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University of Tunis
Institut Supérieur de Gestion

B

E

S

T

M

O

D

Business & Economic Statistics Modeling Laboratory
(BESTMOD)
Thesis submitted to The Department of Management Sciences In Fulfillment of
The Requirements for The Degree of Doctor of Philosophy In Finance

Financial and Statistical Modeling of Forward Risk Premia
In Foreign Exchange Markets: Theory & Applications
By Aziz CHOUIKH
Thesis Jury:
Jury:
♦ Sir. Salaheddine HALLARA

Professor

At the University of Tunis

:Chairman

♦ Sir. Abdelwahed TRABELSI

Professor

At the University of Tunis

:Thesis advisor

♦ Sir. Mohamed AYADI

Professor

At the University of Tunis

:Reading member

♦ Sir. Chokri MAMOGHLI

Professor

At the University of Carthage :Reading member

♦ Sir. Adel KARAA

Professor

At the University of Tunis

Thesis Discussion on Friday, June 13th, 2014.

:Member
1

INTRODUCTION

Motivations
Fields of Risk Premia Studies and Related Literature
Statement of Objectives

2
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Preliminary Notions to Sidestep Misconceptions
The forward
forward--spot differential or the forward premium or the forward
discount (xt,1,i) is the difference at time t between the forward exchange
>0
Rd > Rf: premium.
rate and the spot exchange rate:
rate:
=0
Rd = Rf.
xt,1,i = ft,1,i – St,i
<0

Rd < Rf: discount.

The conditional expected appreciation/depreciation of the foreign
currency is
is::
vt,1,i= E(St+1,i /ψt) - St,i
ψt : the information set available at time t.
According to axiom 5 (1.1.5) the forward exchange rate is the
certainty equivalent (EC) / ft,t,11,i = Pt,t,11,i + E(St+1
t+1,i /ψt) where Pt,1
t,1,i is the
forward risk premium which is the Markowitz risk premium
premium..
3
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

The certainty equivalent, EC, is the
level of wealth such that:
U(EC)=E(U)
)=E(U)..

4
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Modeling Forward Risk Premia

As Observed Components

In this case, it is sufficient to
Model the Forward-spot Differential

As Unobserved Components

We apply the SEBased Approach
We model them
Via Observed
Variables

Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

5

Objectives:
Part I: Is the forward risk premium an unobserved
or observed component? Is an ARMA(
ARMA(p,q
p,q)) model
for the unobserved forward risk premium always
identifiable?? If not, what are the financial and
identifiable
mathematical implications?
Part II:
II: Which observed variables can explain the
unobserved forward exchange risk premia?
premia?
6
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Why ARMA Stochastic Process modeling strategy?
strategy?

Four reasons
reasons::
1/ The aggregation of ARMA models yields ARMA
models..(See Kaiser and Maravall [24
models
24])
])
2/ ARMA modeling is rational in the sense of Feige
and Pearce
Pearce..(See Jacobs [23
23])
])
3/ The application of Ansley, Spivey, and Wrobleski's
(ASW’s) [1] moving average summation theorem is
only compatible with ARMA-based modeling
strategy.
4/ ARMA models are short memory models.
7
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Part I: The Microstructural Foundation of Unobserved
Forward Risk Premia Hypothesis,
Hypothesis, The Statistical
Problem of Model Identification, and The Signal
Extraction(SE)--based Approach
Extraction(SE)

Part II
II:: The Determinants of Risk Premia in Forward
Foreign Exchange (FX) Markets

8
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Part I:
I: The Microstructural Foundation of Unobserved
Forward Risk Premia Hypothesis,
Hypothesis, The Statistical Problem
of Model Identification, and The SESE-based Approach

Chapter 1: The Microstructural Foundation of Unobserved
Forward Risk Premia Hypothesis

Chapter 2: The Statistical Problem of Model Identification
(International Journal of Financial Research:
Research: Vol 5, No
3, July 2014
2014,, Forthcoming
Forthcoming))

Chapter 3: The Signal ExtractionExtraction-based Approach
Chapter 4: Empirical Applications
9
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Part II:
II: The Determinants of Risk Premia in
Forward Foreign Exchange Markets
International Journal of Financial Research:
Research: Vol 5, No 2, April 2014

Chapter 5: A Typology of The Determinants of Forward
Risk Premium Components

Chapter 6: Candidate Models and Their Financial
Foundation

Chapter 7: The Empirical Framework
10
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Part I
The Microstructural Foundation of
Unobserved Risk Premia
Hypothesis,, The Statistical
Hypothesis
Problem of Model Identification,
and The SESE-based Approach

11
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 1
The Microstructural Foundation of
Unobserved Risk Premia
Hypothesis

12
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Figure 1.1:
HOW DOES THE FUNDAMENTAL BECOME BURIED IN THE NOISE?
Information is costly,
and so does arbitrage

Information asymmetry

Dichotomy
Informed or more informed (I) at cost cinf

Uninformed or less
informed (U)

The informed to uninformed plus informed ratio is unknown: λ=I(I+U)-1
Noise arises:
Aggregate demand: x= λ XI+(1- λ)XU
Interference between
the fundamental and the noise
13
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

We have shown (proposition 3 (1.1.3)) in a context
of common value paradigm that the I’s demand
function XI is such that
that::

XI,t,i = Pt,1,i .(a.σ2ε,i)-1

where a is ARA coefficient and σ2ε is the variance of
noise.
14
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Sampled time series
(S1,S2,…,ST)
If we reject(accept,KPSS) the
null at α%

else
test1
Differenced time
series: S

End
The signal
is unobserved

If we reject
the null at α%

test2

Else and the Weakform efficiency ≡ Semistrong efficiency

End
The signal
is unobserved

End
The signal
is observed

Figure 1.2:

THE ALGORITHM (Set of Logical Instructions) FOR TESTING WHETHER
THE FORWARD RISK PREMIUM IS OBSERVED OR NOT.
Test 1: H0: it exists a unit root (ADF or PP) or H0: it does not exist a unit root (KPSS)
Test 2: H0: there are no autocorrelations (Ljung-Box)
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

15

The forward risk premium is observed if and only if
the conditional expected appreciation/depreciation
of the foreign currency equals zero and the weakweakform efficiency coincides with the semi
semi--strong form
efficiency (in the Fama’s sense)
sense)..

The spot exchange rate follows a random walk
stochastic process (a martingale) is a necessary but
an insufficient condition.
condition.
16
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 2
The Statistical Problem of Model
Identification
Article entitled:
entitled: « Modeling Forward Risk Premia in
Forward Exchange Rates As Unobserved
Components:: The Model Identification Problem »
Components
Forthcoming,, International Journal of Financial Research
Forthcoming
Research:: Vol 5, No 3, July 2014

17
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

According to Fama [12] :

ft,1,i = Pt,1,i + E(St+1,i/Ψt)
The market determined
certainty equivalent
=
The forward exchange rate
for the ith foreign currency.

The conditional expectation
of the future spot exchange rate.
Ψt is the information set available
at time t.

The forward risk
premium.
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

18

The signal: It is unobservable.

It follows:
follows:

ft,1,i-St+1,i = Pt,1,i + E(St+1,i/Ψt)-St+1,i

=yt+1,i: the forecast error of using
the forward exchange rate
as a predictor of the future spot
exchange rate. It is observable at t+1.

=εt+1,i: the forecast error
due to market imperfections
and information arrivals.
It is unobservable.

19
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

After showing that εt+1,i is a noise (proposition 1),
we get the signal plus noise model:

yt+1,i = Pt,1,i + εt+1,i
The first source of noise
wherein the signal is buried.
It is unobservable.

The only observed variable
in the model.
The unobservable signal which
contains a second source of noise.

Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

20

1/ Only an ARMA(p1,q1) for the observable time series
(yt+1,i) is identifiable.
2/ In Wolff [38] and Nijman, Palm, and Wolff (NPW)[29],
the noise is hypothesized to be a white noise, i.e
ARMA(0,0). A general ARMA(p3,q3) for the noise can be
undertaken.
3/ Given the ASW’s [1] moving average summation
theorem, we deduce the AR and MA conditions relative to
the unknown ARMA(p2,q2) model for the unobservable
signal.
→ We

define the set of the hypothetical models: H.
21

Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

The model identification problem arises whenever
one of the following items, at least, is not
determined::
determined
1/ (p2,q2), the order of the (Pt,1,i) timetime-series ARMA
process..
process
2/ The autoregressive (AR) and/or moving average
(MA) coefficients of the ARMA(p2,q2) process
process..
3/ The variance of the first source of noise and/or the
variance of the second source of noise
noise..
22
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Hypothetical Models:
Models: Let H be the set of
hypothetical models.
models. We get:
get:
H = {ARMA(p2,q2)/p1=p2+p3 and q1≤Max(p2,q2)}

A/ Identifiable Models:
Models:
ARMA model for the unobserved forward risk
premium is identifiable : H/ q2 = Max(p1,q1) – 1.
B/ Unidentifiable Models:
Models:
ARMA model for the unobserved forward risk
premium is unidentifiable but we point out a Noise
Generating Function (NGF)
(NGF):: HNGF/ q2 > Max(p1,q1)
– 1.
23
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

N.B:
N.B:
A time series has a unique ARMA
representation

24
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

We write both the structural (SF
SF)) and reduced (RF
RF))
form for the observable time series
series::
SF: Φy,i(B)yt+1,i = Θp,i(B)at,i + Φy,i(B) vt+1,i
SF:
RF:: Φy,i(B)yt+1,i = Θy,i(B)ωt+1,i
RF
Let γk be the kk--order autocovariance of the observed
variable / 0 ≤ k ≤ Max(p1,q1). Given that a model is
unique,, we get the following (Yule
unique
Yule--Walker) system
system::
(1) γ0,RF = γ0,SF
(2) γ1,RF = γ1,SF
(Max(p1,q1)+1) γMax(p1,q1)
Max(p1,q1),RF = γMax(p1,q1)
Max(p1,q1),SF
25
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

We get the following matrix form
form::

Such that:
that:
Max(p1,q1) + 1 equations that can be reduced to two
equations
q2 + 2 unknowns
unknowns:: q2 MA coefficients (θ1,P,θ2,P,…,θq2,P)
and the noise variances, σ2ε and σ2a.
Xθp is a 2x2 matrix and Σ is a 2x1 vector.
The right
right--hand side is a 2x1 known vector
vector..
26
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

According to Linear Algebra,
Algebra, we get the following
cases:
q2 = Max(p1,q1) - 1: An ARMA model is identifiable for
the unobserved forward risk premium component
component..
q2 > Max(p1,q1) – 1: (underdetermination
underdetermination)) An ARMA
model for the unobserved forward risk premium
component is unidentifiable but a NGF is pointed out.
out.
q2 < Max(p1,q1) – 1: (overdetermination
overdetermination)) We get
get::
Given that all (Max(p1,q1) + 1) equations are linearly
independent:: No solutions
independent
solutions..
27
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Whenever det
det((Xθp) ≠ 0, we have:

: NGF as functions of the MA
coefficient vector θp= (θ1,P,θ2,P,…,θq2,P).

28
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 3
The Signal ExtractionExtraction-based
Approach

29
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

An observed time
time--series is a sum of a signal(s) plus
an error term (noise)
(noise).. The signal can be a trend
and/or a seasonal and/or a cyclical component(s)
component(s)..

We will extract the signal from the noise using
STAMP (Structural Time
Time--series Analyser, Modeler,
and Predictor)
Predictor) software.
software.
30
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 4
Empirical Applications

31
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Empirical methodology
The source of data
data:: Datastream
Datastream,, Paris Dauphine, the
sample is from 01
01//01
01//1999 to 12
12//01/
01/2006
2006..
To test whether the forward risk premium is observed
or not for EUR/USD, EUR/GBP and EUR/JPY
EUR/JPY..
To identify the suitable structural time series model
for the spot and forward exchange rates time series.
series.
(EUR/USD, EUR/GBP and EUR/JPY)
Choice criteria are
are:: the speed of convergence and the
prediction variance error
error..
To calculate the corresponding signal
signal--to
to--noise ratios as
indicators of the source of time variation
variation..
32
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

What Is a Structural TimeTime-series Model
(STM)?
A basic structural timetime-series model ((BSM
BSM)) is
written as follows
follows::
Observed time seriest = Trendt

+ seasonalt + irregulart

Spot or Forward exchange rates
where the trend, seasonal,
seasonal, and irregular terms are the
unobserved structural components that have a direct
financial interpretation.
interpretation.
33
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

While a local linear trend structural time
time--series
model (LLTM
(LLTM)) is written as follows
follows::
Observed time seriest

=

Trendt

+

irregulart

(See Appendix A12 for more details)

34
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Financial Interpretation of The
Structural Components
Trend: it gives us information about the
Trend:
increasing (bull market = exchange rates are
rising)) or decreasing (bear market = exchange
rising
rates are falling
falling)) movement of exchange rates
rates..
Seasonal:: it gives us information about the
Seasonal
anomalies..
anomalies
Irregular: it gives us information about the first
source of noise related to the observed
component.
35
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Empirical Results
Spot and Forward
Exchange Rates

STM

EUR/USD

LLTM

EUR/GBP

BSM

EUR/JPY

LLTM
36

Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Time Series

Selected ARIMA
Model

Observed or
Unobserved
Signal*

SUSD

ARIMA(0,1,0)

Observed

SGBP

ARIMA(0,1,0)

Observed

SJPY

ARIMA(1,1,0)

Unobserved

*:Given the assumption that the Weak-form Efficiency ≡ Semi-strong efficiency.

37
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Recap I
The algorithm (Set of Logical Instructions) for testin
testing
g whether
or not the forward risk premium is observed.
observed.
The identification of a new class of functions that we call the
noise generating functions (NGF)
(NGF).. An ARMA model for the
unobserved forward risk premia is not always identifiable
identifiable..
We have bettered the upper and lower bounds pointed out by
NPW [29]
29] in a more generalized theoretical framework
(proposition (7) 2.2.3).
We have shown (proposition (12)
12) 2.3.2 and (13
13)) 3.1.1) that the
estimation of the MA(
MA(11) model in Wolff [38]
38] is mathematically
infeasible..
infeasible
The NGF is important in determining the state of the market
noise trading.
trading.
EUR/USD and EUR/JPY evidence a LLTM
LLTM,, whilst EUR/GBP
38
BSM..
evidences a BSM
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Part II
The Determinants of Risk Premia in
Forward Foreign Exchange Markets
International Journal of Financial
Research:: Vol 5, No 2, April 2014
Research

39
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 5

A Typology of The Determinants
of Forward Risk Premium
Components

40
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

The ForwardForward-Spot Differential
The Relative BidBid-Ask Spread
The Exchange Rate Volatility

41
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

1/ The forward discount is measurable at time t and
it expresses the difference between the domestic
(home) and the foreign interest rates
rates::

It contains the forward risk premium.
premium. In fact, we
have::
have

42
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

2/ We will use as a proxy for the liquidity
liquidity,, the
relative bidbid-ask spread
spread,, such that
that::

3/ We will use the conditional heteroskedastic
volatility as a proxy for the underlying foreign
currency risk
risk..
43
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 6

Candidate Models and Their
Financial Foundation

44
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Section 1:
1: Individual Regression Models vs
The Zellner’s SURE (Seemingly Unrelated
REgressions)) Model
REgressions

Section 2:
2: The Univariate ARCH
ARCH--(M)
(AutoRegressive Conditional
Heteroskedasticity--(in mean)) Model
Heteroskedasticity
45
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Chapter 7

The Empirical Framework

46
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Data
Daily Low Bid and High Ask Spot Exchange Rates.
Rates.
Daily Low Bid and High Ask OneOne-Week Forward
Exchange rates.
rates.
Three foreign currencies bilateral to the EUR:
EUR: USD,
GBP, and JPY
JPY..
The source of data
data:: Datastream
Datastream,, Paris Dauphine
Dauphine.. The
sample is from 01
01//01
01//1999 throughout 12
12//01
01//2006.
2006.
The neperian logarithm of exchange rates is used to
avoid the Siegel’s paradox arising from the Jensen’s
inequality..
inequality
47
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

The Empirical Methodology
Step 0: We run the unit root tests (ADF, PP, and KPSS)
KPSS)..
Step 1: We run the estimation of Fama's regressions
regressions..
Step 2: We run the estimation of our alternative regressions
regressions..
Step 3: We introduce ARMA dynamics
dynamics..
Step 4: We introduce ARCH
ARCH--(M) dynamics
dynamics..
Step 5: We introduce the liquidity proxy into ARCHARCH-(M)
models..
models
Step 6: We run the estimation of a SURE model using
Fama's equations.
equations.
Step 7 : We run the estimation of a SURE model using our
alternative equations
equations..
48
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Individual Regression Models vs The
SURE Model
Individual Regressions:
System 1:
1: Fama’s Regressions
System 2:
2: Our Alternative Regressions

49
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications

Notations
YWUSDt+1 = ft,1,USD – St+1,USD = yt+1,USD
YWUSD
XWUSD
XW
USDt = ft,1,USD – St,USD = xt,1,USD
DWSUSD
DWS
USDt+1 = St+1,USD - St,USD
DZUSDt+1= sprt+1 - sprt
The same notations for the GBP and JPY are found
substituting USD, respectively, by GBP and JPY.

50
Financial and Statistical Modeling of Forward Risk Premia in FX Markets: Theory & Applications


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