ACTUARIAL MATHEMATICS I

Chapter 3
Life Insurance
Part 1
Department of Actuarial Science
Faculty of Computer & Mathematical Sciences
Universiti Teknologi MARA, UiTM
Shah Alam, Malaysia
Introduction
OBJECTIVES:
• To develop valuation formulae of traditional insurance benefits for
whole life insurance, term insurance and endowment insurance:
• Continuous future lifetime random variable, 𝑇 𝑥 − 𝑇𝑥
• Curtate future lifetime random variable, 𝐾(𝑥) − 𝐾𝑥

• To identify the random variable representing the present values of the

benefits.
• To derive expressions for moments of these random variables.

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Introduction • Nominal rate of interest
Recap FM: compounded 𝑝 times per year:
𝑖
• Interest theory functions 𝑖 (𝑝) = 𝑝( 1 + 𝑖 𝑝 − 1)
(discount factor): (𝑝) 𝑝
1+𝑖
𝑣 = 1+𝑖 −1
=
1 1+𝑖 =
1+𝑖 𝑝
• Effective rate of discount (yearly):
• The force of interest (yearly): 𝑑 = 1 − 𝑣 = 𝑖𝑣 = 1 − 𝑒 −𝛿
𝛿 = log 1 + 𝑖 • Nominal rate of discount
1 + 𝑖 = 𝑒𝛿
compounded 𝑝 times per yr:
1

𝑣=𝑒 −𝛿 𝑑 (𝑝) = 𝑝 1 − 𝑣 𝑝
𝑝
1−𝑑 (𝑝)
𝑣=
𝑝

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Benefit payable at the moment of death
(continuous insurance)
𝑏𝑡 − benefit paid at time 𝑡 when death occurs at time 𝑡
𝑉𝑡 − present value at issue of a unit amount paid at time t
𝑍𝑡 − PV at issue of benefit, 𝑏𝑡 , paid at time 𝑡 when death occurs at time 𝑡.

𝐴ҧ = 𝐸 𝑍𝑡
𝑏𝑡
= 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑃𝑉 (EPV)
𝑍𝑡 = 𝑏𝑡 𝑉𝑡 𝑥+𝑘 𝑥+𝑘+1 = 𝑁𝑒𝑡 𝑠𝑖𝑛𝑔𝑙𝑒 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
∞
(𝑥 + 𝑡) = ‫׬‬0 𝑍𝑡 𝑔 𝑡 𝑑𝑡
(Death) ∞
𝐸 𝑍𝑡 =? = ‫׬‬0 𝑏𝑡 𝑉𝑡 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

𝑉 𝑍𝑡 = 2𝐴ҧ − 𝐴ҧ 2
2 ҧ
𝐴 = E[𝑍𝑡2 ]
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1. A continuous $𝟏 𝐰hole life insurance on (𝒙)
Provides a benefit of $1 payable at the moment of death (𝑥) at any
time 𝑡 in the future. 𝑡
𝑣

𝑏𝑡 = $1

𝑥 ....... 𝑥+𝑘 𝑥+𝑘+1 .......

𝑍𝑡 = 𝑏𝑡 𝑉𝑡 𝑥+𝑡
(Death)

𝐸 𝑍𝑡 =𝐴ҧ𝑥

𝑏𝑡 = 1 for t > 0
𝑉𝑡 = 𝑣 𝑡 for t > 0
𝑍𝑡 = 𝑏𝑡 𝑉𝑡 = 1. 𝑣 𝑡 for t > 0
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EPV of Benefits, 𝑬 𝒁𝒕 :
∞ ∞ ∞

𝐸 𝑍𝑡 = 𝐴ҧ𝑥 = න 𝑍𝑡 . 𝑔 𝑡 𝑑𝑡 = න 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 = න 1. 𝑣 𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 0 0

∞ ∞

𝐸 𝑍𝑡2 = 2𝐴ҧ𝑥 = න (𝑍𝑡 )2 . 𝑔 𝑡 𝑑𝑡 = න 𝑏𝑡2 𝑉𝑡2 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 0
∞ 2 2𝑡
= ‫׬‬0 1 . 𝑣 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

𝑉𝑎𝑟 𝑍𝑡 = 2𝐴ҧ𝑥 − (𝐴ҧ𝑥 )2

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2. A continuous $𝟏 n-year term Insurance on (𝒙)
Provides a payment of $1 payable at the moment of death of (𝑥) only if
dies within 𝑛 years after issue.
𝑣𝑡
𝑏𝑡 = $1

𝑥 ....... 𝑥+𝑘 𝑥+𝑘+1 .... 𝑥+𝑛−1 𝑥+𝑛

𝑍𝑡 = 𝑏𝑡 𝑉𝑡
𝑥+𝑡
(Death)
𝐸 𝑍𝑡 =𝐴ҧ𝑥:𝑛ȁ
1. 𝑣 𝑡 :0 < 𝑡 ≤ 𝑛
𝑍𝑡 = ቊ
𝑡 0. 𝑣 𝑛 ∶ 𝑡>𝑛
𝑉𝑡 = ቊ 𝑛 : 0 < 𝑡 ≤ 𝑛
𝑣
1 ∶0<𝑡≤𝑛 𝑣 : 𝑡>𝑛 1. 𝑣 𝑡 :0 < 𝑡 ≤ 𝑛
𝑏𝑡 = ቊ =ቊ
0 ∶ 𝑡>𝑛 0 ∶ 𝑡>𝑛

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EPV of Benefits, 𝑬 𝒁𝒕 :
∞ 𝑛 ∞

𝐸 𝑍𝑡 = 𝐴ҧ𝑥:𝑛ȁ = න 𝑍𝑡 . 𝑔 𝑡 𝑑𝑡 = න 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + න 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 0 𝑛

𝑛 ∞

= න 1. 𝑣 𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + න 0. 𝑣 𝑛 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 𝑛

= න 1. 𝑣 𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
0

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 ∞

𝐸 𝑍𝑡2 = 2𝐴ҧ𝑥:𝑛ȁ = න (𝑍𝑡 )2 . 𝑔 𝑡 𝑑𝑡

0
𝑛 2 2 ∞ 2 2
= ‫׬‬0 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + ‫ 𝑡𝑉 𝑡𝑏 𝑛׬‬. 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

𝑛 ∞

= න 12 𝑣 2𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + න 02 𝑣 2𝑛 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 𝑛
𝑛 2 2𝑡
= ‫׬‬0 1 𝑣 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

2 2
ҧ ҧ
𝑉 𝑍𝑡 = 𝐴𝑥:𝑛ȁ − 𝐴𝑥:𝑛ȁ

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Note:
What is the difference between 𝑍𝑡 and 𝐸 𝑍𝑡 ?
• 𝑍𝑡 =Present value of future benefit (real value).
• 𝐸 𝑍𝑡 =Expected value of present value of future benefits.

𝑃[𝑍 < 𝐸 𝑍𝑡 ]= Probability of sufficient fund to pay benefits/claims

𝑃[𝑍 > 𝐸 𝑍𝑡 ]= Probability of insufficient fund to pay benefits/claims

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3. A $𝟏 n-year Pure Endowment Insurance on (𝒙)
Provides an endowment benefit of $1 at the end of 𝑛 years provided
(𝑥) survives to age 𝑥 + 𝑛. The benefits payable upon survives.
𝑣𝑛

𝑏𝑡 = $1

𝑥 ....... 𝑥+𝑛−1 𝑥+𝑛

𝑍𝑡 = 𝑏𝑡 𝑉𝑡

𝑆𝑢𝑟𝑣𝑖𝑣𝑒𝑠
𝐸 𝑍𝑡 =𝐴ҧ𝑥:𝑛ȁ
0. 𝑣 𝑡 :0 < 𝑡 < 𝑛
𝑍𝑡 = ቊ
1. 𝑣 𝑛 ∶ 𝑡≥𝑛
0 ∶0<𝑡<𝑛 𝑡
𝑏𝑡 = ቊ 𝑉𝑡 = ቊ 𝑛 : 0 < 𝑡 < 𝑛
𝑣
=ቊ
0 ∶0<𝑡<𝑛
1 ∶ 𝑡≥𝑛 𝑣 ∶ 𝑡≥𝑛 1. 𝑣 𝑛 : 𝑡≥𝑛

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 ∞
∞ ҧ
𝐴𝑥:𝑛ȁ = ‫׬‬0 𝑍𝑡 . 𝑔 𝑡 𝑑𝑡
∞
𝐸 𝑍𝑡 = 𝐴ҧ𝑥:𝑛ȁ = න 𝑍𝑡 . 𝑔 𝑡 𝑑𝑡 = ‫ 𝑡𝑉 𝑡𝑏 𝑛׬‬. 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
∞
0 = ‫ 𝑛׬‬1. 𝑣 𝑛 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
∞ OR = 1. 𝑣 𝑛
∞
‫𝑥𝜇 𝑥𝑝𝑡 𝑛׬‬+𝑡 𝑑𝑡
= ‫ 𝑡𝑉 𝑡𝑏 𝑛׬‬. 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
∞ 𝑛 ∞
= 1. 𝑣 (‫׬‬0 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
= ‫ 𝑛׬‬1. 𝑣 𝑛 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 𝑛
− ‫׬‬0 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡)
𝑛 ∞
= 𝑣 ‫𝑥𝜇 𝑥𝑝𝑡 𝑛׬‬+𝑡 𝑑𝑡 = 1. 𝑣 𝑛 (1 − 𝑛𝑞𝑥 )
= 𝑣 𝑛 . 𝑛𝑝𝑥 = 1. 𝑣 𝑛 . 𝑛𝑝𝑥
= 𝑛𝐸𝑥
= 𝑛𝐸𝑥

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 ∞
2
𝐸 𝑍𝑡2 = 𝐴ҧ𝑥:𝑛ȁ = න (𝑍𝑡 )2 . 𝑔 𝑡 𝑑𝑡
0
𝑛 2 2 ∞ 2 2
= ‫׬‬0 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + ‫ 𝑡𝑉 𝑡𝑏 𝑛׬‬. 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
𝑛 2 2𝑡 ∞ 2 2𝑛
= ‫׬‬0 0 𝑣 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + ‫ 𝑛׬‬1 𝑣 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
2𝑛 ∞
= 𝑣 ‫𝑥𝜇 𝑥𝑝𝑡 𝑛׬‬+𝑡 𝑑𝑡
= 𝑣 2𝑛 . 𝑛𝑝𝑥

𝑉 𝑍𝑡 = 2𝐴ҧ𝑥:𝑛ȁ − (𝐴ҧ𝑥:𝑛ȁ )2 = 𝑣 2𝑛 . 𝑛𝑝𝑥 − (𝑣 𝑛 . 𝑛𝑝𝑥 )2

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4. A continuous $𝟏 n-year Endowment Insurance on (𝒙)
Provides a benefits of $1 at the moment of death (𝑥) only if death occurs
within n years after issues or, a benefit of $1 at the end of n years if
(𝑥) survives to age 𝑥 + 𝑛. 𝑣𝑛
𝑣𝑡 OR
𝑏𝑡 = $1 𝑏𝑡 = $1

𝑥 ....... 𝑥+𝑘 𝑥+𝑘+1 𝑥+𝑛

𝑍𝑡 = 𝑏𝑡 𝑉𝑡
𝑥+𝑡
𝑑𝑒𝑎𝑡ℎ 𝑆𝑢𝑟𝑣𝑖𝑣𝑒𝑠
𝐸 𝑍𝑡 =𝐴ҧ𝑥:𝑛ȁ

𝑣 𝑡 :0 < 𝑡 < 𝑛 𝑡 :0 < 𝑡 < 𝑛

1 ∶0<𝑡<𝑛 𝑉𝑡 = ቊ 𝑛 1. 𝑣
𝑏𝑡 = ቊ 𝑍𝑡 = ቊ
1 ∶ 𝑡≥𝑛 𝑣 ∶ 𝑡≥𝑛 1. 𝑣 𝑛 ∶ 𝑡≥𝑛

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 ∞ 𝑛 ∞

𝐸 𝑍𝑡 = 𝐴ҧ𝑥:𝑛ȁ = න 𝑍𝑡 . 𝑔 𝑡 𝑑𝑡 = න 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + න 𝑏𝑡 𝑉𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 0 𝑛
𝑛 ∞

= න 1. 𝑣 𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + න 1. 𝑣 𝑛 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡

0 𝑛
𝑛
= ‫׬‬0 1. 𝑣 𝑡 . 𝑡𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + 𝑣 𝑛 . 𝑛𝑝𝑥
= 𝐴ҧ𝑥:𝑛ȁ + 𝐴ҧ𝑥:𝑛ȁ
𝑉 𝑍𝑡 = 2𝐴ҧ𝑥:𝑛ȁ − (𝐴ҧ𝑥:𝑛ȁ )2

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