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Tactical Asset Allocation 2 session 6. Andrei Simonov. 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

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tactical asset allocation 2 session 6

Tactical Asset Allocation 2session 6

Andrei Simonov

Tactical Asset Allocation

agenda
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

Tactical Asset Allocation

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

Akgiray, JB89

Tactical Asset Allocation

it is persistent for different holding periods and asset classes
It is persistent for different holding periods and asset classes

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

Tactical Asset Allocation

volatility clustering
Volatility clustering

Tactical Asset Allocation

kurtosis normal distribution
Kurtosis & Normal distribution
  • Kurtosis=0 for normal dist. If it is positive, there are so-called FAT TAILS

Tactical Asset Allocation

higher moments expected returns
Higher Moments & Expected Returns

Data through June 2002

Tactical Asset Allocation

higher moments expected returns9
Higher Moments & Expected Returns

Data through June 2002

Tactical Asset Allocation

extreme events
Extreme events

Tactical Asset Allocation

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

Tactical Asset Allocation

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

Tactical Asset Allocation

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)=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

Tactical Asset Allocation

simulating arch vs normal
Simulating ARCH vs Normal

ARCH(4)

ARCH(1)

Normal

Tactical Asset Allocation

garch generalized autoregressive heteroskedasticity
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+b1s2(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

Tactical Asset Allocation

persistence
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

Tactical Asset Allocation

forecasting power
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

Tactical Asset Allocation

options implied volatilities
Options’ implied volatilities
  • Are option implicit volatilities informative onfuture realized volatilities? YES
  • If so, are they an unbiased estimate of futurevolatilities? NO
  • Can they be beaten by statistical models ofvolatility behavior (such as GARCH)? I.e. doesone provide information on top of theinformationprovided by the other?
    • Lamoureux and Lastrapes:

ht= w + ae2t-1 + bh t-1+ gsimplied

    • They find gsignificant.

Tactical Asset Allocation

straddles a way to trade on volatility forecast
Straddles: a way to trade on volatility forecast

Profit

  • 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

X

ST

Tactical Asset Allocation

volatility and trade
Volatility and Trade
  • Lamoureux and Lastrapes: Putting volume in the GARCH equiation, makes ARCH effects disappear.

ht= w + ae2t-1 + bh t-1+ gVolume

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

Tactical Asset Allocation

what else matters macroeconomy
What else matters? Macroeconomy

Tactical Asset Allocation

macroeconomic variables 2
Macroeconomic variables (2)

Tactical Asset Allocation

slide26
Stock returns and the business cycle:VolatilityNBER Expansions and ContractionsJanuary 1970-March 1997

Tactical Asset Allocation

predicting correlations 1
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)

Tactical Asset Allocation

predicting correlations 2
Predicting Correlations (2)
  • Use “usual suspects” to predict correlations
  • Simple approach “up-up” vs. “down-down”

Tactical Asset Allocation

predicting correlations 3
Predicting Correlations (3)

Tactical Asset Allocation

chaos as alternative to stochastic modeling
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)

Tactical Asset Allocation

example logistic eq
Example: logistic eq.
  • X(t+1)=4ax(t)(1-x(t))

Tactical Asset Allocation

slide32

A=0.9

A=0.95

Tactical Asset Allocation

hurst exponent
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

Tactical Asset Allocation

long memory
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.

Tactical Asset Allocation

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

Tactical Asset Allocation

skewness expected returns
Skewness & Expected Returns

Data through June 2002

Tactical Asset Allocation

skewness expected returns38
Skewness & Expected Returns

Data through June 2002

Tactical Asset Allocation

skewness or crash premia 1
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 onputs compared to calls

Tactical Asset Allocation

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

Tactical Asset Allocation

slide42

Skewness

See also movie from Cam Harvey web site.

Tactical Asset Allocation

where skewness is coming from
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

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.

Tactical Asset Allocation

co skewness
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.

Tactical Asset Allocation

three dimensional analysis
Three-Dimensional Analysis

Tactical Asset Allocation

slide48

Alternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

Tactical Asset Allocation

slide49

Alternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

Tactical Asset Allocation

slide50

Alternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

Tactical Asset Allocation

slide51

Alternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

Tactical Asset Allocation

slide52

Alternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Figure 5 from Mitchell & Pulvino (2000)

Tactical Asset Allocation

slide53

Alternative Vehicles

Alternate Asset Classes Often Involve Implicit or Explicit Options

6

4

2

0

-15

-10

-5

0

5

10

Event Driven Index Returns

-2

-4

LOWESS fit

-6

Source: Naik (2002)

-8

Russell 3000 Index Returns

Tactical Asset Allocation

slide54

Co-skewness for hedge funds

Source: Lu and Mulvey (2001)

Tactical Asset Allocation