Modelling and forecasting the Volatility Risk Premium
using macroeconomic variables
Master Finance Thesis
Barthold Bekedam
Special research project of dr. Y.R. Kruse
Student number: S2214423
Student name: Barthold Bekedam
Study program: Master of Science in Finance
Supervisor: dr. Y.R. Kruse
Faculty: Faculty of Economics and Business
Field Key Words: Volatility
Special Research Project: Financial Market Data
2
Abstract
This research focuses on the question whether adding (macro)economic variables adds value in modelling and forecasting the volatility risk premium. I approximate the volatility risk premium by computing the difference between realised volatility using returns on a 5-minute window and model-free option-implied volatility of the S&P500, the FTSE100 and the Nikkei225 during the period from January 2000 to December 2015. I compare the model with the best combination of variables with a benchmark model containing only lags of the volatility risk premium itself on explanatory power and forecasting accuracy. Adding (macro)economic variables to a model adds value in modelling the volatility risk premium compared to the benchmark model. When forecasting the volatility risk premium, the models including (macro)economic variables of the UK and Japan show a significant increase in forecasting accuracy compared to the benchmark model. The model for the US also shows improvement but this is not significant. This could be due to structural breaks in the times series.
3
Introduction
In the area of volatility, the volatility risk premium is a quite new concept, even in the academic world. In this research I focus on modelling and forecasting this premium for three different countries using (macro)economic variables. Before going into depth on my approach of modelling and
forecasting this volatility risk premium I shortly explain the concept of the volatility risk premium in this introduction.
The volatility risk premium equals the difference between expected and realised volatility. As
volatility cannot be observed in financial markets, it has to be approximated. Realised volatility can be approximated using different methods, for example the GARCH model. While being a bit of age, GARCH models are still the most well-known and widely used models in modelling and forecasting volatility levels. In short, a GARCH process models the conditional volatility of a time series, the price of an assets for example, based on past sample variances and lags of the conditional variance itself. More information on GARCH models is in the literature review in the next section. Another method to approximate realised volatility is using squared returns. In this method, the sum of squares of returns of an asset equals the realised volatility of the sample period. In recent years this method gained popularity due to the availability of high-frequency returns, which are returns on sample windows of minutes or seconds. I explain this method of realised volatility more into depth in the literature review.
Expected volatility is a bit more complex, it is the expectation of investors on future volatility. This can be approximated using options with a time to maturity equal to the time window of expectation, for example 30 days. Inverting the formula to calculate option prices by using the market price of the option as an input and the volatility level as an output, you are able to imply the level of volatility of a certain option maturing at a certain date corresponding to the current market price. This is the
expectation of volatility of all investors in the market. I explain in the next section how this can be done in detail. Assuming there is no case of asymmetric information and investors can estimate expected volatility correctly, there should not be a difference between expected (option-implied) volatility and realised volatility. There is however a difference between these two, which is called the volatility risk premium.
4 demand to be compensated for this risk (by option buyers). This leads to a premium on the price of options, corresponding to a higher level of (option-implied) volatility.
This paper focuses on explaining this volatility risk premium. The main research question is whether adding (macro)economic variables adds value to modelling and forecasting the volatility risk
premium of the S&P500 (United States), the FTSE100 (United Kingdom) and the Nikkei225 (Japan). To answer this question, I divide it into multiple sub questions as following.
1. Does adding (macro)economic variables add value to modelling the volatility risk premium? a. Do all investors always price volatility risk in the same way? In other words, is the
average volatility risk premium equal over time and between countries?
b. Which variables add value in explaining and modelling the volatility risk premium? Do the results differ between the countries included in the research?
c. Are structural breaks present in the volatility risk premium time series within the sample period?
d. Does changing the sample periods bounded by the structural breaks lead to different outcomes and conclusions?
2. Does adding (macro)economic variables add value to forecasting the volatility risk premium? a. Does changing estimation windows lead to different outcomes?
The remainder of the paper is as follows. In the next section I discuss relevant literature on volatility, the volatility risk premium, different measures of volatility and I state my hypotheses. In the third section I describe the data and variables used, along with the methods of calculating the measures of volatility and of the volatility risk premium and the econometric methods used. In short I build three types of regression models for each country; a ‘basic model’ including only lags, acting as a
benchmark, a ‘kitchen sink model’ including all the variables used in this paper and a ‘best fit model’ which is the best model of all possibilities available. Section 4 describes the empirical results of the different regression models on the three countries included in this research and discusses the results of forecasting the volatility risk premium using the different models. At the end I explain my findings and limitations in the conclusion section. An overview of references and the appendix can be found at the end of the paper. Additional research on realised an implied volatility is in Appendix A5.
5
Literature review
In this section I explain the concept of volatility risk premium in more detail. First I discuss the two parts of the volatility risk premium, realised volatility and expected (option-implied) volatility into depth. Then I explain the volatility risk premium itself in more details.
GARCH models
As mentioned in the introduction, volatility cannot be observed, it has to be computed. Since there exist multiple methods of estimating volatility, there are different ways of computing the same variable. Robert Engle (1982) describes the first method of modelling volatility in his seminal paper about autoregressive conditional heteroscedasticity (ARCH) models. Before the ARCH model, volatility was assumed to be constant, whereas ARCH models use previous values (lags) of the independent variable to model its variance, splitting the variance into a conditional and an
unconditional part. Robert Engle (1993) provides an easy to understand summary in a separate paper. An extension on the ARCH model, and probably even more famous, is the generalised autoregressive conditional heteroscedasticity (GARCH) model by Bollerslev (1986). GARCH models are based on the assumption that the independent variable yt follows a stochastic process dependent on the information set ψt. The addition of GARCH on ARCH models is that the conditional variance in ARCH models is based on past sample variances only whereas the GARCH model also includes a lag of the conditional variance itself. The GARCH(p,q) process is given by
𝓎
𝑡= 𝑥
𝑡′𝑏 + 𝜀
𝑡( 1 )
𝜀
𝑡|𝜓
𝑡−1~𝒩(0, 𝜎
𝑡2) ( 2 )
𝜎
𝑡2= 𝛼
0+ ∑ 𝛼
𝑖𝜀
𝑡−𝑖2 𝑞 𝑖=1+ ∑ 𝛽
𝑖𝜎
𝑡−𝑖2 𝑝 𝑖=1( 3 )
where yt in most cases is the price of an asset, 𝑥𝑡′𝑏 is any explanatory variable with its coefficient, 𝜀𝑡 is the error term conditional on information set 𝜓𝑡−1 and normally distributed with mean 0 and
standard deviation 𝜎𝑡2 as defined in equation (3). A GARCH process defines the conditional variance
𝜎𝑡2 as the sum of p lags of the error term and q lags of the conditional variance itself, with their
coefficients αi and βi respectively. Lags p and q are defined as p>0, when p=0 this would be an ARCH process and q>0, because when p=q=0 then εt would simply be white noise. The
GARCH(p,q) model, usually GARCH(1,1), is a very popular way to approximate the volatility of stocks, but can be used to approximate the variance of any time series.
6 other models to forecast the volatility of the New Zealand stock market and finds ARCH and GARCH to be decent forecasting models.
Volatility from squared returns
Another way of computing the volatility of equity markets is by calculating the variance from the squared returns of these equities, usually called realised volatility. Computing realised volatility from squared returns is a popular method, especially since high frequency data is made available for many countries and many years. Returns with time windows of a couple minutes up to a couple seconds are available for research, so instead of using squared daily returns of stocks, researchers are able to use squared returns of minutes and seconds to compute the realised volatility. Previous literature provides evidence for squared high frequency returns to be a more accurate measure of volatility and a better predictor of future volatility levels compared to historical volatility from GARCH models (Andersen, Bollerslev, Diebold, & Labys, 2003; Paye, 2012). More precisely, Liu, Patton and Sheppard show that the 5-minute window is the most accurate time window amongst high frequency returns (Liu, Patton, & Sheppard, 2015). Poon and Granger (2003) provide an overview of papers on different forecasting methods that have been used in the literature.
Since volatility calculated from squared returns using high frequency data is accepted as a more accurate measure of volatility than GARCH volatility, this method is used in an increasing amount of papers1. Following these papers, I will also approximate volatility using squared returns from a
5-minute window in this research. Exact calculations are in the next section.
Current literature on realised volatility
Modelling and forecasting financial volatility is used in many types of research with many different outcomes. French, Schwert and Stambaugh (1987) were one of the first to lay the link between predicting volatility and expected stock returns. Schwert (1989) shows in his seminal paper that stock market volatility is significantly higher during recessions. The findings of Diebold and Yilmaz (2008) are in line with those of Schwert (1989) and also provide a clear link between volatility in
macroeconomic fundamentals and stock market volatility. The main variables that are found to have a significant impact on stock market volatility are the (volatility of) industrial production growth (Bekaert & Hoerova, 2014; Engle, Ghysels, & Sohn, 2008; Marquering & Verbeek, 2004; Schwert, 1989), the (volatility of the) inflation rate (Diebold & Yilmaz, 2008; Engle et al., 2008; Engle & Rangel, 2008; Paye, 2012; Schwert, 1989) and GDP growth (Diebold & Yilmaz, 2008; Engle & Rangel, 2008). Most of these papers use data on the S&P500 index. These findings provide evidence that during economic downturn, as uncertainty increases, volatility of stock markets tends to rise.
1 A sample of papers using high frequency data consists of (Andersen, Bollerslev, Diebold, & Ebens, 2001; Andersen et al.,
7 The recent papers of Cakmakli and Van Dijk (2010) and Christiansen, Schmeling and Schrimpf (2012) provide a comprehensive study including up to 127 and 38 macroeconomic and financial variables respectively to model volatility. The latter paper also includes financial factors measuring fear or uncertainty in financial markets, which translates in larger spreads between riskless and risky loans. As fear and uncertainty are interesting factors influencing stock market volatility, these spreads could have an impact on the volatility risk premium as well. Variables that are used as fear or
uncertainty measure include the spread between long term loans, 10-year government bonds, and short term loans, 3-month treasury bills, also called the term spread. Christiansen, Schmeling and Schrimpf (2012) also find the spread between the 3-month LIBOR rate and treasury bill rate a significant variable in explaining volatility, called the ted spread. Another variable in this category is the commercial paper spread, the difference between commercial paper rates and the treasury bill rate, which Paye (2012) proves to be significant in explaining volatility. Some research using specific exchange rates also find these variables significant explanatory variables (Engle & Rangel, 2008). Overall, the main variables influencing volatility are proxies of uncertainty, being increasing spreads of loans and volatility of macroeconomic fundamentals.
Option implied volatility
As realised volatility approximated from high frequency squared returns serves as my measure of actual volatility, I only lack a measure of expected volatility to calculate the spread between the two, called the volatility risk premium. This measure of expected volatility is implied volatility. Implied volatility is computed from options trading at market prices, usually with a maturity between 23 and 37 days, so 30 days on average. Therefore, implied volatility is often used as an ex-ante, risk neutral expectation of future risk levels. Ex-ante because it is a measure of what the expectation is for the upcoming 30 days, risk neutral because this volatility is implied from options trading at market prices, assuming the market is arbitrage-free and asset prices reflect the complete market. Option-implied volatility being risk neutral implies that market prices of options are the discounted expected value of the future payoff of the options using a certain measure of volatility, being the option-implied
volatility.
The most well-known method of deriving option implied volatility is the Black-Scholes-Merton model of options pricing. This model is broadly used to calculate option prices. As models of this kind use a number of assumptions of the market, such option pricing models do not calculate the exact option volatility. It is much more accurate to use so-called model-free option implied volatility, which is not bounded by the assumptions of a single model. Britten-Jones and Neuberger (2000) discuss this in their paper and this method is used by Bollerslev, Tauchen and Zhou (2009) and Bollerslev, Gibson and Zhou (2011). This way of calculating option-implied volatility may be expressed as a portfolio of European calls, where Ct(T,K) denotes the price of European call option Ct with strike price K
8
𝐼𝑉
𝑡,𝑡+Δ∗= 2 ∫
𝐶
𝑡(𝑡 + Δ, 𝐾) − 𝐶
𝑡(𝑡, 𝐾)
𝐾
2 ∞ 0𝑑𝐾 ( 4 )
As shown by Britten-Jones and Neuberger (2000), this model-free option-implied volatility equals the risk neutral expectation of volatility 𝐸∗
(∙) based on the information set ℱ
𝑡
, where
ν
t,t+Δrefers to the
realised volatility over the period t, t+Δ, leading to
𝐼𝑉
𝑡,𝑡+Δ∗= 𝐸
∗(𝜈
𝑡,𝑡+Δ
|ℱ
𝑡) ( 5 )
It is common (and probably well-known) that option-implied volatility is somewhat higher than realised volatility. This is not because option traders cannot estimate future volatility levels. The most important reason implied volatility is higher than realised volatility is because option sellers want to be compensated for the risk they bear. Buyers of options have limited risk, while sellers of options have unlimited risk. Option sellers are assumed to be risk neutral as the market is risk neutral, so they want to be compensated for the higher risk they bear on the options they sell. This makes option-implied volatility to a measure of what option sellers expect the volatility to be for the upcoming 30 days including a small premium. The volatility risk premium. This premium can be captured by subtracting the actual volatility from the option-implied volatility over the same period. The volatility risk premium captures this difference assuming the market is arbitrage-free, there is no case of asymmetric information and investors are able to estimate the volatility of the next 30 days.
As measure of the option-implied volatility of the indices used in this research, I use the VIX index of each of the countries. These VIX indices calculate the option-implied volatility of the corresponding index using the aforementioned model-free way of deriving option-implied volatility.
Option-implied volatility in previous research
9 Volatility Risk Premium
In short, the volatility risk premium is the premium that investors demand to be compensated for the risk they bear. This premium cannot directly be observed in the market, it has to be approximated using a proxy. As the volatility risk premium is a premium on top of the actual level of risk, this premium can be approximated by taking the difference between the expected (option-implied) volatility and the actual (realised) volatility. The realised volatility is calculated as the sum of squares of monthly returns, derived from a 5-minute window, as I explain in the part on realised volatility. The expected volatility can be approximated using the option-implied volatility as I explain on the previous page. The difference between option-implied volatility and realised volatility is a proxy, but the most accepted and most used proxy for the volatility risk premium (VRP), calculated as,
VRP
i,t≡ IV
i,t– RV
i,t( 6 )
with RV being the realised volatility of the corresponding index i at time t and IV the option-implied volatility of the corresponding index i at time t measured using the VIX index. Besides the volatility risk premium, VRP sometimes is an abbreviation for the variance risk premium, which is also used in this area of research. This approximation uses the variance, not standard deviation, of the returns and uses the VIX squared and the realised return of, e.g., forward contracts or the expected realised variance. Following the calculations and reasoning of Bollerslev, Tauchen and Zhou (2009), assuming option prices reflect risk-neutral pricing under the Epstein-Zin-Weil (Epstein & Zin, 1991)2 recursive
preference structure, the difference in option prices corresponds with and only with the difference in volatility of the priced asset.
Previous research from Wu (2005) estimates different variance dynamics as well as the variance risk premium from the VIX index and various quadratic variation estimators. Using a raw realised variance estimator from second-to-second sum of squares of returns, an auto-covariance adjusted realised variance estimator and two scale realised variance estimators in a likelihood function, he proves that second-to-second sum of squares of returns contain microstructure effects. To use a realised variance or realised volatility estimator, microstructure effects have to be corrected. Patton and Sheppard (2009) explain how to do this in their paper, which I discuss further in this section. Todorov (2010) also uses high frequency results from the S&P500 in his research towards variance risk premium dynamics. He uses a semiparametric two-factor stochastic volatility model and verifies that the variance risk premium varies significantly over time as well as increases in periods of high volatility and after big jumps in the volatility.
2 Epstein-Zin-Weil recursive preferences are characterized by 𝑈
𝑡= [(1 − 𝛽)𝑐𝑡 𝜌
+ 𝛽𝜇𝑡(𝑈𝑡+1)𝜌] 1/𝜌
10 Carr and Wu (2009) investigate whether they can explain the variance risk premium by classical models such as the CAPM and Fama-French model. They approximate the variance risk premium used in their research as the difference between the 30-day realised variance of forward prices and the VIX index squared which leads to a negative variance risk premium. Carr and Wu (2009) find that the CAPM and the Fama French model fall short in explaining this measure of volatility risk premium and suggest designing stochastic return variance models to capture the term structure of the variance risk premium.
Nyberg and Wilhelmsson (2010) and Drechsler and Yaron (2011) use the variance risk premium, approximated as the VIX squared and the variance of 5-minute returns of the S&P500, and try to measure agent’s perception of risk and uncertainty in the market. They provide more insight into which preferences are able to estimate risk and uncertainty agents perceive in the market.
The most well-known papers in this area of research are by Bollerslev, Tauchen and Zhou (2009) and by Bollerslev, Gibson and Zhou (2011). Both papers link expected returns to the volatility risk premium to try to forecast expected returns. Bollerslev, Tauchen and Zhou (2009) use realised returns from the aggregate S&P 500 composite index on a 5-minute sampling frequency, together with the ‘new’ model-free calculation of the VIX index to approximate the variance risk premium on the period January 1990 to December 2007. They include the P/E ratio, P/D ratio, the default spread, the term spread, the stochastic detrended risk-free rate and the consumption-wealth ratio as explanatory variables in different regressions comparing different time windows. Quarterly return regressions using the P/E value provide the largest degree of return predictability. They perceive the premium as a proxy for the aggregate degree of risk aversion of investors in the market, which might explain the temporal variation in the expected returns.
Bollerslev, Gibson and Zhou (2011) build an investor risk aversion index based on the volatility risk premium. They use the VIX and the realised volatility of the S&P 500 from 1990 to 2005 to construct the volatility risk premium. As in the paper by Bollerslev, Tauchen and Zhou (2009), they use (model-free) high frequency realised return data. These are summations of 5 minute realised returns squared, per every month. Together with the volatility risk premium they use the AAA bond spread, housing start indicator, the P/E ratio, industrial production, inflation rate and the producer price index as explanatory variables to predict returns. From these variables, the P/E ratio of the market, the market volatility, a measure of the credit spread, the industrial production, a variable for housing starts, producer price index and the nonfarm employment are found to be a significant predictors of future returns. They also conclude that the volatility risk premium varies significantly over time, as also found by other researchers in this topic (Bollerslev et al., 2009; Todorov, 2010).
11 and financial instability. For their research they use VIX data and high frequency realised returns of the S&P 500 from 1990 to 2010.
Hypothesis 1b: Based on the current literature, I expect a set of macroeconomic variables, fear indicators and equity market variables to explain volatility and the volatility risk premium.
As macroeconomic variables I expect the volatility of the industrial production growth rate and the volatility of the producer price growth rate to be significant explanatory variables in volatility risk premium modelling.
For risk aversion indicators, I expect the spread between the commercial paper rate and the 3-month T-bill rate, the spread between the LIBOR and the 3-month T-bill rate and the spread between the 10-year government bond and 3-month T-bill spread to have a significant influence on modelling the volatility risk premium.
For equity market variables, I expect the excess return of the market and the P/E ratio of the index to have a significant impact on the explanatory power of the model.
Almost all previous research uses data from the Unites States which makes a comparison with other countries interesting in this case. As the volatility risk premium is expected to differ greatly between countries, I also expect these variables to differ significantly. This leads to ‘best fit’ models being different for every country.
Hypothesis 1: Overall I expect adding (macro)economic variables to add value in modelling the volatility risk premium, based on findings in previous research.
High frequency volatility and volatility jumps
Using high frequency squared returns as approximation of realised volatility has some pros and cons. One advantage is the high accuracy of the data, making it a better proxy of ‘actual’ volatility. Using a too high sampling frequency however, leads to autocorrelation and microstructure effects, such as price discreteness and bid-ask spreads which violate the martingale assumptions of the pricing process (Bollerslev et al., 2009; Patton & Sheppard, 2009). It is necessary to find a balance between high accuracy in data to minimize the estimation error and to decrease ‘noise’ in your estimations. Several researchers (Bollerslev et al., 2009; Patton & Sheppard, 2009) recommend the 5-minute interval a reasonable choice of sampling window.
12 Barndorff-Nielsen and Shephard (2004), who calculate jumps as the difference between realised volatility and the bi-power variation, of which the latter is defined in equation (7). In this equation the return on price P over the period t to t+Δ on n intraday returns on sample windows denoted by j is used to approximate the sample volatility Vs. Todorov (2010) uses another variant of the multipower variation, the tri-power variation, in his research to determine the jump component of the variance. Other stochastic volatility (SV) models are more suitable to separate the continuous part from the discontinuous part of realised volatility than the model I use in this paper. Bollerslev, Tauchen and Zhou (2009) include the jump diffusion model as a robustness test in their research and do not find different results after correcting for volatility jumps. As I mostly follow their calculations, I do not employ this method in this paper.
𝐵𝑉
𝑡,𝑡+𝛥 𝑛≡ ∑ [𝑃
𝑡+𝑛𝑗(𝛥)− 𝑃
𝑡+𝑗−1𝑛 (𝛥)]
𝑛 𝑗=2[𝑃
𝑡+𝑗−1𝑛 (𝛥)− 𝑃
𝑡+𝑗−2𝑛 (𝛥)] → ∫
𝑉
𝑠𝑑𝑠
𝑡+𝛥 𝑡( 7 )
Structural breaksEngle, Ghysels and Sohn (2008), as well as Lamoureux and Lastrapes (1990) show that structural breaks are present in volatility time series. As Schwert (1989) and Diebold and Yilmaz (2008) show that stock market volatility is significantly higher during recessions, these recessions can lead to structural breaks in the time series of the volatility. Two (global) recessions are present within the sample period used, the crash after the ‘dotcom bubble’ which started in 2000 and the ‘US housing recession’ from 2007, which could cause a structural break in the volatility series. Therefore, it could be possible that structural breaks are present in the volatility time series used in this research and possibly also in the volatility risk premium series. This should be taken into account when forecasting volatility, as Chuang, Huang and Lin (2013) find the predictive power of volatility forecasting to decrease significantly in times of global financial crises.
Hypothesis 1c: As structural breaks are present in volatility series I expect these breaks also to be present in volatility risk premium time series.
13 Risk preferences
As there exist differences in risk preferences between countries it could be that this is reflected in the differences between the volatility risk premia of the countries. I assume that option prices reflect the complete market, meaning all investors in the world, so there should on average be no difference between the volatility risk premium of the United States compared to the United Kingdom and Japan. When the average volatility risk premium does differ significantly between countries, this could mean that investors price volatility risk differently due to differences in risk preferences, assuming investors mostly trade domestic options. Investigating differences in risk preferences between countries is beyond the scope of this paper so I do not include any of such variables in the model, but it could be interesting to check whether the difference of the average volatility risk premium is in line with the risk preferences of the corresponding countries. Just for comparison, not for research usage, the table below shows the risk preferences of the United States, the United Kingdom and Japan based on Hofstede’s framework of cultural dimensions ‘uncertainty avoidance’ (Clark, Eckhardt, & Hofstede, 2003) and research from Gándelman and Hernández-Murillo (2015). It is clear that the uncertainty avoidance of Japan is much higher and risk aversion parameter ρ of Japan is much lower than that of the United Kingdom and the United States, indicating lower risk preference or higher risk aversion for Japan compared to these countries.
Table 1: Uncertainty avoidance and risk aversion of the United States, the United Kingdom and Japan
Measurement United States United Kingdom Japan
Uncertainty Avoidance 46 35 92
Relative risk aversion 1.39 1.03 0.44
Note: Table reporting the uncertainty avoidance based on Hofstede’s framework of cultural dimensions (‘Uncertainty avoidance’) and Gandelman and Hernandez calculations of relative Risk Aversion between countries. These variables are just for background information only, they are not used in the research model further in this paper.
Hypothesis 1a: As current research and literature finds that (model free) option-implied volatility is higher than realised volatility, investors do demand to be compensated for future volatility risk. I expect that the volatility risk premium will change over time as the literature explains and to differ on average between countries due to difference in investor risk preferences between countries, with the US having the lowest volatility risk premium and Japan the highest.
14 Hypothesis 2a: Assuming that adding (macro)economic variables to the model improves explanatory power, I expect the ‘best fit’ model to contain (macro)economic variables and to have significant higher forecasting accuracy than the model without these variables.
This research mostly builds on the concepts and calculations of the papers by Bollerslev, Tauchen and Zhou (2009), Bollerslev, Gibson and Zhou (2011) and Wu (2005) and goes beyond the existing literature by mainly two aspects. First, I model and forecast the volatility risk premium itself, instead of using the volatility risk premium to predict future stock returns, as is done in Bollerslev, Gibson and Zhou (2011) and Bollerslev, Tauchen and Zhou (2009). To my knowledge volatility risk premium forecasting using (macro)economic variables is new in this area of research. I investigate whether adding (macro)economic variables adds value in modelling and forecasting the volatility risk premium. This could provide more insight into the effect of macroeconomic circumstances on this volatility risk premium. I use a set of (macro)economic and financial variables as explanatory variables to construct different models. Using 6 variables, as well as 5 lags of the volatility risk premium itself, leading to a maximum of 211=2048 possible models. I investigate the added value in
terms of model improvement and explanatory power of models containing (macro)economic variables and try to find the best combination of variables out of all the 2048 models for each of the three countries. Using multiple models for forecasting, I try to find the best forecasting model and to look differences between the three countries.
15
Data and methodology
This section explains the methods of calculating (approximating) realised volatility, option-implied volatility and the volatility risk premium and discusses the variables I use in this research.
The countries I include in my research satisfy two important conditions. The first condition is that these countries have different cultural and economic aspects from each other. As there could exist structural (macro)economic and cultural differences between countries it would be interesting to see whether those show up in explaining the volatility risk premium of those countries, although this is not a focus area of this research. A second and more practical condition is that these countries have accurate data available of all the variables used and up to at least 15 years in history. I include the United States, together with the United Kingdom and Japan as countries I focus my research on. I use data of the New York S&P500, London FTSE100 and Tokyo Nikkei225 indices and
(macro)economic data from the corresponding countries from the period January 2000 to December 2015. I compute the realised volatility from high frequency returns of these indices, available in the Realized Library of the Oxford-Man Institute of Quantitative Finance (2016). These variances are based on underlying high frequency data obtained through Reuters DataScope Tick History. Following Patton (2015) and Ghysels, Santa-Clara, Valkanov (2006) and Bollerslev, Tauchen and Zhou (2009), I use these returns on a 5-minute time window, as this is the most accurate time window in high frequency data, with the least order of autoregression and microstructure effects.
To compute the realised volatility from these returns, I base my calculations on the method of Bollerslev, Tauchen and Zhou (2009), leading to the following calculation of realised volatility, RV, measured in annualised standard,
𝑅𝑉
𝑖,𝑡= 100√
250
𝑀
𝑡∑ 𝑉𝐴𝑅
𝑖,𝑡+𝑗 𝑛 𝑛 𝑗=1𝑡 = 1, … . , 𝑇 ( 8 )
where VAR is the daily sum of the variances on a 5-minute window of index i composed at day t as,
𝑉𝐴𝑅
𝑖,𝑡+𝑗𝑛≡ ∑ [𝑃
𝑡+𝑛𝑗(Δ)− 𝑃
𝑡+𝑗−1𝑛 (Δ)]
2 𝑛 𝑗=1( 9 )
16 instead of observed and therefore this measure is a proxy of the ‘actual’ volatility. This method of calculation is the most accepted and accurate approximation of ‘actual’ volatility currently available. As risk free expectation of future volatility I use model-free option implied volatility, as I discuss in the literature review of this paper. I use the VIX indices of the corresponding stock indices as a proxy of model-free option implied volatility. The VIX implied volatility is an ex-ante risk neutral
expectation of future volatility over the subsequent 30 days, as it is implied from options with an expiration time between 23 and 37 days. As with realised volatility, also VIX does not cover the exact implied volatility, but the current ‘model-free’ procedure for calculating the VIX is accepted as the industry standard in research in implied volatility, being the most accurate measure available. As VIX is provided as annualised volatility, I annualise the realised volatility measure to make the two better comparable. VIX is calculated as,
𝑉𝐼𝑋
𝑖,𝑡= 100√
𝑁
365𝑁
30[𝑇
1𝜎
12(
𝑁
𝑇2− 𝑁
30𝑁
𝑇2− 𝑁
1) + 𝑇
2𝜎
22(
𝑁
30− 𝑁
𝑇1𝑁
𝑇2− 𝑁
1)] ( 10 )
where the volatility is determined as
𝜎
2=
2
𝑇
∑
Δ𝐾
𝑖𝐾
𝑖2𝑒
𝑅𝑇𝑄(𝐾
𝑖) −
1
𝑇
[
𝐹
𝐾
0− 1]
2 𝑖( 11 )
T is the time to expiration, F the forward index level derived from index option prices, K0 the first strike below the forward index level F, Ki the strike price of the ith out of the money option, ΔKi the interval between strike prices determined as
Δ𝐾
𝑖= 𝐾
𝑖+1−
𝐾
𝑖−12
( 12 )
R is the risk-free interest rate to expiration and Q(Ki) the midpoint of the bid-ask spread for each option with strike Ki. Nx denotes the amount of minutes to settlement of the near term option, the next term option, in 30 days, in 365 days. Detailed calculations of the VIX indices as well as examples of calculation can be found in the whitepapers of the corresponding index3.This leads to the following calculation of the volatility risk premium
VRP
i,t≡ VIX
i,t– RV
i,t( 13 )
where VIXi,t is the implied volatility of index i at time t and RVi,t is the realised volatility of index i at time t.
Figure 1 shows graphs of the option-implied volatility, realised volatility and volatility risk premium of the three indices used in this research and table A1 in the appendix shows descriptive statistics and correlations between the variables of the United States, the United Kingdom and Japan. From the graphs one can notice that the volatility risk premium of all the three countries varies significantly
17 over time. From the descriptive table it can be found that the average volatility risk premium of Japan is almost twice as high as that of the US, which is in line with my expectations based on existing literature and the risk preferences of the countries. Some quick hypotheses test on the mean of the volatility risk premium of the three countries shows that they are significantly different from each other. The results are in table 2 below. As all the three t-tests reject the null of the volatility risk premium having equal means by a probability value of 0.000, I conclude that the average volatility risk premia of the three countries are significantly different from each other. The average volatility risk premium of Japan is almost twice as large as that of the United States, the average volatility risk premium of the United Kingdom is somewhere in between. This is in line with my expectation based on the risk preferences of the countries.
Table 2: Mean volatility risk premium and mean t-test
Equal VRP Mean (t-statistics)
Mean VRP FTSE100 Nikkei225
S&P500 5.728 -3.750*** -11.961***
FTSE100 7.306 7.061***
Nikkei225 10.762
Note: This table summarises the mean volatility risk premium (VRP) of the S&P500 (US), the FTSE100 (UK) and the Nikkei225 (Japan). It also provides the t-statistics of a t-test whether the averages of the volatility risk premium of the countries are equal (null hypothesis) or significantly different. All three tests reject the null with a probability value of 0.000.
Figures A1 to A3 in the appendix include correlograms of the realised volatility, the implied volatility and the volatility risk premium of the indices. There is a form of autocorrelation in realised and implied volatility. This is not present in the volatility risk premium measure however.
Figure 1: Option-implied, realised volatility and the volatility risk premium of the S&P500, FTSE100 and the Nikkei225.
Note: Realised volatility as measured according equation (8), Implied volatility indicating option-implied volatility approximated by the VIX index. Volatility risk premium is defined in equation (13) as the difference between implied volatility and realised volatility. The sample period of the graphs is from January 2000 to December 2015.
0 10 20 30 40 50 60 70 80 90 100 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 S&P500 Index Implied volatility in annualised standard deviation
0 10 20 30 40 50 60 70 80 90 100 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 S&P500 Index Realised Volatility in annualised standard deviation
0 10 20 30 40 50 60 70 80 90 100 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 FTSE100 Index Implied Volatility in annualised standard deviation
0 10 20 30 40 50 60 70 80 90 100 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 FTSE100 Index Realised Volatility in annualised standard deviation
0 10 20 30 40 50 60 70 80 90 100 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 Nikkei225 Index Implied volatility in annualised standard deviation
-40 -30 -20 -10 0 10 20 30 40 50 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 S&P500 Volatility Risk Premium
-40 -30 -20 -10 0 10 20 30 40 50 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 FTSE100 Index Volatility Risk Premium
-40 -30 -20 -10 0 10 20 30 40 50 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 Nikkei225 Index Volatility Risk Premium
0 10 20 30 40 50 60 70 80 90 100 j-00 j-01 j-02 j-03 j-04 j-05 j-06 j-07 j-08 j-09 j-10 j-11 j-12 j-13 j-14 j-15 Nikkei225 Index Realised Volatility in annualised standard
Explanatory variables
Besides including lagged values of the volatility risk premium (VRP) itself, I use a set of
macroeconomic and equity market variables and risk indicators in this research. These variables are based on findings in previous papers, as I discuss in the literature review. The variables can broadly be divided into 3 categories.
1. State of the economy. The volatility of the growth rate of the Producer Price Index (VOLPPIG) and the volatility of the growth rate of the Industrial Production Index (VOLIPG) of the total industry are included to provide information on the state of the economy. These two variables are indicators of the state of the economy and are found to be significant to explain realised volatility, as I discuss in the literature review. In contrast to other macroeconomic variables, such as GDP, these variables respond much faster to changes in the state of the economy and are therefore in favour of ‘slower’ indicators, since I use monthly data in this research. I approximated the volatility of the growth rate of these two variables using a GARCH(1,1) model as specified in equation (1), (2) and (3) in the literature review.
2. Risk indicators. To quantify the level of fear or risk aversion, I include the commercial paper spread (CP), which is the difference between (3-month) commercial paper and the 3-month treasury bill rate. Christiansen, Schmeling and Schrimpf (2012) and Paye (2012) proved that this variable, together with the difference between the 3-month LIBOR and 3-month treasury bill rate (TED) and the difference between 10-year government bond yield and 3-month treasury bill rate (TERMS) have a significant influence on realised volatility and captures risk characteristics of investors.
3. Equity market. To measure the circumstances of the equity market, the Price/Earnings ratio (PE), based on last year’s earnings, is used. This variable is found to be significant in explaining the volatility risk premium in Bollerslev, Gibson and Zhou (2011). I also include the market returns in excess of the risk free rate (ERM), using the 3-month T-bill rate as risk free rate. As it I assume that investors demand higher return when taking increased risk, excess market return should increase when the level of risk increases. This is also proven in the paper of Bollerslev, Gibson and Zhou (2011).
The commercial paper rate of the United Kingdom (UK) is discontinued in 2013, as “the Bank of England is no longer able to collect Euro-commercial paper rates from its external contributors and therefore is no longer able to publish an average representative rate of these.” (CP rates, BoE)
20 demonstrates that this variable is a reasonable substitute for the last 32 months of the commercial paper rate.
The correlation between the TED and CP variables is extremely high (which can be seen in Table A1 in the appendix), leading to a serious problem of multicollinearity. Due to this problem, together with the problem of the commercial paper rate I discuss above, I decide to drop out the commercial paper rate as explanatory variable.
To avoid a look-ahead bias, all variables are lagged by one month in the model, except for the excess return of the market. An overview of data variables, sources and potentially needed calculations is in Table 3 below.
Further in this paper I will refer to all these variables as (macro)economic variables.
Table 3: Overview of variables
Variable Description Source Computation/ transformation
Volatility variables
VRP Volatility Risk Premium Equation (13)
VAR
Variance of index returns on a 5 minute window
Realized Library of Oxford-Man Institute of Quantitative Finance ∑ [𝑃 𝑡+𝑛𝑗(Δ)− 𝑃𝑡+𝑗−1𝑛 (Δ)] 2 𝑛 𝑗=1 IV
Model free, option implied
volatility VIX, Datastream Equation (10), (11) and (12) State of the economy indicators
VOLPPIG
Volatility of the Growth rate of Producer Price Index, Seasonally
adjusted Datastream
Volatility of the logarithmic return, using GARCH(1,1), lagged 1 month
VOLIPG
Volatility of the Growth rate of Industrial Production, Seasonally
adjusted Datastream
Volatility of the logarithmic return, using GARCH(1,1), lagged 1 month
Risk indicators CP
Commercial paper – 3-month T-Bill spread
Bank of England & Datastream;
Lagged 1 month (dropped out due to multicollinearity)
TED LIBOR – 3-month T-bill spread Datastream Lagged 1 month
TERMS
10 year Government bond –
3-month T-Bill spread Datastream Lagged 1 month
Equity market variables
ERM Excess return of the market Datastream
(Annualised) Logarithmic return – 3-month T-Bill rate
21 Econometric specification
To find out whether adding economic variables adds value to modelling the volatility risk premium I compare three models; one containing only the lags as variables (Basic model) and one containing all variables I use in this research (Kitchen Sink model). Subsequently, I compose the best combination of different variables, leading to the ‘Best fit’ model for every country and for all countries together in a panel data. I discuss the model selection approach on how to come to this combination further in this section. This makes three models in total, which I use for each country separately and all three
countries together in a panel data setting.
The benchmark model, ‘Basic model’ (containing only lags) is defined as
𝑉𝑅𝑃
𝑖,𝑡= 𝛼
𝑖+ ∑[𝜑
𝑖𝑉𝑅𝑃
𝑖,𝑡−𝜆]
𝐿
𝜆=1
+ 𝑒
𝑖,𝑡 ( 14 )The ‘Kitchen sink model’ (containing all variables included) is defined as
𝑉𝑅𝑃
𝑖,𝑡= 𝛼
𝑖+ ∑[𝜑
𝑖𝑉𝑅𝑃
𝑖,𝑡−𝜆]
𝐿 𝜆=1+ 𝛩
𝑖+ 𝑒
𝑖,𝑡 ( 15 ) where𝛩
𝑖,𝑡= 𝛽
1,𝑖𝑉𝑂𝐿𝑃𝑃𝐼𝐺
𝑖,𝑡−1+ 𝛽
2,𝑖𝑉𝑂𝐿𝐼𝑃𝐺
𝑖,𝑡−1+ 𝛾
1,𝑖𝑇𝐸𝐷
𝑖,𝑡−1+ 𝛾
2,𝑖𝑇𝐸𝑅𝑀𝑆
𝑖,𝑡−1+ 𝛿
1,𝑖𝑃𝐸
𝑖,𝑡−1+ 𝛿
2,𝑖ERM𝑖,𝑡 ( 16 )The ‘Best fit model’ (differs for every country):
𝑉𝑅𝑃
𝑖,𝑡= 𝛼
𝑖+ ∑[𝜑
𝑖𝑉𝑅𝑃
𝑖,𝑡−𝜆]
𝐿𝜆=1
+ 𝛽′
𝑖𝑋′
𝑖,𝑡−1+ 𝑒
𝑖,𝑡 ( 17 )In these models VRP denotes the volatility risk premium as defined in equation (13), approximated by the difference of realised volatility and implied volatility. φ is the coefficient of the lag of the
volatility risk premium, λ denotes the number of lagged months on volatility risk premium and e is denoted as the error term. When using a panel data model including all countries together, e is specified as ei,t = μi + υit. Equation (16) specifies Θ, in which PE is the Price/Earnings ratio of country
22 Model selection approach
While the ‘basic model’ and the ‘kitchen sink model’ are very straightforward, the ‘best fit model’ is not. My approach in finding the best fit model for every dataset (each country and all countries together in a panel data structure) is to build each of the 211 = 2048 possible econometric models,
using 5 lags and 6 (macro)economic variables, and select the best model according to the BIC
criterion. I determine the model with the lowest BIC value as the best fit model, because BIC provides a relative measure between different models while introducing a penalty term for the number of parameters in the model.
BIC or Schwarz’ criterion, as well as AIC, are often used to compare competing models based on a measure of fit and complexity of the models. They rank these models according to the following equations.
𝐴𝐼𝐶 = −2 × ln 𝐿̂ + 2𝑘 ( 18 ) 𝐵𝐼𝐶 = −2 × ln 𝐿̂ + ln 𝑁 × 𝑘 ( 19 )
Where 𝐿̂ is the likelihood function of the model M, the probability of x given parameter setΘ̂,
𝐿̂ = 𝑝(𝑥|Θ̂, 𝑀) ( 20 ) N is the amount of observations and k is the amount of parameters in the model. The model with the lowest BIC or AIC value has the best fit, after incurring a penalty for parametrization. As in most cases (like in this paper) ln(N) is larger than 2, which means BIC penalizes a higher amount of parameters stronger than AIC. As the models I compare differ in the amount of parameters and BIC includes this stronger ‘penalty’ for heavily parametrized models the BIC criterion is more appropriate than the AIC model selection criterion. Also BIC provides a more relative measure between models, which makes it better for comparing than adjusted R-squared. The limitation of BIC is that the approximation as shown in equation (19) is only valid when N is larger than k. As this is true in all cases, BIC can be used as an adequate criterion for model selection.
I select the model with the lowest BIC value of all 2048 models as the ‘best fit model’, which can be different for every dataset (country). Equation (17) defines these models, where β’ is a vector of the coefficient corresponding to a variable of the vector X’, which is the variable included in the model. For this model I will also perform a multiple structural breakpoint test, which I describe in the next section.
23
Empirical results
24 Table 4: Regression results of all three models of the United States, the United Kingdom, Japan and all three together.
United States United Kingdom Japan All
Basic KS Best fit Basic KS Best fit Basic KS Best fit Basic KS Best fit
α 4.439 2.891 3.727 5.486 4.704 6.251 6.578 9.608 7.721 5.517 4.423 4.332 (0.643)*** (0.864)*** (0.821)*** (0.857)*** (-2.581)* (0.455)*** (1.516)*** (3.503)*** (1.008)*** (0.463)*** (0.582)*** (0.516)*** φ (λ=1) 0.316 0.009 -0.024 0.216 0.127 0.319 0.160 0.294 0.228 0.241 (0.105)*** (-0.093) (-0.116) (0.092)** (-0.066)* (0.104)*** (-0.127) (0.042)*** (0.041)*** (0.038)*** φ (λ=2) -0.092 -0.070 0.031 0.029 0.071 -0.023 0.01 0.028 (-0.117) (-0.070) (-0.103) (-0.066) (-0.073) (-0.090) (-0.042) (-0.040) φ (λ=3) 0.128 (0.038)*** φ (λ=4) -0.144 (0.056)** φ (λ=5) VOLIPGt-1 -4.175 -3.766 0.640 0.068 0.108 (1.376)*** (1.413)*** (0.308)** (-0.054) (-0.063)* VOLPPIG t-1 2.131 3.263 -67.919 -6.09 0.132 (-1.434) (1.500)** (-37.735)* (-10.062) (-0.806) TED t-1 1.526 4.943 3.609 21.452 25.922 1.968 2.212 (-1.094) (0.934)*** (1.126)*** (-12.367)* (9.691)*** (0.679)*** (0.635)*** TERMS t-1 0.849 0.808 0.622 0.93 -2.548 0.409 (0.335)** (0.353)** (0.298)** (0.324)*** (-1.730) (-0.227)* PE t-1 0.042 0.164 0.002 0.001 (-0.024)* (-0.098)* (-0.004) (-0.003) ERM 2.997 3.162 2.810 5.405 2.815 3.185 2.818 2.999 (1.251)** (1.283)** (1.242)** (0.841)*** (1.087)** (1.049)*** (0.371)*** (0.388)*** Observations 190 190 190 189 189 187 190 190 191 569 569 566 F-Statistic 9.377*** 14.868*** 21.541*** 4.951** 9.326*** 26.477*** 13.052*** 7.731*** 24.774*** 33.503*** 22.536*** 39.138*** Adj R-squared 0.081 0.370 0.352 0.040 0.262 0.354 0.113 0.222 0.200 0.186 0.275 0.288 BIC 6.357 6.113 6.075 6.099 5.970 5.751 6.620 6.622 6.511 6.362 6.303 6.249
25 Modelling the Volatility Risk Premium using different models
Table 4 on the previous page reports the results of the regressions using the three different models (Basic, Kitchen Sink (KS) and Best fit) on the three different countries. All F-statistics are highly significant (at the 1% level). Every best fit model includes (macro)economic variables, but differ for every country. The volatility of the Industrial Production Growth (VOLIPG) as well as the volatility of the Producer Price Index Growth (VOLPPIG) are significant for the United States, which is in line with my expectations. These variables however do not show up in the best fit model for the United Kingdom and Japan. For the latter two countries, the excess return of the market (ERM) and the spread between the LIBOR and the 3-month T-bill rate (TED) are (highly) significant variables. Lags of the volatility risk premium differ heavily in significance between the countries, with lags λ=2 and
λ=5 (λ denoting the amount of months for the lag) being significant for the US, λ=4 being significant for the UK and none of the lags being significant for Japan. The only variable that is included in all of the four best fit models and is significant as well is the excess return of the market (ERM). This is in line with my expectations, as the volatility risk premium is able to predict the excess return of the market in other models.
The P/E (PE) value never shows up as explanatory variable in a Best fit model, neither is significant in any Kitchen Sink model. This proves that P/E values, although explaining a part of the future return of equity markets as I discuss in the literature review, does not improve the explanatory power of models explaining the volatility risk premium of any of the countries included in this research. The final three columns in the table, reporting the results of all three countries together in a panel data structure, illustrate that the excess return of the market is a highly significant variable in this setting. Interestingly also the 3-month lag shows up as highly significant variable, while this variable does not show up in any of the countries individually. The spread between the LIBOR and the 3-month T-bill rate (TED) also proves to be a significant variable in the panel data model.
For every country the Kitchen Sink model as well as the Best fit model report significantly higher adjusted R-squared values and lower BIC values compared to the Basic model. This means that these models have higher explanatory power compared to the benchmark model. Only the Kitchen Sink model for Japan reports a slightly higher BIC value than the benchmark Basic model. It is interesting to notice the relatively large differences in adjusted R-squared values between the countries, as the models of the US and the UK have higher explanatory power than the models of Japan.
As all of the Best fit models for the individual countries include (at least) two (macro)economic variables and all three have increased explanatory power compared to the benchmark model, I conclude that using (macro)economic variables in modelling the volatility risk premium does add value compared to a model containing only lags. Besides this I can also conclude that the
26 Structural breaks
As I discuss in the literature review structural breaks exist in volatility time series. These structural breaks can be caused by financial crises and recessions and could also exist in the volatility risk premium, as it is computed using two measures of volatility. As structural breaks can lead to errors in modelling and forecasting I investigate whether structural breaks exist in the datasets of this research and whether they lead to different results.
Andreou and Ghysels (2002)explain how multiple breaks can be detected in financial time series. It would be interesting to see whether the recessions in the time window show up as structural breaks in the time series employed in this research. To uncover these breaks I perform a Bai-Perron multiple breakpoint test on the best fit model of each country separately.
A Bai-Perron test performs parameter-constancy tests4 for different time windows, using the null
hypothesis of no differences between parameters, which would mean there does not exist a break between the subsamples. The null hypothesis states 𝐹𝑇(𝑙 + 1|𝑙), when the test rejects the null this
means a structural break is present in the sample. This test checks for differences in parameters while sequentially increasing l, until it fails to reject the null, meaning there are no structural changes between the different time windows. Bai and Perron (1998) suggest using this sequential method over methods based on information criteria when testing multiple unknown breakpoints in the sample. Table 5 below reports the results of these breakpoint tests as data points. I test each of the sample periods bounded by any of these breakpoints individually using the best fit model. The regression results using the samples from the breakpoint tests are in table 6.
Before using these results to adjust the sample windows it is interesting to note that in the US a structural break exists in November 2008, while according to the National Bureau of Economic Research (Hall et al., 2010) the American crisis started in December 2007 which means that after one year this lead to a structural break in financial market volatility.
Table 5: Structural breakpoints in volatility risk premium series
Country Significant Breakpoints F-Statistic
United States January 2006 13.147**
November 2008 12.690**
March 2011 5.325**
September 2013 5.593**
United Kingdom None
Japan November 2011 5.638**
Note: Table reporting the results of the Bai-Perron multiple breakpoint test to detect structural breakpoints in the Volatility Risk Premium series of every separate country. Bai-Perron test is performed on Best fit model of every country using HAC standard errors, sequential testing, 15% trimming percentage and Bartlett kernel. Breakpoints are reported in dates for every country (none for the UK), with corresponding F-statistic. All F-statistics are significant at the 5% level, as indicated by **.
4 Bai-Perron test is based on the equation 𝑦
𝑡= 𝛽0′𝑧𝑡+ 𝛽1′𝑥𝑡(𝑡 ≤ 𝑇1) + 𝛽2′𝑥𝑡(𝑇1< 𝑡 ≤ 𝑇2) + 𝛽3′𝑥𝑡(𝑡 > 𝑇2) + 𝑒𝑡,
27 Table 6: Regression results using sample periods bounded by structural breaks
United States Japan
Full sample Jan 2000 to Jan 2006 Jan 2006 to Nov 2008 Nov 2008 to March 2011 March 2011 to Sept 2013 Sept 2013 to Dec 2015 Full sample Jan 2000 to Nov 2011 Nov 2011 to Dec 2015 α 3.727 4.135 7.322 16.224 3.771 9.811 7.721 7.439 11.256 (0.821)*** (0.987)*** (1.377)*** (6.649)** (-2.298) (4.143)** (1.008)*** (0.796)*** (3.120)*** φ (λ=1) -0.024 0.267 -0.569 -0.066 -0.097 -0.381 (-0.116) (0.128)** (0.137)*** (-0.105) (-0.117) (0.106)*** φ (λ=2) φ (λ=3) φ (λ=4) φ (λ=5) VOLIPGt-1 -3.766 -0.807 -6.201 -0.224 15.78 -13.163 (1.413)*** (-0.994) (0.248)*** (-2.064) (6.625)** (-7.989) VOLPPIG t-1 3.263 -0.121 -2.155 0.974 2.935 0.914 (1.500)** (-1.712) (-2.055) (-1.807) (-3.271) (-1.021) TED t-1 25.922 26.554 -4.111 (9.691)** (4.603)*** -31.956 TERMS t-1 0.808 0.212 1.587 -2.431 -2.064 -0.335 (0.353)** (-0.412) (0.594)** (-2.153) (0.899)** (-1.434) PE t-1 ERM 3.162 1.363 7.738 3.747 8.642 8.641 3.185 2.996 3.356 (1.283)** (-0.830) (1.458)*** (1.398)** (1.523)*** (2.317)*** (1.049)** (1.131)** (1.582)* Observations 190 71 35 29 31 28 191 142 50 F-Statistic 21.541*** 3.833*** 43.344*** 2.574* 8.553*** 6.878*** 24.774*** 20.149*** 2.363 Adj R-squared 0.352 0.168 0.862 0.219 0.557 0.521 0.200 0.214 0.053 BIC 6.075 5.808 5.422 6.284 5.639 5.842 6.511 6.707 5.827
Note: Regression results on different sample periods with monthly observations using Best fit model, determined for every country separately based on lowest BIC value, as defined in equation (17). Sample periods are determined by the structural breakpoint test of Bai-Perron. Adjusted R-squared as well as BIC values and F-statistics are reported below the regression results. As there are no structural breakpoints for the UK, additional regressions on separate periods are not included for this country. α denotes the constant, φ the coefficient of the corresponding lag indicated by λ, VOLIPG the volatility of the Industrial Production Growth of the corresponding country, VOLPPIG the volatility of the Producer Price Index Growth of the corresponding country, TED the spread between the LIBOR and the 3-month T-bill rate, TERMS the spread between the 10 year government bond rate and the 3-month T-bill rate, PE the Price/Earnings ratio of the corresponding Index and ERM the market return in excess of the corresponding 3-month T-bill rate. Newey-West heteroskedasticity and autocorrelation consistent (HAC) standard errors are reported between parentheses. Significance levels are indicated using *, ** and *** for 10%, 5% and 1% significance levels respectively.
Table 6 above provides the regressions results of the best fit models of the US and Japan over different sample periods, bounded by data from the structural breakpoint test. All periods deliver significant F-statistics, except for the period November 2008 to March 2011 for the US and
28 R-squared value is very high, 0.862 compared to 0.352 over the full sample. Also the BIC value decreases in the individual samples (excluding the period with the insignificant F-statistic) compared to the full sample. These findings indicate that while the variables are not significant in all the individual sample periods, the explanatory power of the model is higher in those periods.
The excess return of the market variable (ERM) is mostly significant in the individual periods. It is interesting that the volatility of producer price index growth (VOLPPIG) is significant for the full sample period, but is not significant in any of the individual sample periods.
For Japan only the regression for the first period (January 2000 to November 2011) is significant. Adjusted R-squared value slightly increases, but BIC value increases. This does not provide strong evidence for an increase in explanatory power during this period compared to the full sample. Mainly due the low amount of observations included in the second period variables are not significant, which prevents drawing conclusions from these results.
The results in this sub part show that the explanatory power of the model differs significantly during individual sample periods. This is in line with my expectations that regressions on sample periods bounded by structural breaks provide different results than the full sample period does. The model has even higher explanatory power in some individual sample periods than in the full sample period.
Forecasting the Volatility Risk Premium
29 𝑄𝐿𝐼𝐾𝐸 = ∑ log 𝑦̂ +𝑡 𝑦𝑦̂𝑡 𝑡 ℎ 𝑇+ℎ 𝑡=𝑇+1 ( 21 ) 𝑀𝑆𝐸 = ∑ (𝑦̂ − 𝑦𝑡 𝑡) 2 ℎ 𝑇+ℎ 𝑡=𝑇+1 ( 22 ) 𝑀𝐴𝐸 = ∑ |𝑦̂ − 𝑦𝑡 𝑡| ℎ 𝑇+ ℎ 𝑡 = 𝑇+1 ( 23 ) 𝑇𝐻𝐸𝐼𝐿 = √∑ (𝑦̂ − 𝑦𝑡 𝑡)2 ℎ 𝑇+ℎ 𝑡=𝑇+1 √∑ 𝑦̂𝑡2 ℎ 𝑇+ℎ 𝑡=𝑇+1 + √∑ 𝑦𝑡2 ℎ 𝑇+ℎ 𝑡=𝑇+1 ( 24 )
In these equations, yt denotes the actual volatility risk premium, 𝑦̂ the forecasted volatility risk 𝑡
premium, h the amount of observations over time T+1 to T+h. Following Patton (2011), I place most emphasis on the QLIKE measure. In forecasting variables using an imperfect proxy, of which volatility is a perfect example, forecast evaluation measures can give a wrong view due to outliers. According to Patton (2011), only QLIKE and MSE measures are robust against noise in the volatility proxy, with QLIKE being less sensitive to extreme observations. As the volatility risk premium contains extreme observations and outliers, I determine the relative quality of the forecast based on QLIKE measure. To statistically prove that adding (macro)economic variables add value to
forecasting volatility risk premium compared to the benchmark model, I perform a Diebold-Mariano test. This test examines whether the accuracy of the forecast (measured using Mean Absolute Error, Mean Squared Error and QLIKE measure) by model 1 (the benchmark, basic model) is significantly different from the forecast by model 2 (the best fit model). H0 states that both forecasts are equal to
each other, H1 that both are significantly different. I perform this test between the ‘basic model’ and
the ‘best fit model’. Both the forecast measurements and the results of the Diebold-Mariano test are provided in table 7.
30 Table 7: Forecast evaluation measures and Diebold-Mariano test results
Country United States United Kingdom Japan
Basic KS Best fit
Best fit
rolling Basic KS Best fit
Best fit
rolling Basic KS Best fit
Best fit rolling QLIKE 2.717 2.626 2.621 2.556 2.973 2.956 2.947 2.952 3.467 3.389 3.351 3.350 MSE 25.888 19.847 20.268 15.465 18.542 18.065 15.470 15.348 33.660 36.813 28.019 27.684 MAE 3.393 2.686 2.748 2.371 2.900 3.083 2.868 2.821 3.682 4.127 3.674 3.681 Theil 0.373 0.324 0.317 0.277 0.275 0.248 0.226 0.226 0.246 0.255 0.234 0.232 Diebold-Mariano Test (Best fit vs Basic model) DM Statistic P-value DM Statistic P-value DM Statistic P-value QLIKE -1.507 0.137 -2.634 0.011 3.524 0.001 MSE 2.592 0.012 0.828 0.411 1.017 0.313 MAE 3.145 0.003 0.105 0.917 0.029 0.977
Note: Table reporting forecasting evaluation measures from volatility risk premium forecasts using different models; Basic model (only lags λ=1 and λ=2) as equation (14) and KS = Kitchen Sink model using all variables as equation (15). Best Fit models are determined for every country separately based on lowest BIC value, as equation (17). Estimation sample period extends from January 2000 to December 2010, the forecasting period lasts from January 2011 to December 2015, both using monthly observations. Best fit rolling reports the evaluation measures for the Best fit model using a fixed, rolling window, the other models use an expanding window. MSE indicates Mean Squared Error, MAE Mean Absolute Error and Theil the Theil inequality coefficient. QLIKE is calculated as in equation (21) following Patton (2011). Diebold-Mariano test reports t-statistic and probability values of Best fit model forecast being significantly different from the Basic model forecast, using HAC standard errors and an expanding estimation window.
Table 7 above reports the results of the forecast evaluation measures. The forecasts use an estimation window from January 2000 to December 2010 and produce forecasts one month ahead over the period January 2011 to December 2015. The Basic, Kitchen sink (KS) and Best fit models use an expanding window, meaning every step (one month) of forecasting the estimation windows increases by one month. The column ‘Best fit rolling’ reports the results of the best fit model on a fixed, rolling estimation window of the past 60 months. The forecasting evaluation measures are defined as in equations (21) to (24). A lower value of the measure indicates higher forecasting accuracy of the model. I focus on the QLIKE measure as I explain above.
The QLIKE measure is lower for the kitchen sink (KS) and best fit models compared to the
benchmark (basic) model for all the three countries. Most of the other accuracy measures (MSE, MAE and Theil), which I report to provide a full picture of the forecasting accuracy, show the same
decrease when comparing kitchen sink and best fit models to basic model.
31 variables in the model from the last 60 months instead of all the observations available (60 months and expanding after every forecast) includes only the most recent observations. As the volatility risk premium time series of the US contains 4 structural breaks, including more recent observations in the estimation window leads to higher forecasting accuracy. Although I do not provide significant evidence of line of reasoning, it is in line with the findings of Pesaran and Timmermann (2007). For both the UK and Japan, the QLIKE measures of the kitchen sink and best fit models are lower compared to the basic model. This indicates that forecasting accuracy improves when
