Transcript Document

Academy of Economic Studies Bucharest
Doctoral School of Finance and Banking
DOFIN
FORWARD DISCOUNT PUZZLE
AN APPLICATION FOR THE USD/GBP EXCHANGE RATE
Supervisor: Professor Moisa ALTAR
MSc student: Anca-Ioana SIRBU
Bucharest, June 2004
CONTENTS
• THE UNCOVERED INTEREST RATE PARITY
• FORWARD RATE UNBIASEDNESS
HYPOTHESIS
• AN EMPIRICAL ANALYSIS
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 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 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
Ft ,k 1  it

St
1  it*
or, a logarithmic approximation
•
If
it 
Ft , k
St
then
(1  it* )  1
 Ft ,k

*
(1  it )  (1  it )

 S t

f t ,k  st  it  it*
COVERED INTEREST RATE PARITY
• Testing for CIP
1. Computing actual deviations from interest parity
2. Regression analysis
f t ,k  st     (it  it* )  ut
Assuming rationale expectations and risk neutrality, we
get
Et (S t 1 )  S t it  it*

St
1  it*
In logs, this relationship is approximately
Et (st 1 )  st  it  it*
FRUH:
Et (st 1 )  f t
FORWARD RATE UNBIASEDNESS
HYPOTHESIS
• 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)
st k     ( f t ,k  st )   t k
FRUH: α = 0, β = 1, and
• Typical finding:
forward discount anomaly
Et(εt+1) = 0
  0 and   1
FORWARD RATE UNBIASEDNESS
HYPOTHESIS
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
f t  st 1  rpt 1  t 1
which leads to the following decomposition
f t  st  st 1  rpt 1  t 1
FORWARD RATE UNBIASEDNESS
HYPOTHESIS
• 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 FRUH
1.OLS regression
st k     ( f t ,k  st )   t k
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))
Long 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
n
  (k )   as n  
k  n
• frequencies’ domain
{Yt} exhibits long memory properties if the spectral density
function f(w) has the following property
f ( w) ~ c w
2 d
as
w  0
• In our analysis we use: GPH estimator, MLP estimator and HURST
exponent
The Bai and Perron Methodology for
estimating 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 DATA
.7
.7
.6
.6
.5
.5
.4
.4
.3
.3
.2
.2
.1
.1
.0
.0
82
84
86
88
90
92
94
96
98
82
84
86
SPOT_RATE
88
90
92
94
96
98
96
98
FORWARD_RATE
15
15
10
10
5
5
0
0
-5
-5
-10
-10
-15
-15
82
84
86
88
90
92
94
96
98
82
84
86
SPOT_DIFFERENCE
88
90
92
94
FORWARD_DIFFERENCE
.4
.0015
.2
.0010
.0
.0005
-.2
.0000
-.4
-.0005
-.6
-.0010
-.8
-.0015
82
84
86
88
90
92
94
96
FORWARD_DISCOUNT
98
82
84
86
88
90
92
PROFIT
94
96
98
EMPIRICAL ANALYSIS
STATIONARITY TESTS
SPOT_DIFFERENCE
ADF Test
PP Test
-15.2345
-15.234
1% level critical value
-2.573652
5% level critical value
-1.942017
10% level critical value
-1.615906
1% level critical value
-2.573652
5% level critical value
-1.942017
10% level critical value
-1.615906
FORWARD_DIFFERENCE
ADF Test
PP Test
-15.19883
-15.19852
1% level critical value
-2.573652
5% level critical value
-1.942017
10% level critical value
-1.615906
1% level critical value
-2.573652
5% level critical value
-1.942017
10% level critical value
-1.615906
MODELS OF COINTEGRATION BETWEEN
st+1 AND ft – 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
Critical values
0,01
-2.5376
0,05
0,1
-1.9420
-1.6519
RESID_COINTEGRATION_1
ADF Test
PP Test
-14.6227
1% level critical value
-2.573685
-14.66957
5% level critical value
-1.942022
10% level critical value
-1.615903
MODELS OF COINTEGRATION BETWEEN
st+1 AND ft – EG Methodology: Step 2
MODELS OF COINTEGRATION BETWEEN
st+1 AND ft – EG Methodology: Step 2
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
0.00989
0.000158
62.40018
0
R-squared
0.970949
Mean dependent var
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
RESID_COINTEGRATION_1
-5.66E-05
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std. Error
t-Statistic
Prob.
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
D(FORWARD(-1))
2.21143
MODELS OF COINTEGRATION BETWEEN
st+1 AND ft – Johansen Methodology
Trend assumption: No deterministic trend
(restricted constant)
Series: SPOT FORWARD
Unrestricted Cointegration Rank Test
Hypothesized
Trace
5 Percent
1 Percent
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
Max-Eigen
5 Percent
1 Percent
Eigenvalue
Statistic
Critical Value
Critical Value
None *
0.05951
16.31917
15.67
20.2
At most 1
0.02083
5.59983
9.24
12.97
No. of CE(s)
Hypothesized
No. of CE(s)
*(**) 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)
MODELS OF COINTEGRATION BETWEEN
st+1 AND ft – Johansen Methodology
.15
.10
.05
.00
-.05
-.10
-.15
82
84
86
88
90
92
94
96
98
Cointegrating relation 1
The cointegrating relation (stationary residuals:ADF and PP)
00
02
MODELS OF COINTEGRATION BETWEEN
st+1 AND ft – Johansen Methodology
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:
SPOT
CointEq1
1.0000
FORWARD(-1)
-1.0000
C
-0.2108
0.0641
MODELS OF COINTEGRATION BETWEEN
st AND ft – 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
Eigenvalue
Statistic
Critical Value
Critical Value
None *
0.058059
23.09491
19.96
24.6
At most 1
0.026018
7.065169
9.24
12.97
Max-Eigen
5 Percent
1 Percent
Eigenvalue
Statistic
Critical Value
Critical Value
None *
0.058059
16.02974
15.67
20.2
At most 1
0.026018
7.065169
9.24
12.97
No. of CE(s)
Hypothesized
No. of CE(s)
*(**) 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)
MODELS OF COINTEGRATION BETWEEN
st AND ft – Johansen Methodology
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]
MODELS OF COINTEGRATION BETWEEN
st AND ft – Johansen Methodology
.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 DISCOUNT
Classic testing of FRUH
st 1     ( f t  st )   t 1
α = -0.398245
(0.262959)
and
β = -1.96552
(0.977023)
Forward discount – AR(1) process
FWD_DISC = -0.21546 + 0.96085*FWD_DISC(-1) + RESID
(0.079487) (0.022346)
[-2.710614]
[42.99797]
FORWARD DISCOUNT
0.0
0.2
ACF
0.4
0.6
0.8
1.0
Series : COINTEGRARE[["forward.discount"]]
0
5
10
15
Lag
20
Long memory in FORWARD DISCOUNT
• GPH estimator
bandwidth
dGPH
p-value
0.625
0.48677
0.045
0.675
0.37448
0.066
0.7
0.32776
0.079
• MLP estimator
dMLP
0.96947784
standard
deviation
0.074574254
• HURST exponent
=
95%confidence
interval
[0.8233123
1.1156434 ]
0.979207261
Structural breaks in FORWARD DISCOUNT
Specification
Zt = {1}
q=1
p=1
h=13
M=5
Tests
SupFT(1)
SupFT(2)
2.6081
6.2952
SupFT(3)
SupFT(4)
SupFT(5)
UDmax
WDmax
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 DISCOUNT
CONCLUSIONS
•
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
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