Evidences of Risk-Return Trade-Off in IBOVESPA Using High

Frequency Data

Breno Pinheiro Néribneri@fgvmail.br www.fgv.br/aluno/bneri

Hilton Hostalácio Notinihilton@fgvmail.br

Escola de Pós-Graduação em EconomiaFundação Getúlio Vargas

April 18, 2023 Risk-Return Trade-Off 2

ICAPM

1 1

0

0

t t t tE R Var R

Introduction Omnibus Definitions Data Results

Merton (1973)Merton (1973)

April 18, 2023 Risk-Return Trade-Off 3

Actually, is there a trade-off?Introduction Omnibus Definitions Data Results

Positive but not statistically significant: Baillie and DeGennaro (1990) French, Schwert and Stambaugh (1987) Campbell and Hentschel (1992)

Negative and statistically significant: Campbell (1987) Nelson (1991)

Depends on the method: Glosten, Jagannathan and Runkle (1993) Harvey (2001) Turner, Startz and Nelson (1989)

April 18, 2023 Risk-Return Trade-Off 4

Mixed Data Sampling (MIDAS)Introduction Omnibus Definitions Data Results

Ghysels, Santa-Clara and Valkanov (2002)Ghysels, Santa-Clara and Valkanov (2002)

1

0

1

0

MAX

m mmt t t

jjm m

jj

jm mm

t jtm

Y B L X

B L B L

L X X

April 18, 2023 Risk-Return Trade-Off 5

Note on notationIntroduction Omnibus Definitions Data Results

1

,

, 11

,

1,1,

,

,*

1,

, 1, 2,..., , 1, 2,...,

: ln , 1,2,..., 1

: ln , 1,2,..., 1 , 1,2,...,

: ln , 1,2,...,

t

t

t

i t t

N tt

N t

i ti t t

i t

N tt

t

P i N t T

PR t T

P

Pr i N t T

P

PR t T

P

April 18, 2023 Risk-Return Trade-Off 6

Note on notationIntroduction Omnibus Definitions Data Results

12 2

1,1

12 * *2

1,1

, 1, 2,...,

, 1, 2,...,

t

t

N

i tti

N

t i t t ti

r t T

Var r Var R E R t T

April 18, 2023 Risk-Return Trade-Off 7

High Frequency DataIntroduction Omnibus Definitions Data Results

São Paulo Stock Exchange Index (IBOVESPA)

01/02/1998 – 07/19/2001 (T=867)Russian and Latin American crises, 1998Blast of the technology-stock market bubble,

1999

10h00 – 18h15, each 15 minMax Nt=34Typical values: 29 – 33350 days (more than 40%) with 29 observationsTotal of observations: 26,030

April 18, 2023 Risk-Return Trade-Off 8

Histogram of NIntroduction Omnibus Definitions Data Results

N

Fre

qu

en

cy

20 22 24 26 28 30 32 34

05

01

00

15

02

00

25

03

00

35

0

April 18, 2023 Risk-Return Trade-Off 9

Typos TreatingIntroduction Omnibus Definitions Data Results

Inverted Digits: 48xx.xx -> 84xx.xx

Missing Digits: 14xx.xx -> 174xx.xx

Missing Decimal Point: 10xxxxx -> 10xxx.xx

Atypical Digit: 67xx.xx -> 97xx.xx

April 18, 2023 Risk-Return Trade-Off 10

Unit Root ADF testIntroduction Omnibus Definitions Data Results

Series Test Statistic P-Value

P1,t -1.5062 0.7873

PNt,t -1.5278 0.7782

Rt+1 -8.2384 <0.01

Rt* -8.5292 <0.01

[σ]t2 -5.8074 <0.01

[σ]t-4.6115 <0.01

April 18, 2023 Risk-Return Trade-Off 11

Descritive StatisticsIntroduction Omnibus Definitions Data Results

Series Rt+1 Rt* [σ]t

2 [σ]t

Mean 0.0003 0.0002 0.0005 0.0195

Variance 0.0080 0.0078 <0.0001 0.0014

Skewness 1.1499 1.2014 6.9387 3.1003

Excess Kurtosis 15.9373 17.5406 66.7872 14.3488

Minimum -0.1723 -0.1723 <0.0001 0.0048

1st Quartil -0.0144 -0.0138 0.0002 0.0128

Median 0.0007 0.0000 0.0003 0.0163

3rd Quartil 0.0147 0.0140 0.0005 0.0219

Maximum 0.2882 0.2919 0.0139 0.1178

Observations 866 867 867 867

April 18, 2023 Risk-Return Trade-Off 12

HistogramsIntroduction Omnibus Definitions Data Results

Histogram of Return

Rt1

De

nsi

ty

-0.2 -0.1 0.0 0.1 0.2 0.3

05

10

15

Histogram of Open-Close Return

Rt*

De

nsi

ty

-0.2 -0.1 0.0 0.1 0.2 0.3

05

10

15

20

Histogram of Realized Variance

[]t2

De

nsi

ty

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

01

00

02

00

0

Histogram of Realized Volatility

[]t

De

nsi

ty

0.00 0.02 0.04 0.06 0.08 0.10 0.12

02

04

06

0

April 18, 2023 Risk-Return Trade-Off 13

No Serial CorrelationIntroduction Omnibus Definitions Data Results

Regression Rt+1=β0+β1Rt+εt+1 Rt*=β0+β1Rt-1

*+εt

F-Statistic (P-Value) 0.309 (0.579) 0.003 (0.960)

OLS Estimator β0 β1 β0 β1

Estimative 0.000 0.019 0.000 -0.002

Standard Error 0.001 0.034 0.001 0.034

T-Statistic 0.306 0.556 0.125 -0.051

P-Value 0.760 0.578 0.900 0.960

April 18, 2023 Risk-Return Trade-Off 14

Risk-Return Trade-OffIntroduction Omnibus Definitions Data Results

Regression Rt+1=β0+β1[σ]t2+εt+1 Rt+1=β0+β1[σ]t+εt+1

F-Statistic (P-Value) 8.181 (0.004) 5.294 (0.022)

OLS Estimator β0 β1 β0 β1

Estimative -0.001 2.876 -0.003 0.188

Standard Error 0.001 1.006 0.002 0.082

T-Statistic -1.076 2.860 -1.803 2.301

P-Value 0.282 0.004 0.072 0.022

April 18, 2023 Risk-Return Trade-Off 15

Risk-Return Trade-OffIntroduction Omnibus Definitions Data Results

Regression Rt*=β0+β1[σ]t-1

2+εt Rt*=β0+β1[σ]t-1+εt

F-Statistic (P-Value) 7.725 (0.006) 4.918 (0.027)

OLS Estimator β0 β1 Β0 β1

Estimative -0.001 2.765 -0.004 0.179

Standard Error 0.001 0.995 0.002 0.081

T-Statistic -1.215 2.779 -1.836 2.218

P-Value 0.225 0.006 0.067 0.027

April 18, 2023 Risk-Return Trade-Off 16

Risk-Return Trade-OffIntroduction Omnibus Definitions Data Results

Regressions 1 and 2 Rt+1=β0+β1[σ]t-

12+εt+1

Rt*=β0+β1[σ]t-2

2+εt

F-Statistic (P-Value) 18.270 (0.000) 18.320 (0.000)

OLS Estimator β0 β1 Β0 β1

Estimative -0.002 4.276 -0.002 4.234

Standard Error 0.001 1.000 0.001 0.989

T-Statistic -1.762 4.275 -1.944 4.280

P-Value 0.078 0.000 0.052 0.000

April 18, 2023 Risk-Return Trade-Off 17

Analyses of ResidualsIntroduction Omnibus Definitions Data Results

0 5 10 20 30

0.0

0.4

0.8

Lag

AC

F

Residuals of Regression 1

0 5 10 20 30

0.0

0.4

0.8

Lag

AC

F

Residuals of Regression 2

0 5 10 20 30

-0.0

50

.05

Lag

Pa

rtia

l AC

F

Residuals of Regression 1

0 5 10 20 30

-0.0

50

.05

Lag

Pa

rtia

l AC

F

Residuals of Regression 2

April 18, 2023 Risk-Return Trade-Off 18

Other AnalysesIntroduction Omnibus Definitions Data Results

We cannot reject (even at 10%) the Ljung-Box and the Box-Pierce tests for independence of the residuals.

Regression of the residuals on its lags are not significant.

Regression of the residuals on the square of its lags are not significant (no ARCH effect).

We reject, at 5%, Teräsvirta and White neural-network tests for nonlinearity.

Information Criteria: two covariates, maximum.

No correlation between Rt+1 e Rt* nor Rt e Rt+1

*.

April 18, 2023 Risk-Return Trade-Off 19

Leverage EffectIntroduction Omnibus Definitions Data Results

Regression [σ]t+1=β0+β1

I{Rt<0}+εt+1

[σ]t+1=β0+β1I{Rt*<0}+εt

+1

F-Statistic (P-Value)

21.070 (0.000) 17.330 (0.000)

OLS Estimator Β0 β1 Β0 β1

Estimative 0.018 0.004 0.018 0.003

Standard Error 0.001 0.001 0.001 0.001

T-Statistic 32.030 4.590 31.990 4.163

P-Value 0.000 0.000 0.000 0.000

April 18, 2023 Risk-Return Trade-Off 20

Conclusion

It has been difficult to find a positive correlation between risk and return in the literature.

MIDAS Regression has been used to find this correlation.

In Brazil, we could find this trade-off by applying OLS to high frequence data.

This maybe due to both the lack of liquidity and the lack of access to intra day data.

The leverage effect is also present.

Next step: is it possible to beat IBOVESPA using this correlation?

Thank you!

Breno Pinheiro Néribneri@fgvmail.br www.fgv.br/aluno/bneri

Hilton Hostalácio Notinihilton@fgvmail.br

Escola de Pós-Graduação em EconomiaFundação Getúlio Vargas