792 0.667 0.664

CAD 加幣 0.001 0.024 0.046 0.083 0.078

MXN 墨西哥披索 0.918 0.399 0.246 0.021 0.033

INR 印度盧比 0.886 0.941 0.940 0.911 0.823

JPY 日圓 0.325 0.402 0.041 0.115 0.129

KRW 韓元 0.142 0.205 0.035 0.001 0.000

SGD 新加坡元 0.005 0.006 0.012 0.087 0.085

THB 泰銖 0.458 0.718 0.835 0.483 0.452

TWD 台幣 0.885 0.009 0.001 0.003 0.007

RUB 俄羅斯盧布 0.508 0.196 0.008 0.001 0.000

EUR 歐元 0.000 0.000 0.000 0.000 0.000

GBP 英鎊 0.027 0.072 0.010 0.019 0.006

CHF 瑞士法郎 0.139 0.140 0.087 0.049 0.007

AUD 澳幣 0.005 0.044 0.171 0.301 0.225

NZD 紐西蘭幣 0.036 0.345 0.381 0.109 0.138

ZAR 南非蘭特 0.470 0.338 0.434 0.252 0.034

Number of Rejecting H0 6 5 8 6 7

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The root mean square errors of currencies from the two model is shown in table 5. It is obvious

that the regression with λi estimated by principle factor solution outperforms the result with λi

estimated by MLE. A reasonable explanation is that returns do not follow a multivariate normal

distribution, and thus the MLE assumption is violated. (The distribution of returns are mostly

thick-tailed and skewed.)

Table 5: Root Mean Square Errors of estimations from Fama-MacBeth Regression

Currency BRL 巴西雷阿爾 CAD 加幣 MXN 墨西哥披索 INR 印度盧比 JPY 日圓 KRW 韓元 SGD 新加坡元 THB 泰銖

PFS 0.007 0.005 0.004 0.004 0.004 0.005 0.004 0.004

MLE 0.011 0.007 0.006 0.004 0.005 0.005 0.004 0.003

Currency TWD 台幣 RUB 俄羅斯盧布 EUR 歐元 GBP 英鎊 CHF 瑞士法郎 AUD 澳幣 NZD 紐西蘭幣 ZAR 南非蘭特

PFS 0.003 0.011 0.007 0.008 0.010 0.004 0.005 0.005

MLE 0.003 0.021 0.008 0.008 0.015 0.007 0.008 0.008

The time series plots of estimated λi are shown in the following figure. There are two important

observations. First, each λi is actually time-dependent. Second, they all have similar features of

financial returns - thick tails, volatility clustering, and so on. This is reasonable because we regard each

λi as a specific risk premium, which is also a type of financial returns.

Figure 6: Estimated λ1, λ2, and λ3

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Conclusion and Discussion

In this homework assignment, I use the daily return data in New York foreign exchange market to

present an empirical test of arbitrage pricing theory. The main findings are shown below.

1. From the orthogonal factor model, one can find three common factors of FX returns. The first

common factor represents returns of well-developed country such as JPY, SGD, NZD, AUD,

EUR, GBP, and CHF. The second factor captures market dynamic of currencies of developing

countries such as MXN, ZAR, BRL, INR, CAD, RUB, and THB. The third common factor

measures the foreign exchange rate movements of Taiwan and Korea. The first and second factors

explain approximately 18-19% of the total variance, while the third factor explains approximately

7-8% of the total variance.

2. The estimated expect return R∗ = R− L̂F̂ has significant difference from the real standardized

return (with maximum RMSE approximately 1.78 and minimum 1.21. This result suggests that

the foreign exchange market is not in equilibrium. This result may derive from the intervention

from the central bank of each country.

3. There is no strong evidence that could reject the arbitrage pricing theory. On the other hand, the

Fama-MacBeth regression suggests us to use the principle factor solution instead of MLE under

normality assumption to construct the arbitrage pricing model. Finally, each λi is

time-dependent with stylized facts of financial returns, and thus it is reasonable to interpret each

λi as a specific market risk premium.

In future research, one can focus on applying the arbitrage pricing model to predict returns of

foreign exchange rates. The main problem is the way to forecast the unobserved factor loadings and

the market risk premiums.

Citation of the Source

The data set is downloaded from the TEJ database(台灣經濟新報資料庫). Each student or

faculty can get a free access to this database.

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References

1. Fama, E. F., and MacBeth, J.D. ”Risk, Return, and Equilibrium: Empirical Tests.” Journal of

Political Economics (1973): 607 - 636.

2. Florin Dan Pieleanu. ”Empirical Testing of the APT Model with Pre-Specifying the Factors in

the Case of Romanian Stock Market.” International Journal of Advances in Management and

Economics (2012) 1(2): 27 - 41.

3. Huberman, G., and Zhenyu, W. ”Arbitrage Pricing Theory.” Staff Report, Federal Reserve Bank

of New York (2005) 216.

4. Roll, R., and Ross, S. A. ”An empirical investigation of the Arbitrage Pricing Theory.” Journal of

Finance (1980) 35(6): 1073 - 1103.

5. Ross, S. A. ”The Arbitrage Pricing Theory of capital asset pricing.” Journal of Economic Theory

(1976) 13(4): 341 - 360.

6. Yli-Olli, P., and Virtanen, I. ”Some Empirical Tests of the Arbitrage Pricing Theory Using

Transformation Analysis.” Empirical Economics (1992) 17: 507 - 522.

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