Mihály Ormos and Dávid Zibriczky

Department of FinanceBudapest University of Technology and Economics

5th international ECEE series conference "Economic Challenges in Enlarged Europe„

Tallinn, Estonia, June 16-18, 2013

Risk vs. expected return (risk premium)

Standard deviation (Markowitz, 1952)

Assumes normal distribution

CAPM beta (Sharpe, 1964)

Requires market portfolio

Assumes linearity

Measures systematic risk only

Problems:

Assumptions

Low explanatory power

Alternative single risk measure?

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mathematically-defined quantity that is generally used for characterizing the probability of outcomes in a system through a process

Main applications:

Thermodynamics: Clausius (1867) – distribution of inner energy

Statistical Mechanics: Boltzmann (1872) – molecular disorder

Information Theory: Shannon (1948) – message compression

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Generalized discrete formula: 𝐻𝛼 𝑋 =1

1−𝛼log 𝑖 𝑝𝑖

𝛼

𝑝𝑖: probability of discrete outcome Xi

Special cases 𝛼 = 1: Shannon entropy (L'Hôpital's rule)

𝛼 = 2: Rényi entropy

Problem: Value of return is continuous, discrete formula cannot be applied

Continuous formula: 𝐻𝛼 𝑋 =1

1−𝛼ln 𝑓 𝑥 𝛼𝑑𝑥 (differential entropy)

𝑓 𝑥 :probability function

𝑓 𝑥 = ?

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Density estimation: 𝑓𝑛 𝑥 ~ 𝑓 𝑥

Methods

Histogram (fixed bin width)

Kernel density estimation (sum of core weights)

Sample spacing (fixed number of elements in one bin)

Histogram Kernel Sample spacing

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Original formula: 𝐻𝛼 𝑋 =1

1−𝛼ln 𝑓 𝑥 𝛼𝑑𝑥

Entropy estimation in single formula using histogram*:

Shannon entropy: 𝐻1,𝑛 𝑋 = −1

𝑛 𝑗 𝑣𝑗 ln

𝑣𝑗

𝑛ℎ

Rényi entropy: 𝐻2,𝑛 𝑋 = −ln 𝑗 ℎ𝑣𝑗

𝑛ℎ

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Entropy risk measure**: 𝜿𝑯𝜶𝑺𝒊 = 𝒆

𝑯𝜶,𝒏 𝑹𝒊−𝑹𝑭

* 𝑣𝑗: number of elements falling into the jth bin, h: bin size

** 𝜿: risk measure, 𝑺𝒊: Security i, 𝑹𝒊 − 𝑹𝑭: Risk premium6

Standard deviation (Markovitz):

𝜿𝝈 𝑺𝒊 = 𝝈 𝑹𝒊 − 𝑹𝑭

Beta (CAPM):

𝜿𝜷 𝑺𝒊 =𝐜𝐨𝐯 𝑹𝒊−𝑹𝑭,𝑹𝑴−𝑹𝑭

𝝈𝟐 𝑹𝑴−𝑹𝑭

Shannon entropy:

𝜿𝑯𝟏𝑺𝒊 = 𝒆

𝑯𝟏 𝑹𝒊−𝑹𝑭

Rényi entropy:

𝜿𝑯𝟐𝑺𝒊 = 𝒆

𝑯𝟐 𝑹𝒊−𝑹𝑭

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Source: The Center for Research in Security Prices (CRSP)

Series: Daily return

Market return (value weighted)

Risk free rate (1-month T-bill)

150, randomly selected securities from the components of S&P500 index

Period: 1985-2011 (27 years or 6810 days)

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0

1

2

3

4

5

6

7

8

1 10 100

Averag

e r

isk

Number of securities in portfolio

Shannon Rényi StDev

0%

10%

20%

30%

40%

50%

1 10 100

Averag

e r

isk r

ed

ucti

on

Number of securities in portfolio

Shannon Rényi StDev

9

-0,02

0

0,02

0,04

0,06

0,08

0,1

0 2,5 5 7,5 10 12,5 15

E(r

p-r

F)

Risk (H1)

Shannon entropy

n=1

n=2

n=5

n=10

10

-0,02

0

0,02

0,04

0,06

0,08

0,1

0 2,5 5 7,5 10 12,5 15

E(r

p-r

F)

Risk (H1)

Shannon entropy

n=1

n=2

n=5

n=10

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Evaluation method

Long term (1985-2011), for 150 random securities

Explanatory variable (X): Risk

Target variable (Y): Expected risk premium

Linear regression (X,Y)

Explanatory power: Goodness of fit of regression line (R2)

Result

Higher explanatory power

Risk measure R2 long

Standard deviation 7.83%Beta 6.17%Shannon entropy 12.98%

Rényi entropy 15.71%

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-0,02

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0 1 2 3 4 5

E(r

i-r

F)

Risk (std)

Standard deviation

E(ri-rF) = 0.0170 + 0.0085*std

R² = 7.83%

-0,02

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0 0,5 1 1,5 2

E(r

i-r

F)

Risk (beta)

Beta

E(ri-rF) = 0.0209 + 0.0151*beta

R² = 6.17%

-0,02

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0 2,5 5 7,5 10 12,5 15

E(r

i-r

F)

Risk (H1)

Shannon entropy

E(ri-rF) = 0.0091 + 0.0034*H1

R² = 12.98%

-0,02

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0 2,5 5 7,5 10 12,5

E(r

i-r

F)

Risk (H2)

Rényi entropy

E(ri-rF) = 0.0059 + 0.0049*H2

R² = 15.71%

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0

0,05

0,1

0,15

0 10 20 30 40 50 60 70 80 90 100

Exp

lan

ato

ry p

ow

er (

R-s

qu

ared

)

Number of securities in portfolio

StDev Beta Shannon Rényi

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-0,3

-0,2

-0,1

0

0,1

0,2

0 5 10 15 20

E(r

i-r

F)

Risk (H1)

Shannon entropy

bear market

E(ri-rF) = 0.0818 - 0.0150*H1

R² = 0.3961

-0,05

0

0,05

0,1

0,15

0,2

0 5 10 15

E(r

i-r

F)

Risk (H1)

Shannon entropy

bull market

E(ri-rF) = -0.0116 + 0.0103*H1

R² = 0.4345

Risk measure R2 long R2 bull R2 bear

Standard deviation 7.83% 33.9% 36.7%Beta 6.17% 36.7% 43.7%Shannon entropy 12.98% 43.5% 39.6%

Rényi entropy 15.71% 42.4% 38.6%

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Evaluation method

Estimating risk based on a 5-year period (short term)

Predicting average risk premium for the next 5-year period

Applying this on several periods

Predicting power: Average goodness of fit (R2) based on tested periods

Reliability: Standard deviation of R2 values (lower is better)

Result

Better explanatory power for the same first 5 years (R2 short)

Better predicting power for the next 5 years (R2 pred)

Higher reliability (σ)

Risk measure R2 long R2 short R2 pred σ short σ pred

Standard deviation 7.83% 7.94% 9.70% 0.73 0.63Beta 6.17% 13.31% 6.45% 0.95 0.99Shannon entropy 12.98% 13.38% 10.15% 0.67 0.62

Rényi entropy 15.71% 12.82% 9.34% 0.62 0.60

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No assumptions concerning the returns

Entropy estimation doesn’t require an undefined market portfolio

Characterizes specific risk and captures diversification effect

More efficient and reliable risk estimate compared to the standard methods

If the trend is identified entropy based equilibrium model behaves similarly to the

standard models

Thank you for paying attention!

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