Ch.14: Introducing Variety in Risk Management

From “The Best of Wilmott 1” (Wiley, 2004)

Yoshiharu SatoUniversity of Warsaw

(https://sites.google.com/site/yoshi2233/)

Authors

・ Fabrizio LilloPhD in Physics (Palermo, Italy)Professor of Quantitative FinanceResearch Focus:・ Market Microstructure・ High Frequency Finance

・ Rosario MantegnaPhD in Physics (Palermo, Italy)Professor of Applied PhysicsPioneer in Econophysics

Authors (cont.)

・ Jean-Philippe BouchaudPhD in Physics (ENS, France)Professor of Statistical PhysicsPioneer in EconophysicsChairman of CFM

・ Marc PottersPhD in Physics (Princeton, USA)CEO of CFM

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 changeif 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%?

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

Variety vs Volatility

・ Variety is not the volatility of the index

・ Volatility refers to the amplitude of the fluctuations of theindex from one day to the next, not the dispersion of theresult between different stocks

・ Consider a day where the market has gone down 5% witha variety of 0.1% – that is, all stocks have gone down bynearly 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

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)

One-Factor Model (cont.)

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 degreeof 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

Correlation between Stocks

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 stocksperformed better than the market, the median is larger thanthe 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 ofreturns (= a few stocks do exceptionally well) whereas large negative days show the opposite behavior

Thanks