Volatility, the Size Premium, and the Information Quality of the VIX and VIX Futures: New Evidence

Lorne N. Switzer and Qianyin Shan

Concordia University, John Molson School of Business

2015 Morton Topfer Chair Lecture Series

New York University

Polytechnic School of Engineering

Department of Finance and Risk Engineering

Six Metrotech Center, RH Dibner Auditorium

Tuesday, February 17th 4:30PM.

Abstract

This paper examines the futures market efficiency of the VIX and the relative merits of the VIX and VIX futures contracts in forecasting future S&P 500 excess returns the future Russell 2000 excess returns, and the future small-cap premium. We find that the current VIX is significantly negative related to S&P 500 index excess returns and positively related to the Russell 2000 index excess returns. These results suggest that the VIX predicts asset returns based on size based portfolios asymmetrically – with higher (lower) values of the VIX associated with lower (lower) values of large-cap (small-cap) returns in the future. The VIX and VIX modeled by an ARIMA process are significantly positive related to future values of the small-cap premium consistent with a fundamental default risk explanation. In addition, VIX futures show forecasting prowess for the S&P 500 excess return, the Russell 2000 excess return and the small-cap premium. The results for the speculative efficiency of the VIX futures contracts are mixed, however. Overall, the analyses support the hypothesis of informational advantages of the futures markets relative to the spot market in the price discovery process not just for sized based asset returns, but on the size premium as well.

Agenda

Section I Section II Literature review Section III Modeling Issues Section IV Data Description Section V Empirical results Section VI Conclusion

Motivation

Motivation

Markets are volatile Volatility affects investor (Switzer et al

2015) Volatility affects asset prices (Black and

Scholes, 1973, Ross, 1976, ensuing generations of option pricing models

Is VIX as a leading indicator for returns as well as the small cap premium? Is the market efficient?

US Market Volatility 1996-2013 (plot from Switzer et al 2015)

Volatility affects investor behavior

Motivation, cont’d

Perhaps the most widely quoted measure of market volatility is the CBOE Volatility Index (the VIX) The VIX is meant to capture the market’s expectation of stock market volatility over the next 30 calendar days, and has been disseminated by the CBOE on a real-time basis since 1993, as a weighted blend of prices for a range of options on the S&P 500 index. The VIX is quoted in percentage points and translates, roughly, to the annualized expected movement in the S&P 500 index over the upcoming 30-day period.

Many investors consider the VIX Index to be the world's premier barometer of investor sentiment and market volatility, and VIX options are very powerful risk management tools

Is VIX as a leading indicator for returns as well as the small cap premium? Is the market efficient?

How the VIX is calculated

The VIX formula

Calculating the VIX, cont’d

Trading on the VIX

On March 26, 2004, trading in futures on the VIX began on CBOE Futures Exchange (CFE). On February 24, 2006, options on the CBOE Volatility Index® (VIX®) began trading on the Chicago Board Options Exchange. The VIX options contract is the first product on market volatility to be listed on an SEC-regulated securities exchange: the most successful new product in CBOE history. In just ten years since the launch, combined trading activity in VIX options and futures has grown to over 800,000 contracts per day.

Many investors consider the VIX Index to be the world's premier barometer of investor sentiment and market volatility, and VIX options are very powerful risk management tools

The VIX presumably captures fear

The VIX is often referred to as the fear index or the fear gauge, since high levels of VIX typically coincide with high degrees of market turmoil (Whaley, JPM 2000)

“At the first whiff of bad news, stocks tend to plummet and the CBOE Volatility Index (VIX) tends to spike.” Steven Sears in Barrons, January 7, 2015

VIX (the so-called fear guage) as a forecast of market returns

07/07/14 01:34 PM "Front month VIX testing its all-time lows," wrote Hedgeye CEO Keith McCullough in today's Morning Newsletter. "Closing at 10.32 (it has never held below 10, sustainably) – contrarian signal: e.g. the adage "When the VIX is high, it's time to buy.…VIX is one of the best contrarian indicators in the business.” (Volnado trades)

But the analysts are not in agreement…..

“Surprise! VIX is too LOW for a Market Crash” http://finance.yahoo.com/news/surprise-vix-too-low-market-032813345.html

July 25, 2014

Too low for a crash? Oops

VIX and current markets

“U.S. stocks closed at highs on Friday, with the Dow above 18,000 and S&P 500 setting a new record as firming oil prices sent the energy sector higher……The CBOE Volatility Index (VIX), widely considered the best gauge of fear in the market, traded near 15..” CNBC News: http://www.cnbc.com/id/102424287., Feb. 13, 2015

Inefficient pricing of the VIX (weak form sense)?

“When the VIX fell as low as 16.6 on July 6 last year, Money Morning Capital Wave Strategist Shah Gilani warned investors to protect themselves against potential volatility..The low VIX creates an excellent opportunity for you to buy put protection at reasonable prices," Gilani said."In the face of future unknowns, and as long as implied volatility is low, you should take advantage of cheap puts to add some portfolio protection … just in case. Exactly two days later the VIX began a relentless climb. Investors who heeded his advice were glad they did. The VIX jumped to 48 and remained above 40 throughout October as the S&P 500 tumbled to a yearly low of around 1100. Meanwhile, Gilani had also advised his Capital Wave Forecast subscribers to hedge with put options in May and June. On Aug. 8- the day the Dow Jones Industrial Average plunged 635 points – subscribers to Capital Wave Forecast locked in gains of 456%, 455%, 371%, and 197% on four of those holdings….” Don Miller, Monday Morning,”The VIX Indicator: What this Contrarian Index is Telling Us Now,” March 7, 2012 .

VIX and the Size Premium?

Sparse evidence to date Unanswered question: perhaps the VIX affects

the size premium as a proxy for (or reflection of) fundamental default risk

.0

.1

.2

.3

.4

.5

.6

.7

.8

90 92 94 96 98 00 02 04 06 08 10 12

VIX: 1990 - July 2013

Figure 1. Weekly data

-.12

-.08

-.04

.00

.04

.08

90 92 94 96 98 00 02 04 06 08 10 12

small cap premium: 1990-Jul. 2013

A few related academic papers

Banerjee et al.(2007) and Kanas(2013) find that VIX predicts returns on large-cap stock market indices (S&P 500).

Small Cap Premia in the Literature

Risk/Return Profile Studies:• Fama and French, 1993 – introduce size premium as

a risk factor, Dimson and Marsh (1999), Switzer and Fan (2007), Switzer and Tahaoglu (IRFA,2015) – spanning, recursive cointegration

• Performance studies: Switzer (2010, 2012) looks at alphas for several countries: Small cap anomaly appears for the post 2001 period for many countries Governance factors (Switzer and Tang (2009)); Innovation (Switzer (2012))

24

Small cap premium and default risk

Key finding of previous studies: In countries with common law jurisdictions, the US default risk, which is uncorrelated with local default risk is found to be priced in many international markets where protection of shareholders and creditors in bankruptcy states is limited. The default risk factors are distinct from local business cycle turning points per se (term structure and inflation risk are related to the business cycle though: Switzer and Picard (2015) – Markov switching model))

Link between this study and the size premium literature

Hypothesize in this paper: just as default risk as measured by yield spreads can explain the small cap premium in a consistent way, so should the VIX

Modeling Approach

Britten-Jones and Neuberger (2000) derive the “model-free” implied volatility from current option prices.

Jiang and Tian (2005) extend the model-free implied volatility to asset price processes with jumps.

Carr and Wu (2009) develop a direct and robust method for quantifying the return variance risk premium on financial assets - simple principle to develop estimating equations; no arbitrage dictates that the variance swap rate equals the risk-neutral expected value of the realized variance

Modeling, cont’d

excess return (Realized annualized return variance RV) vs. the fixed variance swap rate SW) can be measured by a CAPM model:

Modeling, Cont’d

Realized volatility can be efficiently measured from implied volatility, hence:

2 2, ,'RV T IV t (5)

where 2,IV t is the implied volatility at time t prior to time T. Combining equation (4) and

(5), the relation between excess return and implied variances can be written as:

2, ,( ' ) / ( 1) / /mt T IV tr (6)

which can be rewritten as

* * 2 *, ,

mt T IV tr (7)

Modeling, cont’d

Link (7) to a GARCH-M model, as per Kanas (2013)

1

l

t i t i t t ti

r c a r h h

(8)

with the conditional variance equation following GARCH (1, 1) by including the squared

implied volatility as an exogenous variable.

2 21 1 1(1,1) : t t t tGARCH h h gVIX (9)

Rt is the excess total market return, ht is the conditional variance and λ is the risk-return parameter.

Modeling: approach of this paper

Extend model – look at returns and size premium, and use Asymmetric GARCH:

2000 500 2000 500

1

( )l

Russell sp Rusell spt t i t i t i t t t

i

R R c a R R h h

(10)

For the conditional variance equation, we allow the squared VIX as an exogenous

variable, and consider the GJR asymmetric GARCH (1, 1) specification:

2 2 21 1 1 1 1t t t t t th h I gVIX (11)

where 1tI if 0t and 0 otherwise.

Tests for Speculative Efficiency- Fama (1984) approach

We implement Fama’s (1984) regression approach to test whether the basis at any

period contains information about future spot prices or contains information about the

risk premium at the expiration of the contract. We estimate two equations:

1 1 1 1, 1( )t t t t tS S F S (12)

1 2 2 2, 1( )t t t t tF S F S (13)

Tests for Speculative Efficiency Park and Switzer(1997)

Second, we test market efficiency by examining the prowess of futures prices relative

to random walk predictors using daily data. As per Park and Switzer (1997) we examine

0 1 , 2i i i iT t T t tS F MAT (14)

where iTS is the prevailing spot price for contract i at time T (when contract i matures);

,it TF is the futures price of contract i at time t; MAT is the number of days for contract i to

mature as of time t, and it is the error term. If 1 is found to be significantly different

from 0, then the current contract prices are good predictors of future spot prices.

Modeling, cont’d

To eliminate the effect of a momentum factor in the VIX on the estimation result, we

also model VIX as an ARMA (p, q) process:

1 1 2 2 1 1... ...

t t

t t t p t p t t q t q

VIX c u

u u u u

Data Description

We obtain data for S&P 500, Russell 2000, and VIX index from Bloomberg. The sample period is from January 1990 to July 2013. Dividend yields for the S&P 500 and Russell 2000 are also obtained from Bloomberg for the same sample period.

To construct the excess returns series of both S&P 500 and Russell 2000, we obtained the data of 3 month Treasury bill of the total sample period from Federal Reserve Bank Reports.

Data Characteristics

Returns not normally distributed, but they appear stationary

Data Distributions

Return-SP500 Return-Russell 2000

Small cap premium

VIX VIX^2

Mean

Daily 0.0004 0.0005 0.00008 0.20 0.047

Weekly 0.0020 0.0021 0.00016 0.20 0.048

30 day 0.0083 0.0092 0.00083 0.20 0.047

Standard deviation

Daily 0.0117 0.0137 0.0068 0.082 0.05

Weekly 0.023 0.029 0.015 0.08 0.05

30 day 0.043 0.056 0.033 0.078 0.043

JB test

Daily 18574* 8886* 3868* 16496* 304856*

Weekly 1529* 1050* 485* 3519* 64615*

30 day 32.7* 26.8* 250* 310* 3499*

ADF unit root test

Daily -57.89 -77.82 -77.10 -4.77 -5.54

Weekly -38.21 -35.58 -35.86 -4.86 -6.31

30 day -15.6 -14.78 -18.06 -4.87 -5.52

Empirical Results: VIX as an explanatory factor

The squared current VIX in the conditional variance equation is a significant explanatory factor for the small cap size premium. The coefficient λ is positive and significant for 30 day return, and the coefficient g of VIX in the conditional variance equation is significant for all the returns of different horizons.

Estimation Results: Small Cap Premium

Table 2: Estimation results for the small cap size premium with squared current VIX in the conditional variance equation for the sample period from 1990 to July 2013. (***, **, * denote significant at 1%, 5%, and 10% level, respectively. t statistics in parentheses) daily return Weekly return 30 day return Conditional mean equation parameters c 0.0000(0.53) 0.0009(1.20) -0.01**(-2.05) a1 -0.02(-1.32) -0.06*(-1.91) -0.06(-1.19) a2 0.005(0.36) 0.03(0.86) 0.07(1.13) a3 -0.003(-0.18) 0.05*(1.80) -0.06(-1.14) a4 0.01(0.60) 0.01(0.35) -0.16***(-2.69) a5 -0.001(-0.10) -0.04(-1.36) -0.04(-0.63) a6 -0.02(-1.55) -0.05(-1.51) -0.16***(-2.71) a7 -0.02(-1.35) -0.003(-0.105) 0.002(0.03) a8 0.003(0.27) -0.003(-0.09) 0.04(0.49) λ 1.17(0.35) -3.36(-0.90) 0.38**(1.97) Conditional variance equation parameters ω (*10, 000) 0.006***(4.07) 0.138**(2.47) 0.029(0.23) α 0.063***(7.99) 0.114***(3.22) -0.087***(-167.3) β γ

0.88***(75.30) 0.016(1.63)

0.692***(8.95) -0.005(-0.125)

0.995***(56.91) 0.147***(5.45)

g (*10, 000) 0.35***(6.47) 6.17***(3.24) 2.86*(1.85)

Estimation Results: Small Cap Premium- cont’d

Table 3: Estimation results for the small cap size premium with squared VIX modeled by ARMA (5, 3) process in the conditional variance equation for the total sample period from 1990 to July 2013. (***, **, * denote significant at 1%, 5%, and 10% level, respectively. t statistics in parentheses) daily return Weekly return 30 day return Conditional mean equation parameters c 0.0002(1.33) 0.0006(0.77) -0.003(-1.49) a1 -0.02(-1.53) -0.06*(-1.88) -0.16***(-3.57) a2 0.004(0.32) 0.04(1.18) -0.03(-0.52) a3 -0.007(-0.50) 0.04(1.45) -0.12**(-2.26) a4 0.008(0.58) 0.01(0.38) -0.10*(-1.92) a5 -0.0003(-0.02) -0.06*(-1.91) -0.05(-0.80) a6 -0.02(-1.63) -0.05*(-1.67) -0.15***(-2.81) a7 -0.02(-1.23) 0.02(0.67) -0.01(-0.10) a8 0.004(0.30) 0.006(0.19) 0.07(1.00) λ -2.87(-0.87) -1.22(-0.32) 0.14**(2.55) Conditional variance equation parameters ω (*10, 000) 0.05***(23.33) 1.14***(7.22) -10.14***(-350.7) α 0.056***(9.77) 0.095***(4.66) -0.051***(-4.67) β γ

0.92***(149.4) 0.025***(3.21)

0.858***(37.62) 0.015(0.67)

1.007***(4766.8) 0.090***(4.00)

g (*10, 000) -1.11***(-20.5) -26.18***(-6.32) 251.97***(348.9) Log likelihood 20988 3386 571 Durbin-Watson stat 2.00 1.94 1.86

VIX as a predictor

VIX Basis is not an unbiased predictor of Futures Spot Change (eqn 12), time varying risk premium (eqn 13)

Table 12. Results of Fama (1984) Model

Estimated period: April 2004-July 2013

1

1

F-Stat

Regression (12)

1 1 1 1, 1( )t t t t tS S F S 0.148349

[0.574634] -0.264397 [0.181089]

2.131705

2 2 F-Stat

Regression (13)

1 2 2 2, 1( )t t t t tF S F S -0.148349 [0.574634]

1.264397* [0.181089]

48.75081*

Note: Robust standard errors are reported inside parentheses. The * denotes significance at a 1% level.

Test of Market Efficiency of VIX –time varying risk premium supported(eqn. 13)

Table 13. Wald Test Results of the Fama (1984) Model

Estimated period: April 2004-July 2013

1 10, 1

1 1

1 0

Regression (12)

1 1 1 1, 1( )t t t t tS S F S 27.18831* [0.0000]

48.75081* [0.0000]

0.066648 [0.7968]

2 20, 1 2 1 2 0

Regression (13)

1 2 2 2, 1( )t t t t tF S F S 1.104387 [0.3352]

2.131705 [0.1473]

0.066648 [0.7968]

Note: F values reported. p-values reported in parentheses.

VIX Futures as Predictor of Future Spot VIX

Table 14. VIX Futures Contracts as Predictors of Futures Spot VIX: Daily Data

Independent Variable Coefficient t-Statistics

OLS estimates of 0 1 , 2i i i iT t T t tS F MAT

Estimation period: April 2004-December 2012

,it TF 0.999243**

[0.013601] 73.47

MAT -0.019037* [0.011416]

-1.67

0 0.115214 [0.377704]

0.31

F-statistic 2714.689 Prob(F-statistic) 0.0000

Note: ** denotes significance at a 1% level. * denotes significance at 10%. Robust standard errors are

reported in parentheses. iTS is the prevailing spot price for contract i that matures at time T; ,

it TF is the

futures price of contract i at time; MAT is the number of days for contract i to mature as of time t, and it

is the error term.

Lagged futures VIX vs. of lagged current VIX and the small-cap premium

We replace the lagged VIX by lagged VIX futures price in equation (11). The estimation sample period is from April 2004 to July 2013. Table 15 provides the result of the estimation of future small-cap size premium with the squared current VIX futures price in the conditional variance equation. Including squared current VIX futures in the conditional variance equation does improve the prediction of future small cap size premium. The coefficient of λ is significantly positive at 1% level for 30 day returns, and significantly negative at 1% level for weekly returns.

Estimation Results: VIX futures and the small cap premium

Table 15: Estimation result for the future small cap premium with squared current VIX futures price in the conditional variance equation for the sample period from April 2004 to July 2013. (***, **, * denote significant at 1%, 5%, and 10% level, respectively. t statistics in parentheses) daily return Weekly return 30 day return Conditional mean equation parameters c -0.0000(-0.01) 0.007***(3.96) -0.115***(-13.38) a1 -0.03(-1.44) -0.15***(-3.52) -0.8***(-158.5) a2 -0.02(-1.06) -0.04(-0.76) -0.44***(-5.78) a3 0.01(0.30) -0.08*(-1.80) -0.01(-0.09) a4 -0.03(-1.51) -0.05(-1.12) 0.36***(3.64) a5 -0.03(-1.52) -0.01(-0.26) 0.04***(2047.8) a6 -0.06***(-2.73) -0.09**(-1.97) 0.15*(1.72) a7 -0.01(-0.69) -0.05(-0.94) -0.21**(-2.15) a8 0.01(0.66) 0.02(0.47) 0.13(1.24) λ 3.77(0.72) -35.16***(-3.20) 5.68***(90.2) Conditional variance equation parameters ω (*10, 000) 0.009***(3.02) 1.59***(5.18) 2.11***(20.9) α 0.026**(2.09) -0.074**(-2.49) 0.159***(263.8) β γ

0.873***(44.16) 0.099***(4.72)

-0.788***(-10.77) 0.05(1.07)

0.611***(200812) 0.059***(8.28)

g (*10, 000) 0.242***(2.99) 30.02***(3.71) -135.5***(-18466) Log likelihood 8565 1411 -3759 Durbin-Watson stat 2.11 1.95 0.41

VIX Futures as a predictor of future spot VIX

Table 14. VIX Futures Contracts as Predictors of Futures Spot VIX: Daily Data

Independent Variable Coefficient t-Statistics

OLS estimates of 0 1 , 2i i i iT t T t tS F MAT

Estimation period: April 2004-December 2012

,it TF 0.999243**

[0.013601] 73.47

MAT -0.019037* [0.011416]

-1.67

0 0.115214 [0.377704]

0.31

F-statistic 2714.689 Prob(F-statistic) 0.0000

Note: ** denotes significance at a 1% level. * denotes significance at 10%. Robust standard errors are

reported in parentheses. iTS is the prevailing spot price for contract i that matures at time T; ,

it TF is the

futures price of contract i at time; MAT is the number of days for contract i to mature as of time t, and it

is the error term.

Conclusions

VIX predicts asset returns on size based portfolios asymmetrically for daily returns. VIX is significantly negative related to S&P 500 excess returns and positively related to Russell 2000 excess returns.

VIX futures show forecasting prowess for small-cap premium and asset returns on size based portfolios. VIX futures are negatively related to these series. It supports the hypothesis of informational advantages of the futures markets relative to the spot market in the price discovery process.

The speculative efficiency results for the VIX futures are mixed. Further work to be explored – 1. directly link fundamental default risk and

other risk factors to the VIX – e.g. Market liquidity issues (Switzer and Picard (2015) ; 2. Look at effects of changes in estimation of VIX in 2014– more precise estimates of future 30 day volatility with the introduction of SPX Weeklys allows the VIX Index to be calculated with S&P 500 Index option seriesthat most precisely match the 30-day target timeframe for expected volatility that the VIX Index is intended to represent.

Short term vs. long term investment horizon

Thank you!