portfolio selection using intraday data -...

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

2

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

3

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.

4

5

6

7

8

9

10

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

11

12

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

14

Distribution of Realized Beta

• Barndorff-Nielsen and Shephard (2004) showthat

• From this we have that

• So that

15

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

16

Earnings Surprise

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

• And forecast dispersions

17

Event Time Beta

18

Event Beta by Earnings Surprise

19

Event Beta by Forecast Dispersion

20

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.

21

Third Application:Do Realized Skewness and Kurtosis

Predict the Cross-Section ofEquity Returns?

Diego AmayaPeter Christoffersen

Kris JacobsAurelio Vasquez

22

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.

23

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.

24

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.

25

Daily Firm-Specific Sample Moments

• Realized Daily Variance

• Realized Daily Skewness

• Realized Daily Kurtosis

26

Weekly Firm-Specific Moments

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

27

Distribution of RVol(two million firm-week observations)

28

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

Distribution of RSkew

29

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

Distribution of RKurt

30

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

RVol and the Cross Sectionof Stock Returns

31

No action here…

RSkew and the Cross Sectionof Stock Returns

32

Very large premium on negative skewness

RKurt and the Cross Sectionof Stock Returns

33

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

35

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.

36

37

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

38

First sorton otherfactor andthen onskewness.Then checkskewnesspremium ineachquintile ofthe otherfactor

39

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.

40

Continuous Time Limits under SVJ• Consider the realized moment estimators

• Taking limits

• Gives (for j=2,3,4):

41

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.

42

RM(2) Signature Plot

43

The sampling frequency is in seconds.

RM(3) Signature Plot

44

The sampling frequency is in seconds.

RM(4) Signature Plot

45

The sampling frequency is in seconds.

Returns on High Minus Low Skewnessusing Jump Robust RV Measures

46

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.

47

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.

48