steiner 11 geometric and arithmetic volatility

Electronic copy available at: http://ssrn.com/abstract=1948795

© 2011, Andreas Steiner Consulting GmbH. All rights reserved. 1 / 7

Research Note

Andreas Steiner Consulting GmbH

October 2011

Geometric And Arithmetic Volatility

Introduction

Volatility as a risk measure has been criticized and dismissed on various grounds. Yet,

volatility is the dominating risk measure in the investment management industry and has

been carved in stone by several regulators. For example, the Committee of European

Securities Regulators (CESR) requires the risks and rewards of UCITS funds to be

expressed through a Synthetic Risk and Reward Indicator (SRRI). When looking at the

requirements in more detail, one finds that the indicator is mainly based on historic volatility

calculated using five years of return data.

At the same time, it is current industry practice (see for example the Global Investment

Performance Standards, GIPS) to use discrete returns, chain-linked annualized discrete

returns and geometric average discrete returns in investment reporting. Information on

continuous returns is clearly the exception.

In this research note, we show that there exist geometric and arithmetic versions of

volatility. We discuss how they relate and how they should be used. The relevance of the

topic arises from the fact that the current industry practice is to calculate arithmetic volatility

from discrete returns, which can distort reported volatility figures under certain conditions.

Arithmetic and Geometric Moments

Volatility in a time series context is defined as the standard deviation sA of continuous

returns…

n

i

AicontA mrn

s1

2

,

1

Standard deviation is the square root of the mean square deviation from the arithmetic

mean mA calculated as…

n

i

icontA rn

m1

,

1

The deviation measured in volatility is the arithmetic difference of continuous returns from

the mean, the arithmetic mean.

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Electronic copy available at: http://ssrn.com/abstract=1948795


Remember that continuous returns are called such because they assume continuous

compounding…

TcontT rMVMV ,0,0 exp

Discrete returns, on the other hand, assume a discrete compounding model…

TdiscT rMVMV ,0,0 1

The choice of compounding model does not affect wealth…

TdiscTcontT rMVrMVMV ,0,0,0,0 1exp

This simplifies to…

TdiscTcont rr ,0,,0, 1exp

From the above expression, we can derive well-known conversion formulas which allow the

quick and convenient conversion between continuous and discrete rates…

1exp ,, icontidisc rr idiscicont rr ,, 1ln

If we analyze continuous and discrete returns in a multi-period setting, we see that discrete

compounding is a multiplicative i.e. geometric process and continuous compounding an

additive i.e. arithmetic process…

ttdisctt rMVMV ,1,1 1

TTdiscdiscdiscT rrrMVMV ,1,2,1,1,0,0 1...11

TcontT rMVMV ,0,0 exp

TTcontcontcontTTcontcontcontT rrrMVrrrMVMV ,1,2,1,1,0,0,1,2,1,1,0,0 ...expexp...expexp

Note that there is no such thing as “additive” or “geometric” returns: additive and geometric

describe operations that in principle can be applied to any type of returns. Obviously, it

does not make sense to compound discrete returns in an additive fashion or to apply

geometric operations to continuous returns; but in principle, it could be done. That it should

not be done is rather well understood in the calculation of multi-period and average returns:

most performance analysts nowadays are aware that discrete returns require “chain-

linking”, the industry’s preferred expression for geometric compounding.

From the above analysis, it follows immediately that the geometric mean mG return is

defined as…

11

1

1

,

nn

i

idiscG rm

Substituting the definition of continuous returns and taking logarithms…


ncontcontcont

nn

i

icontG rrrn

rm ,2,1,

1

1

, expln...explnexpln1

expln1ln

The above expression simplifies to the arithmetic mean return...

A

n

i

icontG mrn

m 1

,

11ln

…respectively…

1exp AG mm ...and… GA mm 1ln

Therefore, the conversion formulas introduced above are not only applicable for single

period returns, but also average returns. Substituting the definition of discrete returns into

the volatility formula…

n

i

GidiscA mrn

s1

2

, 1ln1ln1

…and applying the exponential function to both sides of the above expression and

subtracting one…

G

n

iG

idiscn

iGidiscA s

m

r

nmr

ns

11

1ln

1exp11ln1ln

1exp1exp

1

2

,

1

2

,

…results in what is known as geometric volatility sG. From the above expressions, we see

that geometric volatility can be calculated directly as…

11

1ln

1exp

1

2

,

n

iG

idisc

Gm

r

ns

…or indirectly from arithmetic volatility…

1exp AG ss GA ss 1ln

We see that the conversion formulas not only apply to return and average returns, but also

standard deviation. In fact, it can be shown that the conversion formulas apply to any

moments1.

1 The central moments, to be more precise. The conversion formulas cannot be applied to the sample skewness and

excess kurtosis directly.


Note that the geometric standard deviation is not equal to the arithmetic standard deviation

calculated from discrete returns…

n

i

AidiscG mrn

s1

2

,

1 …with…

n

i

idiscA rn

m1

,

1

Applying an arithmetic operation to geometric data leads to results which do not have a

meaningful financial interpretation. This is most obvious when trying to interpret the

arithmetic mean for discrete returns. Unfortunately, this calculation has been implemented

in many spreadsheets and systems.

Analysis and Assessment

There is a simple check for illustrating the plausibility of the derived results: Volatility is

supposed to measure dispersion around the mean. The higher the volatility, the more

“scattered” the returns. Volatility itself is measured in “returns squared”, which is a rather

abstract unit of measurement. A more intuitive approach is to express dispersion as the

values that lie k-times the standard deviation from the mean. For continuous returns, this

is…

AAA skmlb AAA skmub

If mA = 10%, sA = 25% and k = 2, we get…

%40%252%10 Alb %60%252%10 Aub

For discrete returns, geometric versions of the above formulas have to be used…

1

1

1

k

G

GG

s

mlb 111

k

GGG smub

Applying the formulas to our example…

%5171.101%10exp Gm %4025.281%25exp Gs

%9680.32

%4025.281

%5171.1012

Glb

%2119.82%4025.281%5171.1012Glb

If the formulas are consistent, we would expect that…

1exp AG lblb 1exp AG ubub

…and…

GA lblb 1ln GA ubub 1ln

This is the case in our example because…

%9680.321ln%40

%2119.821ln%60


How relevant is the difference between arithmetic and geometric volatility? The smaller the

period of time over which returns are measured, the smaller the difference between

continuous and discrete returns. Therefore, the higher the data frequency, the smaller the

difference between geometric and arithmetic volatility. For example, when calculating daily

equity volatility…

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

18

.01

.51

18

.01

.54

18

.01

.57

18

.01

.60

18

.01

.63

18

.01

.66

18

.01

.69

18

.01

.72

18

.01

.75

18

.01

.78

18

.01

.81

18

.01

.84

18

.01

.87

18

.01

.90

18

.01

.93

18

.01

.96

18

.01

.99

18

.01

.02

18

.01

.05

18

.01

.08

18

.01

.11

S&P 500 Arithmetic and Geometric VolatilityDaily Volatilities

Arithmetic Volatility Geometric Volatility

The line for arithmetic and geometric volatility overlap, i.e. no relevant difference exists. If

we calculate 250 day volatilities from the same data, we get relevant differences…

0.00%

10.00%

20.00%

30.00%

40.00%

50.00%

60.00%

70.00%

18

.01

.51

18

.01

.54

18

.01

.57

18

.01

.60

18

.01

.63

18

.01

.66

18

.01

.69

18

.01

.72

18

.01

.75

18

.01

.78

18

.01

.81

18

.01

.84

18

.01

.87

18

.01

.90

18

.01

.93

18

.01

.96

18

.01

.99

18

.01

.02

18

.01

.05

18

.01

.08

18

.01

.11

S&P 500 Arithmetic and Geometric VolatilityAnnualized Rolling 250-Day Volatilities

Arithmetic Volatility Geometric Volatility


We see that the difference between geometric and arithmetic volatility only becomes

relevant if the figures are “large”. This can be seen when plotting the conversion formulas…

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

140.00%

160.00%

180.00%

200.00%

0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 120.00%

Geo

metr

ic S

tan

dard

Dev

iati

on

Arithmetic Volatility

It follows that, the riskier the investment strategy analyzed (e.g. equity versus money

market), the more relevant the difference between geometric and arithmetic volatility.

Both the exponential and logarithmic functions are positive monotone. It can be shown that

this mathematical property implies that rankings are not affected whether they are based on

geometric or arithmetic volatilities. For example, if all UCITS IV funds report their SRRI

expressed in terms of arithmetic volatility, it would not change the relative riskiness of a

UCITS fund if the regulator would switch to geometric volatilities.

While geometric and arithmetic moments can be converted directly, this is not possible with

Sharpe ratios…

1exp AG SharpeSharpe

Sharpe ratio rankings are unaffected by whether the measurement is arithmetic or

geometric (quotient of two positive monotone functions is positive monotone). In fact, the

Sharpe Ratio values do not change at all…

G

G

G

A

A

A

AA Sharpe

s

m

s

m

s

mSharpe

1exp

1exp

Interestingly, Sharpe ratio rankings are also not affected if returns are measured

geometrically and volatilities arithmetically (or vice versa). Of course, rankings are affected

when geometric Sharpe ratios is compared to arithmetic Sharpe ratios.

The arithmetic mean is always larger than the geometric mean. The same applies to

arithmetic standard deviation; it is always larger than its geometric counterpart. Individuals

typically like return and dislike volatility risk, which would explain the strong interest in

reporting geometric averages of discrete returns and absence of interest in reporting proper

geometric volatility figures.


Summary

Arithmetic and geometric averages are part of the established body of knowledge of the

investment industry. We have shown that there also are arithmetic and geometric

volatilities. The choice affects reported figures when they are large, and they never affect

the conclusions derived from comparing figures over time or portfolios: geometric and

arithmetic calculations do not affect the ranking of returns, volatilities or Sharpe ratios.

Comparisons may be distorted when mixing geometric and arithmetic calculations or

applying them in the wrong context. Unfortunately, the current industry practice of reporting

geometric average returns and arithmetic volatilities has the potential to distort risk and

return analysis. Discrete returns should be described with geometric statistics, continuous

returns with arithmetic statistics. The choice of the calculation methodology has to be

aligned with the compounding model chosen. The compounding model itself neither affects

investor wealth nor the performance of investment managers; it is merely a reporting

convention.

steiner 11 geometric and arithmetic volatility

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