Portfolio Selection using Intraday Data

Peter ChristoffersenRotman School of Management, University of Toronto,

Copenhagen Business School, andCREATES, University of Aarhus

11st Lectureon Friday

Overview: Sample Applications ofIntraday Data for Daily Portfolio Mgmt.• 1) Portfolio allocation with realized volatility and

covariance. Fleming, Kirby and Oestdiek (JFE,2002).

• 2) Realized beta. Patton and Verardo (RFS, 2013)• 3) Cross-sectional asset pricing with realized

volatility, skewness and kurtosis. ACJV (WP, 2011)

• Think about other applications of intraday data inrisk management

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First Application (FKO, 2002):Portfolio Allocation with RV and RCov

• S&P500, Treasury Bond, and Gold futurescontracts. 1984-2000. Remainder in cash.

• 5 minute returns. Linear interpolation. Biascorrections of RV and Rcov.

• Min portfolio vol subject to target expectedreturn gives weights

• Realized daily quadratic utility

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Moment Estimation

• Assume the mean is constant over time. Noestimation risk: ex-post mean return is known.Alternative: bootstrapping returns imposing

• Covariance matrix from exponential smoothing ofV=RCov,

• Bias corrections to correct for asynchronicity etc.

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Dynamic Daily and Realized outcomes versus mean-variancefrontier based on ex-post optimal static allocations

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Discussion

• Redo FKO using current state-of-the-art RCovtechniques (ABCD survey).

• Asyncronicity issues are key. Barndorff-Nielsen,Lunde, Hansen, Shephard (JEconm, 2011).

• Allow for non-normal distribution?• CRRA versus quadratic preferences• Non-myopic investors• How valuable is RV for longer horizons?• See Hautch, Kyj and Malec (SSRN, 2011) for a

recent portfolio application.13

Second Application: Realized BetaPatton and Verardo (RFS, 2013)

• Define realized beta (see ABD and Wu (AER,2005)) as

• And define integrated beta as

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Distribution of Realized Beta

• Barndorff-Nielsen and Shephard (2004) showthat

• From this we have that

• So that

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Inference on Realized Beta

• The previous result can be used to motivaterunning regressions on realized beta to try tocapture the dynamics in integrated beta.

• Patton and Verardo consider changes in betasaround earnings announcements

• Where I is an indicator for an announcement dayand D is a year dummy

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Earnings Surprise

• Use analyst forecasts to construct expectedearnings which are used to define earningssurprises by

• And forecast dispersions

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Event Time Beta

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Event Beta by Earnings Surprise

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Event Beta by Forecast Dispersion

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Summary of Results

• Beta increases on days of earningsannouncements and revert 2-5 days later.

• Beta increases more for large positive ornegative earnings surprises.

• Beta increases more for announcements thatresolve greater uncertainty.

• Beta increases more for more liquid and morevisible stocks.

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Third Application:Do Realized Skewness and Kurtosis

Predict the Cross-Section ofEquity Returns?

Diego AmayaPeter Christoffersen

Kris JacobsAurelio Vasquez

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Yes!• This week’s realized skewness and kurtosis

predicts next week’s stock return in the cross-section.– We find a strong negative cross sectional relationship

between this week’s realized skewness and nextweek’s returns.

– We find a positive relationship between this week’srealized kurtosis and next week’s return.

• We do not find a robust bivariate relationshipbetween return and realized volatility, BUT:– Skewness and volatility interact to form interesting

risk-return relationships conditional on the level ofskewness.

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Harnessing Big Data is the Big Idea• Skewness and kurtosis are difficult to estimate

reliably from low-frequency returns.• We use key insights in high-frequency financial

econometrics to help us learn about lowfrequency returns in the cross-section of equities.

• We use more than two million firm-weekobservations.

• Each firm-week realized moment is computedfrom roughly four hundred 5-minute returnsobserved during market open in the week.

• Upshot: Intraday returns make weekly momentsvirtually observable.

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Data

• Every listed stock in TAQ from January 4, 1993to September 30, 2008.

• NYSE, American Stock Exchange, NASDAQ,SmallCap.

• Five minute log returns from 9:30 to 4:00 EST.• We require a minimum of 80 daily

transactions in each stock on each day. Oursubsequent results robust to using 100, 250and 500 transaction minimum.

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Daily Firm-Specific Sample Moments

• Realized Daily Variance

• Realized Daily Skewness

• Realized Daily Kurtosis

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Weekly Firm-Specific Moments

• Our cross-sectional asset pricing analysis isdone at the weekly frequency (daily returnsare noisy) so we construct weekly samplemoments

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Distribution of RVol(two million firm-week observations)

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Weekly RVol averaged Across Firms Moving 3-month Average ofPercentiles across firms

Distribution of RSkew

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Weekly RSkew averaged Across Firms Moving 3-month Average ofPercentiles across firms

Distribution of RKurt

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Weekly RKurt averaged Across Firms Moving 3-month Average ofPercentiles across firms

RVol and the Cross Sectionof Stock Returns

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No action here…

RSkew and the Cross Sectionof Stock Returns

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Very large premium on negative skewness

RKurt and the Cross Sectionof Stock Returns

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Large positive premium on kurtosis

Fama and McBeth (1973) Regressions

• Each week t we estimate the following cross-sectional regression across firms, i,

and we then report the time series averagesof the coefficients

• Z is a vector of firm characteristics and othercontrol variables

• Note that returns are one-week-ahead34

Fama / McBeth Results

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Regression (5) includes: Book to market, beta, historical skewness,idiosyncratic vol, co-skewness, max monthly return, number ofanalysts, illiquidity, number of intraday transactions.

Skewness and Volatility Interaction

• The weak relationship between average returnand RVol is surprising.

• But because skewness varies across firms wewant to see if the traditional risk-returnrelationship varies by skewness level

• It does.

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A Range of Robustness Checks• Subtract return drift• Subsamples• Alternative Moment Measures

– Quantile based moments (Bowley, 1920, Moors, 1988)– Average RV Style Estimators from ZMA (2005)

• Other firm characteristics– Double sorts on skewness and size, etc

• Monthly returns• Alternative skewness measures

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First sorton otherfactor andthen onskewness.Then checkskewnesspremium ineachquintile ofthe otherfactor

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Toy SVJ Price Process

• Forget about the cross-sectional properties andassume that the log price of a stock follows anaffine SVJ jump diffusion

• The two BMs are correlated. The jumps arePoisson with constant intensity.

• We derive the first four realized moments as thesampling frequency gets infinitely high.

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Continuous Time Limits under SVJ• Consider the realized moment estimators

• Taking limits

• Gives (for j=2,3,4):

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Adding Market Microstructure Noise

• Assume that we observe

• Where u is i.i.d. nomal noise with zero mean.• Parameterize as in Ait-Sahalia and Yu (2009)• Decimalization in 2001. In second simulation

we assume prices are only observed in $1/16increments.

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RM(2) Signature Plot

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The sampling frequency is in seconds.

RM(3) Signature Plot

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The sampling frequency is in seconds.

RM(4) Signature Plot

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The sampling frequency is in seconds.

Returns on High Minus Low Skewnessusing Jump Robust RV Measures

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Summary• This week’s realized skewness and kurtosis predicts

next week’s stock return in the cross-section.• We find a strong negative cross sectional relationship

between this week’s realized skewness and next week’sreturns.

• We find a positive relationship between this week’srealized kurtosis and next week’s return.

• We do not find a strong bivariate relationship betweenreturn and realized volatility, BUT:

• Skewness and volatility interact to form interestingrisk-return relationships conditional on the level ofskewness.

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Discussion• In Particular:

– Investigate alternative measures of realized skewnessand kurtosis. Neuberger (WP, 2011).

– Investigate realized co-skewness. Kraus andLitzenberger (1976), Harvey and Siddique (2000).Asynchronicity issues are crucial.

• In General:– The interfaces between 1) high-frequency returns, 2)

derivatives prices, and 3) equity returns are fruitful forfurther research in my view.

– We will discuss the interface between 2) and 3)tomorrow.

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