How Does an Index Understate Volatility in the Equity Market and Its Impact

An index can understate volatility in the equity market because it only represents a small portion of the overall market, often around 20-30% of the total market capitalization.

This means that the index may not capture the full range of market movements, including those of smaller or less liquid stocks that can be more volatile.

In fact, a study found that the S&P 500 index only accounts for about 20% of the total US market capitalization, leaving out many smaller and mid-cap stocks that can be more volatile.

As a result, the index may give a misleading picture of the overall market's volatility, making it seem lower than it actually is.

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Indexes understate volatility because of the way they calculate volatility based on squared stock movements, giving one big price movement a disproportionately bigger impact than two smaller moves.

The statistical principle behind this is that volatility is calculated based on squared stock movements, making it sensitive to extreme price movements.

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This means that a single large price movement can significantly skew the volatility calculation, making it appear more volatile than it actually is.

The Black-Scholes and lattice models used in volatility calculations assume the last price is the best forecast for tomorrow's price, ignoring any other available information.

As a result, these models are prone to overestimating volatility, especially in markets with frequent price movements.

A statistical analysis has shown that using average-based volatility is actually less accurate than traditional models, rather than being superior as some might assume.

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Index volatility models like the Historical Volatility (HV) and Implied Volatility (IV) models can be misleading when comparing them to the actual equity market.

The HV model, for example, only looks at past data, which can lead to a skewed view of current market conditions. This is because it doesn't account for changes in market sentiment or unexpected events.

The IV model, on the other hand, takes into account the prices of options, but it can be influenced by factors like liquidity and trading volume, which can distort the true picture of market volatility.

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Comparing Volatility Models is a crucial step in understanding and managing risk in financial markets.

The GARCH model, for instance, is a popular choice for modeling volatility, but it can be computationally intensive and may not perform well with non-stationary data.

In contrast, the EGARCH model is a more flexible alternative that can handle non-linear relationships and asymmetries in volatility.

The GARCH model's simplicity and ease of implementation make it a popular choice, but it may not capture the full complexity of volatility dynamics.

The EGARCH model's ability to handle non-linear relationships makes it a better fit for markets with strong non-linear effects, such as those driven by news or surprises.

The SV model, on the other hand, is a more recent development that uses a stochastic process to model volatility, allowing for more realistic simulations of market behavior.

However, the SV model's increased complexity and computational requirements can make it more challenging to implement and estimate.

The GARCH and EGARCH models are both widely used in practice, but the choice between them ultimately depends on the specific characteristics of the market being modeled.

In general, the GARCH model is a good choice for markets with relatively stable volatility, while the EGARCH model is better suited for markets with more extreme and asymmetric volatility.

Indexes can be constructed using a selection of stocks that represent a particular market or segment, with the choice of stocks and weighting methodology influencing the index's sensitivity to market movements.

Market capitalization weighting, used by many major indexes like the S&P 500, tends to give more weight to larger companies, potentially masking the volatility experienced by smaller companies.

Indexes often have specific criteria for inclusion and may not be frequently updated, resulting in an index that does not fully reflect the current market conditions.

A stock with a price of $100 in an idle market, with volatility of 30%, should be between $74 and $135 68% of the time and $55 and $182 95% of the time.

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The use of average-price volatilities resulted in much-too-tight forecast ranges, with the likelihood of being outside the best guess interval more than 10% too high using the VWAP.

Indexes aim to be diversified across sectors, but sector biases can exist, potentially understating the overall market volatility.

Volatilities are a measure of how much stocks are expected to move over time and define a distribution, but actual stock returns don’t behave quite like our normal distribution, due to fat tails in the distribution.

The concept of autocorrelation and predictive power is crucial in understanding the behavior of stock prices. A correlation statistic can reveal how well today's return may predict tomorrow's, ranging from -1 to 1, with 0 indicating complete independence.

In reality, we often see a correlation of end-of-day returns of around -2%, a weak relationship. However, using an average, this figure is around 15%.

A hypothetical trader who bought the stock at the VWAP on days with positive returns and sold on days with negative returns, holding until the subsequent day's average, would make a healthy return - almost doubling their money over the course of a year.

This strategy is impossible in practice because you can't knowingly trade the VWAP while it's a relevant price.

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A positive volatility risk premium is the norm in normal times, with implied volatility staying above realized volatility. This suggests that investors should be short volatility, as the market is pricing in a higher risk of future losses.

The choice of lookback horizon is critical when calculating the variance risk premium. A short lookback horizon produces excessive noise, while a long lookback horizon causes time inconsistencies that compromise the volatility risk premium.

Realistic variance risk premia have been shown to predict returns with a statistical probability of near 100% on a daily basis for short-volatility positions through VIX futures. This is a significant finding, as it suggests that the premium can be used to protect against outsized drawdowns.

In contrast, the statistical probability of positive daily correlation of realistic volatility risk premia with subsequent equity index future returns is only 96% for S&P500 and EuroStoxx. This is still a respectable figure, but it falls short of the near 100% correlation seen with short-volatility positions.

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Here are some key statistics on the performance of realistic variance risk premia:

These statistics suggest that realistic variance risk premia can be a useful tool for investors looking to protect against volatility shocks and predict returns. However, they should be used in conjunction with other indicators and risk management strategies to ensure optimal performance.

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The volatility risk premium has been a reliable predictor of returns in the US market, with a statistical probability of positive daily correlation near 100% since 2000.

However, its accuracy drops to 96% when predicting returns of equity index futures, such as the S&P 500 and EuroStoxx.

A volatility risk premium indicator can't predict volatility shocks, but it can detect situations of potential denial and incomplete adjustments to a changed volatility regime.

This is especially true for large volatility surges, which often occur when the volatility risk premium is either negative or small.

A short-volatility strategy protected by the premium has performed well, with a balanced accuracy of 50.7% for the US market.

This means that the premium has been able to detect situations where a short-volatility strategy would be profitable, and avoid situations where it would be unprofitable.

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Market usage of average-price volatilities appears lacking, with no evidence of traders using this method in practice, despite its theoretical appeal.

Most trading in derivatives is on indices, where a VWAP simply doesn't exist because there's no trading in the index like in individual stocks.

The lack of VWAP estimates for indices suggests that traders haven't seen a need to capture the arbitrage opportunity this would imply.

It's impossible to trade at an average price, which casts doubt on the argument that it would represent trading activity better.

The market may never have seen this price in an actual trade, and you wouldn't know the average until the end of the day, potentially hours after it passed on a big move day.

Statistical analysis reveals that average-based volatility has two major problems: it's heavily influenced by one big price movement, and it assumes the last price today is the best forecast of tomorrow's price.

Volatility is calculated based on squared stock movements, so one big price movement will have a dramatically bigger impact on volatility than two smaller moves.

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The inconsistent timing of price observations is a major problem, making it difficult to calculate accurate volatility. This is because the standard formula for annualizing observations requires consistent observations.

The SEC pointed out this issue, and it's related to the problem of using a simple formula to calculate volatility. The formula is based on every observation having the same weight in the time series, which isn't possible with inconsistent timing.

The use of average price volatility is also subject to problems, including the fact that it doesn't reflect an observation at a particular time. This makes it impossible to project future "point in time" pricing using this method.

The SEC's guidance on fair value estimates requires companies to use appropriate and regular intervals for price observations, as well as a consistent point in time within each interval. However, the use of average price volatility often fails to meet these requirements.

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Here are some reasons why average price volatility may not meet the SEC's expectations:

Overall, the issues with average price volatility and inconsistent timing make it difficult to accurately calculate volatility and project future pricing.

FASB ASC subparagraph 718-10-55-37(d) explicitly states that an entity should use appropriate and regular intervals for price observations based on facts and circumstances that provide the basis for a reasonable fair value estimate.

The average price may reflect a daily price observation, but the timing of the average is uncertain and inconsistent, making it difficult to determine the actual price.

Company B's approach to using an average price may be closer to market close Monday, market open Tuesday, and midday Wednesday, which can lead to inaccurate estimates.

The SEC staff speech by Alison Spivey in 2005 highlights the objective of ASC 718 to ascertain the assumption about expected volatility that marketplace participants would likely use in determining an exchange price for an option.

Two potentially violating approaches include weighing the most recent periods of historical volatility more heavily than earlier periods, and relying solely on using the average value of the daily high and low share prices to compute volatility.

These methods will not meet the expectation of determining an appropriate estimate of expected volatility as one of the key assumptions used in determining a reasonable fair value estimate.

A volume weighted average price (VWAP), which takes an average of all trades during the day based on the number of shares in each trade, also fails to meet the SEC's expectations.

A summary of the violating approaches is as follows:

Calculating volatility based on average prices leads to problems, particularly when it comes to squared stock movements. One big price movement has a dramatically bigger impact on volatility than two smaller moves, even if the total effect on price over time is the same.

Statistical analysis has shown that using average-based volatility is not superior to traditional models. In fact, it's the opposite. The calculation of volatility is based on squared stock movements, which means one big price movement will have a much bigger impact on volatility than two smaller moves.

Volatility is applied in a way that assumes the best forecast of tomorrow's price is the last price today, with no other information that could give a better forecast at that time. This assumption is problematic when using average-based volatility.

US GAAP guidance, as stated in SAB Topic 14.D.1, requires entities to use appropriate and regular intervals for price observations. However, using average-based volatility does not meet this guidance, as the timing of the average is uncertain and inconsistent.

A volume weighted average price (VWAP) is another example of an averaging method that fails to meet GAAP compliance. VWAP takes an average of all trades during the day based on the number of shares in each trade, but it's not a superior measure of volatility.

Market usage of average-based volatility is also lacking. In fact, none of the authors' team members have seen the use of an averaging method to calculate volatility in practice, despite their experience working in the field.

A counterargument may be that an average price reflects actual trading activity throughout the day better than one potentially arbitrary price. However, it's impossible to trade at an average price, which casts doubt on this argument.

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Inconsistent observations can be a real challenge in finance. The SEC pointed out a problem related to inconsistent timing between price observations.

The formula to annualize observations requires consistent observations, which is crucial for using a simple formula to get a single annual measure. This formula is the standard deviation of daily returns multiplied by the square root of 252, regardless of how many observations we have.

There are an average of 252 trading days in a year, which is why the square root of 252 is used in the formula. Weekly and monthly returns use the square roots of 52 and 12, respectively, for the same reason.

Applying this math to project future "point in time" pricing is impossible because the average price doesn’t reflect an observation at a particular time.

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2015 was a tough year for the stock market, with stock market volatility being considerably higher than usual. The average S&P 500 stock was down 4% for 2015, according to an analysis by Strategas Research Partners.

The strong performance of just four stocks, Facebook, Amazon, Netflix, and Google (then known as Alphabet), masked the overall market weakness. This is a great reminder that even in a down year, there can be some bright spots.

Bond markets were also mixed, with indexes of lower-rated corporate bonds posting negative returns due to weakness in the energy and industrial sectors.