1) The document presents a study comparing entropy-based risk measures to traditional measures like standard deviation and beta. 2) Entropy measures calculate risk based on the probability distribution of returns rather than assumptions of normality. 3) The study finds entropy measures have higher explanatory power of expected returns, especially Rényi entropy, and better predict risk over multiple time periods. 4) Entropy measures provide a potential improved alternative to traditional risk measures by not requiring assumptions, better capturing diversification, and producing a more reliable estimate of risk.

1. Mihály Ormos and Dávid Zibriczky
Department of Finance
Budapest University of Technology and Economics
5th international ECEE series conference "Economic Challenges in Enlarged Europe„
Tallinn, Estonia, June 16-18, 2013

2.  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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3. 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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4.  Generalized discrete formula: 𝐻 𝛼 𝑋 =
1
1−𝛼
log 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: 𝐻 𝛼 𝑋 =
1
1−𝛼
ln 𝑓 𝑥 𝛼
𝑑𝑥 (differential entropy)
 𝑓 𝑥 : probability function
 𝑓 𝑥 = ?
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5.  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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6.  Original formula: 𝐻 𝛼 𝑋 =
1
1−𝛼
ln 𝑓 𝑥 𝛼
𝑑𝑥
 Entropy estimation in single formula using histogram*:
 Shannon entropy: 𝐻1,𝑛 𝑋 = −
1
𝑛 𝑗 𝑣𝑗 ln
𝑣 𝑗
𝑛ℎ
 Rényi entropy: 𝐻2,𝑛 𝑋 = −ln 𝑗 ℎ
𝑣 𝑗
𝑛ℎ
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 Entropy risk measure**: 𝜿 𝑯 𝜶
𝑺𝒊 = 𝒆
𝑯 𝜶,𝒏 𝑹 𝒊−𝑹 𝑭
* 𝑣𝑗: number of elements falling into the jth bin, h: bin size
** 𝜿: risk measure, 𝑺𝒊: Security i, 𝑹𝒊 − 𝑹 𝑭: Risk premium
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7.  Standard deviation (Markovitz):
𝜿 𝝈 𝑺𝒊 = 𝝈 𝑹𝒊 − 𝑹 𝑭
 Beta (CAPM):
𝜿 𝜷 𝑺𝒊 =
𝐜𝐨𝐯 𝑹 𝒊−𝑹 𝑭,𝑹 𝑴−𝑹 𝑭
𝝈 𝟐 𝑹 𝑴−𝑹 𝑭
 Shannon entropy:
𝜿 𝑯 𝟏
𝑺𝒊 = 𝒆
𝑯 𝟏 𝑹𝒊−𝑹 𝑭
 Rényi entropy:
𝜿 𝑯 𝟐
𝑺𝒊 = 𝒆
𝑯 𝟐 𝑹𝒊−𝑹 𝑭
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8.  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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9. 0
1
2
3
4
5
6
7
8
1 10 100
Averagerisk
Number of securities in portfolio
Shannon Rényi StDev
0%
10%
20%
30%
40%
50%
1 10 100
Averageriskreduction
Number of securities in portfolio
Shannon Rényi StDev
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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(rp-rF)
Risk (H1)
Shannon entropy
n=1
n=2
n=5
n=10
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11. -0,02
0
0,02
0,04
0,06
0,08
0,1
0 2,5 5 7,5 10 12,5 15
E(rp-rF)
Risk (H1)
Shannon entropy
n=1
n=2
n=5
n=10
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12.  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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13. -0,02
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
0 1 2 3 4 5
E(ri-rF)
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(ri-rF)
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(ri-rF)
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(ri-rF)
Risk (H2)
Rényi entropy
E(ri-rF) = 0.0059 + 0.0049*H2
R² = 15.71%
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14. 0
0,05
0,1
0,15
0 10 20 30 40 50 60 70 80 90 100
Explanatorypower(R-squared)
Number of securities in portfolio
StDev Beta Shannon Rényi
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15. -0,3
-0,2
-0,1
0
0,1
0,2
0 5 10 15 20
E(ri-rF)
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(ri-rF)
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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16.  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 0.63
Beta 6.17% 13.31% 6.45% 0.95 0.99
Shannon 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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17.  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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