econometrics forecasting financial markets-chaos theory
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Econometrics Forecasting Financial Markets-Chaos Theory - Presentation-pptTRANSCRIPT
Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 1
Forecasting Financial Markets
Chaos Theory
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OverviewFractals
Self-similarityFractal Market HypothesisLong Term Memory Processes
Rescale Range Analysis Biased Random walkHurst Exponent
CyclesPhase SpaceChaos & the Capital Markets
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The Chaos GamePlot point P at randomRoll diceProceed halfway from P to point labeled with rolled number & plot new pointRepeat 10,000 times
P
A (1, 2)
B (3, 4)C (5, 6)
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The Sierpinksi Triangle
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ChaosLocal randomness, global determinism
Apparently random process may contain deterministic pattern
Stable, self-similar structureSierpinksi Triangle
Plot order impossible to predictBut odds of plotting each point are not equal• Empty spaces in each triangle have zero probability• Local randomness does not equate to independence
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Characteristics of FractalsSelf-similarity
The part is similar to the whole• Precise similarity in case of Sierpinski triangle
Scale InvarianceSub-parts not to same scale as parent
DimensionEuclidean space features integer dimensionsFractals occupy fractional dimension• E.g dimension of Sierpinski triangle is more than a
line but less than a plane (1 < d < 2)
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Fractal Time Series
Dimension measures how “jagged” series isStraight line has fractal dimension of 1Random time series has fractal dimension of 1.5• 50% chance of rising or falling
A line can have fractal dimension between 1 and 2At values ≠ 1.5 series is less or more jagged than a random series• Non-Gaussian
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Non-Gaussian Properties of Financial Markets
Distribution of ReturnsHigher peak at mean than NormalFatter tails• Uniformly fatter
– As many observations at 4σ away from mean as at 2σ
Markets tend to stay still or make major moves more often than theory predicts• Reflected in option volatility smiles
Term Structure of VolatilityScales at faster rate than T1/2
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Example: Returns on the DJIADow Jones Industrials Returns
0.00
0.10
0.20
0.30
0.40
0.50
0.60
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Standard Deviations
Normal
30 Day Returns
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Volatility Term StructureDJIA Volatility Term structure 1888-1900
1000 Days
-2.5
-2.0
-1.5
-1.0
-0.5
0.00.0 1.0 2.0 3.0 4.0 5.0
Log(Days)
Log(
SD)
Actual
Theoretical
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Regression Analysis of Term Structure
Chart indicates clear breakdown in volatility term structure after n = 1,000 daysRegression analysis confirms this:
Days < 1,000
Days > 1,000
Intercept -1.95 -1.38Slope 0.53 0.31R2 99.3% 47.2%F 1823.0 5.4p 0.00% 5.99%
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Conclusion Re term Structure
Risk-ReturnRiskier to invest for period < 4 yearsIncreasingly less risk incurred beyond 4 yearsTied to business cycle?
Sharpe RatioGets larger for longer time horizons
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Sharpe Ratio & Time Horizon
DJIA Sharpe Ratio and Time Horizon
y = 9E-08x2 - 0.0001x + 1.1433R2 = 0.7465
0.00.51.01.52.02.53.03.54.04.55.0
0 1000 2000 3000 4000 5000 6000 7000
Days
Sha
rpe
Ratio
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Fractal Theory of MarketsStable markets
Investors with many investment horizons• Ensures liquidity
Information set depends on investment horizonShort term: market sentiment & technical factorsLong term: fundamental analysis
Unstable marketsOccur when LT traders exit market or trade ST
Prices set by combination of ST & LT valuationST trends are noise. LT trends tied to economic cycles
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Rescaled Range AnalysisDeveloped by H.E. Hurst 1950’sBrownian Motion
Distance traveled R ∝ T0.5
Hurst Exponent(R/S)T = cTH
• H is the Hurst Exponent• c is a constant• T is # observations• (R/S)T is the rescaled range, a standardized measure
of distance traveled• Note for random time series H = 0.5
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Hurst Exponent & Market Behavior
H measures persistenceCorrelation C = 2(2H-1) - 1White Noise: H = 0.5, C = 0Black Noise: 0.5 < H < 1 , 0 < C < 1
Persistent, trend reinforcing series“Long memory”
Pink Noise: 0 < H < 0.5, C < 0Antipersistent, mean-revertingChoppier, more volatile than random series
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White Noise ProcessFractal Random Walk
-140
-120
-100
-80
-60
-40
-20
0
20
H = 0.5
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Black Noise ProcessFractal Random Walk
-600
-500
-400
-300
-200
-100
0
100
H = 0.9• Smoother series• Trend
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Fractal Random Walk
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
Pink Noise Process
H = 0.1• More volatile• Antipersistent
– Mean reverting
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Simulating a Fractal Random WalkFeder (1988):
• Ei is a strict white noise process, No(0, 1)• M is the number of periods for which long memory
is generated• n is set to 5• t is set to 1• H is Hurst exponent
[ ]⎭⎬⎫
⎩⎨⎧
−++×⎟⎟⎠
⎞⎜⎜⎝
⎛+Γ
=∆ ∑ ∑=
−
=−+−+
−−−++
−− nt
i
Mn
iitMn
HHiMn
HH
H EiinEiHnty
1
)1(
1))1(1(
)5.0()5.0())1(1(
)5.0( )()5.0(
)(
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Calculating (R/S)Form series of returns
rt = Ln(Pt / Pt -1) for t = 1, 2, . . . , TDivide into A contiguous sub-periods
Length n, such that An = TCompute average for each sub-periodForm cumulative series
Define range Ra = Max(Xk,a) - Min(Xk,a)
∑=
=n
kaka rr
1
( )∑=
−=k
iaiaka rrX
1
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Calculating (R/S)Compute standard deviation for each subperiod
Calculate average R/S for each n
Use OLS Regression to Estimate HLn(R/S)n = Ln(c) + H Ln(n)
( )2/12
1
1⎥⎥⎦
⎤
⎢⎢⎣
⎡−= ∑
=
n
kakaa rr
nS
∑=
=A
aaan SR
ASR
1)/(1)/(
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Example: RS WorksheetGenerate random sequence using RAND() fn.
Periods of length n = 10, 20, . . ., 100Calculate mean and SD for each sub-periodForm cumulative series Calculate R, R/S and Ln(R/S) for each sub-periodRepeat 10 timesPlot Ln(n) against average Ln(R/S)
Fit linear trend OLS slope estimate = H
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Example: RS AnalysisRescaled Range Analysis
y = 0.507x + 0.0572R2 = 0.9619
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-1.0 0.0 1.0 2.0 3.0 4.0 5.0
Ln(N)
Ln(R
/S)
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Hurst Exponent for Random Series1,000 Simulations
Simulated Values of the Hurst Exponenent
0
50
100
150
200
250
300
350
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
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Testing (R/S): E(R/Sn)
Anis & Lloyd (1976)
For large n (>350) Use Stirling Function
∑−
=
−Γ
−Γ=
1
1)5.0()]1(5.0[)/(
n
in i
inn
nSREπ
∑−
=
− −⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −
=1
1
5.0
25.0)/(
n
in i
innn
nSRE π
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Testing (R/S)
E(H) = Ln(E[R/Sn]) / Ln(n)Var(H) = 1 / T
If underlying process is random Gaussian, H will be Normally distributed
V-StatisticRecall (R / Sn) = cnH
V-Statistic• Divide by √n• V(n) = (R / Sn) / √n = cn(H-0.5)
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V-Statistic
V-Statistic V(n) = (R / Sn) / √n = cn(H-0.5)
For Persistent Series H > 0.5V(n) is increasing fn. of n
For Random Process H = 0.5V(n) is constant
For Antipersistent process H < 0.5V(n) is declining fn. of n
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V Statistic for E(R/Sn)V Statistic for E(R/Sn)
0.8
0.9
0.9
1.0
1.0
1.1
1.1
1.2
1.2
1.3
1.3
2.0 3.0 4.0 5.0 6.0 7.0 8.0Log(n)
V n =
E(R
/Sn)
/ n0.
5
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V-Statistic for Random SeriesV-Statistic
0.0
0.5
1.0
1.5
2.0
2.0 3.0 4.0 5.0
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Example: Long memory Process in DJIA (20-day Returns)
R/S Analysis DJIA (20 Day Returns)
y = 0.6119x - 0.148R2 = 0.9898
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.0 0.5 1.0 1.5 2.0 2.5 3.0Log(n)
Log(
R/S
)
DowE(R/S)Linear (Dow)
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Example: Long memory Process in DJIA (20-day Returns)
V-Statistic
n=52
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
0 100 200 300 400 500 600 700n
V
DowE(R/S)
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Dow R/S Regression AnalysisEstimated Hurst Exponent
H = 0.62 • OLS estimate of regression slop coefficient
Indicates fractal persistent memory process10 < n < 52
H = 0.71, R2 = 99.9%• Highly persistent Hurst process
52 < n < 650H = 0.49, R2 = 99.8%• White noise process
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Dow: Conclusions
R/S analysis indicates long memory processAverage cycle length approx 4 years
Tied to economic cycleEvents occurring today affected by events up to 4 years ago
Long memory effects dissipated after 4 years
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R/S Analysis of Stocks
Hurst CycleExponent (months)
IBM 0.72 18Xerox 0.73 18Apple 0.75 18Coca-Cola 0.70 42McDonald’s 0.65 42Con Edison 0.68 90
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R/S Analysis: Conclusions About Stock
Innovative, high growth firmsHave high H and short cycles
Stable, low growth firmsHave low H and long cycles
Implications for riskHigh H firms are less risky
• Less noise in series• Contradicts standard theory
What about diversification?• Dow index has one of the highest H exponents
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Lab: R/S Analysis of S&P 500 IndexMonthly returns April 75 to Feb 99
Detrended to remove short term memory effectsCalculate
RS, E(R/S), v-statistic (actual and expected)Plot
Ln(n) vs. Ln(R/S)N vs. V-statistic
EstimateCycle lengthHurst exponents pre and post cycle
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Lab: R/S Analysis of S&P 500 IndexMonthly S&P500 Index Returns Apr 75 - Feb 99
-25.00%
-20.00%
-15.00%
-10.00%
-5.00%
0.00%
5.00%
10.00%
15.00%
Apr-
75
Apr-
77
Apr-
79
Apr-
81
Apr-
83
Apr-
85
Apr-
87
Apr-
89
Apr-
91
Apr-
93
Apr-
95
Apr-
97
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Solution: R/S Analysis of S&P 500 Index - Ln(R/S) Plot
R/S Analysis S&P500 Monthly Returns
y = 0.5153x - 0.0637R2 = 0.9692
1.01.21.41.61.82.02.22.42.62.8
2.0 3.0 4.0 5.0Ln(n)
Ln(R
/S)
SP500
E(R/S)
Linear (SP500)
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Solution: R/S Analysis of S&P 500 Index - V-Statistic
V-Statistic
0.65
0.75
0.85
0.95
1.05
1.15
1.25
0 20 40 60 80 100 120 140n
V
SP500E(R/S)
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SP500 Regression Analysis
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Economic Indicators
Industrial Production IndexH = 0.79• Based on monthly data, Jan 1946 - Jan 1999• Strongly persistent
Cycle 42 months• Shorter than 4-year cycle accepted by economists
Ties in with S&P 500 Index
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R/S Analysis: CurrenciesTrue Hurst process: no cycle length
R/S continues scaling at rate H indefinitely with nInfinite memory processNot tied to economic cycles No “fundamental” valuation of currencyLess persistent, more volatile than stocks
H Exponents (based on daily data) • Yen: H = 0.64• GBP: H = 0.63• DM: H = 0.62
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R/S Analysis: Other Financial MarketsTreasury Bonds
H = 0.68• Based on daily yields, Jan 1950-Dec 1989
Cycle length 5 yearsGold
Some evidence of 4-year cycleH = 0.58, but not significant
VolatilityA true pink noise antipersistent processH = 0.31 for S&P 500 Index vol. (realized)
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Chaos Theory & Financial Markets -Summary
R/S Analysis confirms aperiodic cycles in many series
Stocks, stock indices, bonds, economic indicatorsCycle length related to economic cycle
Currencies are true Hurst processScale indefinitely
Volatility is only known antipersistent financial time series (apart from Wheat futures!)
Mean-reverting?