In a couple of days, Benoit will give a talk at the 9th Conference on Public Economics At the Regional and Local level (PEARL) in Helsinki, in Finland organized by the VATT (Government Institute for Economic Research). The slides are now available on the blog, and the paper is still online here
Category Archives: Research
Talk on multivariate comonotonicity and risk measures, JDS in Brussels
Today is the last day of the Journées de Statistique, in Brussels, http://jds2012.ulb.ac.be/. Alfred gave a survey on “Multivariate comonotonicity, stochastic orders and risk measures” in plenary session, this morning (as invited speaker). I have uploaded the slides.
Basketball: score dynamics and game theory
Tomorrow morning, I will be giving a talk at Mont Tremblant, for the Journées de la Société Canadienne de Sciences Economiques. I will present a joint work – in progress – with Nathalie Colombier and Romuald Elie. Since the working paper is not online yet, I will wait a little bit before uploading the slides. But they will be online, someday (hopefully soon)…
“An important aspect of the strategy of most organizations is the provision of incentives to the employees to meet the organization’s objectives. Typically this implies tying pay to performance (see Prendergast, 1999). In order to reward employees for their effort, firms spend considerable resources on performance evaluations. In many cases, evaluation consists of comparing actual performance to a pre-defined individual target. Another frequently used format is relative performance evaluation. Relative performance evaluation may motivate employees to work harder.But it may also be demoralizing and create an excessively competitive workplace, which may hinder overall performance; see Lazear (1989). Determining the overall impact of relative performance evaluation is crucial for companies. Economic research on relative performance evaluation has mainly focused on the comparison of final performances between competitors,like in tournament theory, and on quantitative and subjective performance ratings (Lazear and Gibbs, 2009). In contrast, what happens during a competition and the impact of feedback frequency on effort have so far received little attention. Following Berger and Pope (2011), we decided to use a basketball application to get a better understanding of the role of the feedback information. Sports datasets allow to observe score and team behavior continuously (during a game but also during the season) which can be use as a proxy of the effort. Berger an Pope (2010) asked ”can loosing lead to winning ?” looking at the impact of the halftime score difference on winning probability in NCAA (college) and NBA(pro) games. More precisely, they studied whether a team loosing at halftime is more likely to win than expected using a logit model. They find that usually the higher the score difference the more likely the are to win. But if the halftime score difference is around 0 they observe a discontinuity: loosing with a small difference (e.g. down by 1 point) can lead to increase the effort and win the game. In this paper we try answer the question ”when loosing lead to winning ?”.“
Correlations, dimension, and risk measure
Yesterday, while I was attending the IFM2 conference, at HEC Montreal, I heard a nice talk about credit risk, and a comparison between contagion (or at least default correlation), for corporate and retail companies (in the US). And it was mentioned that default correlation was much lower for retail companies than it could be for corporate risk. In a discussion that followed those slides, it was mentioned that banks in the US should actually have been working more with those small firms, since contagion risk was much lower.
A problem here is that the link between correlation, risk and dimension is rather complicated:
- corporate means a small number of firms, high correlation (and possible large individual losses)
- retail means a large number of firms (even perhaps extremely large), lower correlation (and small individual losses)
A simple model for default models is based on the assumption that we deal with an exchangeable portfolio (as in a previous post). With the following code, given an (individual) default probability, a default correlation, and a number of firms, it is possible to calculate the probability to have more than a given number of defaults.
proba=function(s,a,m,n){ b=a/m-a choose(n,s)*integrate(function(t){t^s*(1-t)^(n-s)* dbeta(t,a,b)},lower=0,upper=1,subdivisions=1000, stop.on.error = FALSE)$value} CDF=function(x=10,r=.4,m=.1,n=50){ a=m*(1-r)/r ; V=rep(NA,n+1) for(i in 0:n){ V[i+1]=proba(i,a,m,n)} V=V/sum(V); return(sum(V[1:(x+1)])) }
It is possible to calculate, for a large range of correlations, the probability to have – at least – 20% of default in the portfolio (in order to compare things that are comparable).
R=seq(.01,.99,by=.01) VQ=matrix(NA,length(A),2) for(i in 1:length(A)){ VQ[i,1]=1-CDF(r=A[i],x=4,n=20); VQ[i,2]=1-CDF(r=A[i],x=200,n=1000)}
With 20 firms (corporate) we want to have at least 4 defaults, while with 1000 firms (retail) there should be 200 defaults. As mentioned in the previous post, the relationship between correlation and quantiles of sums is not simple. Hence, it might not be monotone. The dotted line is the probability to have at least 4 defaults when default correlation is 50% (around 10%). The plain line is the probability to have at least 200 defaults, as a function of the correlation,
plot(A,1-VQ[,2],type="l",col="red",ylim=c(0,.22)) abline(h=1-VQ[50,1],lty=2,col="red")
In that case, with only a correlation of 10% among retail firms, the probability of having 20% defaults is the same as the same probability for corporate, but with 50% correlation… One should remember that in portfolio analysis, the links between correlation, dimension and risk measure is a sensitive issue…
Talk on bivariate count times series in finance and risk management
I will be giving a talk on May 4th, at the Mathematical Finance Days, at HEC Montréal, on multivariate dynamic models for counts. The conference is organized by IFM2 (Institut de Finance Mathématique de Montréal). I will be chairing some session and I will give a talk based on the joint paper with Mathieu Boudreault.
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The slides can be downloaded from the blog,
“In various situations in the insurance industry, in finance, in epidemiology, etc., one needs to represent the joint evolution of the number of occurrences of an event. In this paper, we present a multivariate integer‐valued autoregressive (MINAR) model, derive its properties and apply the model to earthquake occurrences across various pairs of tectonic plates. The model is an extension of Pedelis & Karlis (2011) where cross autocorrelation (spatial contagion in a seismic context) is considered. We fit various bivariate count models and find that for many contiguous tectonic plates, spatial contagion is significant in both directions. Furthermore, ignoring cross autocorrelation can underestimate the potential for high numbers of occurrences over the short‐term. An application to risk management and cat‐bond pricing will be discussed.”
Local utility and multivariate risk aversion
Marc will give a talk today at the European Center for Advanced Research in Economics and Statistics (ECARES) today, at ULB in Brussels, based on some joint work with also Alfred (the paper can be found online on http://papers.ssrn.com/).
Short versus long papers, in academic journals
This Monday, during my talk on quantile regressions (at the Montreal R-meeting), we’ve seen how those nice graphs could be interpreted, with the evolution of the slope of the linear regression, as a function of the probability level. One illustration was on large hurricanes, from Elsner, Kossin & Jagger (2008). The other one was on birthweight, from Abrevaya (2001).
It is also to illustrate that technique to academic publication, e.g. the length of papers, over time. Actually, the data we can extract from Scopus are quite similar to the ones uses on hurricanes. For several journals, it is possible to look at the length of articles. Since Scopus is quite expensive ($60,000 per year for the campus, as far as remember, so I can imagine the penalty I might have to pay for sharing such a dataset)
base=read.table("/home/scopus.csv", header=TRUE,sep=",") pages=base$Page.end-base$Page.start year=base$Year
Again, a first idea can be to look at boxplots, and regression on (nonparametric) quantiles, here for Econometrica,
boxplot(pages~as.factor(year),col="light blue") Q=function(p=.9) as.vector(by(pages,as.factor(year), function(x) quantile(x,p))) u=1:16 points(u,Q(p),pch=19,col="blue") abline(lm(Q(p)~u,weights=table(year)),lwd=2,col="blue")
Consider now (as in the slides in the previous post) a quantile regression (instead of a regression on quantiles), for instance in the Annals of Probability,
library(quantreg) u=seq(.05,.95,by=.01) coefstd=function(u) summary(rq(pages~year, tau=u))$coefficients[,2] coefest=function(u) summary(rq(pages~year, tau=u))$coefficients[,1] CS=Vectorize(coefstd)(u) CE=Vectorize(coefest)(u) k=2 plot(u,CE[k,],ylim=c(min(CE[k,]-2*CS[k,]), max(CE[k,]+2*CS[k,]))) polygon(c(u,rev(u)),c(CE[k,]+1.96*CS[k,], rev(CE[k,]-1.96*CS[k,])), col="light green",border=NA) lines(u,CE[k,],lwd=2,col="red") abline(h=0)
We have the following slope, for the year, as a function of the probability level,
The slope is always positive, so size of papers is increasing with time, short and long papers. But the influence of time is much larger for long paper than short one: for short papers (lower decile) every year, the size keeps increasing, with one more page every three years. For long paper (upper decile), it is two more pages every three years.
If we look now at the Annals of Statistics, we have
and for the evolution of the slope of the quantile regression,
Again the impact is positive: papers are longer in 2010 than 15 years ago. But the trend is the reverse: short papers (lower decile) are much longer, almost one more page every year, with long paper increase only by one more page every two years… Initially, I want to run such a study on a much longer term, with quantile regressions and splines to see when there might have been a change, both in lower and upper tails. Unfortunately, as suggested by some colleagues, there might have been some changes in the format of the journal (columns, margins, fonts, etc). That’s a shame, because I rediscover nice short papers of 5-10 pages published 20 or 30 years ago. They are nice to read (and also potentially interesting for a post on the blog). 5 pages, that’s perfect, but 40 pages, that’s way too long. I wonder if I am the only one having this feeling, missing those short but extremely interesting papers….
Talk on quantiles at the R Montreal group
This afternoon, I will be giving a two-hour talk at McGill on quantiles, quantile regressions, confidence regions, bagplots and outliers. Before defining (properly) quantile regressions, we will mention regression on (local) quantiles, as on the graph below, on hurricanes,
In order to illustrate quantile regression, consider the following natality database,
base=read.table( "http://freakonometrics.free.fr/natality2005.txt", header=TRUE,sep=";")
We can use it produce those nice graphs we can find in several papers, modeling weight of newborns,
u=seq(.05,.95,by=.01) coefstd=function(u) summary(rq(WEIGHT~SEX+ SMOKER+WEIGHTGAIN+BIRTHRECORD+AGE+ BLACKM+ BLACKF+COLLEGE,data=base,tau=u))$coefficients[,2] coefest=function(u) summary(rq(WEIGHT~SEX+ SMOKER+WEIGHTGAIN+BIRTHRECORD+AGE+ BLACKM+ BLACKF+COLLEGE,data=base,tau=u))$coefficients[,1] CS=Vectorize(coefstd)(u) CE=Vectorize(coefest)(u)
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The slides can be downloaded on the blog, as well as the R-code.
Miami, Public Choice conference
Last week, my co-author Benoît Le Maux gave a talk in Miami at the Second World Congress of the Public Choice Societies (http://www.pubchoicesoc.org/), on natural catastrophes and government intervention (see here for the paper). The slides can be found on the blog.
Second Workshop on Insurance Mathematics
This Friday, the Second Québec-Ontario Workshop on Insurance Mathematics (WIM) will take place in Toronto, at the Fields Institute. Mathieu will give a talk on “Multivariate integer-valued autoregressive models applied to earthquake occurrences“. The paper can still be downloaded on http://arxiv.org/ and the slides can be downloaded here.
Talk on bivariate count times series to model eathquake dynamics
Monday afternoon, I will be giving a talk at UQAM on very recent work done with Mathieu Boudreault, on bivariate counting processes, applied to model earthquakes dynamics (see here). The slides can be downloaded on the blog,

Local Utility and Multivariate Risk Aversion
The paper with Alfred Galichon and Marc Henry, on Local Utility and Multivariate Risk Aversion is now available online on http://papers.ssrn.com/,
“The present paper We revisit Machina’s local utility as a tool to analyze attitudes to multivariate risks. Using martingale embedding techniques, we show that for non-expected utility maximizers choosing between multivariate prospects, aversion to multivariate mean preserving increases in risk is equivalent to the concavity of the local utility functions, thereby generalizing Machina’s result in Machina (1982). To analyze comparative risk attitudes within the multivariate extension of rank dependent expected utility of Galichon and Henry (2011), we extend Quiggin’s monotone mean and utility preserving increases in risk and show that the useful characterization given in Landsberger and Meilijson (1994) still holds in the multivariate case“
Natural Catastrophe Insurance: How Should the Government Intervene?
An updated version of the joint paper with Benoit Le Maux is online on http://papers.ssrn.com/.
“The present paper develops a new theoretical framework for analyzing the decision to provide or buy insurance against the risk of natural catastrophes. In contrast with conventional models of insurance, the insurer has a non-zero probability of insolvency that depends on the distribution of the risks, the premium rate, and the amount of capital in the company. Among several results, we show that risk-averse policyholders will accept to pay higher rates for a government-provided insurance with unlimited guarantee. However, depending on the correlation between and within the regional risks, a government program can be more attractive to high-correlation than to low correlation areas, which may lead to inefficiencies if the insurance ratings are not appropriately chosen.”
BINAR processes and earthquakes
With Mathieu Boudreault, we finally uploaded our working paper on multivariate integer-valued autoregressive models applied to earthquake counts onhttp://hal.archives-ouvertes.fr/ and on http://arxiv.org/.
“In various situations in the insurance industry, in finance, in epidemiology, etc., one needs to represent the joint evolution of the number of occurrences of an event. In this paper, we present a multivariate integer-valued autoregressive (MINAR) model, derive its properties and apply the model to earthquake occurrences across various pairs of tectonic plates. The model is an extension of Pedelis & Karlis (2011) where cross autocorrelation (spatial contagion in a seismic context) is considered. We fit various bivariate count models and find that for many contiguous tectonic plates, spatial contagion is significant in both directions. Furthermore, ignoring cross autocorrelation can underestimate the potential for high numbers of occurrences over the short-term. Our overall findings seem to further confirm Parsons & Velasco (2001).“
The starting point of our paper with Mathieu was the paper on the absence of remotely triggered large earthquakes beyond the main shock region, by Thomas Parsons and Aaron Velasco published in May 2011 in Nature Geoscience. I was supposed to present this work at the Geotop seminar last week, but the seminar has been canceled and I will probably present it this Winter. Slides as well as R code will be uploaded for the seminar.
Multivariate comonotonicity and multivariate dispersion orderings
Marc will give a talk at the Kyoto Symposium in 京都市 on November 29th, on “Multivariate comonotonicity and multivariate dispersion orderings” based on some joint work with also Alfred (the paper can be found online on http://papers.ssrn.com/).