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## Tactical Asset Allocation 2 session 6

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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

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

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Volatility Clustering, rt=ln(St/St-1).

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Volatility clustering

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Kurtosis & Normal distribution

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

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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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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

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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 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.

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

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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

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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

- 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

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Macroeconomic variables (2)

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

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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)

- Use “usual suspects” to predict correlations
- Simple approach “up-up” vs. “down-down”

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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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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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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

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Skewness or ”crash” premia (2)

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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.

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Conditional Skewness, Bakshi, Harvey and Siddique (2002)

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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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Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

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Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

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Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

Tactical Asset Allocation

Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Naik (2002)

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Alternate Asset Classes Often Involve Implicit or Explicit Options

Source: Figure 5 from Mitchell & Pulvino (2000)

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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

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