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

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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 : n − Ax : 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 ( T ≤ x + 1 | T > x )

˚ e 0 := E ( T ) =

0

s ( x ) dx (average life span)