steiner 12 annualized volatility

8/9/2019 Steiner 12 Annualized Volatility

1/8Electronic copy available at: http://ssrn.com/abstract=2007620

2011, Andreas Steiner Consulting GmbH. All rights reserved. 1 / 8

Research Note

Andreas Steiner Consulting GmbH

February 2012

Annualized Volatility

Introduction

In this research note, we compare S&P 500 volatility figures calculated with the popular

square-root-n rule to volatility figures derived from time-aggregated daily returns and try to

reconcile the differences with popular time-series models featuring serial correlation inreturns or volatilities.

We show that the deviations from the square-root-n rule cannot be explained with serial

correlation in returns, rather with a GARCH model. We conclude that volatility figures

annualized with the square-root-n rule should not be interpreted as accurate estimates for

true annual volatility. The square-root-n rule is also not suitable to standardize volatility

figures for reporting purposes.

The Square-Root-N Rule

Volatility is financial market lingo for standard deviation of continuous returns1 and

generally used as a measure for return dispersion over time and a proxy for risk,

respectively for uncertainty.

If continuous returns are modeled as stochastic processes, volatility can be interpreted as

one particular characteristic of the process. Justified by the idea that price changes are

largely driven by unexpected changes in valuation factors, it is typically assumed that price

changes2of financial assets are independent over time. In fact, this assumption is nothing

else than assuming that markets exhibit weak-form information efficiency.

It can be shown formally that this assumption leads to the so called square-root-n rule for

annualizing volatilities3

(1) nCalculatedAnnualized

with Calculatedas the standard deviation measured from a time series of continuous returns

and n being the data frequency of this time series expressed in periods per annum. For

example, if we are calculating volatility of 6% from a monthly return time series, the square-

root-n rule tells us that the annualized volatility is

(2) %78.2012%6 Annualized
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The assumptions underlying the rule imply that it should not make a difference whether we

calculate annual volatility from annual, monthly, daily etc. returns. Therefore

(3) 412250,,,

QuarterlyCalculatedMonthlyCalculatedDailyCalculatedAnnualized

More generally, the square-root-n rule can be used to convert volatilities calculated from

returns with a certain frequency to volatility figures for returns over a longer time period. For

example, quarterly volatility can be derived from monthly volatilities as follows

(4) 3, MonthlyCalculatedQuarterly

Ironically, the square-root-n rule is used every day by thousands of active investment

managers in ex ante as well ex post portfolio analytics, i.e. investment managers which can

only create value-added for their clients if markets are at least informationally inefficient of

the weak form.

Annualized S&P500 Volatilities

In order to assess the validity of the square-root-n rule, we have analyzed a data set

consisting of S&P 500 daily continuous price returns from January 5, 1987 to November 20,

2011 (data source: Yahoo Finance).

We calculate annualized S&P500 volatility figures with two methods

Square-Root-N rule: Using daily volatility as a starting point, we calculate volatility

figures for various longer time periods up to one year (250 trading days)

Time-Aggregated Returns: Instead of converting volatilities, we convert the return

data. For example, weekly continuous returns can be calculated from daily returns

by simply adding five daily observations. The weekly return series generated this

way can then be used to calculate weekly volatility.

An alternative way to time-aggregated returns would be to resample/bootstrap from the

daily return data set. Nave resampling procedures destroy any dependence of returns or

volatilities over time. More sophisticated resampling procedures trying to preserve

dependence patterns exist, but introduce additional assumptions into the analysis. As

intertemporal dependence patterns are at the core of the issues discussed here, we prefer

to calculate volatilities from historical time-aggregated returns as described above.

The results from the above calculations4can be summarized graphically


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0.00%

2.50%

5.00%

7.50%

10.00%

12.50%

15.00%

17.50%

20.00%

22.50%

25.00%

0 50 100 150 200 250

Annualized

Volatility

Time Period [Expressed in Number of Periods p.a.]

Time-Aggregated Returns

Square-Root-N Rule

As we can see, volatilities calculated with the square-root-n rule generally do not coincide

with true historical volatilities calculated from time-aggregated returns. In this particular

case, the square-root-n rule seems to overestimate volatility on average by approximately

15%, in some cases by almost 25%.

The above result that returns are not independent is not really new, but has been reportedin numerous empirical studies. The more interesting question is: which kind of

independence over time causes the square-root-n to fail? This question is of practical

relevance, since certain dependence structures allow to filter the original data such that

the square-root-n rule can still be used.

Dependence in Returns and Volatilities

The simplest way to model dependence of returns over time is to assume that returns are

linearly autocorrelated5. Various explanations exist for autocorrelation in returns. For

example, it is rather typical to find autocorrelation in returns on markets with pricing issues

(e.g. fixed income, private equity, real estate).

Autocorrelation patterns in returns can be detected in a chart plotting autocorrelation

coefficients for returnsat various lags. For our S&P500 data


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-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

1 815

22

29

36

43

50

57

64

71

78

85

92

99

106

113

120

127

134

141

148

155

162

169

176

183

190

197

204

211

218

225

232

239

246

Autocorrelation

Coefficient

Lag

The red dotted lines define a confidence band; autocorrelation coefficients outside the band

can be considered statistically significant.

As we can see, autocorrelation coefficients for daily S&P500 returns are generally small,

and not statistically significant most of the time. This picture is rather typical for a rather

efficient market; price changes seem to be unpredictable, unsuitable for simple shorts-term

momentum or contrarian trading strategies.

One popular method to remove autocorrelations is the Blundell-Ward filter. This filter iscapable of removing first-order autocorrelations (i.e. autocorrelations calculated at lag one).

If we apply the Blundell-Ward filter to our S&P500 return series and then recalculate

volatilities, we get

0.00%

2.50%

5.00%

7.50%

10.00%

12.50%

15.00%

17.50%

20.00%

22.50%

25.00%

0 50 100 150 200 250

Annualized

Volatilit

y


Time-Aggregated Returns

Square-Root-N Rule


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As we can see, the Blundell-Ward filter removing first-order autocorrelations does not

improve the performance of the square-root-n rule significantly. The square-root-n rule still

systematically overestimates true volatilities.

Interestingly, autocorrelations for the squared returns of the same return time series

look very different

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

35.00%

1 815

22

29

36

43

50

57

64

71

78

85

92

99

106

113

120

127

134

141

148

155

162

169

17

6

183

190

197

204

211

218

225

232

239

246

Autocorrelatio

n

Coefficients

Lags

Autocorrelations in squared returns are clearly much larger (as high as 30%), statistically

significant and exhibit a decaying pattern with lag size. This pattern is typical for first-order

autoregressive processes. How do we interpret autocorrelations in squared returns

economically? This can be seen from the definition of standard deviation if we assume that

the arithmetic mean is so small that it can be neglected

(5)

N

i

ir

N

i

i rN

rrN 1

2

01

2

1

1

1

1

If the average return is small (and average daily returns are small indeed), then the

squared return can be interpreted as a contribution to volatility, respectively a volatility

innovation. Autocorrelations in squared returns can, therefore, be interpreted as

autocorrelation in volatility.

The most popular model exhibiting autocorrelation in volatility is GARCH(1,1), a specific

parameterization of a more general class of generalized autoregressive conditional

heteroscedastic models. Estimating the GARCH(1,1) allows one to compute short-term

volatilities, which are assumed to exhibit first-order autocorrelation.


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It looks like the GARCH(1,1) model explains autocorrelation in squared returns rather well.

In the chart below, we applied the square-root-n rule and volatility calculations based on the

time-aggregated returns to the GARCH(1,1)-standardized return

-

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

0 50 100 150 200 250

Annualized

Volatility


The GARCH(1,1) model nicely removes the systematic bias: the differences between the

volatilities calculated from time-aggregated returns and with the square-root-n rule are notonly smaller, but their signs seem to vary randomly.

Unfortunately, the volatility figures calculated from GARCH(1,1) standardized returns no

longer have an economic meaning. This is not a surprise, because GARCH(1,1) not only

feature first-order autocorrelation of volatilities, but also mean reversion. For example, in

turbulent times with high realized volatilities, volatilities have a tendency to revert back to a

lower long-term value. Annualizing with a square-root-n rule becomes meaningless, as we

would either need to predict future volatility innovations or work with expected volatility

innovations (which are zero by definition). Annualizing in a GARCH world is really

calculating forward volatilities, given current volatility levels and some GARCH parameters.6

Conclusions

Unlike returns, true volatility is a characteristic of a financial asset or portfolio which

cannot be measured directly, but always has to be estimated7. This statement applies to ex

ante as well as ex post analysis. While ex post return measurement is mainly an

accounting exercise, measuring ex post return volatility is about statistical inference,

requiring a rather different skill set from the person executing the calculations.

The square-root-n rule is a formula based on very specific assumptions. Unfortunately, the

rule is used by many practitioners without questioning whether data characteristics justifythe underlying assumptions. Additionally, many systems used by practitioners do not


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generate indicators which would allow users to assess the quality of the annualized

volatility figures calculated.

We do not recommend annualization of volatility figures for reporting purposes, as the

operation is likely to introduce systematic biases when comparing results for different return

time series. We recommend calculating annualized volatility from time-aggregated (i.e.annualized) returns instead. If the square-root-n rule has to be used, for example, due to

lack of access to the return series, we recommend to disclose additional information like

the data frequency at which the original volatility calculations were performed or statistical

indicators for autocorrelation in returns.

1See our research note on how to calculate discrete volatility.

2And therefore returns, which are nothing else than relative price changes.

3There exist other simple scaling rules if we make stronger assumptions about the stochastic

process generating returns. For example, if we assume that returns follow a so-called stable

distributions. But as financial returns generally exhibit finite variances, stable distributions are

generally not relevant for financial time series.

4All calculations in this research note were performed using our Advanced Portfolio Analytics

Excel Add-In, seewww.andreassteiner.net/apalibfor more information.

5Some authors use the expression serial correlation instead of autocorrelation.

6The formulas to calculate expected forward volatiles can be found in the GARCH literature.

7Alternatively, volatility can be inferred from markets: ex ante volatility is traded on rather liquid

markets (e.g. VIX). Ex post volatility is traded on (rather illiquid) OTC markets for certain

derivative instruments (e.g. variance swaps).
http://www.andreassteiner.net/apalibhttp://www.andreassteiner.net/apalibhttp://www.andreassteiner.net/apalibhttp://www.andreassteiner.net/apalib

steiner 12 annualized volatility

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