Predictability and mispricing/good deals in currency
markets
Richard Levich
(New York University)
Valerio Potì
(Dublin City University)
Presented by Valerio Poti’
Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale
Superiore, Pisa
12 November 2010
DCU Business School
www.dcu.ie/dcubs
FX determination puzzle
• Meese and Rogoff (JIE, 1983): exchange rate
disconnect puzzle (‘Houston, we have a problem’)
• Engel and West (2004, 2005): disconnect not as bad
as you think, fundamentals do not need to forecast
exchange rates (‘Hang on Houston, problem solved!’)
• Lyons and Moore (JIMF, 2006), Brennan and Xia
(RFS, 2006): disconnect is bad if you compare
volatilities of fundamentals and exchange rates
(‘Sorry Houston, we still have a problem’)
DCU Business School
Flexible-Price Monetary Model
Excess Exchange Rate Volatility
• Volatility of the US$/GB£ exchange rate vs. the volatility of
the US-GB money growth rate differential:
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
1984:05
1986:11
1989:05
US$/GB£ vol.
DCU Business School
1991:11
1994:05
1996:11
Money growth diff. () vol.
1999:05
What we are asking
• We want to make inferences on mispricing in FX
markets (and once we’ve fine tuned the techniques,
also in other ones)
• The benchmark for no mispricing is pricing in
accordance with RE and ‘plausible’ assumptions
about decision making under uncertainty,
– That is, we look at the strong-form EMH and the in-sample
exchange rate properties it implies
– Why? We want to tell you guys to what extent (if any) RE
gets and E(U) get it wrong so you can devise a better
theory
DCU Business School
A peek preview of the main results
• Some mispricing but…nope, prices (exchange rates)
are not really badly out of whack with RE, at least
when you make realistic assumptions about the
environment in which trading takes place
• They wander a bit though around a RE benchmark,
and you’d wonder what this wandering is
about…learning?, search for the right heuristic?,
cycles in ‘supply’ of skilled investors?
DCU Business School
The pricing problem…a potentially ill posed one
• From no arbitrage, mt+1 > 0 exists s.t.
Pt = Et(mt+1xt+1) = Et*(xt+1)
m(s) = q(s)/p(s)
This can be solved for the expected excess-return that rational investors
should equate to their own required rate of return, and in so doing they fix
the state prices q(s) and thus, given p(s), Pt and mt+1
Et rt 1   (1  R f ,t )Covt rt 1 , mt 1   Covt rt 1 , mt 1 
The problem is that a pile of combinations of Q(s) and P(s)
are observationally equivalent, i.e. give you the same traded
price Pt given payoff xt+1!!!
DCU Business School
The tight link between predictability and
discount factor volatility: an upper bound
• Straight from Pt = Et(mt+1xt+1),
Variance of conditional mean excess-returns
Coefficient of determination of
‘true’ (consistently estimated)
reduced form representation of
DGP
2


t 1 
2
R  2
  2 rt 1 , mt 1  2 mt 1 
 rt 1 
Variance of excess-returns on asset with
payoff xt+1
We don’t know everything about mt+1 but we do know a
thing or two, so we can look at predictability to make
inferences on EMH/mispricing
DCU Business School
Deriving the bound I
• From Pt = Et(mt+1xt+1),
Et rt 1   (1  R f ,t )Corrt rt 1 , mt 1  t rt 1  t mt 1 

t 1   t rt 1 , mt 1  t rt 1  t mt 1 
(1)
 2 t 1   E t21    2 rt 1 , mt 1  2 rt 1  2 mt 1 
(2)
DCU Business School
Deriving the bound(s) II
• Dividing through by the variance of the asset excessreturn,
2


t 1 
2
R  2
  2 rt 1 , mt 1  2 mt 1 
 rt 1 
• And, by the Cauchy–Schwarz inequality:
R 
2
DCU Business School
 
 2 t 1
 rt 1 
2
2
mt 1 
What about the kernel?
• Two possibilities:
– We know the strategies/factors that span the MV frontier,
in which case m is any kernel that prices them
– We know the IMRS of the marginal trader and use the fact
that, as shown by Ross (2005), σ(m) ≤ σ(IMRS) for the least
volatile m
• In either case we rule out “good deals”, e.g. Cochrane
and Saà Requeio (JPE, 2000), Cerný and Hodges
(2001).
DCU Business School
The currencies we consider
• Exchange rates against the US Dollar of :
– Australian and Canadian Dollar (AUD and CAD,
respectively)
– the Euro (denoted as ECU/EUR because we combine data
on the ECU before the introduction of the Euro and on the
latter after its launch)
– Japanese Jen (JPY)
– British Pound (GPB)
– Swiss Franc (CHF)
• Our benchmark dataset uses currency futures prices
on these currencies, but we also consider spot rates
DCU Business School
We assume (hope) DGP can be represented as
simple ARMA(p,q) reduced-form model
• DGP:
rt 1  t  ut
t  E (rt | I t 1 )
• Reduced form model:
rt  t  ut  a  b1rt 1  ....  b p rt  p  c1ut 1  ....  cq ut 1  ut
ˆ t  aˆ  bˆ1rt 1  ....  bˆp rt  p  cˆ1uˆt 1  ....  cˆquˆt 1
DCU Business School
The estimated reduced-form models
AUD
CAD
JPY
Panel A
(p = 1, q = 2)
1,2
1,2
0.35
1.50
20.49
42.43
GBP
CHF
ECU/
EUR
p,q
R2
SR p.a.
1,2
0.98
34.29
1,2
1.56
43.27
1,2
2.00
48.99
1,2
2.52
54.99
p,q
R2
SR p.a.
Panel B
(AIC-based model selection)
5,2
3,2
4,4
2,2
5.81
4.10
6.06
3.29
83.50
70.14
85.28
62.83
1,1
2.57
55.53
2,4
3.43
64.16
DCU Business School
From estimated mean returns to Max
SRs strategies
• We can attain the max SR by using the estimated t
as a filter, constructing an inter-temporal portfolio
with ‘time weights’ wt
wt 
t
 2 (rt 1 )
w   1
DCU Business School
 02 (r1 )
0
0
 12 (r2 )
...
..
..
0
..
ET 1 (rT )
..
E0 (r1 )
E1 (r2 )
  E2 (r3 )

..
0
..
 22 (r3 ) 0
..
0
0
...
..
0  T21 (rT )
Max SR2 is an upper bound to R2
• R2 is approx. the squared max SR
 2 ( t )   / T  (  i / T ) 2
 
1







2
2
2
 (rt 1 )
 (rt 1 )
T (rt 1 )
• We also prove that the excess-return on an asset
satisfies the predictability bound for a given pricing
kernel iff that kernel prices the Max SR strategy
R   rt 1 , mt 1  mt 1  
2
DCU Business School
2
2
E (r mt 1 )  0
R2
i ,t
Max SR strategies show large alphas…
(1988-2006, t.c. = 2 bps)
Model
FF
FF+Bond
FF
FF+Bond
DCU Business School
AUD
CAD
JPY
GBP
CHF
ECU/
EUR
ravg
AFX
Carry
0.04
(1.37)
(0.086)
0.04
(1.23)
(0.110)
0.00
(0.63)
(0.263)
0.00
(0.32)
(0.375)
Panel A
(p = 1, q = 2)
0.05
0.05
(2.29)
(2.61)
(0.011) (0.004)
0.06
0.05
(2.80)
(2.07)
(0.003) (0.019)
0.08
(2.10)
(0.018)
0.08
(2.10)
(0.018)
0.15
(1.94)
(0.026)
0.15
(2.02)
(0.022)
0.05
(3.74)
(0.000)
0.05
(3.71)
(0.000)
2.67
(5.08)
(0.000)
2.69
(5.52)
(0.000)
5.48
(2.19)
(0.014)
5.85
(2.45)
(0.007)
Panel B
(AIC-based model selection)
1.59
2.33
1.65
1.44
(3.69)
(4.55)
(1.29)
(1.93)
(0.000) (0.000) (0.098) (0.027)
1.35
2.35
1.15
1.53
(2.83)
(4.65)
(1.12)
(2.15)
(0.002) (0.000) (0.131) (0.016)
2.17
(2.49)
(0.006)
2.26
(2.19)
(0.014)
69.16
(2.10)
(0.018)
69.81
(2.44)
(0.007)
12.61
(2.30)
(0.011)
12.64
(2.67)
(0.004)
2.67
(5.08)
(0.000)
2.69
(5.52)
(0.000)
5.48
(2.19)
(0.014)
5.85
(2.45)
(0.007)
…but not huge SRs
(1988-2006, t.c. = 2 bps)
Model
FF
FF+Bond
FF
FF+Bond
DCU Business School
AUD
CAD
JPY
GBP
Panel A
(p = 1, q = 2)
9.98
10.92
(-1.82) (-1.98)
(0.035) (0.024)
(-3.09) (-3.34)
(0.001) (0.000)
CHF
ECU/
EUR
AFX
Carry
12.40
(-1.91)
(0.029)
(-3.52)
(0.000)
10.10
(-1.66)
(0.049)
(-2.69)
(0.004)
13.19
(-1.30)
(0.098)
(-2.51)
(0.006)
14.60
(-1.57)
(0.059)
(-3.13)
(0.001)
10.54
(-1.15)
(0.125)
(-0.63)
(0.265)
21.86
(0.00)
(0.500)
(-0.15)
(0.440)
22.66
(0.13)
(0.447)
(-1.13)
(0.130)
Panel B
(AIC-based model selection)
20.21
28.13
11.77
11.39
(-0.21)
(0.98) (-1.79) (-1.43)
(0.416)
(0.165) (0.037) (0.077)
(-1.35)
(-0.35) (-1.23) (-1.85)
(0.089)
(0.363) (0.110) (0.033)
12.76
(0.00)
(0.499)
(0.00)
(0.500)
10.54
(-1.15)
(0.125)
(-0.63)
(0.265)
21.86
(0.00)
(0.500)
(-0.15)
(0.440)
Max SR strategies vs. F-F factors
(1988-2006, t.c. = 2 bps)
ri ,Rt   i   i ,m rm ,t   i , SMB SMBt   i , HML HMLt   i ,t
2
AUD
Excess-returns on predictabilitybased max SR strategies for each
currency, strategies that mimic the
trading action of a trader endowed
with RE whose aim is to generate
the largest attainable SR by
trading a currency at a time
CAD
JPY
GBP
CHF
ECU-EUR
DCU Business School
SR
alpha
Rm-Rf
(AIC-based model selection)
78.49
2.17
-1.12
*(5.91)
*(3.42)
(-0.83)
(0.38)
*(2.49)
(-1.44)
70.01
1.46
-1.43
*(4.68)
*(3.01)
(-1.37)
(-0.75)
*(3.91)
*(-1.72)
97.44
2.32
0.52
*(8.66)
*(4.18)
(0.44)
*(2.89)
*(4.51)
(0.50)
40.77
1.12
-0.97
(0.43)
(1.57)
(-0.63)
*(-4.63)
(1.32)
(-0.88)
39.46
1.57
1.81
(0.24)
*(2.00)
(1.08)
*(-4.80)
*(2.12)
(1.30)
44.20
69.16
111.22
(0.93)
*(1.69)
(1.29)
*(-4.17)
*(2.10)
(1.39)
SMB
-0.92
(-0.54)
(-0.85)
0.65
(0.50)
(0.46)
-1.21
(-0.81)
(-0.81)
2.09
(1.09)
(0.57)
-2.64
(-1.25)
(-1.24)
-126.09
(-1.16)
(-1.49)
HML
1.69
(1.09)
(1.41)
-2.00
*(-1.68)
*(-2.43)
-2.21
(-1.61)
*(-2.11)
0.00
(0.00)
(0.00)
-1.44
(-0.75)
(-1.21)
-30.89
(-0.32)
(-1.17)
Overall max SR strategy tracks well the AFX
index of Lequeux and Acar (EJF , 1998)
150%
100%
50%
0%
-50%
-100%
-150%
85
86
87
88
89
90
91
92
93
94
95
96
SR AFX
DCU Business School
97
98
SR r*
99
00
01
02
03
04
05
06
Overall max SR strategy vs. ‘carry trade
index’ of Lustig et al. (2007)
200%
150%
100%
50%
0%
-50%
-100%
-150%
85
86
87
88
89
90
91
92
93
94
95
96
SR HML
DCU Business School
97
98
SR r*
99
00
01
02
03
04
05
06
Excess-predictability relative to
• Meese and Rogoff (JIE, 1983): exchange rate
disconnect puzzle (‘Houston, we have a problem’)
• Engel and West (2004, 2005): disconnect not as bad
as you think, fundamentals do not need to forecast
exchange rates (‘Hang on Houston, problem solved!’)
• Lyons and Moore (JIMF, 2006), Brennan and Xia
(RFS, 2006): disconnect is bad if you compare
volatilities of fundamentals and exchange rates
(‘Sorry Houston, we still have a problem’)
DCU Business School
Lots of time-variation in excess-predictability
(net of sampling error)
50
50
50
40
40
40
30
30
30
20
20
10
10
20
10
0
0
-10
-10
-20
-20
-10
-30
-30
-20
-40
-40
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01
AUD - se 95%
0
-30
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01
12 per. Mov. Avg. (AUD - se 95%)
CAD - se 95%
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01
12 per. Mov. Avg. (CAD - se 95%)
JPY - se 95%
50
50
50
40
40
40
30
30
20
20
10
10
0
0
-10
-10
-20
-20
-30
30
20
10
0
-10
-30
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01
GBP - se 95%
12 per. Mov. Avg. (GBP - se 95%)
12 per. Mov. Avg. (JPY - se 95%)
-20
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01
CHF - se 95%
12 per. Mov. Avg. (CHF - se 95%)
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01
ECU/EUR - se 95%
12 per. Mov. Avg. (ECU/EUR - se 95%)
Note: Not as bad as it looks for EMH! Consecutive excess-predictability episodes are
rare, i.e. the market corrects within a few months at most.
DCU Business School
Stylized facts summary
• Evidence of statistically significant excesspredictability in terms of ‘alphas’ but not so much
SRs
– This is economically significant net of realistic
transaction costs, i.e. a RE investor would find it
attractive
– The marginal currency trader seeks SRs but leaves
‘alphas’ on the table
– Why?
DCU Business School
A ‘Limits to Speculation’ story?
• We conjecture ‘limits to speculation’ à la Lyon
• Gathering and exploiting available public AND private information
about exchange rate fundamentals requires specialization to exploit
economies of scale and scope
• It means that traders cannot take ‘marginal’ positions in currencies,
they must invest in ‘size’ and thus take on diversifiable risk as well
as undiversifiable risk (hence, the emphasis on the SR)
• These currency traders are likely capital-constrained due to
incomplete contracting and agency costs (hence, max SR should
vary inversely with the availability of risk capital)
DCU Business School
Does risk-capital matter?
•
Swamy’s (1970) ‘Random Coefficient’ panel regressions of the BVIs against their own
lags and three alternative sets of regressors (Panel A, B and C):
Const.
3.91
4.69
(0.403)
-1.42
0.95
(0.132)
-0.84
0.90
(0.350)
-1.26
0.74
(0.088)
Trend
-0.03
0.023
(0.172)
0.00
0.00
(0.553)
0.00
0.00
(0.853)
0.00
0.00
(0.571)
BVIt-1
0.19
0.035
(0.000)
0.24
0.02
(0.000)
0.25
0.02
(0.000)
0.25
0.02
(0.000)
DCU Business School
BVIt-2
-0.00
0.036
(0.973)
0.05
0.02
(0.078)
0.05
0.02
(0.042)
0.04
0.02
(0.052)
BVIt-3
0.01
0.036
(0.690)
0.08
0.02
(0.001)
0.08
0.02
(0.000)
0.08
0.02
(0.000)
BVIt-4
AUMt-60
sentot-60
|sentot-60|
rrel-60
TED-60
VIX-60
DW
0.04
0.035
(0.186)
Panel A
15.15
0.59
29.02
1.76
(0.601)
(0.737)
0.59
1.76
(0.737)
0.05
0.02
(0.052)
Panel B
-31.40
-1.14
11.34
0.54
(0.005)
(0.033)
0.18
0.74
(0.808)
1.98
0.05
0.02
(0.027)
Panel C
-28.21
11.22
(0.011)
-0.83
0.57
(0.149)
1.99
0.05
0.02
(0.031)
Panel D
-30.89
-1.03
11.29
0.40
(0.006)
(0.010)
-1.05
0.61
(0.087)
3.24
2.81
(0.250)
0.05
0.10
(0.625)
1.86
1.99
Conclusions
• Lots more ‘alpha’ for longer periods, evidence that
FX traders seek reward for total not systematic risk.
• Excess-predictability is declining but still present,
despite profitability of popular trading rules has
disappeared (?) for main currencies, e.g. Taylor
(2005).
– Markets more weak-form efficient but not more strong
form efficient? This needs further investigation.
• Most obvious patterns are cyclicality, in agreement
with Lo’s (JPM, 2004) AMH, and role of risk-capital.
DCU Business School
Extensions I (boring )
• Can we explain excess-predictability further?
– Economic cycle, microstructure issues and/or order-flow
(e.g. COT reports)?
– Central Banks? LeBaron (JIE, 1998) found that excessprofitability of MA trading rules disappears when Fed is
not active, what about more general forms of mispricing?
Opportunity to compare impact of Fed and ECB
DCU Business School
Extensions II (exciting )
• Lo’s (2004) AMH vs. Fama’s (1970) EMH:
– Need to formally test whether excess-predictability is
trending downwards, e.g. Neely, Weller and Ulrich (JFQA,
forthcoming)
– Can we match bursts of excess-predictability with regime
changes, e.g. along the lines of Killeen, Lyons and Moore
(2000)?
– Learning or reflexivity?
DCU Business School
Concluding thoughts: learning vs.
reflexivity
• What if traders do not recurringly re-learn to process information after a
regime change but instead every so often they ‘imagine’ new regimes?
• Financial media and think tanks as possible intermediaries of reflexive
influences between traders, economists and the economy
– It basically means that it is much harder to define such thinks as
“fundamentals”, “fair value”, “long tern economic value”, etc.
– Implications for ‘synchronization risk’ studied by HF literature
– Soros’ (2009) General Theory of Reflexivity (!) radical take
– In this setup, neither financial markets nor the economic profession and the media are
side shows
– Lots of scope for interdisciplinary discourse/research here
• A case of extreme ‘coordination failure’ and ‘non anchored expectations’
or is it ‘reality’ reminding us dismal scientists about what being humans is
all about? 
DCU Business School
Appendix
DCU Business School
Joint hypothesis problem
• The null is typically EMH + asset pricing/FX determination
model,
– i.e. H0 : ‘μt+1  E(rt+1|It) = kt+1’
• What about H0 : EMH + ‘|kt+1| ≤ boundk’?
– i.e., H0 : ‘|μt+1| ≤ boundk’?
• First, will recast ‘|μt+1| ≤ boundk’ as
R 2  bound R 2
DCU Business School
What we are not asking
• Not concerned with out-of-sample forecastability, we
don’t need it
– Cochrane (2005, p. 396), “‘excess volatility’ is exactly the
same thing as (abnormal) return predictability”
– The point is that RE investors know the DGP
• Why not look at out-of-sample forecastability too,
since we are at it?
– Because of our (of us econometricians) coarse information
set, that would be at best a test of weak-form EMH and
plenty has already been done on it, e.g. Neely et al. (2007)
DCU Business School
A curious finding! 
• The international finance/asset pricing disconnect
puzzle
– The connection between excess-predictability and possible
currency mispricing was well known to earlier researchers,
e.g. Obstfeld and Rogoff (2000).
– Recent work in the broad international finance domain,
with the emphasis placed on out-of-sample forecast when
making inferences about the EMH, seems less aware of this
connection.
DCU Business School
Some perspective
• Economy Max SR:
2
2
 (mt )  RRArepr.

(rm,t 1 )
inv.
• CAPM world (RRArepr. inv = 2.5):
 (mt )  (2.5) 2 (0.16) 2  2.5  0.16  40 % p.a.
• Higher-moment CAPM or ICAPM (RRArepr. inv = 5):
 (mt )  52 (0.16) 2  5  0.16  80 % p.a.
DCU Business School
Transaction costs
• Not all predictability is exploitable, not even by an
investor endowed with RE, due to transaction costs.
• Turns out that the maximal SR attainable by
exploiting predictability is approximately the
maximal R2 of predictive regressions
• We can assess implications of transaction costs by
looking at impact on SRs of max SR strategies.
– We find that, unlike daily predictability, monthly
predictability is ‘robust’ to transaction costs
DCU Business School
‘Time-weights’ of CAD max SR
strategies (95-06)
1.5%
1.0%
Daily:
0.5%
0.0%
(Strategy drowns
in transaction
costs)
-0.5%
-1.0%
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
150%
100%
Monthly:
(Transaction
costs not much
of a problem)
50%
0%
-50%
-100%
1996
DCU Business School
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Transaction costs of maximal SR
strategies (1995-2006)
Transaction
costs (bps)
AUD
CAD
JPY
GBP
CHF
ECU/EUR
AUD
CAD
JPY
GBP
CHF
ECU/EUR
DCU Business School
0
2
3
5
*57.3
**130.1
*49.1
*48.4
**104.3
*82.1
Daily
(1995-2006)
17.4
-2.3
*47.1
5.62
-11.0
-41.1
20.0
5.7
*47.7
19.6
22.7
-7.1
-41.7
-77.4
-101.2
-22.7
-36.7
-66.6
*81.5
*72.0
**106.4
**78.6
*56.0
*63.9
Monthly
(1995-2006)
*79.2
*78.0
*68.1
*66.1
**105.0 **104.4
*75.0
*73.3
*54.9
*54.4
*62.2
*61.3
*75.7
*62.0
**103.0
69.8
*53.4
*59.7
25
*51.9
20.3
**89.6
34.1
*43.7
*42.8
Bound
RRAV
= 2.5
Bound
RRAV
= 5.0
46.0
88.0
37.0
74.0
Sampling error estimate
• We compute a lower bound on the 5th percentile of the
distribution of estimated excess-predictability.
• This is done by subtracting, from the point estimate of
the latter, (an upper bound of) the 95th percentile of the
R2 distribution under OLS assumptions and the null of
no explanatory power:
(T  K  1) R 2 (T  K  1)
R

~ FK ,T  K 1
K
(1  R 2 ) K
K
 R52%  R 2 
F95%;K ,T  K 1
(T  K  1)
2
DCU Business School
Open Issue
• Little question: how do you capture very high order lag autocorrelation? (plot refers to serial correlation of CAD at lags 1-200)
– Fractionally integrated ARMA (ARFIMA), else?
1.00
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
0
DCU Business School
25
50
75
100
125
150
175