Presentation foundation of-actuarial_science

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

7. 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

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

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

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

35. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary Actuarial Valuation of Risks Pricing Reserving Solution - Cont'd Premium 1(4) (6) (6) 5| A34.58¯ :15 0.06 3 ≡ P34.58¯ · 2| ¨34.58¯ :10 0.06 3 a 3 1(4) (6) 5| A34.58¯ :15 0.06 3 P34.58¯ 3 = (6) = 0.238799% a 2| ¨34.58¯ :10 0.06 3 1 Actual bi-monthly premium = ($100, 000.00 × 0.238799%) 6 = $39.80

40. Introduction Motivational Examples Generalized Traditional Life Insurance Valuation Summary Actuarial Valuation of Risks Pricing Reserving Solution - Cont'd Reserve 1(4) 8 V34.58¯ 3 max{5−8,0}| A34.58¯+8 : min{max{15−8,0},15} 0.06 := 3 (6) − 0.238799% · max{2−8,0}| ¨a 34.58¯+8 : min{max{10−8,0},10} 0.06 3 = 0.67677% th Actual 8 -year-reserve-level = $100, 000 × 0.67677% = $676.77

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 

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 )

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 }

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

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

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