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Knowledge is the only good; ignorance the only evil. Socrates

Convex Considerations blogs anything I find interesting or want to remember.

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About

about

About me and the ConvexConsiderations blog

Reading in 2024

about

vsi

reading

2024: the year of the Very Short Introduction!

Classifying and Identifying Trees

notes

trees

Introduction to multipart-post on tree classification and Introduction.

Galaxies VSI

about

vsi

reading

The age of the Universe is determined by studying the largest things using general relativity. The age of star comes from studying the smallest using quantum mechanics. The consistency between the…

Extreme Tails of the Normal Distribution

notes

probability

Estimating extreme quantiles of the normal distribution.

Chemistry VSI

about

vsi

reading

Bonds: ionic, covalent, metallic; reactions: proton transfer, electron transfer, radical, Lewis acids

Particle Physics VSI

about

vsi

reading

There are seventeen fundamental particles: six quarks, six leptons, and five bosons.

Plate Tectonics VSI

about

vsi

reading

Lithosphere takes tens of millions of years to warm up when subducted into the asthenosphere.

What I Ask GPT

notes

llm

How GPT helped me cluster the questions I’ve asked it over the last year.

Potted Measure Theory

notes

llm

mathematics

probability

A quick summary of measure theory on the real line.

Energy Intensity of Internet Traffic

notes

llm

How much energy is required to transfer 1GB of data over the internet from London to Paris?

Geography VSI

about

vsi

llm

reading

Space, place, and environment.

Science and Pseudo-science

notes

llm

This post is another comparison of LLMs. I asked for a summary of Popper’s philosophy distinguishing science from pseudo-science.

The Forgetting Curve

notes

llm

The forgetting curve is a concept that describes how the ability to recall information decreases over time when no effort is made to retain it. This theory was first proposed by the German…

Trees

notes

llm

LLM guided tree classification.

Defining Bullshit

notes

llm

LLM definitions of Bullshit.

Joplin-Evernote Migration Policy

about

llm

LLM’s offer advice on maintaining historical notes.

Non-Measurable Sets

notes

set theory

mathematics

Detailed construction of a non-Lebesgue measurable set, per Halmos.

Doob’s Upcrossing Lemma

notes

llm

probability

Doob’s optional stopping theorem is a fundamental result in probability theory, particularly in the theory of martingales. The theorem addresses the expected value of a martingale at a stopping…

Geometric Brownian Motion, Yet Again!

notes

probability

A heuristic explanation of geometric Brownian motion

Waiting Time for a Random Walk to Hit One

notes

Let \(X_n\) be a symmetric random walk moving up or down by one at each step with equal probability. Let \(T\) be the waiting time until \(X_n=1\). Then \[ \def\E{\mathsf E} \def\P{\mathsf…

The Incurred Loss Process

notes

llm

probability

insurance

mathematics

modeling

research

risk

Towards a theory of incurred loss processes, starting with a review of stochastic processes.

Reinsurance Decision Making

A 20-Year Evolution

presentations

insurance

risk

pricing

What’s wrong the the constant cost of capital assumption

Why Go Spectral?

notes

presentations

insurance

pricing

risk

probability

Why and how to use insurance spectral pricing methods.

Compensators

notes

llm

probability

mathematics

Given an exponential random variable \(T\) with the hazard function \(\alpha(t) = t\), the compensator \(A_t\) is defined as \(A_t = \min(t, T)\). Observe that \(N_t\) is either 0 or 1, where \(N_t…

Human Evolution VSI

about

vsi

reading

llm

Out of Africa or multiregional?

Keynes VSI

about

vsi

reading

llm

Economists should strive to be as useful as dentists.

Stars and The Sun VSIs

about

vsi

reading

llm

We are all made of atoms created in stars!

Conditional Expectations Given The Sum of Independent Random Variables with Regularly Varying Densities

notes

Summary of results and examples from a recent paper of Denuit et al.

The First World War VSI

about

vsi

reading

llm

“At the end of 1917 the prospects for the Allies still looked grim.”

Climate and Climate Change VSIs

about

vsi

reading

“IPCC 2021 states that the evidence for climate change is unequivocal, and there is a very high confidence that this warming is due to human emissions of GHGs.”

Hosting Documentation on GitHub Pages

meta

llm

Another comparison of LLMs.

File Consolidator

meta

Rationalizing years of file archives

LLM Comparison: Managing Sublime Text Projects

notes

llm

Gemini, GPT, and Claude answer questions about Sublime Text

LLM Comparison: Why Buy Insurance?

notes

llm

insurance

Editing and tidying up transcribed text using LLMs.

Large Cardinals and Set Theory

notes

llm

set theory

mathematics

What can “large cardinals” have to do with Lebesgue measurable sets? I asked Chat GPT to find out more. Here’s what I learned.

MJ Images

meta

Retrieving and processing Midjourney images.

Markdown snippets

meta

llm

blog

Handy Quarto markdown tips and techniques that are easy to forget (=I forget them).

Using TikZ in a blog page

meta

blog

Using great2 to process tikz into svg files for blog posts.

Conditional Probability

notes

probability

mathematics

\[ \def\E{\mathsf E} \def\A{\mathscr A} \def\B{\mathscr B} % for general sigma alg \def\C{\mathscr C} \def\EE{\mathscr E} \def\F{\mathscr F} \def\G{\mathscr G} \def\H{\mathscr H} \def\S{\mathscr…

Gas

notes

llm

The ideal gas model and when it fails.

Weather VSI

about

vsi

reading

Different types of cloud.

How a Pneumatic Tire Works

about

llm

A car tire supports the weight of the car by the difference in tension in the top and bottom tire walls.

Isotopes VSI

about

vsi

reading

llm

Same on the outside but different on the inside; detect one atom in \(10^{-15}\); cosmogenic creation

Accounting VSI

about

vsi

reading

The earliest human written records are all accounts; expenses use assets so are entered on the left.

Earth System Science VSI

about

vsi

reading

llm

The Earth has experienced huge changes over very long time frames but feedback induced stability over shorter time frames.

Geophysics VSI

about

vsi

reading

Global Catastrophes VSI

about

vsi

reading

Meteorites will kill us…or maybe a volcano or our own stupidity. Landslides could flood the US East Coast.

Ice

about

llm

Table showing different types of ice.

Insurance in the Arts

about

llm

Cases where insurance could change the plot in famous literature.

Magna Carta VSI

about

vsi

reading

llm

Charters were common; John was a really bad king; original MC included enforcement by 25 barons.

Midjourney Prompts

about

llm

Mention specific art styles, techniques, artists, or art movements.

Newton VSI

about

vsi

reading

llm

Newton: scientist, alchemist, anti-Trinitarian, MP, Master of the Mint.

Planets VSI

about

vsi

reading

llm

The asteroid Apophis will have a close encounter with the Earth miss on 2029-04-13, which is a Friday!

Risk VSI

about

vsi

reading

Normative - descriptive - prescriptive: What should we do? What do we do? How can we improve?

Statistics VSI

about

vsi

reading

The technology of extracting meaning from data and of handling uncertainty. Mechanistic vs empirical models.

The Earth VSI

about

vsi

reading

llm

The Earth was very different a long time ago; microbes under the seafloor; paleomagnetism and reversals of magnetic field.

The Ice Age VSI

about

vsi

reading

Ice bergs or ice sheets debate; seabed and ice cores; Milankovitch’s three cycles; global impact of Himalayas.

The Laws of Thermodynamics VSI

about

vsi

reading

llm

Temperature: most probable distribution over available states; conservation of energy; entropy never decreases in an isolated system; as the temperature of a system approaches absolute zero, the…

The Period Table VSI

about

vsi

reading

llm

Potassium-Argon reversal. Where should hydrogen appear? Why the actinides and lanthanides appear along the bottom. Quantum basis for the length of each row.

Evensong Order of Service

about

Church of England Evensong Order of Service

Hyphens

about

llm

Chat GPT’s instructions on the use of hyphens.

Words

about

vsi

logophilia

Technical words from VSIs

Knowledge in Diagrams

about

A collection of explanatory diagrams

Ages of the Earth

about

llm

vsi

Earth and human timelines.

VSI Series

about

vsi

reading

List of Very Short Introduction Books.

Numpy meshgrid

Python

programming

This always confuses me. Here’s the help on meshgrid:

Matplotlib: tips and tricks

programming

Python

llm

Here are several things I always forget when using matplotlib. This post written with help from Gemini and ChatGPT4.0.

Podcast and Video Appearances

about

presentations

I have appeared as a guest on several podcasts and videos. See also my YouTube videos.

Reading Since 1990

about

reading

A searchable table of all the books I have read since 1990.

Quarto YAML and Markdown: tips and tricks

meta

programming

blog

Python

What I learned today by RTFD.

Chat GPT and Google Gemini Answer Programming Questions

notes

programming

Python

llm

I believe that large language models (LLMs) are the most amazing thing that has happened in my lifetime—frankly, by a long stretch¹. While they are a bit of a cheat, because they incorporate…

Effective Python: Classes

programming

Python

effective python

llm

Object oriented programming in Python

My Shelf of Favorite Books

about

Contents and history of my shelf of favorite books since 2003.

Catastrophes Ranked by Energy

notes

insurance

llm

catastrophe

Add summary here…

Code Snippets

programming

Python

int('000000007fff4070', base=16)

Effective Python: R Style Regression

programming

Python

effective python

I use R, but I love Python. However, let’s face it, basic linear regression in R is very straightforward. A few clear and intuitive lines of R code produce textbook¹ output that is informative and…

The Cost of Goods Sold

presentations

insurance

risk

pricing

John Major and Stephen Mildenhall, CARe Meeting, June 9, 2021

Presentations Since 1996

about

presentations

Conference, meeting, and online talks and presentation since 1997.

Publications

about

publications

Journal articles, books, popular press, and working papers.

Resume

about

Professional and academic experience.

Abstracts from Mario Ghossoub’s Papers on Risk

notes

risk

Ghossoub’s Website

Effective Python: Floats

programming

Python

effective python

I studied math(s) for 9 years as a undergraduate and graduate student. I never came across a non-associative operator. Standard IEEE floating point addition is non-associative, as the next example…

Effective Python: Building a Test DataFrame

programming

Python

effective python

It is often helpful to have a test dataframe. This function makes one with several optional variants. I use it surprisingly often. It is part of great2.

Effective Python: Introspection

programming

Python

effective python

Get the arguments of a function. Used in aggregate.distr.Aggregate.

Effective Python: Munging Fitbit Data with Pandas

programming

Python

effective python

Pain lies at the intersection of the computer and the real-world. Real-world data is messy, missing, inconsistent. Transforming or munging it into an analysis dataset is and art and a science and…

Reinsurance Optimization Literature Review

notes

risk

reinsurance

Catastrophe reinsurance provides insurance companies protection against excessive losses caused by catastrophes such as hurricanes, earthquakes, severe convective storms, flood, and winter freeze.…

Effective Python: Projects

programming

Python

effective python

The best way to learn Python is to use it to solve a real-life problem you face. I’m often asked to suggest example projects that are good for beginners. That’s a tricky question because we all have…

Bon Pint

actuarial review

The nerdier regulars at the Bon Pint pub enjoy a peculiar number game. They start with an integer \(n\) and expand it in powers of 2. Then, they expand all the exponents in powers of two, and so on…

Academic Descriptions of Risk

notes

risk

Equal probability lotteries \([a, b]\) describe outcomes \(a\) and \(b\), which can be fixed or random.

Reinsurance Capital or Credential

notes

risk

reinsurance

insurance

Reinsurance: Capital or Credential?

Effective Python: Importing Data into Pandas

programming

Python

effective python

Problems arise when computers interact with the real world. You step out of a controlled environment into one with bizarre, uncontrolled user inputs. Even structured (read tabular) real world data…

ZFC Axioms

notes

The Zermelo-Frankel axioms of set theory: extension, foundation, pairing, union, power, infinity, separation, replacemeht, choice.

Non Bayesian Risk

notes

bayesian

pricing

insurance

risk

underwriting

Distortion risk measures

David Wright Notes

notes

What is the insurance business? What do you manufacture? I asked that question to a lot of people, “What do you manufacture?” You manufacture decisions. It’s no different than the investment…

When is Premium Riskier Than Loss Slides

presentations

insurance

pricing

PIRC presentation comparing volatility from premium and loss uncertainty.

CEOs Talk About Risk

notes

risk

insurance

In my judgment, the calendar year or published combined ratio is the primary measure of underwriting performance that investors and management should focus on, because natural catastrophes are a…

Effective Python: Converting Datatypes in Pandas

programming

Python

effective python

Series method astype it is already a number and you want to change the type. Will not coerce.

Towards the Quantitative Analysis of Quantitative Factors

notes

pricing

insurance

risk

underwriting

process risk; param risk; winner’s curse; PAT; ambiguity; morale hazard; info asymmetry

Effective Python: Dates

programming

Python

effective python

Problems arise when computers interact with the real world. You step out of a controlled environment into one with user inputs, paper jams, invalid filenames, and dates. Dates are on the front-line…

Effective Python: `requests`, `BeautifulSoup`, and web scraping

programming

Python

effective python

The requests library does for web interaction what pathlib.Path path does for files.

Risk and Return

notes

insurance

risk

There are no universally true statements about risk.

Effective Python: Introspection with Pandas

programming

Python

effective python

This code defines a function that extracts all an objects properties into a pandas dataframe. It is created as a row so that each column can have the appropriate datatype. This function is part of gr…

Effective Python: Strings and Unicode

programming

Python

effective python

Ever cut and paste text between applications and get unexpected results? Particularly when the source is a webpage? For example, the sentence¹

Effective Python: `zip`, `enumerate`, and `iter`

programming

Python

effective python

This short post looks at three built-in Python functions: zip, enumerate, and iter.

Effective Python: Files

programming

Python

effective python

If you write a program of any length, you will likely have to manipulate files and filenames. Common file-related tasks include:

Hash Magic Encryption and Digital Signatures

notes

Hash functions are miraculous, helpful tools we have no right to expect to exist. They seem far too convenient! They have many uses, including determining if two files are the same, creating unique…

Creating TeX Format Files

meta

blog

Creating a TeX format file for faster processing.

Cartograms

notes

programming

Wordle the Best Start

notes

probability

bayesian

Wordle is a popular word game. You have six chances to guess a five-letter word. For each guess, you learn how many letters are in the right place, and how many are in the word but in a different…

ConvexConsiderations Quarto Blog Setup

meta

programming

blog

Defines containing folder

Active and Inactive Research

about

research

programming

insurance

Effective Python Outline

programming

Python

effective python

A series of posts about effective Python programming for the intermediate user.

Effective Python: Regular Expressions

programming

Python

regular expressions

Regular expressions are a wonderful shorthand way to select and manipulate strings.

Effective Python: Setting up Python Environments

programming

Python

Notes I found helpful setting up my Python environments.

Markdown

meta

The text in this box becomes the description for the page.

Modes of Convergence

notes

probability

Random variables can converge in several different ways. Here is a brief introduction, highlighting examples of the different behaviors that can occur. Many of the examples below are taken from [1].

Pricing Insurance Risk Book

publications

pricing

insurance

risk

pir

Pricing Insurance Risk (PIR), written with John Major, describes the last mile of underwriting. Actuaries and accountants have determined the cost of goods sold: the expected loss cost, direct and…

Pricing Seasonal Peril Catastrophe Bonds: A Simplified Approach

publications

insurance

pricing

catastrophe

We show a catastrophe bond’s economically meaningful cash flows are the same as a fully continuous term life insurance policy. As a result we obtain a simple catastrophe bond pricing formula…

Similar risks have similar prices: A useful and exact quantification

publications

research

insurance

pricing

This paper introduces an straightforward algorithm to determine a range of prices consistent with complete information about the risk but only partial information about the pricing risk measure. In…

Pricing Insurance Risk Course (PIRC)

insurance

pricing

insurance

risk

pir

This course explains modern approaches to pricing insurance risk, the process of setting a technical premium, including a target risk margin or profit load. PIR is the last mile of underwriting…

The Aggregate Package

programming

Python

probability

The aggregate Python package makes working with aggregate loss distributions easy.

When is Premium Riskier Than Loss

An analysis of US statutory insurance data since 1992 shows that premium is more uncertain and harder to forecast than loss for several casualty lines! Property lines have loss-driven volatility…

A Modern Approach to Pricing for Risk

presentations

Liberty Mutual, November 2021

Bodoff’s Percentile Layer of Capital-Method

research

pricing

Bodoff’s paper¹ describes a new approach to capital allocation; the catalyst for this new approach is a new formulation of the meaning of holding Value at Risk (VaR) capital. This new formulation…

Equilibrium Risk Pools

research

insurance

risk

pricing

presentations

This research investigates equilibrium risk pools in a market with risk-based solvency regulation and costly capital. It considers a market with two classes of risk, each having different aggregate…

Minimum Bias and Exponential Family Distributions

research

probability

minimum bias

glm

exponential family

This post introduces four short expository notes describing exponential family distributions that are often used to model insurance losses. The exponential family underlies generalized linear models…

Bailey Simon Minimum Bias Reexamined

actuarial review

publications

programming

minimum bias

glm

exponential family

modeling

Part I of a four part series “Minimum Bias and Exponential Family Distributions”.

Exponential Family Distributions

research

probability

insurance

probability

minimum bias

glm

exponential family

Part II of a four part series “Minimum Bias and Exponential Family Distributions”.

Probability Models for Insurance Losses

research

probability

insurance

minimum bias

glm

exponential family

Part III of a four part series “Minimum Bias and Exponential Family Distributions”.

The Tweedie Power Variance Function Family

research

probability

insurance

minimum bias

glm

exponential family

Part IV of a four part series “Minimum Bias and Exponential Family Distributions”.

Charging for Diversifiable Risk and Proud to Do It

presentations

John Major and Stephen Mildenhall, CAS Webinar, September 24, 2020.

Teaching

notes

teaching

From 2016-2020 I was Assistant Professor in the School of Risk Management, Insurance, and Actuarial Science at the Tobin Business School, St. John’s University in New York City. I taught the…

How to Allocate Capital if You Really Must

presentations

John Major and Stephen Mildenhall, CAS Spring Meeting, May 13, 2020

Spectral Risk Measures and Applications in Insurance ERM

presentations

John Major and Stephen Mildenhall, ASTIN Special Interest Webinar, March 25, 2020

ERM Saves the Day for the CEA

research

pricing

Abstract. Since its emergence in the 1990s enterprise risk management (ERM) has struggled to define itself and to prove its value. Numerous statistical studies suggest ERM does add value, but at the…

ERM Saves the Day for the CEA v2

research

pricing

Abstract. The California Earthquake Authority (CEA) became operational December 1, 1996. It was created in response to an insurance crisis precipitated by the 1994 Northridge Earthquake. Its…

AI and Insurance

presentations

New AI techniques help computers solve problems that people find easy but that computers have historically found difficult, such as image or voice recognition. The techniques discover hidden…

Life Insurance Backdating

research

Assuming continuous rating the insured is always worse off with backdating on a net basis. A net basis assumes products are priced using the equivalence principle, i.e. at expected cost.

Programming Your Career

actuarial review

publications

programming

I love to program, and I am convinced programming has been beneficial to my career. Other actuaries proudly proclaim their reluctance or inability to program, and a recent Wall Street Journal…

Blockchain and Insurance

presentations

Blockchain solves trust problems. If you trust your counter-parties, or operate in a country with a well developed legal system, do you need to put your data in a tank?

Business Risk and Analytics

notes

risk

There is a synergistic relationship between business, analytics, and technology (the BAT triangle) that applies to all business, but is particularly powerful in the case of a property-casualty…

Frequency Distributions

notes

probability

A random variable \(N\) is \(G\)-mixed Poisson if \(N\mid G\) has a Poisson \(nG\) distribution for some fixed non-negative \(n\) and a non-negative mixing distribution \(G\) with \(\mathsf{E}(G)=1\)…

Summary of Insurance Priority Laws and Statutes

notes

insurance

Upon the issuance of a proper court order placing a domestic insurer in receivership or placing a foreign insurer in ancillary receivership for rehabilitation or liquidation, pursuant to this…

In Praise of VaR Part II

actuarial review

publications

modeling

risk

“Can he have everything louder than everything else?” Ian Gillan, Deep Purple, Made In Japan

The Insurance Stack in a Disrupted World

research

insurance

pricing

DRAFT / work in progress!

In Praise of VaR Part I

actuarial review

publications

modeling

risk

VaR can misbehave,
Hiding dragons in the tail.
Many views reveal.

FinTech and InsurTech

presentations

Billions of dollars are flowing into new FinTech and InsurTech start-ups. How do these companies view the insurance stack? Will they revolutionize insurance?

Bowls

about

woodwork

I have turned 149 bowls from a variety of North American woods.

Woodworking

about

woodwork

Between 2004 and 2013, I was a sporadically active woodworker with a basement shop.

Learning Objectives

notes

teaching

Learning objectives have three components:

Election Day Lessons for the Data Scientist

actuarial review

publications

modeling

risk

Kim Hypothetical, FCAS, is a pricing actuary in the Large Accounts department at Hypothetical Insurance (no relation). It is just before lunch. Kim wonders “Is traditional actuarial work the best…

The Black-Scholes Option Pricing Formula

presentations

insurance

pricing

risk

I had been fascinated by Black-Scholes since a fellow student at the University of Chicago math department at told me that you could use PDEs to price options. How was that possible?! I wanted to…

Wang’s Proportional Hazard Transform

presentations

pricing

insurance

Early description and exploration of the exciting (then) new Wang transform to my colleagues at CNA Re. Includes a list of desirable properties of premium calculation principles:

Modeling Loss Ratios for Aggregate Features

research

insurance

risk

Very early work on aggregate loss distributions! From the introduction:

Premium Margins and Managing the Catastrophe Risk

research

insurance

risk

I applied the method to different lines of business, based on A.M. Bests’ aggregates and averages data, producing broadly reasonable results. Not shown, I also used losses by line by state by day to…

A Note on Elkies’ Proof

notes

research

This is a short three-page note on Elkies’ proof that an elliptic curve has infinitely many supersingular primes that I wrote at the University of Chicago in 1991. Unfortunately, the same proof was…