Microlending got its start in Bangladesh originally, championed by Muhammad Yunus, via the Grameen Bank (in Bengali: গ্রামীণ বাংক). Yunus was awarded the Nobel Peace Prize in 2006, in recognition of his extraordinary achievement in the promotion and proof of concept of Microlending.

What we will discuss is how to model, using mathematics, the working of Microlending. In other types of finance, particularly stock market high finance, mathematics has led to major advances. Not much mathematics has been devoted to the analysis of Microlending, and there are perhaps several reasons: the theory to date is not martingale based, which is a beautiful theory amenable to high finance. The second is that not a lot of money is involved, and researchers typically like to “follow the money,” in choosing their objects of study.

The two primary mathematical tools used to model Microlending are the theory of Markov chains and Markov processes, ands game theory. Our emphasis will be on the former, and we will explain the basics of the theory of Markov chains and Markov processes in the course. Some of the issues will be making sure the system works, so that the bank does not go bankrupt or lose enough money to be put into jeopardy. What is clever about the system Yunus has developed is that it gives a method to lend small amounts of money without having first to evaluate the credit risk of the borrowers. This is important in some regions of the world where a system of credit risk, highly developed in North America and Europe, is poorly developed or nonexistent among the poorer segments of the population, as was the case in Bangladesh.

The loans typically are small and of short duration, and by repaying the loans as promised, the borrowers earn their credit rating, and obtain sequential financing. To make this idea precise, one needs mathematical models.

We will also treat the cost effectiveness of the Grameen Bank, the canonical example of microfinance.

We will keep the mathematical level at a low level to make it accessible to the audience, but sophisticated enough to lend insight into how the entire system works. Mathematical models also give the benefit of making results quantifiable.

Extreme value statistics helps protecting us from devastating waves, floods, and windstorms, and contributes to material science, bioinformatics, medicine, and traffic safety – and is widely used for risk management in finance and insurance.

Participants in this course will learn about established extreme value methods for risk handling, get an introduction to the hot research area of multidimensional extreme value theory, and an overview of existing program packages in the area. After the course you will be able to use the latest extreme value statistics technology to help handling risks in finance and insurance – and you will have a starting point if you want to enter into the exciting extreme value statistics research area.

The first part of the course will introduce the well-established and much used statistical theory for extremes of one-dimensional variables. Topics include the block maxima and peaks over thresholds methods; threshold choice; maximum likelihood methods; and model diagnostics.

The second part will survey some of the intensive research in multivariate extreme value statistics which happens right now. Multivariate block maxima methods have so far seen the most development. However, in more than one dimension, block maxima hide information of whether extremes occur at the same time or not, and likelihoods often become unwieldy in dimensions higher than 3 or 4. Instead peaks over threshold methods keep track of whether extremes occur at the same time or not, and are often at the center of interest for finance. I will show how multivariate peaks over thresholds models based on the multivariate generalized Pareto distribution can be more useful for financial risk handling than currently available one-dimensional estimates. These models, perhaps surprisingly, have simpler and tractable likelihoods, and permit use of the entire standard maximum likelihood machinery for estimation, testing, and model checking.

The course will include many examples on how to use the methods in finance and insurance: risk estimation for wind storm insurance; computation of Value at Risk and Expected Shortfall in finance; portfolio risk estimation, …

Finally, I will introduce some of the existing program packages which make Extreme Value Statistics practically useful, including extRemes, SpatialExtremes, WAFO, RandomFields, and some MATLAB routines. Throughout, an important issue is how estimated risk should be presented and understood.