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Formalizing Actuarial Mathematics: Determining Insurance Prices using Coq, Study notes of Mathematics

This document, written by Yosuke ITO from Sompo Himawari Life Insurance Inc., discusses the formalization of actuarial mathematics using Coq. It covers the basics of life insurance mathematics, pricing pure endowments, whole life annuities, and term life insurance. The document also touches upon actuarial notations and formulas, survival functions, and life tables.

Typology: Study notes

2021/2022

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Formalizing Actuarial Mathematics
Yosuke ITO
Sompo Himawari Life Insurance Inc.
November 21st, 2021
Yosuke ITO (Sompo Himawari Life) Formalizing Actuarial Mathematics November 21st, 2021 1/ 32
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Download Formalizing Actuarial Mathematics: Determining Insurance Prices using Coq and more Study notes Mathematics in PDF only on Docsity!

Formalizing Actuarial Mathematics

Yosuke ITO

Sompo Himawari Life Insurance Inc.

November 21st, 2021

Disclaimer

  • The contents presented here are solely the speaker’s opinions and do not reflect the views of Company.
  • There are some inaccuracies in explaining actuarial mathematics due to the priority on intuitive understanding.

Self Introduction

Professional Experience

  • (^) Sompo Himawari Life Insurance Inc., December 2020 – Present. I (^) Aggregates the business results of life insurance products.
  • (^) Meiji Yasuda Life Insurance Company, April 2014 – November 2020. I (^) Revised the reinsurance contracts. I (^) Determined the prices of life insurance products. I (^) Attended the approval negotiations with Financial Services Agency. I (^) Qualified as an actuary (Fellow of the Institute of Actuaries of Japan). I (^) Aggregated the business results of group life insurance. I (^) Calculated retirement benefit obligations of client enterprises. I (^) Validated the financial soundness of Employees’ Pension Plans. Education
  • (^) Nagoya University I (^) Master of Mathematical Sciences, March 2014.
  • (^) The University of Tokyo I (^) Bachelor of Science, Mathematics Course, March 2012.

Contents

1 Formalizing Actuarial Mathematics

2 Minimal Introduction of Life Insurance Mathematics

3 The Actuary Package

Formalizing Actuarial Mathematics

  • (^) The most traditional area of actuarial mathematics is life insurance mathematics.
  • It deals with I (^) how to determine the prices of life insurance products, I (^) the estimation of loss reserves – “the amount an insurer would need to pay for future claims on insurance policies it underwrites”. [1]
  • (^) I formalized the basic part of life insurance mathematics in Coq. I (^) GitHub: Yosuke-Ito-345/Actuary https://github.com/Yosuke-Ito-345/Actuary I (^) How to install: opam install coq-actuary (thanks to Karl Palmskog)
  • I delivered a presentation of this work in the annual conference of the Institute of Actuaries of Japan in November 5th, 2021. (The proceeding will be published in 2022.)

Contents

1 Formalizing Actuarial Mathematics

2 Minimal Introduction of Life Insurance Mathematics

3 The Actuary Package

Pricing a Pure Endowment II

Assumption

  • (^) amount insured: $
  • entry age: 30 years old
  • (^) policy period: 10 years
  • probability that the insured person will survive for 10 years: 90%
  • (^) annual interest rate: 2%
  • (^) The expected payment after 10 years is

$ 10000 × 90 % = $ 9000.

Question

Do you really need $9000 now?

Pricing a Pure Endowment III

Assumption

  • amount insured: $
  • (^) entry age: 30 years old
  • (^) policy period: 10 years
  • probability that the insured person will survive for 10 years: 90%
  • (^) annual interest rate: 2%
  • (^) If the insurance company earns 2% investment yield annually, the required amount for this insurance can be discounted:

$ 9000 ( 1 + 2 %)^10

Pricing a Whole Life Annuity I

Whole Life Annuity: “a financial product sold by insurance companies; it gives out monthly, quarterly, semi-annual, or annual payments to a person for as long as they live, beginning at a stated age” [2]

Assumption

  • amount insured: $
  • (^) frequency of payment: yearly
  • entry age: 60 years old
  • (^) annual mortality rates: (^) { qx = 0. 1 if x < 99 qx = 1 if x = 99
  • annual interest rate: 2%
  • The present value of the expected payment after k years is

$ 1000 × A 60 : k^1 = $ 1000 × (^) k p 60 vk^.

Pricing a Whole Life Annuity II

  • (^) The present value of this annuity is

∑^39

k = 0

$ 1000 × A 60 :^1 k =

∑^39

k = 0

$ 1000 × (^) k p 60 vk

∑^39

k = 0

$ 1000 ×

k ∏− 1

j = 0

( 1 − q 60 + j )

vk

∑^39

k = 0

$ 1000 × 0. 9 k^ ·

) k

= $ 1000 ×

1 − ( 0. 9 / 1. 02 )^40

Pricing a Term Life Insurance I

Term Life Insurance: “a type of life insurance that guarantees payment of a stated death benefit if the covered person dies during a specified term” [2]

Assumption

  • amount insured: $
  • entry age: 30 years old
  • (^) policy period: 10 years
  • annual mortality rate: 0.
  • (^) annual interest rate: 2%
  • (^) The present value of the expected payment after k years is

$ 10000 × (^) k − 1 p 30 · q 30 +( k − 1 ) · vk^.

Here, the death benefit is supposed to be paid at the end of the year of death.

Pricing a Term Life Insurance II

  • (^) The present value of this insurance is

∑^10

k = 1

$ 10000 × (^) k − 1 p 30 · q 30 +( k − 1 ) · vk

∑^10

k = 1

$ 10000 × 0. 99 k −^1 · 0. 01 ·

) k

= $ 10000 × 0. 01 ·

1 − ( 0. 99 / 1. 02 )^10

Actuarial Notations and Formulas

  • (^) These kind of symbols are called “actuarial notations” and commonly used in various countries. I (^) INTERNATIONAL ACTUARIAL NOTATION https://www.casact.org/sites/default/files/database/ proceed_proceed49_49123.pdf
  • In life insurance mathematics, the relations between the actuarial symbols are well examined:

A^1 x : n = 1 − iv ¨ ax : nAx : 1 n.

  • (^) Actuaries use these symbols efficiently to calculate prices of products, reserves of the company, etc.

Survival Function

Definition

Let T be a random lifetime variable, and define s ( x ) := P ( T > x ) for an age x. The function s is called the “survival distributive function”.

x

y

O

y = s ( x )

Example

npx =^ P ( T^ >^ x^ +^ n^ |^ T^ >^ x ) qx = P ( Tx + 1 | T > x )

˚ e 0 := E ( T ) =

0

s ( x ) dx (average life span)