VIX-Yield Curve Cycles Within Regime Changes

Darin Tuttle, CFA - Tuttle Ventures LLC.

Brandon La Bella - UC Riverside A. Gary Anderson School of Management

12/9/2022

Abstract

Advances in recent macroeconomic research have shown there is a cyclical relationship between implied volatility on equity index options (represented by the VIX index) and the U.S. Treasury Yield Curve^[1].

We define the VIX-Yield Curve Cycle (VYCC) and suggest evidence to categorize VIX-Yield Curve Cycles (VYCCs) and their predictive power for recessions above and beyond other leading economic indicators.

Our research paper builds on recent work to ask two fundamental questions:

1. What are the defined stages present across a VIX-Yield Curve Cycle (VYCC)?
2. What are potential models to represent the relevant nonlinear features of the VYCC?

We accomplish this by:

1) We identify four regimes within full VYCC iterations, using time series analysis and stratified group sampling.

Using cross-sectional analysis from data by the Chicago Fed’s Adjusted National Financial Conditions Index^[2] (ANFCI) we classify four regimes within the VIX-Yield Curve Cycle (VYCC) and define notable characteristics in each macroeconomic market condition.

Namely: Goldilocks, Stabilization, Fallout, and Overweight.

Explanatory commentary is given for each stage along with justification and rationale.

Then, we use the regime classifications as a baseline reference guide to test for and evaluate different macroeconomic models.

2) We explore the best type of model to evaluate the dependent variable (recessionary regime shifts), volatility, and the treasury yield curve.

Economic models generally consist of a set of mathematical equations that describe a theory of economic behavior.

What is difficult about the relationship of the VYCC is that it is not a linear relationship.

The specific functional forms used in nonlinear models imply that in general the function that generates the relationship is different from the one implied by traditional linear methods.

For this reason, models outside of economics may be able to characterize nonlinear behavior.  

We propose certain exponential models that may have the ability to approximate any predictable patterns of continuous function and its derivatives arbitrarily well when separated by stages^[3] [4].

The model presented here is an introductory concept based on the principle that all VYCCs follow the shape of a counter-clockwise loop when graphically represented on an x-y plane.

However, it may be better to visualize this relationship in other models as well.

It was our expectation that VIX Yield Curve Cycles have no statistical significance in our set of modeled macroeconomic observations, however we did find some exponential functions with some significance depending on the stage defined.

Here is an outline of our Data & Calculations, and Exhibits:

Introduction

In recent years, there has been a great deal of focus on the potential predictive power of the VIX-Yield Curve Cycle (VYCC) for future macroeconomic conditions.

To examine this question, we first need to define what a VYCC is and how it works.

In broad terms, VIX-Yield-Curve Cycles capture the interplay between financial markets and the stance of monetary policy^[5].

The VYCC is a cyclical relationship between implied volatility on equity index options (represented by the VIX index) and the yield curve.

The way the YYCC works is we take the co-movement to variables; the first one  is the two-year moving average of the VIX Index,  and the second one is the two-year moving average of the 30-year-3-month treasury spread. Economists can then  determine which macroeconomic regime we are in currently.

A closer look reveals a fascinating connection between the VIX index, which measures market volatility, and the yield curve -- two seemingly unrelated financial indicators.

There are different phases that we categorize by using the VYCC:

1. Goldilocks (G)
2. Fallout (F)
3. Overweight (O)
4. Stabilization (S)

G - F - S - O

(1 Full Cycle)

The first one is the Goldilocks period, this part of the cycle happens when economic conditions are ideal. During this period, The VIX is below fifteen and capital markets are not turbulent, in conjunction with that, the 30 year - 3 month treasury spread is wide. Over time that widening slope begins to narrow. A two year moving average is used for both variables to smooth out any seasonality or outliers.

We then move on to the Fallout period; this is the phase of the economic cycle where we see elevated levels of implied volatility in the equity index^[6] (VIX). The second component of Fallout is  seeing the 30 year - 3 month treasury spread stays range bound in a narrow range. This occurs when the U.S. Central bank is tightening financial conditions to reach some stability in the economy and the longer end of the yield curve is lower than normal.

The next phase is the Stabilization period, that occurs when the VIX stays at an elevated range and treasury spreads begin to widen as a result of the central bank loosening financial conditions.

The cycle then loops back around counter clockwise to Overweight, once financial conditions are tightened enough, it appears that the VIX begins to subside, while the yield spread remains wide.

Since 1990 there have been three successive cycles^[7]:

Cycle 1 1990 - 1999, S - O - G - F

Cycle 2 2000 - 2008, S - O - G - F

Cycle 3 2009 - 2022, S - O - G - F

The cycle repeats back to the economic peak, which was the Goldilocks period, where VIX is below fifteen again, treasury spreads are wide, and the cycle continues. This is consistent with the work from Hansen: Predicting Recession Using VIX-Yield Curve Cycles has shown as well.    

Besides monitoring monetary policy action, we use regression analysis and cross-sectional comparison to measure the adequacy of this cycle predictor.

Based on our observations, while displaying some consistent themes, every VYCC also responds to different external forces. For example, the COVID-19 pandemic was a significant external factor that the model captured in the Fallout stage during a much more acute period compared to other Fallout stages. Our review of the financial conditions indicators did not show any specific reason for this phenomenon.

Overall, we believe the VIX and treasury spread have a significant relationship which builds the case of validity to use it. The main caveat with using the VIX for this cycle predictor is the length of time that it has existed, the VIX data only goes back to the early months of 1990 to present day so that is a limitation to our research.

If we apply this model through that time-span, that means we are only looking at four recessions to observe. We hope testing the predictive power of the cycle prediction model in the future will provide an adequate way to scale macroeconomic conditions.

Methods

First, we performed a t-Test Two-Sample assuming unequal variances between both variables.  We did this to determine whether there is any indication of a difference between the means of the two different populations.

This statistical evaluation produced the following results:

We observed the statistics and can conclude that since the p-value is smaller than an alpha of 0.05 and 0.01, there was a significant difference in the means of each sample.

This is important so that we could consider other statistical relationships between the two variables.

Next, we developed a process to identify stages across the Yield Curve Cycle.

To classify the stages, we looked over the period from data provided by Chicago Board of Options Exchange (CBOE) since 2004 and created quartile ranges for both the 2-year Moving Average 3m-30 year and the 2 year Moving Average VIX.

The reason why we chose the quartile ranges is because we wanted to understand the spread of the middle half of the distribution, but also wanted to group categories that were statistically large. This made the stages unique enough to include a relevant sample size of n>30 across the sample periods to perform additional statistical analysis and set reasonable parameters.

From the quartile ranges, we determined the stages based off of historical analysis and the Adjusted National Financial Conditions Index (ANFCI) levels over time.

The ranges were selected between the 50th percentile on the 2 year Moving Average VIX at a level of 16.28 and a wide yield curve spread.

Exponential equations are calculated across each stage and r-squared is calculated.

Results

According to Estrella and Mishkin, the slope of the yield curve can be used to predict future macroeconomic growth. A wide yield curve spread is representative of positive economic conditions. ^[8] 

Using the VYCC, we can chronicle the different stages that align with the economic cycle to better understand the current conditions for the US economy. We believe this cycle model demonstrates the capability to outline future expansion and contraction.  

We use regression model tests using various lagging and leading indicators to measure the significance of the data and the reliability of using the VIX as a dependent variable.

While there is some overlap at the upper right range, this appears to be a possible transitional period into the next phase iterative Stabilization period and has some spurious properties.

We define the Goldilocks VIX-Yield Curve Stave with the exponential function.

y=0.3424e^0.1539x

Exponential functions are based on relationships involving a constant multiplier^[9]. The constant multiplier here is 0.3424   e^ 0.1539 multiplied by the VIX.

Table 1.2

As you can see during the Goldilocks stage, there is a clear exponential relationship and an R-squared of 0.9276 which indicates a good fit. By using this equation, it is also possible to identify when the Goldilocks period ends, because the multiplier is longer in line with other observations.

This chart plots the number of periods when the treasury was wide spread, and the VIX was moving at “normal” levels. These are times when the is in good conditions. Implied option volatility is often nicknamed the “fear index.” This rationale of uncertainty is driven by investor’s anxiety. This falls in line with expectation theory, which implies that the slope of the yield curve forecasts future interest rates and what investors expect them to be in the short-term, medium term, and long-term horizon.

When economic conditions are good, then volatility is low, and the 3 month 30 year spread is wide, because market participants do not expect there to be instability in the near term.

Illustrated in the chart above is a sampled Goldilocks stage placed in two different periods, the stage is defined as a phase when treasury spreads are wide and the VIX is trending at “lower than normal” levels. The blue bars are displaying short-term periods when the Goldilocks stages are occurring. From January 2004 to Early March 2004, and early September 2004 to April 2005. This happened at a time when fed fund rates were at 1-1.6. So the expected rates on longer-term bonds were higher than current rates. Financial conditions were very loose during this time.

Table 1.3

In the stabilization stage, we can see there is a clear pathway in differentiating the cyclical relationship of volatility and the expected rates of treasuries. With an R squared of 0.6593, this chart indicates somewhat of a good fit. The stabilization stage implies that treasury spreads are widening again, while the VIX is relatively higher than average.  This is exhibited in Table 1.3.

In the Fallout stage, you will notice that it is in the 3rd tier of the VIX levels. You can also see that there are multiple somewhat contradictory levels of the relationship. The yield curve here is somewhat narrow and flat.  A flat yield curve is observed when all maturities have similar yields, whereas a humped curve results when short-term and long-term yields are equal and medium-term yields are higher than those of the short-term and long-term^[10]. A flat curve sends signals of uncertainty in the economy. This dichotomy is one of the key characteristics of the Fallout stage. It is not clear and increasing in volatility but not at a constant time forward path. This is illustrated in Table 1.4.  

Table 1.4

In Table 1.4, it is an image of the cycle loop collapsing on itself.  Which is a directional trend of overweighting volatility when long-term expected rates are dragging down. This is the Fallout stage, this marks a transition from Goldilocks to destabilization, where you see declining levels in treasury rates which coincides with increased levels of implied volatility.  

The sampled fallout stage includes both the 2008 Global Financial Crisis and the COVID-19 Pandemic.

Table 1.5

In the overweight stage, we categorize this as the recessionary phase of the cycle. The 2 year moving average of the VIX and 3 month - 30 year Treasury spread shows a significant relationship when volatility moves upwards and the 3 month - 30 year spread flattens to signify the suppression of financial conditions and the U.S. economy. In Table 1.5, it shows the two variables have an R squared of 0.9491. This reiterates the exponential fit between the treasury spread and the VIX.  

Second we use the Adjusted National Financial Conditions (ANFCI) to set parameters and build our econometric approach in application to VIX-Yield Curve Cycle (VYCCs).  

The ANFCI can be used as a dummy variable to create a non- linear model that moves in line with the VIX and the treasury yield curve. We can see in exhibit 1.2 that at some time pre-pandemic, there is a jumpout of the VIX during the goldilocks period and then it transitions into a new but brief period of Fallout.

The reason we use the Adjusted National Conditions (ANFCI) index is because U.S. economic and financial conditions tend to be highly correlated.

This index isolates a component of financial conditions uncorrelated with economic conditions to provide an update on financial conditions relative to current economic conditions^[11].

The ANFCI is constructed to have an average value of zero and a standard deviation of one over a sample period extending back to 1971.

Positive values of the ANFCI have been historically associated with financial conditions that are tighter than what would be typically suggested by prevailing macroeconomic conditions, while negative values have been historically associated with the opposite^[12].

We believe our research will further add to the expanding field of macroeconomic analysis and provide a robust framework from which other researchers can evaluate in the future.

The results of our analysis showed that there are indeed potential common catalysts which have explanatory power across the VIX-Yield Curve Cycle (VYCC) over time.

Specifically, we found that regime changes in the Federal Reserve's monetary policy stance are potential lead indicators of turning points in the VYCC and explained by exponential functions.

This finding is important because it provides further evidence that the VYCC contains predictive power for future macroeconomic conditions. It also suggests that central banks may be able to use their monetary policy tools to influence the course of the VYCC and potentially mitigate some of its adverse effects.

During the process, we used a cross-sectional analysis of the Adjusted Financial Conditions Index (ANFCI) and the two-year moving average of the CBOE Volatility Index (VIX) to construct a nonlinear time series across the most recent.

In the ANFCI chart, it illustrates a “Goldilocks” period of low interest rates and low implied volatility at the beginning, however there starts to be a turning point at March 2020, when the COVID-19 Pandemic becomes apparent across the globe. This super-event triggered a massive trend of Fallout that only occurred during a short period of time.

We believe that the Federal Reserve policy decisions and the fiscal interventions of the U.S Government played a primary role in driving a “cool off” of the Fallout period following their actions.  From there on, you can see in the ANFCI chart that there is a run-off of a new regime of stabilization of low volatility and low interest rates, however it begins to shift back to Fallout. This can be seen as aligning with the timeline when central banks started to tighten their balance sheets and the federal government stopped providing stimulus funds to the public. Thus, bringing back Fallout.

Discussion

In conclusion, our research provides evidence that there are potential common catalysts which have explanatory power across the VIX-Yield Curve Cycle (VYCC) over time. This finding is important because it provides further evidence that the VYCC contains predictive power for future macroeconomic conditions.

It also suggests that central banks may be able to use their monetary policy tools to influence the course of the VYCC and potentially mitigate some of its adverse effects.

If this current trend holds, then we could intuitively see a new Goldilocks regime on the horizon in the early part of 2023. Based on the two-year moving average of the VIX, this would remove March 2020 from the rollover period, thus, we can begin to identify the potential catalyst of a new regime restarting.

References

1. Ang, A., M. Piazzesi, and M. Wei (2006), "What Does the Yield Curve Tell Us About GDP Growth?," Journal of Econometrics 131(1-2), pp. 359-403.

2. Chicago Fed Letter, No. 386, 2017 Introducing the Chicago Fed’s New Adjusted National Financial Conditions Index By Scott A. Brave , David Kelley

3. Damiano Brigo; Fabio Mercurio (2001). Interest Rate Models – Theory and Practice. Springer. ISBN 978-3-540-41772-9.

4. Dueker, Michael J. "Strengthening the Case for the Yield Curve as a Predictor of U.S. Recessions," Review, Federal Reserve Bank of St. Louis (March/April 1997), pp. 41-51.

5. Economic Perspectives, Vol. 35, No. 1, 2011 Monitoring Financial Stability: A Financial Conditions Index Approach By Scott A. Brave , R. Andrew Butters

6. Estrella, Arturo and Frederick S. Mishkin. "The Yield Curve as a Predictor of U.S. Recessions," Current Issues in Economics and Finance, Federal Reserve Bank of New York (June 1996).

7. Goldstein, Daniel G.; Taleb, Nassim Nicholas (28 March 2007). "We Don't Quite Know What We are Talking About When We Talk About Volatility". Journal of Portfolio Management. 33 (4). doi:10.3905/jpm.2007.690609. S2CID 153535794. SSRN 970480

8. Hansen, Anne Lundgaard, Predicting Recessions Using VIX-Yield Curve Cycles (March 8, 2022). Available at SSRN: https://ssrn.com/abstract=3943982 or http://dx.doi.org/10.2139/ssrn.3943982

9. Jackson, John E., and Ken Kollman. “Modeling, Measuring, and Distinguishing Path Dependence, Outcome Dependence, and Outcome Independence.” Political Analysis, vol. 20, no. 2, 2012, pp. 157–74. JSTOR, http://www.jstor.org/stable/23260170. Accessed 9 Dec. 2022.

10. J H M Darbyshire (2017). Pricing and Trading Interest Rate Derivatives (2nd ed. 2017 ed.). Aitch and Dee Ltd. ISBN 978-0995455528.

11. Kuepper, Justin. "CBOE Volatility Index (VIX) Definition". Investopedia. Retrieved 2020-04-10.

12. National Financial Conditions Index (NFCI) Background - Federal . Retrieved December 2, 2022, from https://www.chicagofed.org/research/data/nfci/background

13. Uri Benzion, Alon Granot, Joseph Yagil, The valuation of the exponential function and implications for derived interest rates, Economics Letters, Volume 38, Issue 3, 1992, Pages 299-303, ISSN 0165-1765, https://doi.org/10.1016/0165-1765(92)90074-9.