Variety is a measure for quantifying the dispersion around a change in the market

1. Ch.14: Introducing Variety
in Risk Management
From “The Best of Wilmott 1” (Wiley, 2004)
Yoshiharu Sato
University of Warsaw
(https://sites.google.com/site/yoshi2233/)

2. Authors
・ Fabrizio Lillo
PhD in Physics (Palermo, Italy)
Professor of Quantitative Finance
Research Focus:
・ Market Microstructure
・ High Frequency Finance
・ Rosario Mantegna
PhD in Physics (Palermo, Italy)
Professor of Applied Physics
Pioneer in Econophysics

3. Authors (cont.)
・ Jean-Philippe Bouchaud
PhD in Physics (ENS, France)
Professor of Statistical Physics
Pioneer in Econophysics
Chairman of CFM
・ Marc Potters
PhD in Physics (Princeton, USA)
CEO of CFM

4. It's All About Change
・ On April 14, 2015, the S&P 500 rose 3.41 points, or 0.16%
= Daily Change
・ In 2014, the standard deviation σ of S&P 500's daily changes
was 0.72%
= Daily Volatility
・ σ x √252 = 0.72% x 15.87 = 11.43%
= Annual Volatility
・ But how can we quantify the dispersion around a change
if S&P 500 went up by 3% in one day and if half the stocks
went up by 5% and the other half went down by 3%?

5. Variety
・ We introduce a new quantitative measure called 'variety'
・ If the variety is 0.1%, then most stocks have indeed made
between 2.9% and 3.1%. But if the variety is 10%, then stocks
followed rather different trends during the day and their
average happened to be positive, but this is just an average
information

6. Variety vs Volatility
・ Variety is not the volatility of the index
・ Volatility refers to the amplitude of the fluctuations of the
index from one day to the next, not the dispersion of the
result between different stocks
・ Consider a day where the market has gone down 5% with
a variety of 0.1% – that is, all stocks have gone down by
nearly 5%. This is a very volatile day, but with a low variety
・ Low variety means that it is hard to diversify since all stocks
behave the same way

8. One-Factor Model
・ Theoretical relation between variety and volatility can be
obtained within the framework of the one-factor model, which
suggests a positive correlation between volatility and variety
・ Idiosyncratic return ϵi(t): the part of the excess return (=
difference between an asset's return and the risk-free rate)
that is not explained by common factors (= elements of return
that influence many assets; e.g., size, valuation)

9. One-Factor Model (cont.)

10. One-Factor Model (cont.)
・ One-factor model assumes that the idiosyncratic part is
independent of the market return. In this case, the variety of
idiosyncratic terms ν(t) is constant in time and independent
from rm
・ However, the empirical results show that a significant
correlation between ν(t) and rm(t) indeed exists. The degree
of correlation is different for positive and negative values of
the market average
・ Best linear least-squares fit between ν(t) and rm(t) provides
different slopes when the fit is performed for positive (slope
+0.55) or negative (slope −0.30) value of the market average

12. Correlation between Stocks

14. Asymmetry
・ A(t) = rm(t) − r*(t)
・ Median r* is, by definition, the return such that 50% of the
stocks are above, 50% below. If more than 50% of the stocks
performed better than the market, the median is larger than
the average, therefore A is negative and vice versa
・ Asymmetry is also correlated with the market factor. Large
positive days show a positive skewness in the distribution of
returns (= a few stocks do exceptionally well) whereas large
negative days show the opposite behavior

16. Thanks