Academy of Economic Studies Bucharest Doctoral School of Finance and Banking DOFIN

Academy of Economic Studies BucharestAcademy of Economic Studies BucharestDoctoral School of Finance and BankingDoctoral School of Finance and Banking

DOFINDOFIN

FORWARD DISCOUNT PUZZLEFORWARD DISCOUNT PUZZLEAN APPLICATION FOR THE USD/GBP EXCHANGE RATEAN APPLICATION FOR THE USD/GBP EXCHANGE RATE

Supervisor: Professor Moisa ALTARSupervisor: Professor Moisa ALTARMSc student: Anca-Ioana SIRBUMSc student: Anca-Ioana SIRBU

Bucharest, June 2004Bucharest, June 2004

CONTENTSCONTENTS

• THE UNCOVERED INTEREST RATE PARITY

• FORWARD RATE UNBIASEDNESS HYPOTHESIS

• AN EMPIRICAL ANALYSIS

UNCOVERED INTEREST RATE PARITY(UIP)UNCOVERED INTEREST RATE PARITY(UIP)

• UIP is the cornerstone of international finance (it appears as a key behavioral relationship in almost all models of exchange rate determination)

• UIP states that if risk-neutral market hypothesis holds, then the expected foreign exchange gain from holding one currency rather than another must be offset by the opportunity cost of holding funds in this currency rather than the other-the interest rate differential

• Since UIP reflects the market’s expectations of exchange rate changes, it represents the starting point for any analysis which depends on future exchange rate values.

• That is why, if there are reasons to believe UIP will not hold precisely, an investor must be able to identify the source of deviation and respond accordingly.

Notations usedNotations used• St – nominal spot exchange rate at time t expressed as the

price, in “home-country” monetary units, of foreign exchange (USD against GBP);

• Ste – expected nominal spot exchange rate at time t;

• Ft – forward rate at time t;

• it, it* , nominal interest rate at time t in home country,

respectively in the foreign country.

• E(.) - expectation conditional on information available at time t.

• Small letters denote the nominal logarithm of the variable

COVERED INTEREST RATE PARITYCOVERED INTEREST RATE PARITY

• In the absence of arbitrage barriers across international financial markets, the arbitrage should ensure that the interest rate differential on two assets, identical in any relevant aspect, except currency of denomination, adjust to cover the movement of currencies at the maturity of the underlying assets in the forward market

or, a logarithmic approximation

• If

then

*

,

1

1

t

t

t

kt

i

i

S

F

1)1( *, tt

ktt i

S

Fi

)1()1( *,

ttt

kt iiS

F

*, tttkt iisf

COVERED INTEREST RATE PARITYCOVERED INTEREST RATE PARITY

• Testing for CIP 1. Computing actual deviations from interest parity 2. Regression analysis

Assuming rationale expectations and risk neutrality, we

get

In logs, this relationship is approximately

FRUH:

ttttkt uiisf )( *,

*

*1

1

)(

t

tt

t

ttt

i

ii

S

SSE

*1)( ttttt iissE

ttt fsE )( 1

FORWARD RATE UNBIASEDNESS FORWARD RATE UNBIASEDNESS HYPOTHESISHYPOTHESIS

• FRUH stipulates that under the joint hypothesis of risk neutrality and rational expectations, the current forward rate is an unbiased predictor of the future spot rate

• Bilson(1981) and Fama(1984)

FRUH: α = 0, β = 1, and Et(εt+1) = 0

• Typical finding: forward discount anomaly

kttktkt sfs )( ,

10 and


ENGEL(1996)

• the existence of a foreign exchange risk premium; • a peso problem;• irrational expectations;• international financial market inefficiency from various

frictions.

Fama(1984): omitted variables

which leads to the following decomposition

111 tttt rpsf

111 ttttt rpssf


• Frankel and Froot (1987): excess returns are due to systematic forecast errors – participants form expectations in an irrational manner

* irrational agents earn higher expected returns because they bear higher risk; * rational agents, being more risk-averse, are not necessarily able to drive the first group out of the market by aggressively betting against them.

• Baillie and Bollerslev (1994,2000): time series statistical properties, that is the long memory behavior exhibited by the forward discount, which results in an unbalanced regression.

• Granger (1999): structural changes or regime switches can generate spurious long memory behavior in an observed series

Techniques for analyzing FRUHTechniques for analyzing FRUH

1.OLS regression

2.Cointegration techniques: *long-run relation (cointegration between st+1 and ft)

*short-run relation (cointegration between st and ft)

Zivot(2000), Guerra(2002)

3.Fractional integration Structural changes (Bai and Perron (2001))

kttktkt sfs )( ,

Long Memory ProcessesLong Memory Processes • time domain

{Yt} a covariance stationary process exhibits long memory in the time domain if its autocorrelations ( ρ(k)) exhibit slow decay and persistence

• frequencies’ domain {Yt} exhibits long memory properties if the spectral density

function f(w) has the following property

• In our analysis we use: GPH estimator, MLP estimator and HURST

exponent

naskn

nk

)(

0~)(2

waswcwfd

The Bai and Perron Methodology for The Bai and Perron Methodology for estimating structural breaksestimating structural breaks

• estimation of single and multiple structural breaks in dynamic linear regression models

• estimates the unknown break points given T observations by the least squares principle

• provide general consistency and asymptotic distribution results under fairly weak conditions

*serial correlation *heteroskedasticity• considers the simple structural change in mean model

• pure and partial structural change models

EXCHANGE RATE DATAEXCHANGE RATE DATA

.0

.1

.2

.3

.4

.5

.6

.7

82 84 86 88 90 92 94 96 98

SPOT_RATE

.0

.1

.2

.3

.4

.5

.6

.7

82 84 86 88 90 92 94 96 98

FORWARD_RATE

-15

-10

-5

0

5

10

15

82 84 86 88 90 92 94 96 98

SPOT_DIFFERENCE

-15

-10

-5

0

5

10

15

82 84 86 88 90 92 94 96 98

FORWARD_DIFFERENCE

-.8

-.6

-.4

-.2

.0

.2

.4

82 84 86 88 90 92 94 96 98

FORWARD_DISCOUNT

-.0015

-.0010

-.0005

.0000

.0005

.0010

.0015

82 84 86 88 90 92 94 96 98

PROFIT

EMPIRICAL ANALYSISEMPIRICAL ANALYSISSTATIONARITY TESTSSTATIONARITY TESTS

SPOT_DIFFERENCE

ADF Test -15.2345 1% level critical value -2.573652

5% level critical value -1.942017


PP Test -15.234 1% level critical value -2.573652



FORWARD_DIFFERENCE







MODELS OF COINTEGRATION BETWEEN MODELS OF COINTEGRATION BETWEEN sst+1t+1 AND f AND ftt – EG Methodology: Step 1 – EG Methodology: Step 1

• st+1 = 0.02395420+ 0.951353*ft + et

(0.899071) (0.019174) [2.664328] [49.61727]

MacKinnon critical values for cointegration

Level 0,01 0,05 0,1

Critical values -2.5376 -1.9420 -1.6519

RESID_COINTEGRATION_1





White Heteroskedasticity-Consistent Standard Errors & Covariance

Variable Coefficient Std. Error t-Statistic Prob.

D(SPOT) -0.026943 0.014258 -1.889652 0.0599

D(FORWARD(-1)) -0.029963 0.013787 -2.173368 0.0306

RESID_COINTEGRATION_1 0.00989 0.000158 62.40018 0

R-squared 0.970949 Mean dependent var -5.66E-05

Adjusted R-squared 0.970728 S.D. dependent var 0.030691

S.E. of regression 0.005251 Akaike info criterion -7.64961

Sum squared resid 0.007251 Schwarz criterion -7.6092

Log likelihood 1020.398 Durbin-Watson stat 0.07203

White Heteroskedasticity-Consistent Standard Errors & Covariance

Variable Coefficient Std. Error t-Statistic Prob.

D(FORWARD(-1)) 0.00204 0.001337 1.525582 0.12830

D(SPOT) 1.000032 0.001288 776.3671 0.00000

R-squared 0.99971 Mean dependent var -0.00015

Adjusted R-squared 0.999709 S.D. dependent var 0.03067

S.E. of regression 0.000523 Akaike info criterion -12.26529

Sum squared resid 7.26E-05 Schwarz criterion -12.23842

Log likelihood 1639.416 Durbin-Watson stat 2.21143

MODELS OF COINTEGRATION BETWEEN MODELS OF COINTEGRATION BETWEEN sst+1t+1 AND f AND ftt – Johansen Methodology – Johansen Methodology

Trend assumption: No deterministic trend (restricted constant)

Series: SPOT FORWARD

Unrestricted Cointegration Rank Test

Hypothesized Trace 5 Percent 1 Percent

No. of CE(s) Eigenvalue Statistic Critical Value Critical Value

None * 0.05951 21.91900 19.96 24.6

At most 1 0.02083 5.59983 9.24 12.97

Hypothesized Max-Eigen 5 Percent 1 Percent


None * 0.05951 16.31917 15.67 20.2

At most 1 0.02083 5.59983 9.24 12.97

*(**) denotes rejection of the hypothesis at the 5%(1%) level

Trace test indicates 1 cointegrating equation(s) at the 5% level

Max-eigenvalue test indicates 1 cointegrating equation(s) at the 5% level

st+1 =- 0.008327+1.022440ft + et+1

(0.00296) (0.00634)


-.15

-.10

-.05

.00

.05

.10

.15

82 84 86 88 90 92 94 96 98 00 02

Cointegrating relation 1

The cointegrating relation (stationary residuals:ADF and PP)


Vector Error Correction Estimates

Cointegration Restrictions:

B(1,1)=1,B(1,2)=-1

Restrictions identify all cointegrating vectors

LR test for binding restrictions (rank = 1):

Chi-square(1) 7.603624653

Probability 0.005825108

Cointegrating Eq: CointEq1

SPOT 1.0000

FORWARD(-1) -1.0000

C -0.2108

0.0641

MODELS OF COINTEGRATION BETWEEN MODELS OF COINTEGRATION BETWEEN sstt AND f AND ftt – Johansen Methodology – Johansen Methodology

Trend assumption: No deterministic trend (restricted constant)

Series: SPOT FORWARD

Lags interval (in first differences): No lags

Hypothesized Trace 5 Percent 1 Percent


None * 0.058059 23.09491 19.96 24.6

At most 1 0.026018 7.065169 9.24 12.97

Hypothesized Max-Eigen 5 Percent 1 Percent


None * 0.058059 16.02974 15.67 20.2

At most 1 0.026018 7.065169 9.24 12.97

*(**) denotes rejection of the hypothesis at the 5%(1%) level

Trace test indicates 1 cointegrating equation(s) at the 5% level

Max-eigenvalue test indicates 1 cointegrating equation(s) at the 5% level

st =- 0.821933+1.022254ft + et

(0.00637) (0.29862)


Vector Error Correction Estimates

Cointegration Restrictions:

B(1,1)=1,B(1,2)=-1

Convergence achieved after 3 iterations.

Restrictions identify all cointegrating vectors

LR test for binding restrictions (rank = 1):

Chi-square(1) 8.858151

Probability 0.002918

Cointegrating Eq: CointEq1

SPOT(-1) 1

FORWARD(-1) -1

C -0.00214

-0.00067

[-3.18061]


-.006

-.004

-.002

.000

.002

.004

.006

82 84 86 88 90 92 94 96 98 00 02

Cointegrating relation 1

The cointegrating relation (stationary residuals:ADF and PP)

FORWARD DISCOUNTFORWARD DISCOUNT

Classic testing of FRUH

Forward discount – AR(1) process

11 )( tttt sfs

α = -0.398245 and β = -1.96552 (0.262959) (0.977023)

FWD_DISC = -0.21546 + 0.96085*FWD_DISC(-1) + RESID(0.079487) (0.022346)

[-2.710614] [42.99797]

FORWARD DISCOUNTFORWARD DISCOUNT

Lag

AC

F

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

Series : COINTEGRARE[["forward.discount"]]

Long memory in FORWARD DISCOUNTLong memory in FORWARD DISCOUNT

• GPH estimator

• MLP estimator

• HURST exponent = 0.979207261

dMLPstandard deviation

95%confidence interval

0.96947784 0.074574254[0.8233123 1.1156434 ]

bandwidth dGPH p-value0.625 0.48677 0.045

0.675 0.37448 0.066

0.7 0.32776 0.079

Structural breaks in FORWARD DISCOUNTStructural breaks in FORWARD DISCOUNT-

Specification

Zt = {1} q=1 p=1 h=13 M=5

Tests

SupFT(1) SupFT(2) SupFT(3) SupFT(4) SupFT(5) UDmax WDmax

2.6081 6.2952 18.8677** 13.1897*** 18.8941*** 18.8941*** 32.6855***

SupFT(2/1) SupFT(3/2) SupFT(4/3) SupFT(5/4)

15.5539** 15.5539** 40.5748*** 40.5748***

Number of breaks selected

Sequential 0

BIC 4

Break Point Dates (with Confidence Interval)

C1 0.833018(0.023847) T1 1984:09:00 [21.0000 39.0000]

C2 0.014112(0.008896) T2 1986:03:00 [49.0000 73.0000]

C3 -0.065802(0.010645) T3 1988:06:00 [71.0000 83.0000]

C4 -0.006225(0.013110) T4 1992:10:00 [124.0000 132.0000]

C5 - 0.017911(0.004947)

Structural breaks in FORWARD DISCOUNTStructural breaks in FORWARD DISCOUNT

CONCLUSIONSCONCLUSIONS• We find evidence of a negative β for 1982/01:2004/05, which suggests that the risk

premium is negatively correlated with the expected depreciation, which may explain the negative slope coefficient and can therefore explain the puzzle.

• Using cointegration techniques, we find that in the long run there is mixed evidence regarding the FRUH, as we can accept the unbiasedness, finding that the coefficients are close to their theoretical values, even though by imposing a priori restrictions, we reject the unbiasedness assumption.

• The short-run investigation clearly rejects the FRUH

• A possible explanation for the FRUH not to hold may be that the forward discount is a fractionally integrated process, so it exhibits long memory, which makes the classical regression unbalanced

• Part of the long-memory behavior turns out to be due to structural breaks. We identify four such structural break points for the analyzed period.

• Further analysis should identify how much of the long memory behavior may be explained by the existence of structural breaks

BIBLIOGRAPHYBIBLIOGRAPHY

• Bai, J. and P. Perron (2001): “Computation and Analysis of Multiple Structural Change • Models”, Journal of Applied Econometrics• Baillie, T. R. and T. Bollerslev, (1994): “Cointegration, Fractional Cointegration and • Exchange Rate Dynamics”, The Journal of Finance,vol. 49, no 2, 737-745.• Baillie, T. R. and T. Bollerslev, (2000), “The Forward Premium Anomaly is not as Bad • as you Think,” Journal of International Money and Finance, 19, 471-488.• Bilson, John F.0. (1981): “The ‘speculative efficiency’ hypothesis,” Journal of Business,• 54, 435-51.• Enders,W. (2000), “Applied Econometric Time Series”, in John Wiley & Sons.• Engel, C., (1996): The forward discount anomaly and the risk premium: A survey of • Recent Evidence Journal of Empirical Finance 3, 123–192• Engle, R.F. and C.W Granger., (1987): Cointegration and error correction: representation,• estimation and testing Econometrica 55, 251–276.• Evans, Martin D.D. and Karen Lewis (1995): “Do long-term swings in the dollar affect• estimates of the risk premia?” Review of Financial Studies, Vol. 8, No. 3, • 709-742.• Fama, E., (1984): Forward and spot exchange rates. Journal of Monetary Economics 14, • estimation and testing Econometrica 55, 251–276 • Froot, Kenneth A. and Jeffrey A. Frankel(1987): “Using Survey Data to Test Standard • Propositions Regarding Exchange Rate Expectations”, The American Economic • Review 77, no. 1, 133-153.• Froot, Kenneth A. and Jeffrey A. Frankel (1989): “Forward discount bias: is it an • exchange risk premium?” Quarterly Journal of Economics, 104, 139-61

BIBLIOGRAPHYBIBLIOGRAPHY

• Guerra, Roger (2002): “Forward Premium Unbiasedness Hypothesis: Old Puzzle, • New Results”, University of Geneva, Department of Economics, working • paper no. 02.02. • Hamilton, J., 1993. Time Series Analysis. Princeton University Press, Princeton (NJ).• Maynard A., Phillips P.C.B. 2001. Rethinking an old empirical puzzle: econometric • evidence on the forward discount anomaly. Journal of Applied Econometrics • 16: 671-708.• Sakoulis G., Zivot E. 2001. Time variation and structural change in the forward discount:• Implications for the forward rate unbiasedness hypothesis. Working Paper, • Department of Economics, University of Washington• Sarno, L. and M.Taylor(2000): “The Economics of Exchange Rates”, Cambridge • University Press.• Zivot, E., 1999. The power of single equation tests for cointegration when the • cointegrating vector is prespecified. Econometric Theory (in press) Department of

• Economics, University of Washington, Seattle.• Zivot E. 2000. Cointegration and forward and spot exchange rates. Journal of• International Money and Finance 19: 785-812• Zivot E, J. Wang(2003), Modeling Financial Time Series with S-Plus, Springer

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