Tactical Asset Allocation 2
session 6
Andrei Simonov
Tactical Asset Allocation
1
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Agenda

Statistical properties of volatility.
– Persistence
– Clustering
– Fat tails


Is covariance matrix constant?
Predictive methodologies
– Macroecon variables
– Modelling volatility process: GARCH process and related
methodologies
– Volume
– Chaos

Skewness
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Volatility is persistent

Returns2 are
MORE
autocorrelated
than returns
themselves.
Volatility is
indeed persistent.
Akgiray, JB89
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It is persistent for different
holding periods and asset
classes
LAG
FT All Share
Daily r
Daily |r|
Weekly r
Weekly |r|
GB£/US$ Daily r
Daily |r|
1
0.193
0.347
2
0.020
0.292
3
0.031
0.256
4
0.045
0.260
5
0.018
0.222
10
0.070
0.225
0.086
0.255
0.144
0.192
0.021
0.195
0.076
0.210
-0.052
0.207
-0.037
0.137
-0.022
0.133
-0.003
0.120
-0.008
0.068
-0.006
0.052
0.024
0.114
0.002
0.076
Sources: Hsien JBES(1989), Taylor&Poon, JFB92
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Volatility Clustering, rt=ln(St/St-1).
US$/SEK ContCompRate
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
198702-02
198802-02
198902-02
199002-02
Tactical Asset Allocation
199102-02
199202-02
199302-02
199402-02
199502-02
199602-02
199702-02
199802-02
199902-02
200002-02
200102-02
5
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Volatility clustering
S&P100 ContComp Returns
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
feb87
feb88
feb89
feb90
Tactical Asset Allocation
feb91
feb92
feb93
feb94
feb95
feb96
feb97
feb98
feb99
feb00
feb01
6
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Kurtosis & Normal distribution
4

2


x j  x 
n(n  1)
3
(
n

1
)


 
KURTOSIS  


x)   (n  2)( n  3)
 (n  1)( n  2)( n  3)  std (Kurtosis


Kurtosis=
0 for
normal
dist. If it
is
positive,
there are
so-called
FAT
TAILS
3.46
3.5
2.65
3
2.5
1.96
2
1.93
1.68
1.5
1
0.5
Tactical Asset Allocation
0
US $TO UK £ NOON
NY
US $ TO SWEDISH
KRONA (JPM)
US $ TO SWISS
FRANC (JPM)
S&P 100
MSCI SWEDEN
7
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us
t
A ralia
us
Be tri
l a
Ca gium
D na
en da
m
Fi ark
n
Fr land
G an
H erm ce
on a
g ny
K
Ire ong
la
n
Ita d
N J ly
N ethe apa
ew rl n
Ze and
a s
N l and
o
Po rwa
rtu y
g
Sp al
Sw Sw ain
it z ede
er n
la
nd
U
K
W
U
or W S
ld or
ex ld
EAUS
FE
A
Higher Moments & Expected Returns
Average Excess Kurtosis in Developed Markets
6
5
4
3
2
1
0
-1
Data through June 2002
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rg
e
Banti n
hr a
Br a in
a
Chz il
Cz
i
e c Co Chi l e
l
h om na
Re b
pu ia
b
Eg l ic
G yp
H ree t
un c e
ga
In In ry
do di
ne a
Is sia
Jo rael
rd
M Ko an
ala re
a
M ysi a
M ex
or ic
o o
N cco
ig
er
Pa Om i a
ki an
Ph sta n
ili Pe
pp ru
i
Po nes
Sa R lan
ud u d
i A ssi
So Sl rab a
ut ov i a
h a
Sr Afr ki a
i L ica
Ta a nk
Th iw a
a an
Tuila n
V rk d
e
Zi nez e y
Comba uela
m bw
po e
sit
e
A
Higher Moments & Expected Returns
Average Excess Kurtosis in Emerging Markets
6
5
4
3
2
1
0
-1
Tactical Asset Allocation
Data through June 2002
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Extreme events
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Normal distribution:
Only 1 observation in 15800 should be
outside of 4 standard deviations band from
the mean.
 Historicaly observed:

– 1 in 293 for stock returns (S&P)
– 1 in 138 for metals
– 1 in 156 for agricultural futures
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What do we know about returns?
Returns are NOT predictable (martingale
property)
 Absolute value of returns and squared
returns are strongly serially correlated and
not iid.
 Kurtosis>0, thus,returns are not normally
distributed and have fat tails
 -’ve skewness is observed for asset returns

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ARCH(1)
volatility at time t is a function of volatility
at time t-1 and the square of of the
unexpected change of security price at t-1.
Ret(t)=f Ret(t-1)+et
e(t)= s(t) z(t)
s2(t)=a0+a1e2(t-1), z~N(0, 1)
 If volatility at t is high(low), volatility at
t+1 will be high(low) as well
 Greater a1 corresponds to more
persistency

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Simulating ARCH vs Normal
1
ARCH(4)
0.8
0.6
0.4
0.2
ARCH(1)
-0.2
493
481
469
457
445
433
421
409
397
385
373
361
349
337
325
313
301
289
277
265
253
241
229
217
205
193
181
169
157
145
133
121
109
97
85
73
61
49
37
25
13
1
0
Normal
-0.4
Tactical Asset Allocation
Normal
ARCH(1)
ARCH(4)
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GARCH=Generalized Autoregressive
Heteroskedasticity

volatility at time t is a function of volatility at time t-1
and the square of of the unexpected change of security
price at t-1.
s2(t)=a0+b1 s2(t-1)+ a1e2(t-1), e~N(0, s2t)

If volatility at t is high(low), volatility at t+1 will be
high(low) as well
The greater b, the more gradual the fluctuations of
volatility are over time
Greater a1 corresponds to more rapid changes in
volatility


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Conditional and Unconditional Swedish 360days T-Bill Volatility
1.4
Standard Deviation
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1992-02-10 1994-01-10 1995-12-11 1997-11-10 1999-10-11 2001-09-10
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S&P return volatility
0.16
0.14
0.12
0.10
Conditional
Unconditional
0.08
0.06
0.04
0.02
Tactical Asset Allocation
feb-01
feb-00
feb-99
feb-98
feb-97
feb-96
feb-95
feb-94
feb-93
feb-92
feb-91
feb-90
feb-89
feb-88
feb-87
feb-86
feb-85
feb-84
feb-83
feb-82
0.00
Moving Av
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Persistence




If (a1+b)>1, then the shock is persistent (i.e.,
they accumulate).
If (a1+b)<1, then the shock is transitory and
will decay over time
For S&P500 (a1+b)=0.841, then in 1 month only
0.8414=0.5 of volatility shock will remain, in 6
month only 0.01 will remain
Those estimates went down from 1980-es (in
1988 Chow estimated (a1+b)=0.986
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Forecasting power
GARCH forecast is far
better then other forecasts
 Difference is larger over
high volatility periods
 Still, all forecasts are not
very precise
(MAPE>30%)
 xGARCH industry

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Options’ implied volatilities



Are option implicit volatilities informative on
future realized volatilities? YES
If so, are they an unbiased estimate of future
volatilities? NO
Can they be beaten by statistical models of
volatility behavior (such as GARCH)? I.e. does
one provide information on top of the
information provided by the other?
– Lamoureux and Lastrapes:
ht = w + ae2t-1 + bh t-1 + gsimplied
– They find g significant.
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Which method is better?
(credit due: Poon & Granger, JEL 2003)
Number % of
of studies studies
HISTVOL> GARCH
GARCH> HISTVOL
22
17
56%
44%
HISVOL> ImpVol
ImpVol > HISVOL
8
26
24%
76%
GARCH> ImpVol
ImpVol > GARCH
1
17
6%
94%
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Straddles: a way to trade on
volatility forecast
Straddles delivers profit
if stock price is moving
outside the normal range
 If model predicts higher
volatility, buy straddle.
 If model predicts lower
volatility, sell straddle

Tactical Asset Allocation
Profit
X
ST
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Volatility and Trade

Lamoureux and Lastrapes: Putting volume in the
GARCH equiation, makes ARCH effects
disappear.
ht = w + ae2t-1 + bh t-1 + gVolume
GARCH11
GARCH11+Vol


Mean
Median
Mean
Median
Alfa1
0.102
0.066
0.057
0.037
Beta
0.626
0.716
0.016
0.000
Gamma
0.671
0.668
Beta+Alfa1
0.728
0.782
0.073
0.037
Heteroscedastisity is (at least, partially) due to
the information arrival and incorporation of this
information into prices.
Processing of information matters!
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What else matters? Macroeconomy
US Div Yield (t-1)
Unconditional
0.08
Conditional-"less than mean"
0.07
Conditional-"more than mean"
0.06
0.05
0.04
0.03
0.02
0.01
0
USA
AUSTRALIA
Tactical Asset Allocation
CANADA
GERMANY
JAPAN
U.K.
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Macroeconomic variables (2)
Volatility condit on US Ret(t-1) and US yield curve(t-1)
0.1
Unconditional
Conditional-"less than mean"
0.09
Conditional-"more than mean"
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
USA
AUSTRALIA
Tactical Asset Allocation
CANADA
GERMANY
JAPAN
U.K.
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Stock returns and the business cycle:
Expansion
Tactical Asset Allocation
USA
UK
Sw itzerland
Sw eden
Spain
Singapore
Norw ay
New Zealand
Netherlands
Japan
Italy
Ireland
Hong Kong
Germ any
France
Finland
Denm ark
Canada
Belgium
Austria
Australia
EAFE
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
AC World
Volatility
NBER Expansions and Contractions
January 1970-March 1997
Contraction
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Predicting Correlations (1)


Crucial for VaR
Crucial for Portfolio
Management
– Stock markets crash
together in 87 (Roll) and
again in 98...
– Correlations varies
widely with time, thus,
opportunities for
diversification (Harvey
et al., FAJ 94)
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Predicting Correlations (2)
Correlation M atrix
--- Conditioned on:
US Return (t-1)
&
US Yield Curve (t-1)
Use “usual
suspects” to
predict
correlations
 Simple
approach “upup” vs. “downdown”

Remove Black Monday ?
yes
USA
AUSTRALIA
CANADA GERMANYJAPAN
U.K.
1) Unconditional
0.40
0.69
0.31
0.24
0.47
2) Conditional-"less than mean"
0.61
0.79
0.37
0.35
0.56
3) Conditional-"more than mean"
0.19
0.60
0.28
0.18
0.26
Australia
CANADA GERMANYJAPAN
U.K.
1) Unconditional
0.55
0.24
0.27
0.44
2) Conditional-"less than mean"
0.65
0.40
0.40
0.63
3) Conditional-"more than mean"
0.46
0.06
0.21
0.41
CANADA
GERMANYJAPAN
U.K.
1) Unconditional
0.27
0.26
0.49
2) Conditional-"less than mean"
0.37
0.40
0.60
3) Conditional-"more than mean"
0.29
0.24
0.41
GERMANY
JAPAN
U.K.
1) Unconditional
0.36
0.40
2) Conditional-"less than mean"
0.44
0.53
3) Conditional-"more than mean"
0.33
0.26
JAPAN
U.K.
1) Unconditional
0.35
2) Conditional-"less than mean"
0.44
3) Conditional-"more than mean"
0.32
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Predicting Correlations (3)
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Chaos as alternative to
stochastic modeling





Chaos in deterministic non-linear dynamic system that
can produce random-looking results
Feedback systems, x(t)=f(x(t-1), x(t-2)...)
Critical levels: if x(t) exceeds x0, the system can start
behaving differently (line 1929, 1987, 1989, etc.)
The attractiveness of chaotic dynamics is in its ability
to generate large movements which appear to be
random with greater frequency than linear models
(Noah effect)
Long memory of the process (Joseph effect)
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Example: logistic eq.

X(t+1)=4ax(t)(1-x(t))
0.8
0.7
0.6
0.5
0.4
0.3
a=0.5
a=0.75
0.2
0.1
0
1
6
11
Tactical Asset Allocation
16
21
26
31
36
41
46
31
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1
0.9
0.8
0.7
0.6
A=0.9
0.5
0.4
0.3
0.2
0.1
0
1
1
6
11
16
21
26
31
36
41
46
1
6
11
16
21
26
31
36
41
46
0.9
0.8
0.7
0.6
A=0.95
0.5
0.4
0.3
0.2
0.1
0
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Hurst Exponent
Var(X(t)-X(0)) t2H
 H=1/2 corresponds to “normal” Brownian motion
 H<(>)1/2 – indicates negative (positive) correlations of
increments
 For financial markets (Jan 63-Dec89, monthly returns):
IBM
0.72
Coca-Cola
0.70
Texas State Utility
0.54
S&P500
0.78
MSCI UK
0.68
Japanese Yen
0.64
UK £
0.50

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Long Memory
Memory cannot last forever. Length of memory is finite.
 For financial markets (Jan 63-Dec89, monthly returns):
IBM
18 month
Coca-Cola
42
Texas State Utility
90
S&P500
48
MSCI UK
30
 Industries with high level of innovation have short cycle (but
high H)
 “Boring” industries have long cycle (but H close to 0.5)
 Cycle length matches the one for US industrial production
 Most of predictions of chaos models can be generated by
stochastic models. It is econometrically impossible to
distinguish between the two.

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Correlations and Volatility:






Predictable.
Important in asset management
Can be used in building dynamic trading strategy
(“vol trading”)
Correlation forecasting is of somewhat limited
importance in “classical TAA”, difference with
static returns is rather small.
Pecking order: expected returns, volatility,
everything else…
Good model: EGARCH with a lot of dummies
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Smile please!
Black- Scholes implied volatilities (01.04.92)
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us
t
A ralia
u
Be stri
lg a
Ca ium
D na
en da
m
Fi ark
nl
Fr and
G a
H erm nce
on a
g ny
K
Ire ong
la
n
Ita d
N J ly
N ethe apa
ew rl n
Ze and
a s
N l an
or d
Po wa
rtu y
g
Sp al
Sw Swe ain
it z de
er n
la
n
Ud
K
W
U
or W S
ld or
ex ld
EAUS
FE
A
Skewness & Expected Returns
Average Skewness in Developed Markets
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
Data through June 2002
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rg
e
Banti n
hr a
Br a in
a
Chz il
Cz
i
e c Co Chi l e
h lo na
Re m
pu bia
b
Eg l ic
G yp
H ree t
un c e
ga
In In ry
do di
ne a
Is sia
Jo rae
rd l
M Ko an
ala re
a
M ysi a
M ex
or ic o
o
N cco
ig
er
PaOm i a
ki an
Ph sta n
ili Pe
pp ru
i
Po nes
Sa
la
ud Ru nd
i A ss
So Sl rabi a
ut ov i a
h a
Sr Afr ki a
i L ica
Ta a nk
Th iw a
a i an
la
T
V ur nd
e
Zi nez ke y
Comba uela
m bw
po e
sit
e
A
Skewness & Expected Returns
Average Skewness in Emerging Markets
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
Data through June 2002
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Skewness or ”crash” premia (1)
Skewness premium =Price of calls at
strike 4% above forward price/ price of
puts at strike 4% below forward price 1
 The two diagrams following show:

That fears of crash exist mostly since the 1987
crash
 This shows also in the volume of transactions on
puts compared to calls

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Skewness or ”crash” premia (2)
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Skewness
Skewness
2
1
0
Variance
-1
-20
5
10
15
12.5
10
RF
Expected Return
7.5
5
See also movie from Cam Harvey web site.
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Where skewness is coming
from?
Log-normal distribution
 Behavioral preferences (non-equivalence
between gains and losses)
 Experiments: People like +’ve skewness
and hate negative skewness.

Tactical Asset Allocation
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Conditional Skewness, Bakshi, Harvey
and Siddique (2002)
For 1996
f 5skew
7. 00
5. 44
3. 89
2. 33
0. 78
- 0. 78
- 2. 33
- 3. 89
- 5. 44
- 7. 00
- 0. 55
0. 31
1. 16
2. 02
2. 88
3. 73
0. 000
4. 59
0. 053
0. 106
5. 45
0. 158
l ogsi ze 6. 31
0. 211
0.
264
7. 16
0. 317
0. 370
8. 02
0. 422book_m
kt
0. 475
8. 88
0. 528
9. 74
0. 581
0. 634
10. 59
0. 686
11. 45 0. 739
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What can explain skewness?


Stein-Hong-Chen:
imperfections of
the market cause
delays in
incorporation of
the information
into prices.
Measure of info
flows – turnover or
volume.
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Co-skewness
Describe the probability of the assets to
run-up or crash together.
 Examples: ”Asian flu” of 98,” crashes in
Eastern Europe after Russian Default.
 Can be partially explained by the flows.
 Important: Try to avoid assets with +’ve
co-skewness. Especially important for
hedge funds
 Difficult to measure.

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Three-Dimensional Analysis
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Alternative Vehicles
Alternate Asset Classes Often Involve Implicit or Explicit Options
7
6
5
4
3
2
1
0
-1
-2
-3
-4
S&P 500
Global Macro
1
2
3
4
Source: Naik (2002)
Tactical Asset Allocation
5
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Alternative Vehicles
Alternate Asset Classes Often Involve Implicit or Explicit Options
8
6
4
2
S&P 500
Trend Followers
0
-2
-4
-6
-8
1
2
3
4
Source: Naik (2002)
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5
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Alternative Vehicles
Alternate Asset Classes Often Involve Implicit or Explicit Options
7
6
5
4
3
2
1
0
-1
-2
-3
-4
S&P 500
FI Arb
1
2
3
4
5
Source: Naik (2002)
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Alternative Vehicles
Alternate Asset Classes Often Involve Implicit or Explicit Options
2
1.5
1
0.5
Delta(BAA-10yTBond)
x10
FI Arb
0
-0.5
-1
-1.5
-2
1
2
Source:
Naik (2002)
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4
5
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Alternative Vehicles
Alternate Asset Classes Often Involve Implicit or Explicit Options
.1
Panel B: PRAM Returns, 1990 - 1998
Risk Arb Return - Risk-free Rate
.08
.06
9002
9811
9602
9504
9706
9011
9410 9704
9107
9010
9004
9304
9607 9411
9407
9712
9310
9311
9312 9105
9308
9710
9402
9801
9804
9610
9006
9606
9101
9508
9201
9207
9604
9301
9408
9611 9005
9208
9306
9106 9807
9608
9303
9409 9702
9605
9507
9404
9509
9505
9812
9203
9412
9405
9506
9502
9806 9701
9112
9802
9210
9307
9708 9206
9501
9603
9512
9510
9205
9209
9202
9511
9612
9109
9108
9805 9104
9810
9003
9609 9705
9711
9110
9302
9103
9211
9305
9707
9403 9406
9212
9309
9401
9204
9503
9709 9102
9001 9703
9007
9601 9803
9111
.04
.02
0
-.02
9008
9012
9809
9808
-.04
-.06
9009
-.08
-.1
-.2
-.16
-.12
-.08
-.04
0
.04
.08
.12
.16
.2
Market Return minus Risk-free Rate
Tactical Asset Allocation
Source: Figure 5 from Mitchell & Pulvino (2000)
52
3/22/2016
Alternative Vehicles
Alternate Asset Classes Often Involve Implicit or Explicit Options
6
4
Event Driven Index Returns
2
0
-15
-10
-5
0
5
10
-2
-4
LOWESS fit
-6
Source: Naik (2002)
-8
Russell 3000 Index Returns
Tactical Asset Allocation
53
3/22/2016
Co-skewness for hedge funds
Co-Skewness Measure (Definition 2)
(Total of 42 Funds, over Jan 1997 - Feb 2001)
4
Mean Returns (Geometric)
3.5
3
2.5
2
1.5
1
0.5
0
-0.4
-0.3
-0.2
-0.1
-0.5 0
0.1
0.2
0.3
Coskew ness
Source: Lu and Mulvey (2001)
Tactical Asset Allocation
54
3/22/2016