Finance and the Future

Finance and the FutureIn this great future you cant forget your past

by David Pollard1Financial forecastingMany reasons for forecasting financial dataSpeculative tradingPuntersSpeculators who work on instinct apparently without a systematic methodRisk managementForecasting downside scenarios & probabilities Asset allocationModern Portfolio TheoryForecasts of asset prices & volatilityConstruction of diversified portfolios

2What price in 6 months time?3 And the answer is!You can even use Astrology for your predictions but what if you want to use Maths ? 470Comment aside about Prof. Kibbles (Higgs-Kibble Boson) ideas on theory selection: What works!4

ARMAGARCHAR modelMA modelMoving AverageAuto RegressiveGeneralised Auto Regressive Conditional Heteroscedasticity!5Time Series Modeling5History = time seriesPrice vs. Time or FX Rate vs. Time graphBenchmarkDaily, closing price / rate dataLook out forOther periodicity e.g. GASCI data are weeklyRegularity E.g. TTSE changed from thrice weekly to daily in 2008Practical pointData storage in DatabasesBeyond Excel spreadsheets

6What's predictable?Ultimately we want to forecast prices and volatilitiesShould we work with the price time-series directly?No! Statistics not usually stationaryConsider price returns insteadStatistics more likely to be stationary (and so tractable)Recall that price returns

7Time series modelsTime series models can produce sequences that look like return graphsGeneral form

f is a function of prior values of the observed returnPrevisibleFunction can also depend on other variables e.g. prior volatilities More about volatilities laterError term often assumed to be Normally Distributed with zero mean

Return at time tError / noise termFunction we can modelUnivariateonly!

8The Bell CurveMoving averagesMA series is the weighted sum of (prior) returns from some other series

Effectively it smooths the other seriesMA can be a filter of the other seriesWith appropriate weights wLet other series simply be prior errorsMA(p)

9correlationVariance is volatility () squaredIt measures average, squared deviations from the mean

The correlation coefficient is given by

The Correlation of an asset with itself = 110

Measures the extent to which deviations in 2 series match each otherCorrelation - visuallyUn-correlated series

Correlated series

11AutoregressionWhat if we looked at the correlation between one time-series and a second one that was simply a time shift of the first?

Auto-correlation!Auto-Correlation function (ACF)Correlation of X with X-1 is 1st auto-correlation coefficientCorrelation of X with X-2 is 2nd auto-correlation coefficient If auto-correlation is significant the series is said to be Autoregressive

12XX-1timeX-2Auto Correlation FunctionsNasdaq: ACF

UCL: ACF

13The future aint what is used to beYogi BerraIf X is correlated with X+1 then our history (X) tells us about our future (X+1) AR ModelsTime series equation for an Autoregressive process AR(q)

AR(1) example

AR(2) example (graphed below)

14

ClusteringReversion

Explain how both t-1 and t-2 terms contribute to reversion. Comment about restrictions on a1, a2 for boundedness14ARMA modelsAuto Regressive + Moving Average = ARMA

So ARMA(p,q) model equation

Will see a real life example in the case study that follows15

Auto regressive partMoving average partNoisemaths vs. man - wco case study West Indian Tobacco Company (WCO)Trinidadian equivalent of Demerara Tobacco Company (DTC)

ProcedureCompute and analyse daily returnsCompute Auto Correlation Function (ACF / PACF)Evidence of Auto Regressive behaviour?Choose an ARMA specificationFit the model only keep statistically significant termsUse (computer) simulation to produce a Forecast Fan16

17WCO: time series fitBig returns during GFC!Some auto-correlation?Return forecast mostly flat

18WCO: building a forecastFind paths of Median, Upper Decile (0.9) and Lower Decile (0.1)

WCO: 6 month forecast

19Median forecast 70.7Actual 70.1What about the Volatility?ExpectationTaking Expectation is equivalent to averagingVariance is Expectation of squared deviations

Conditional ExpectationIn a time series context what we know changes as time evolvesWhat is left as random (the error / noise term) also evolves so how we compute averages (expectations) also evolves in time20

20Time series varianceConsider our time series model equations

Then the conditional expectation of one step ahead returns

Which is what we used when forecastingSimilarly for conditional variance we have

21

Conditional variance of returns is determined by the noise / error termRemind audience that variance is volatility squared21financial Volatility: nasdaq

22ClusteringHeteroscedasticityNon-normal NoiseVolatility & return acfs

23Squared returnsReturnsPartial ACFGARCH!Generalised Auto-Regressive Conditional HeteroscedasticityInsightIntroduce an explicit volatility multiplier for the error / noise termThat (conditional) volatility will need to be heteroscedasticreflecting observed, empirical featuresUse an auto-regressive time series model for the conditional variance

GARCHRecall our time series model

Instead now use

24Robert EngleEconomics Nobel Prize 2003

GARCH: variance equation25Regression on squared returnsAuto-regression on previous conditional varianceSo for GARCH(1,1)

For GARCH(p,q) the variance equation generalises

Nasdaq: garch variance26

CrisisDateBlack MondayOct 1987Asian CrisisOct 1997LTCM/Russian CrisisAug 1998Dot-com BubbleApr 2000VIX is the fear gaugeARMA(1,1) meanGARCH(1,1) - varianceStudents t NoiseMondays are specialA pause for breathLet us review the path takenMoving Average (MA) modelsSmooth randomness revealing trendAutoregressive (AR) modelsCapture statistical relations between current and recent historyAutoregressive Moving Average (ARMA) modelsCombine AR and MA featuresCan produce convincing forecastsGeneralisd Autoregressive Conditional Heteroscedasticity (GARCH) modelsInclude volatility modellingWidely accepted volatility forecasting capabilities27QuizWhich time-series model uses the longest history?A) ARMA(1,2)

B) GARCH(2,2)

C) MA(2)

D) AR(3)28QuizWhich one of the following is not true of the Auto Correlation Function?A) Its value is always 1

B) Its value is always between -1 and +1

C) A value above (or below) the level of significance indicates auto-regression

D) It is an important tool in the analysis of time series data29QuizIn time series modeling what does the acronym GARCH mean?A) Growing auto regression for controlling homogeneity

B) Growing and regressing classical homeothapy

C) Generalised auto regression conditioned with heteroscedasticity

D) Generalised auto regressive conditional heteroscedasticity 30ClosePrediction is very difficult, especially about the futureNeils Bohr, Physicist Mathematical forecasting uses statistics to find links between past and future then builds models that capture those links The approach can be startlingly successful at times despite the fundamental impossibility of what is being attempted31Next week :Finance for the FutureToolsBooksTime Series Analysis, James Hamilton, 1994Time Series Models, Andrew Harvey, 1993Econometric Analysis, William H. Greene, 7th Ed., 2011SoftwareR (www.r-project.org)OxMetrics (www.oxmetrics.net)Mathematica (www.wolfram.com/mathematica)MatLab (www.mathworks.com)32End33No Woman No Cry (sample)Bob Marley & The WailersNatty Dread, track 2/10, disc 1/11974Reggae10501.216eng - iTunPGAP0eng - iTunNORM 00000679 000006AB 00005ABC 000061B1 0000132F 0000132F 00008552 00008358 00002249 000020A7eng - iTunSMPB 00000000 00000210 00000B30 00000000000703C0 00000000 00032415 00000000 00000000 00000000 00000000 00000000 0000000012 Etudes, Op. 10: No. 11 in E FlatMaurizio PolliniMaurizio PolliniFrdric ChopinChopin: Etudes, Opp. 10 & 2512 Etudes, Op. 101985-01-09T08:00:00Zpollard.byfrons@btinternet.com 1984 Deutsche Grammophon GmbH, Hamburg2009-10-03 13:00:44