Building on the work of Robert Shiller, in recent posts I investigated the use of the CAPE ratio to predict future stock market performance and examine for the structural change in market valuation over time. This work revealed that stock market returns depend significantly on valuation and are surprisingly predictable in the long term based on these simple measures.
While the relationship between valuation and returns is well documented in the finance literature, what is less well understood is the relationship between valuation and return volatility. As we observed in “Stock Market Valuation and the 2020’s in R”, high valuations are strongly associated with low future returns typically unfolding over several years. It is also commonly accepted that poor returns are associated with higher volatility. Therefore, it would stand to reason that higher valuations should also be associated with more volatile returns. If true, then high valuations would imply low and volatile future returns; a perfect storm for investors!
In this post, I will explore the relationship between the CAPE ratio (as our preferred valuation metric) and return volatility. Along the way I will present all of my R code for you to follow along. All of the data used in this study can be freely obtained from Robert Shiller’s website here. The time period and frequency under consideration is monthly from January 1970 through June 2020.
Defining Volatility
To begin, it’s important to define the nature of the volatility under investigation. In my previous posts on performance and valuation we defined the performance metric as the annualized, geometric return for stocks over a period of either 3 or 5 years in the future. The fact that we are using the geometric return as opposed to the more common arithmetic average return is an important distinction. Let’s quickly review the difference.
Arithmetic Return
An arithmetic average is simply the sum of a series of numbers divided by the count of that series of numbers. This is the notion of average that most everyone knows and is comfortable with. The formula for the average is defined as follows:
Where:
- N = Number of periods
- Ai = The return over period ‘i’
Geometric Return
The geometric mean for a series of numbers is calculated by taking the product of these numbers and raising it to the inverse of the length of the series. The geometric average is less uniformly used, but (as we shall see) is generally more reflective of reality. The formula is as follows:
Where:
- N = Number of periods
- Xi = The return over period ‘i’
Example
In order to illustrate the difference, I’ll use the following extreme and rather unrealistic, but very instructive example for the case of a 2-period return.
Let’s suppose that you’ve held TSLA stock in your portfolio for 2 months and over this two-month period you have the following sequence of returns:
- Month 1: 100%
- Month 2: -50%
On second thought, in the case of TSLA, maybe this example isn’t as unrealistic as I thought! Now let’s calculate the arithmetic average return over your holding period.
If you’re using the arithmetic return, then it appears that you’ve done really well: your investment is up 25% in just two months! Now let’s take a look at your geometric return:
Based on the geometric return, you’ve done a lot is a lot worse: you haven’t made any money and are right back where you started!
There is no set standard in finance for how to measure return, but you can see the big difference it makes. Unfortunately, in this case, the geometric return represents the reality.
Volatility
To summarize, in previous posts our focus was on using the CAPE ratio today to forecast returns either 3 or 5 years in the future. I will be taking the same approach here: using today’s CAPE to forecast future realized volatility. Volatility will be defined as the standard deviation of the annualized, geometric rate of return.
The volatility of an investment is of obvious interest to investors and the notion of volatility that we are using here describes how your future average return can fluctuate from one month to the next. Consider the following example:
You expect an investment to produce an average annual return of 5% over the next 3-years. Consider how this 5% might accrue to you. It could produce 5% each year with no deviation so your geometric average return would appear as follows:
Or you could make 10% in Year 1, -8% in Year 2 and 14% in Year 3 and your geometric average return would be as follows:
The end result is the same, but the path to get there is not.
Consider a second example which is the motivation behind our study. Suppose it is March 1998 and you are considering investing in the market. Tech stocks have been doing really well and you want to get in on the action! However, you don’t know if you should invest your money all at once, dollar-cost-average in over the next several months or try to time in at some point in the future.
If you had invested all of your money in March 1998, then over the next 3 years your average yearly return would have been 7.33%. Nothing crazy, but a tidy profit indeed. Now, let’s say that you decided to wait and try and buy in at a dip. In March, the S&P 500 was at 1,074 points and continued to climb steadily upward during the summer to reach 1,156 at the end of July, then, in August, it dipped back down to 1,076. Perfect, this is basically the level it was at in March and you decide to buy in. Had you done so, your average return over the next 3 years would have been a paltry .38%.
That’s a big difference in realized return just for waiting a few months to buy! Contrast the 1998 experience with the following: an investment made in October 2003 would have produced an average return of 8.31% in the subsequent 3 years. A few months later in March 2004, that same investment would have returned 7.58% in the next 3 years. These two returns are much closer and the investor is not penalized for waiting or timing in incorrectly.
Quantifying how different future returns can be month-to-month should be of interest to any investor and will be the focus of our study. Using the CAPE ratio, we will try to forecast the volatility of future geometric rates of return for the S&P 500. By doing so we hope to gain better insight into the “point in time” risk/reward trade off of investing in the market.
CAPE Ratio and Volatility Forecasting
The below plot shows the rolling annualized volatility for the S&P 500 from January 1970 through June 2020 at 1-, 3-, 5- and 10-year intervals. We observe that returns can vary greatly on a yearly basis, but that volatility declines steadily as we expand the window to 3- ,5- or 10-years. This observation lends some credence to our hypothesis that returns are more forecastable in the long run.
For this study I will focus on forecasting future volatility at 3- and 5- years. The general model specification will be as follows and will use simple linear regression:
3-Year Forecast
The following R Code runs the model, builds a prediction interval and plots the results with ggplot2.
We can see that the coefficient for the CAPE ratio is positive which implies that an increase in valuation is associated with an increase in future volatility . Furthermore, the coefficient is highly significant which shows that current valuation is an important determinant of future returns. Both of these results conform with our expectations (thank goodness!). The R-squared is ~29%, which indicates that valuation is able to explain a reasonable percentage of the total variability in realized volatility. The explanatory power is pretty good for a univariate model, but clearly there is also a lot that is left unexplained.
Below I plot of the results along with the prediction interval:
The regression line shows a general increase in volatility as the CAPE increase. However, the prediction interval is rather wide, indicating substantial disparity. Curiously, at “extreme” valuations (i.e. <15 and >30) the impact of valuation on volatility appears more consistent and “tight” around the mean. It is at average levels of valuation that the impact appears more defuse. At average valuations, future realized volatility has spanned anywhere from 2% to 10%. This is a very wide range and it’s not immediately clear how we would use this information in practice.
5-Year Forecast
Turning to the 5-year forecast:
The results from the regression are very encouraging. Again, the CAPE ratio is positive and highly statistically significant, but moreover the coefficient of .001046 is very close to the estimate from the 3-year model. The R-squared of ~.25 is also similar to the estimate from the 3-year model. Taken together these results demonstrate consistency between the models over two different time periods and suggests robustness to our results.
Taking a look at the plot, we observe the same general upward trending relationship between volatility and valuation. Similar patterns are also evident: at the extremes the relationship appears tighter with low valuation closely associated with low volatility and high valuation associated with high volatility. In the middle, the picture is more muddled. In fact, there exists a cluster of points that appears distinct from the rest of the group. Let’s label these points:
We observe that all of these points are between 1993 and 1995. As any good market historian knows, 5 years hence was the peak and subsequent collapse of the Tech Bubble; indeed, during this period there was substantial volatility.
What Does this Mean for Stocks Today?
While not the only factor, the results from our model indicate that current market valuation is an important determinant for forecasting future market volatility. In particular, when valuation, as measured by the CAPE ratio, is “extreme” the model appears to produce a tighter forecast than when valuation is closer to its long run average. These results are of obvious utility to investors who need to decide where to allocate future capital.
With that said, what does the model say about today? At 33.5, the CAPE ratio is certainly getting close to the extreme range. Our 5-year forward prediction is calculated as follows:
If history is any guide, then this level of future 5-year average vol. is pretty high. For context, the realized volatility for the period beginning January 2015 was 2.3% so we are forecasting volatility that is over double that which we have observed over the past 5-years, which in many respects has not been a smooth ride.
Taken in conjunction with the performance forecast of 1.8% from “Structural Change in Stock Market Valuation” investors can expect low and volatile returns over the next 5 years.
This does not mean that 2021 will not be a great year for stocks. With easy monetary policy, vaccine distribution and broad scale economic reopening I think it’s quite possible that 2021 ends up being a good year. It is the middle part of the decade that, in my view, we should be concerned with.
Concluding Remarks
In this post I built a case for investors to consider the volatility of future annualized geometric returns when making investing decisions. To that end, I built a simple model which proposed that current valuation is an important contributing factor to future stock market volatility. Using the CAPE ratio we found that valuation does partially explain future vol. Furthermore, the relationship was strong when valuation approached extreme levels. However, substantial variability in realized vol. was left unexplained by the model which provides fruitful ground for research.
I hope you all found these findings as interesting as I did! Until next time, thanks for reading!
-Aric Lux.
Originally published at https://lightfinance.blog on January 2, 2021.
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