5/23/2015 Tactical Asset Allocation 1 Tactical Asset Allocation 2 session 6 Andrei Simonov.

04/18/23Tactical Asset Allocation1

Tactical Asset Allocation 2Tactical Asset Allocation 2sessionsession 6 6

Andrei Simonov


AgendaAgenda 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


Volatility is persistentVolatility is persistent

Returns2 are MORE autocorrelated than returns themselves. Volatility is indeed persistent.

Akgiray, JB89


It is persistent for different It is persistent for different holding periods and asset holding periods and asset classesclasses

LAG 1 2 3 4 5 10FT All Share Daily r 0.193 0.020 0.031 0.045 0.018 0.070

Daily |r| 0.347 0.292 0.256 0.260 0.222 0.225

Weekly r 0.086 0.144 0.021 0.076 -0.052 -0.037Weekly |r| 0.255 0.192 0.195 0.210 0.207 0.137

GB£/US$ Daily r -0.022 -0.003 -0.008 -0.006 0.024 0.002Daily |r| 0.133 0.120 0.068 0.052 0.114 0.076

Sources: Hsien JBES(1989), Taylor&Poon, JFB92


Volatility Clustering, Volatility Clustering, rrtt=ln(S=ln(Stt/S/St-1t-1).). US$/SEK ContCompRate

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1987-02-02

1988-02-02

1989-02-02

1990-02-02

1991-02-02

1992-02-02

1993-02-02

1994-02-02

1995-02-02

1996-02-02

1997-02-02

1998-02-02

1999-02-02

2000-02-02

2001-02-02


Volatility clusteringVolatility clusteringS&P100 ContComp Returns

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

feb-87

feb-88

feb-89

feb-90

feb-91

feb-92

feb-93

feb-94

feb-95

feb-96

feb-97

feb-98

feb-99

feb-00

feb-01


Kurtosis & Normal distributionKurtosis & Normal distribution

Kurtosis=0 for normal dist. If it is positive, there are so-called FAT TAILS

)3)(2(

)1(3

)()3)(2)(1(

)1( 24

nn

n

xstd

xx

nnn

nnKURTOSIS j

1.68

1.96 1.93

3.46

2.65

0

0.5

1

1.5

2

2.5

3

3.5

US $TO UK £ NOONNY

US $ TO SWEDISHKRONA (JPM)

US $ TO SWISSFRANC (JPM)

S&P 100 MSCI SWEDEN

Kurtosis


Higher Moments & Expected ReturnsHigher Moments & Expected Returns

Data through June 2002

-1

0

1

2

3

4

5

6

Australi

a

Austria

Belg

ium

Canad

a

Den

mar

k

Finlan

d

France

Ger

man

y

Hong K

ong

Irelan

d It

aly

Japan

Nether

lands

New

Zea

land

Norway

Portugal

Spain

Swed

en

Switzer

land UK US

World

World

ex-U

S

EAFE

Average Excess Kurtosis in Developed Markets


Higher Moments & Expected ReturnsHigher Moments & Expected Returns


-1

0

1

2

3

4

5

6

Argen

tina

Bahrai

n

Brazil

Chile

China

Colombia

Czech

Rep

ublicEgy

pt

Greece

Hunga

ry

India

Indo

nesia

Israe

l

Jord

an

Korea

Mala

ysia

Mex

ico

Mor

occo

Nigeria

Oman

Pakist

an

Peru

Philipp

ines

Poland

Russia

Saudi

Arabia

Slovak

ia

South

Africa

Sri Lan

ka

Taiw

an

Thaila

nd

Turke

y

Venez

uela

Zimba

bwe

Compo

site

Average Excess Kurtosis in Emerging Markets


Extreme Extreme eventsevents




Normal distribution: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


What do we know about returns?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


ARCH(1)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)= Ret(t-1)+t

(t)= (t) z(t)2(t)=0+12(t-1), z)

If volatility at t is high(low), volatility at t+1 will be high(low) as well

Greater 1 corresponds to more persistency


Simulating ARCH vs NormalSimulating ARCH vs Normal

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1

13 25 37 49 61 73 85 97

109

121

133

145

157

169

181

193

205

217

229

241

253

265

277

289

301

313

325

337

349

361

373

385

397

409

421

433

445

457

469

481

493

Normal ARCH(1) ARCH(4)

Normal

ARCH(1)

ARCH(4)


GARCH=GARCH=Generalized Autoregressive Generalized Autoregressive HeteroskedasticityHeteroskedasticity

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.

2(t)=0+1 2(t-1)+ 12(t-1), 2t)

If volatility at t is high(low), volatility at t+1 will be high(low) as well

The greater , the more gradual the fluctuations of volatility are over time

Greater 1 corresponds to more rapid changes in volatility


Conditional and Unconditional Swedish 360days T-Bill Volatility

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1992-02-10 1994-01-10 1995-12-11 1997-11-10 1999-10-11 2001-09-10

Stan

dard

Dev

iation


S&P return volatility

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16feb

-82feb

-83feb

-84feb

-85feb

-86feb

-87feb

-88feb

-89feb

-90feb

-91feb

-92feb

-93feb

-94feb

-95feb

-96feb

-97feb

-98feb

-99feb

-00feb

-01

Conditional

Unconditional

Moving Av


PersistencePersistence

If (1+)>1, then the shock is persistent (i.e., they accumulate).

If (1+)<1, then the shock is transitory and will decay over time

For S&P500 (1+)=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 (1+)=0.986


Forecasting powerForecasting 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


Options’ implied volatilitiesOptions’ 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 = + 2t-1 + h t-1 + implied

– They find significant.


Which method is better?Which method is better?(credit due: Poon & Granger, JEL 2003)(credit due: Poon & Granger, JEL 2003)

Number of studies

% of studies

HISTVOL> GARCH 22 56%GARCH> HISTVOL 17 44%

HISVOL> ImpVol 8 24%ImpVol > HISVOL 26 76%

GARCH> ImpVol 1 6%ImpVol > GARCH 17 94%


Straddles: a way to trade on Straddles: a way to trade on volatility forecastvolatility forecastStraddles 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

ST

X

Profit


Volatility and Trade Volatility and Trade Lamoureux and Lastrapes: Putting volume in the

GARCH equiation, makes ARCH effects disappear.

ht = + 2t-1 + h t-1 + Volume

Heteroscedastisity is (at least, partially) due to the information arrival and incorporation of this information into prices.

Processing of information matters!

Alfa1 Beta Gamma Beta+Alfa1GARCH11 Mean 0.102 0.626 - 0.728

Median 0.066 0.716 - 0.782GARCH11+Vol Mean 0.057 0.016 0.671 0.073

Median 0.037 0.000 0.668 0.037


What else matters? MacroeconomyWhat else matters? Macroeconomy

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

USA AUSTRALIA CANADA GERMANY JAPAN U.K.

US Div Yield (t-1)

Unconditional

Conditional-"less than mean"

Conditional-"more than mean"


Macroeconomic variables (2)Macroeconomic variables (2)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

USA AUSTRALIA CANADA GERMANY JAPAN U.K.

Volatility condit on US Ret(t-1) and US yield curve(t-1)

Unconditional

Conditional-"less than mean"

Conditional-"more than mean"


Stock returns and the business cycle:Stock returns and the business cycle:VolatilityVolatility

NBER Expansions and ContractionsNBER Expansions and ContractionsJanuary 1970-March 1997January 1970-March 1997

00.050.1

0.150.2

0.250.3

0.350.4

0.45

AC

Wo

rld

EAFE

Au

str

alia

Au

str

ia

Be

lgiu

m

Can

ada

De

nm

ark

Fin

lan

d

Fran

ce

Ge

rman

y

Ho

ng

Ko

ng

Ire

lan

d

Ital

y

Jap

an

Ne

the

rlan

ds

Ne

w Z

eal

and

No

rway

Sin

gap

ore

Sp

ain

Sw

ed

en

Sw

itze

rlan

d

UK

US

A

Expansion Contraction


Predicting Correlations Predicting Correlations (1)(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)



Use “usual suspects” to predict correlations

Simple approach “up-up” vs. “down-down”

Correlation Matrix --- Conditioned on: US Return (t-1) &

US Yield Curve (t-1)

Remove Black Monday ?yes

USA AUSTRALIACANADA GERMANY JAPAN U.K. 1) Unconditional 0.40 0.69 0.31 0.24 0.472) Conditional-"less than mean" 0.61 0.79 0.37 0.35 0.563) Conditional-"more than mean" 0.19 0.60 0.28 0.18 0.26

Australia CANADA GERMANY JAPAN U.K. 1) Unconditional 0.55 0.24 0.27 0.442) Conditional-"less than mean" 0.65 0.40 0.40 0.633) Conditional-"more than mean" 0.46 0.06 0.21 0.41

CANADA GERMANY JAPAN U.K. 1) Unconditional 0.27 0.26 0.492) Conditional-"less than mean" 0.37 0.40 0.603) Conditional-"more than mean" 0.29 0.24 0.41

GERMANY JAPAN U.K. 1) Unconditional 0.36 0.402) Conditional-"less than mean" 0.44 0.533) Conditional-"more than mean" 0.33 0.26

JAPAN U.K. 1) Unconditional 0.352) Conditional-"less than mean" 0.443) Conditional-"more than mean" 0.32




Chaos as alternative to Chaos as alternative to stochastic modelingstochastic 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)


Example: logistic eq.Example: logistic eq.

X(t+1)=4ax(t)(1-x(t))

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 6 11 16 21 26 31 36 41 46

a=0.5 a=0.75


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 6 11 16 21 26 31 36 41 46

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 6 11 16 21 26 31 36 41 46

A=0.9

A=0.95


Hurst ExponentHurst 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 0.72Coca-Cola 0.70Texas State Utility 0.54S&P500 0.78MSCI UK 0.68Japanese Yen 0.64UK £ 0.50


Long MemoryLong Memory Memory cannot last forever. Length of memory is finite. For financial markets (Jan 63-Dec89, monthly returns):IBM 18 monthCoca-Cola 42Texas State Utility 90S&P500 48MSCI 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.


Correlations and Volatility: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


Smile please!Smile please!Black- Scholes implied volatilities (01.04.92)Black- Scholes implied volatilities (01.04.92)


Skewness & Expected ReturnsSkewness & Expected Returns

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Australi

a

Austria

Belg

ium

Canad

a

Den

mar

k

Finlan

d

France

Ger

man

y

Hong K

ong

Irelan

d It

aly

Japan

Nether

lands

New

Zea

land

Norway

Portugal

Spain

Swed

en

Switzer

land UK US

World

World

ex-U

S

EAFE

Average Skewness in Developed Markets



Skewness & Expected ReturnsSkewness & Expected Returns

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Argen

tina

Bahrai

n

Brazil

Chile

China

Colombia

Czech

Rep

ublicEgy

pt

Greece

Hunga

ry

India

Indo

nesia

Israe

l

Jord

an

Korea

Mala

ysia

Mex

ico

Mor

occo

Nigeria

Oman

Pakist

an

Peru

Philipp

ines

Poland

Russia

Saudi

Arabia

Slovak

ia

South

Africa

Sri Lan

ka

Taiw

an

Thaila

nd

Turke

y

Venez

uela

Zimba

bwe

Compo

site

Average Skewness in Emerging Markets



Skewness or ”crash” premia Skewness or ”crash” premia (1)(1)

Skewness premium =Price of calls at strike 4% above forward price/ price of puts at strike 4% below forward price1

The two diagrams following show:That fears of crash exist mostly since the 1987

crashThis shows also in the volume of transactions on

puts compared to calls


Skewness or ”crash” premia Skewness or ”crash” premia (2)(2)



SkewnessSkewness

05 10 15

Variance

- 2- 1012

Skewness

5

7.5

10

12.5

Expected Return

RF

05 10 15

Variance

5

7.5

10

12.5

Expected Return

See also movie from Cam Harvey web site.


Where skewness is coming Where skewness is coming from?from?Log-normal distributionBehavioral preferences (non-equivalence

between gains and losses)Experiments: People like +’ve skewness

and hate negative skewness.


Conditional Skewness, Bakshi, Harvey Conditional Skewness, Bakshi, Harvey

and Siddique (2002)and Siddique (2002) For 1996

0. 7390. 686

0. 6340. 581

0. 5280. 475

0. 4220. 370

0. 3170. 264

0. 2110. 158

0. 1060. 053

0. 000

book_mkt

11. 4510. 59

9. 748. 88

8. 027. 16

6. 315. 45

4. 593. 73

2. 882. 02

1. 160. 31

- 0. 55

l ogs i ze

f 5s kew

- 7. 00

- 5. 44

- 3. 89

- 2. 33

- 0. 78

0. 78

2. 33

3. 89

5. 44

7. 00


What can explain skewness?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.


Co-skewnessCo-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.


Three-Dimensional AnalysisThree-Dimensional Analysis


Alternative VehiclesAlternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

-4-3-2-101234567

1 2 3 4 5

S&P 500

Global Macro

Source: Naik (2002)




-8

-6

-4

-2

0

2

4

6

8

1 2 3 4 5

S&P 500

Trend Followers

Source: Naik (2002)




-4-3-2-101234567

1 2 3 4 5

S&P 500

FI Arb

Source: Naik (2002)




-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1 2 3 4 5

Delta(BAA-10yTBond)x10

FI Arb

Source: Naik (2002)




Panel B: PRAM Returns, 1990 - 1998

Ris

k A

rb R

etu

rn -

Ris

k-f

ree

Rat

e

Market Return minus Risk-free Rate

-.2 -.16 -.12 -.08 -.04 0 .04 .08 .12 .16 .2

-.1

-.08

-.06

-.04

-.02

0

.02

.04

.06

.08

.1

9808

9008

9001

9009

9607

9703

9106

9403

9111

9411

9708

97109004

94069805

9304

98079203

9402920892069409

93119010

9007

961295109109

9606900692019702930791049309

98019306

93029205

9508

9405

9804

9404960394129210

9002

9209

96109301951292029204

9410

9602

9110

9712

9212

9310

9501

9312

9003

9604

9504

9108

9503

96059303

9012

96019103

950697119305

9407

96089505

9401

980695099502

91059507

93089207

9211

9704

9511

9408

970691079101

9803

9701

9609

9709

9811

9812

9809

9011

9611

970598029810

91029707

90059112

Source: Figure 5 from Mitchell & Pulvino (2000)




-8

-6

-4

-2

0

2

4

6

-15 -10 -5 0 5 10

Russell 3000 Index Returns

Eve

nt D

riven

Inde

x R

etur

ns

LOWESS fit

Source: Naik (2002)


Co-skewness for hedge fundsCo-skewness for hedge funds

Co-Skewness Measure (Definition 2)(Total of 42 Funds, over Jan 1997 - Feb 2001)

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

Coskewness

Mea

n R

etu

rns

(Geo

met

ric)

Source: Lu and Mulvey (2001)

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5/23/2015 Tactical Asset Allocation 1 Tactical Asset Allocation 2 session 6 Andrei Simonov.

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