This document outlines the valuation of traditional life insurance policies and time-valued risks. It begins with an introduction and overview of key concepts. Motivational examples are then provided to illustrate time value of money calculations like present value and actuarial present value. The document then generalizes the traditional approach to valuing life insurance, covering the problem, model, evaluation, and solution. It concludes with a summary of the key points. The overall aim is to introduce foundational actuarial valuation concepts.
The document provides an overview of the Integrity with GFD solution, a new wealth accumulation and protection solution for business owners. It addresses main challenges business owners face like not having enough money for retirement or rewarding family. The Integrity solution offers accelerated wealth accumulation, additional life insurance protection, and significant tax advantages. It works by providing an upfront loan that is deposited into a universal life insurance policy to grow tax-deferred. Business owners can then enjoy multiple exit strategies like selling the business, using it for retirement income, or leaving a financial legacy.
A New Arrow for The Pension Practitioners Quiver: Pension Risk TransferJay Dinunzio
Webinar Presentation Slides
Gone are the days of group annuity contracts only being able to satisfy the plan termination objectives of a pension plan sponsor. Today, there are a wide variety of useful applications for guaranteed institutional annuity contract structures to provide an alternative to traditional fixed income investments. Are you or your pension clients:
•Struggling with cost and volatility issues surrounding a defined benefit pension plan?
•Considering a liability driven investment strategy that will de-risk the plan investment and allow for stable, predictable funding?
•Limited by fixed income funds that only allow for simple duration matching, and expose the plan to cash flow mismatch risks?
•Unaware of the variety of customized institutional insurance contract structures available?
•Lacking a fiduciary process for evaluating and monitoring the attractiveness of insured pension solutions?
The document summarizes research on the quality of life insurance advice provided by agents in India. It finds that agents overwhelmingly recommend whole life policies over term life policies, even when term policies better meet customers' needs for risk coverage at low cost. Agents are motivated to recommend whole life due to higher commissions. The research tests whether the quality of advice improves when disclosure requirements make agency problems more transparent or when competition is increased. It uses an audit study approach with standardized customer profiles to evaluate agent recommendations across conditions.
Smart Ambient Learning with Physical Artifacts Using Wearable Technologies - ...IstvanKoren
Presentation we gave at iLRN 2015 conference in Prague on July 13th, 2015 together with my colleague Ralf Klamma. We present our initial work about a construction exhibition and advertise Open Standards, Open Documentation and Open Source!
The document provides background information on lean manufacturing and the Toyota Production System. It summarizes that traditional manufacturing systems have high inventory levels and long lead times, while lean systems based on Toyota aim to reduce waste and optimize flow. The core of lean is the Toyota Production System (TPS) which treats all processes as experiments to continuously improve. TPS creates a "community of scientists" using the scientific method to solve problems at all levels.
This document discusses accident causation and workplace safety. It outlines several accident causation models from simple linear models to complex non-linear models. It also discusses factors that can lead to workplace accidents like lack of training or unsafe conditions. The document notes that accident investigations are important to identify causes and implement corrective actions to reduce risks. Finally, it discusses employees' rights to compensation and benefits for workplace injuries under relevant laws and regulations.
Insurance Planning is a very vital component of Financial Planning. This presentation is all about various types of Life and General Insurance available in India. It also talks about the benefit of Term Insurance Policy. One should only go for Term Insurance Policy as it is best cover for Protection. This presentation also talks about the Mis-selling happening in India when it comes to insurance selling.
The document provides an overview of the Integrity with GFD solution, a new wealth accumulation and protection solution for business owners. It addresses main challenges business owners face like not having enough money for retirement or rewarding family. The Integrity solution offers accelerated wealth accumulation, additional life insurance protection, and significant tax advantages. It works by providing an upfront loan that is deposited into a universal life insurance policy to grow tax-deferred. Business owners can then enjoy multiple exit strategies like selling the business, using it for retirement income, or leaving a financial legacy.
A New Arrow for The Pension Practitioners Quiver: Pension Risk TransferJay Dinunzio
Webinar Presentation Slides
Gone are the days of group annuity contracts only being able to satisfy the plan termination objectives of a pension plan sponsor. Today, there are a wide variety of useful applications for guaranteed institutional annuity contract structures to provide an alternative to traditional fixed income investments. Are you or your pension clients:
•Struggling with cost and volatility issues surrounding a defined benefit pension plan?
•Considering a liability driven investment strategy that will de-risk the plan investment and allow for stable, predictable funding?
•Limited by fixed income funds that only allow for simple duration matching, and expose the plan to cash flow mismatch risks?
•Unaware of the variety of customized institutional insurance contract structures available?
•Lacking a fiduciary process for evaluating and monitoring the attractiveness of insured pension solutions?
The document summarizes research on the quality of life insurance advice provided by agents in India. It finds that agents overwhelmingly recommend whole life policies over term life policies, even when term policies better meet customers' needs for risk coverage at low cost. Agents are motivated to recommend whole life due to higher commissions. The research tests whether the quality of advice improves when disclosure requirements make agency problems more transparent or when competition is increased. It uses an audit study approach with standardized customer profiles to evaluate agent recommendations across conditions.
Smart Ambient Learning with Physical Artifacts Using Wearable Technologies - ...IstvanKoren
Presentation we gave at iLRN 2015 conference in Prague on July 13th, 2015 together with my colleague Ralf Klamma. We present our initial work about a construction exhibition and advertise Open Standards, Open Documentation and Open Source!
The document provides background information on lean manufacturing and the Toyota Production System. It summarizes that traditional manufacturing systems have high inventory levels and long lead times, while lean systems based on Toyota aim to reduce waste and optimize flow. The core of lean is the Toyota Production System (TPS) which treats all processes as experiments to continuously improve. TPS creates a "community of scientists" using the scientific method to solve problems at all levels.
This document discusses accident causation and workplace safety. It outlines several accident causation models from simple linear models to complex non-linear models. It also discusses factors that can lead to workplace accidents like lack of training or unsafe conditions. The document notes that accident investigations are important to identify causes and implement corrective actions to reduce risks. Finally, it discusses employees' rights to compensation and benefits for workplace injuries under relevant laws and regulations.
Insurance Planning is a very vital component of Financial Planning. This presentation is all about various types of Life and General Insurance available in India. It also talks about the benefit of Term Insurance Policy. One should only go for Term Insurance Policy as it is best cover for Protection. This presentation also talks about the Mis-selling happening in India when it comes to insurance selling.
Revised Version of Risk Management - a conceptual frameworkSVS College
This document provides a conceptual framework for risk management. It defines risk as the possibility of loss and classifies risks into pure risks and speculative risks. Pure risks only carry the potential for loss, while speculative risks may result in losses or profits. The document outlines various principles of risk management like utmost good faith, insurable interest, indemnity, subrogation, and contribution. It also discusses types of pure and speculative risks, as well as approaches to handling risks like risk retention, risk transfer, risk avoidance, and diversification. The key steps in the risk management process are identified as setting objectives, identifying risks, evaluating risks, selecting policies, developing strategies, and organizational control.
This document discusses various topics related to health care planning, including identifying sources of health care plans and types of coverage provided. It describes major provisions of plans and policies, as well as the purpose of disability income insurance. The document provides tips for reducing personal health care costs through prevention. It also discusses health insurance as financial protection and as part of risk management. Various types of health care providers, coverage, and policies are defined.
This document provides an introduction to life insurance, including the different types of life insurance policies and how to take out a life insurance policy. It discusses key concepts in life insurance including claim settlement processes, surrender and paid up values. The document concludes by recommending additional resources to learn more about insurance.
Insurance involves pooling and transferring risks from individuals to insurers. It has key characteristics like pooling losses, paying fortuitous losses, risk transfer, and indemnification. Insurance requires an insurable risk to have a large number of exposure units and losses must be accidental, determinable, and not catastrophic. Common insurance products include life, health, property, vehicle, and travel insurance. Life insurance provides protection for dependents and has various types like term, whole, universal, and variable policies.
It's a form of coverage that provides financial protection in the event of head injuries sustained while wearing a helmet. Helmet insurance can cover medical expenses, lost income, and rehabilitation costs.
Study of salary administration for delta l ife insurance company ltdSupa Buoy
Hi Friends
This is supa bouy
I am a mentor, Friend for all Management Aspirants, Any query related to anything in Management, Do write me @ supabuoy@gmail.com.
I will try to assist the best way I can.
Cheers to lyf…!!!
Supa Bouy
The document provides background information on Delta Life Insurance Company Ltd (DLIC), a leading life insurance company in Bangladesh. It discusses how DLIC was established in 1986 and has since introduced innovative life insurance schemes and products, including being the first to offer micro-insurance. The summary also provides an overview of DLIC's business performance, noting its total revenue increased 16.21% in 2002 with gross premium up 13.25% and investment income rising 32.78% from the previous year.
More Related Content
Similar to Presentation foundation of-actuarial_science
Revised Version of Risk Management - a conceptual frameworkSVS College
This document provides a conceptual framework for risk management. It defines risk as the possibility of loss and classifies risks into pure risks and speculative risks. Pure risks only carry the potential for loss, while speculative risks may result in losses or profits. The document outlines various principles of risk management like utmost good faith, insurable interest, indemnity, subrogation, and contribution. It also discusses types of pure and speculative risks, as well as approaches to handling risks like risk retention, risk transfer, risk avoidance, and diversification. The key steps in the risk management process are identified as setting objectives, identifying risks, evaluating risks, selecting policies, developing strategies, and organizational control.
This document discusses various topics related to health care planning, including identifying sources of health care plans and types of coverage provided. It describes major provisions of plans and policies, as well as the purpose of disability income insurance. The document provides tips for reducing personal health care costs through prevention. It also discusses health insurance as financial protection and as part of risk management. Various types of health care providers, coverage, and policies are defined.
This document provides an introduction to life insurance, including the different types of life insurance policies and how to take out a life insurance policy. It discusses key concepts in life insurance including claim settlement processes, surrender and paid up values. The document concludes by recommending additional resources to learn more about insurance.
Insurance involves pooling and transferring risks from individuals to insurers. It has key characteristics like pooling losses, paying fortuitous losses, risk transfer, and indemnification. Insurance requires an insurable risk to have a large number of exposure units and losses must be accidental, determinable, and not catastrophic. Common insurance products include life, health, property, vehicle, and travel insurance. Life insurance provides protection for dependents and has various types like term, whole, universal, and variable policies.
It's a form of coverage that provides financial protection in the event of head injuries sustained while wearing a helmet. Helmet insurance can cover medical expenses, lost income, and rehabilitation costs.
Study of salary administration for delta l ife insurance company ltdSupa Buoy
Hi Friends
This is supa bouy
I am a mentor, Friend for all Management Aspirants, Any query related to anything in Management, Do write me @ supabuoy@gmail.com.
I will try to assist the best way I can.
Cheers to lyf…!!!
Supa Bouy
The document provides background information on Delta Life Insurance Company Ltd (DLIC), a leading life insurance company in Bangladesh. It discusses how DLIC was established in 1986 and has since introduced innovative life insurance schemes and products, including being the first to offer micro-insurance. The summary also provides an overview of DLIC's business performance, noting its total revenue increased 16.21% in 2002 with gross premium up 13.25% and investment income rising 32.78% from the previous year.
Similar to Presentation foundation of-actuarial_science (10)
1. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Foundations of Actuarial Science
A Generalized Introduction to Actuarial Valuations
Chisomo Makombo Sakala
Truman State University
May 7, 2012
2. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
THE ACTUARY!
3. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Outline
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
4. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Outline
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
5. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Outline
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
6. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Outline
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
7. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Some Terminology
Actuary
Financial Security System
Economic Risks
8. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Some Terminology
Actuary
Financial Security System
Economic Risks
9. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Some Terminology
Actuary
Financial Security System
Economic Risks
10. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems & Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
11. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
12. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
13. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
14. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
15. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
16. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
17. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Examples of Financial Security Systems Their Risks
Insurance
Property and Liability InsuranceWarranty
Life Insurance and Annuities
Health Insurance
Employee Benets
Retirement Benets and Pensions
Health and Welfare Benets
Group Benets
18. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Time Value of Money
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
19. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Time Value of Money
Example 1 - Present Value (PV)
An individual is obliged to make a payment of 150, nine months
from now and receive another payment in the amount of 73, ve
years from now. Assuming an interest rate of 9%, and hence a
discounting factor of v = 1.09−1 , determine the present value of
this cashow stream.
20. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Time Value of Money
PV Solution
PV [(0.75, 150) , (5, −73)] = 150v 0.75 − 73v 5
−1 0.75 −1 5
= 150 1.09 − 73 1.09
= $ 93.17
21. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Time Value of Money
PV Solution
PV [(0.75, 150) , (5, −73)] = 150v 0.75 − 73v 5
−1 0.75 −1 5
= 150 1.09 − 73 1.09
= $ 93.17
22. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Time Value of Money
PV Solution
PV [(0.75, 150) , (5, −73)] = 150v 0.75 − 73v 5
−1 0.75 −1 5
= 150 1.09 − 73 1.09
= $ 93.17
23. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Time Value of Money
PV Solution
PV [(0.75, 150) , (5, −73)] = 150v 0.75 − 73v 5
−1 0.75 −1 5
= 150 1.09 − 73 1.09
= $ 93.17
24. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
25. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
Example 2- Actuarial Present Value (APV)
Suppose a (nancial security system's) provider is under the
obligation to compensate a participant a benet of $ 15, 000 for an
economic loss. It is known that this loss will almost surely occur at
some time-point in the future and that the probability distribution
for a random variable T that models the year in which the loss
occurs is:
t 1 2 3 4 5
Pr [T = t ] 0.25 0.18 0.15 0.19 0.23
Assuming an 8% interest rate, determine the APV.
26. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
APV Solution
APV [(T , 15, 000)] = E 15, 000 · v T
5
= 15, 000 Pr [T = t ] · v t
t =1
= 15, 000 Pr [T = 1] · v + Pr [T = 2] · v 2
+ Pr [T = 3] · v 3 + Pr [T = 4] · v 4 + Pr [T = 5] · v 5
−1 −2
= 15, 000 0.25 · (1.08) + 0.18 · (1.08)
−3 −4 −5
+0.15 · (1.08) + 0.19 · (1.08) + 0.23 · (1.08)
= $ 12, 016.01
27. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
APV Solution
APV [(T , 15, 000)] = E 15, 000 · v T
5
= 15, 000 Pr [T = t ] · v t
t =1
= 15, 000 Pr [T = 1] · v + Pr [T = 2] · v 2
+ Pr [T = 3] · v 3 + Pr [T = 4] · v 4 + Pr [T = 5] · v 5
−1 −2
= 15, 000 0.25 · (1.08) + 0.18 · (1.08)
−3 −4 −5
+0.15 · (1.08) + 0.19 · (1.08) + 0.23 · (1.08)
= $ 12, 016.01
28. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
APV Solution
APV [(T , 15, 000)] = E 15, 000 · v T
5
= 15, 000 Pr [T = t ] · v t
t =1
= 15, 000 Pr [T = 1] · v + Pr [T = 2] · v 2
+ Pr [T = 3] · v 3 + Pr [T = 4] · v 4 + Pr [T = 5] · v 5
−1 −2
= 15, 000 0.25 · (1.08) + 0.18 · (1.08)
−3 −4 −5
+0.15 · (1.08) + 0.19 · (1.08) + 0.23 · (1.08)
= $ 12, 016.01
29. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
APV Solution
APV [(T , 15, 000)] = E 15, 000 · v T
5
= 15, 000 Pr [T = t ] · v t
t =1
= 15, 000 Pr [T = 1] · v + Pr [T = 2] · v 2
+ Pr [T = 3] · v 3 + Pr [T = 4] · v 4 + Pr [T = 5] · v 5
−1 −2
= 15, 000 0.25 · (1.08) + 0.18 · (1.08)
−3 −4 −5
+0.15 · (1.08) + 0.19 · (1.08) + 0.23 · (1.08)
= $ 12, 016.01
30. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Valuation of Time-Valued Risks
APV Solution
APV [(T , 15, 000)] = E 15, 000 · v T
5
= 15, 000 Pr [T = t ] · v t
t =1
= 15, 000 Pr [T = 1] · v + Pr [T = 2] · v 2
+ Pr [T = 3] · v 3 + Pr [T = 4] · v 4 + Pr [T = 5] · v 5
−1 −2
= 15, 000 0.25 · (1.08) + 0.18 · (1.08)
−3 −4 −5
+0.15 · (1.08) + 0.19 · (1.08) + 0.23 · (1.08)
= $ 12, 016.01
31. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Actuarial Valuation of Risks
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
32. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Actuarial Valuation of Risks
Example 3 - Pricing Reserving
Policy on Woman aged 34 years and 7 months
Benet
$100,000 death benet
payable at end of the quarter of death
No benet if death within either 5 years of inception, or later
than the 20 years after issue
Funding
Level bi-monthly contributions
starting 2 years after the contract's inception
ending either when she dies or in 10 years (whichever
occurs earlier).
Assume 6% interest rate
Determine minimum bi-monthly premium
Determine minimum reserve 8 years
to be held after policy
issue
33. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Actuarial Valuation of Risks
Example 3 - Pricing Reserving
Policy on Woman aged 34 years and 7 months
Benet
$100,000 death benet
payable at end of the quarter of death
No benet if death within either 5 years of inception, or later
than the 20 years after issue
Funding
Level bi-monthly contributions
starting 2 years after the contract's inception
ending either when she dies or in 10 years (whichever
occurs earlier).
Assume 6% interest rate
Determine minimum bi-monthly premium
Determine minimum reserve 8 years
to be held after policy
issue
34. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Actuarial Valuation of Risks
Pricing Reserving Solution
Model Parameters
x = 34.58¯ t = 8 i = 0.06
3
m1 = 4 k1 = 5 n1 = 15
m0 = 6 k0 = 2 n0 = 10
and AAA 2001 CSO Survival Distribution - Female Composite
44. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Problem
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
45. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Problem
Generalized Life Insurance Valuation Problem
Problem Statement
An insurer oers (x ) a k1 -year-deferred, n1 -year-temporary life
insurance policy with a benet in the amount of a unit that is paid
in the
th
m1 in which (x ) fails to survive. Suppose the terms of the
policy are that the benet must be funded by a k0 -year-deferred,
n0 -year-temporary life annuity in which level contributions are made
th
each m0 until (x ) fails to survive. Suppose inception is at time x ,
then under the Principle of Actuarial Equivalence and assuming a
thly
compound interest rate of i, determine a nominal m0 premium
(m )
Px 0 that is actuarially fair. In addition, for each nonnegative
integer t , determine the minimum reserve t Vx that must be held at
time x + t.
46. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Model
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
47. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Model
Developing the Model
Let m ∈ N, x ≥ 0 and dene Jxm) : (x , +∞) −→ m ∈ R | r ∈ Z so
( r
that Jxm)(g ) := min m ∈ [g − x , +∞) |r ∈ Z for all g x . Then for
( r
Age at Death random variable G ,
thly
The present value of a unit of death benet and of the unit m0 funding
contributions
m1 ) G )
v Jx if G ∈ (x + k1 , x + k1 + n1 ]
(
(
Z x (G ) =
0 otherwise
k0 1−v n0
v · 1 if G x + k0 + n0
m0 1−v m0
(m )
Yx(m0 )(G ) = k0 1−v Jx 0 (G1 k0
)−
v · m 1−v m0 if G ∈ (x + k0 , x + k0 + n0 ]
0
0 otherwise
48. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Model
Developing the Model
Let m ∈ N, x ≥ 0 and dene Jxm) : (x , +∞) −→ m ∈ R | r ∈ Z so
( r
that Jxm)(g ) := min m ∈ [g − x , +∞) |r ∈ Z for all g x . Then for
( r
Age at Death random variable G ,
thly
The present value of a unit of death benet and of the unit m0 funding
contributions
m1 ) G )
v Jx if G ∈ (x + k1 , x + k1 + n1 ]
(
(
Z x (G ) =
0 otherwise
k0 1−v n0
v · 1 if G x + k0 + n0
m0 1−v m0
(m )
Yx(m0 )(G ) = k0 1−v Jx 0 (G1 k0
)−
v · m 1−v m0 if G ∈ (x + k0 , x + k0 + n0 ]
0
0 otherwise
49. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Model
Developing the Model
Let m ∈ N, x ≥ 0 and dene Jxm) : (x , +∞) −→ m ∈ R | r ∈ Z so
( r
that Jxm)(g ) := min m ∈ [g − x , +∞) |r ∈ Z for all g x . Then for
( r
Age at Death random variable G ,
thly
The present value of a unit of death benet and of the unit m0 funding
contributions
m1 ) G )
v Jx if G ∈ (x + k1 , x + k1 + n1 ]
(
(
Z x (G ) =
0 otherwise
k0 1−v n0
v · 1 if G x + k0 + n0
m0 1−v m0
(m )
Yx(m0 )(G ) = k0 1−v Jx 0 (G1 k0
)−
v · m 1−v m0 if G ∈ (x + k0 , x + k0 + n0 ]
0
0 otherwise
50. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Evaluation
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
51. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Evaluation
Evaluation of Generalized Model
Dene SG : R −→ [0, 1] so that SG (g ) := Pr [G g ] for all g ∈ R.
Then,
1(m1 )
k1 | Ax :n1 := E [Zx (G ) | G x ]
m1 ·n1 −1 S x + k1 + m1 − SG u + rm11
r +
= v k1 + rm11 · G
+
r =0 SG (x )
and
(m )
k0 | ¨x :n0 := E Yx 0 (G ) | G x
(m )
a 0
m0 ·n0 −1 r
SG x + k0 + m0
r
= v k0 + m0 ·
r =0 SG (x )
52. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Evaluation
Evaluation of Generalized Model
Dene SG : R −→ [0, 1] so that SG (g ) := Pr [G g ] for all g ∈ R.
Then,
1(m1 )
k1 | Ax :n1 := E [Zx (G ) | G x ]
m1 ·n1 −1 S x + k1 + m1 − SG u + rm11
r +
= v k1 + rm11 · G
+
r =0 SG (x )
and
(m )
k0 | ¨x :n0 := E Yx 0 (G ) | G x
(m )
a 0
m0 ·n0 −1 r
SG x + k0 + m0
r
= v k0 + m0 ·
r =0 SG (x )
53. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Evaluation
Evaluation of Generalized Model
Dene SG : R −→ [0, 1] so that SG (g ) := Pr [G g ] for all g ∈ R.
Then,
1(m1 )
k1 | Ax :n1 := E [Zx (G ) | G x ]
m1 ·n1 −1 S x + k1 + m1 − SG u + rm11
r +
= v k1 + rm11 · G
+
r =0 SG (x )
and
(m )
k0 | ¨x :n0 := E Yx 0 (G ) | G x
(m )
a 0
m0 ·n0 −1 r
SG x + k0 + m0
r
= v k0 + m0 ·
r =0 SG (x )
54. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Solution
1 Introduction
2 Motivational Examples
Time Value of Money
Valuation of Time-Valued Risks
Actuarial Valuation of Risks
3 Generalized Traditional Life Insurance Valuation
The Problem
The Model
The Evaluation
The Solution
4 Summary
55. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Solution
Solution to the Generalized Traditional Life Insurance
Valuation Problem
1(m1 ) (m0 ) (m )
The nominal mthly0 premium saties k1 | Ax :n
1
≡ Px a 0
· k0 | ¨x :n0 .
Therefore,
1(m1 )
(m ) k1 | Ax :n1
Px 0 = (m )
a 0
k0 | ¨x :n0
In addition, the reserve at time t≥0 is the (net) Actuarial Present
Value of future contingent cashows, hence
1(m1 )
t Vx max{k1 −t ,0}| Ax +t : min{max{n −t ,0},n }
=
1 1
(m0 ) (m0 )
−Px a
· max{k0 −t ,0}| ¨
x +t : min{max{n0 −t ,0},n0 }
56. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Solution
Solution to the Generalized Traditional Life Insurance
Valuation Problem
1(m1 ) (m0 ) (m )
The nominal mthly0 premium saties k1 | Ax :n
1
≡ Px a 0
· k0 | ¨x :n0 .
Therefore,
1(m1 )
(m ) k1 | Ax :n1
Px 0 = (m )
a 0
k0 | ¨x :n0
In addition, the reserve at time t≥0 is the (net) Actuarial Present
Value of future contingent cashows, hence
1(m1 )
t Vx max{k1 −t ,0}| Ax +t : min{max{n −t ,0},n }
=
1 1
(m0 ) (m0 )
−Px a
· max{k0 −t ,0}| ¨
x +t : min{max{n0 −t ,0},n0 }
57. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
The Solution
Solution to the Generalized Traditional Life Insurance
Valuation Problem
1(m1 ) (m0 ) (m )
The nominal mthly0 premium saties k1 | Ax :n
1
≡ Px a 0
· k0 | ¨x :n0 .
Therefore,
1(m1 )
(m ) k1 | Ax :n1
Px 0 = (m )
a 0
k0 | ¨x :n0
In addition, the reserve at time t≥0 is the (net) Actuarial Present
Value of future contingent cashows, hence
1(m1 )
t Vx max{k1 −t ,0}| Ax +t : min{max{n −t ,0},n }
=
1 1
(m0 ) (m0 )
−Px a
· max{k0 −t ,0}| ¨
x +t : min{max{n0 −t ,0},n0 }
58. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
59. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
60. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
61. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
62. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
63. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
64. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Real-world Extensions to ideal Actuarial Model
ExpensesAcquisition Costs
Lapse, Withdrawal
Dynamic Interest Rates
Financial Economics
Ination, Exchange, Tax Rates
... Many more
65. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary
Conclusion
Professional Ethics: Actuaries serve to optimize intersts of both the
participants and providers of a nancial security
system