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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)