**Executive Summary**

- Market risk, that is to say the variability of returns from one period to the next, or volatility, is one of the many risks faced by investors in equities, bonds and currencies. In the case of U.S. equities, the VIX by the Chicago Board Option exchange (CBOE) measures implied volatility, i.e. the level of volatility implied by the prices of S&P options. It is seen as a forward looking measure, in contrast to realized (or actual) volatility, which measures the variability of historical (or known) prices.
- In this paper, we introduce perceived volatility, an exponential and dynamically-weighted combination of past returns on the S&P 500. We find that it is closely correlated with implied volatility, showing that the VIX may just be a (smart) reflection of the past instead of a real forward-looking measure, as generally claimed.
- Just like implied volatility, perceived volatility has some predicting power of future volatility (27% standard estimation error), but the model is unstable and limited to short forecasting horizons (38-trading day is the optimal forecasting horizon, very close to the VIX promise).
- This matters because volatility — be it historical or implied — is widely used to calibrate risk-taking in the financial services industry, from volatility-targeting strategies, to collateral requirements estimation, to prudential regulation. As such, it is a potential endogenous source of man-made unknown unknowns.

**Market volatility: What a maze!**

No pain, no gain! No risk, no return! Simple as it is, this basic idea underpins both the theory and the practice of investing. But what is risk and how should it be measured? Investment risk is a hydra that manifests itself in different ways. For instance, the buyer of a corporate bond may or may not get in due time all the money owed to him by the issuer — this is credit risk. Similarly, the owner of a stock may or may not find it easy to sell her holdings without depressing market prices — this is liquidity risk. The price in a week’s time of a share, a bond or a currency may or may not be very different from today’s price: this is market risk, the subject matter of this paper.

Historical volatility relates to the variability of past returns. For example, in the case of the S&P 500 from January 3rd, 1986 to December 6th, 2019 (i.e. 8,844 equally-weighted daily observations), the average daily (log) return has been 0.03% while its daily standard-deviation has been 1.11%. Had we conducted this exercise on 32 equally-weighted annual observations, we would have found that the average annual return has been 7.98%, while its annual standard-deviation has been 15.63%. That the historical daily volatility of the S&P 500 has been 1.11% over the last 33 years (or 15.63% at annual rate) may be a valuable statistical observation for an historian. But what matters to an investor is the volatility of returns over the next 30, 90, 180 days … depending on his investment horizon. How confident can he be that the past is any guide to the future?

Now, hardly a day or a week goes by without pundits suggesting that the CBOE’s Vix index captures U.S. equity market risk. When they do bother to define this index, they are content with repeating - time and again - how conventional wisdom claims the Vix should be interpreted. To wit:

“Wall Street’s so-called fear gauge […], the CBOE’s Vix index, […] gauges investor expectations of short-term volatility in US stocks. The Vix index is the most closely watched gauge of implied volatility in the world’s biggest stock market. It reflects the cost of buying short-term options on the S&P 500”;

or, from other recent example:

“The Vix volatility index – a measure of expected swings in the S&P over the next 30 days – has slumped to 12.78, not far off its lowest levels of the year, and well below its 30-year average of around 19.”

To be fair, prominent practitioners lend their credibility to such statements. To wit:

“Implied volatilities are used to monitor the market’s opinion about the volatility of a particular stock. Whereas historical volatilities are backward looking, implied volatilities are forward looking.” ;

Another example:

“[Implied volatilities] are based on prevailing market prices rather than on the past history of returns and, therefore, they are forward-looking measures of volatility.”

These four statements create a kind of maze of seemingly important concepts: backward looking historical volatilities, forward-looking implied volatilities, volatility index, investor expectations of short-term volatility and Wall Street’s fear gauge.

How to walk through this maze, how to connect these dots, is the subject matter of the present investigation. We will start by defining historical as well as implied volatility. We will then argue that implied volatility is not as forward-looking as generally believed. Provided historical returns are properly weighted, implied volatility is indeed strongly correlated with historical volatility. Furthermore, the causality runs both ways, from implied to historical volatility, but also the other way around. This calls into question the basis of some of the most closely watched barometers of market uncertainty, and has consequences for and volatility-targeting strategies.

**Historical volatility is...volatile**

The essential characteristic of the daily, weekly, monthly… historical (or past) returns on bonds and equities is that they are not constant through time. They can be positive or negative, and more or less so in either direction. In other words, they are volatile. In first approximation at least, the distributions of historical asset returns are close to the famous bell-curve distribution, with a large bump in the middle and progressively thinner and thinner symmetric tails on either side.

At the center of the distribution lies the average historical return; the larger the absolute deviation from the average return, the lower its frequency. Under the controversial assumption that empirical distributions follow the bell-curve pattern, one single statistical parameter – the standard-deviation of returns – suffices to characterize the dispersion of returns around their average, as shown in Table 1 In finance, the standard deviation of historical returns is called historical volatility. On it depend both the range and the frequency of potential outcomes.