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1 Lesson 1 - Probability Review 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8 1.9 1.10 1.11 1.12 1.13

𝜇2 = 𝜇′2 − 𝜇 2 𝜇3 = 𝜇′3 − 3𝜇2′ 𝜇 − 2𝜇 3 𝑉𝑎𝑟(𝑋) = 𝐸[𝑋 2 ] − 𝐸[𝑋]2 𝑉𝑎𝑟(𝑎𝑋 + 𝑏𝑌) = 𝑎2 𝑉𝑎𝑟(𝑋) + 2𝑎𝑏𝐶𝑜𝑣 (𝑋, 𝑌) + 𝑏 2 𝑉𝑎𝑟(𝑌) 𝑉𝑎𝑟(∑𝑛𝑖=1 𝑋𝑖) = 𝑛𝑉𝑎𝑟(𝑋) ∑𝑛 𝑋 𝑛𝑉𝑎𝑟(𝑋) 𝑉𝑎𝑟(𝑥) = 𝑉𝑎𝑟(𝑋 ) = 𝑉𝑎𝑟 ( 𝑖=1 𝑖) = 2 𝑛

Bayes Theorem Pr(𝐵 |𝐴)Pr(𝐴) Pr(𝐴|𝐵) =

𝑓𝑥 (𝑥|𝑦) =

𝑛

𝑛

Pr(𝐵)

𝑓𝑦( 𝑦|𝑥 )𝑓𝑥 (𝑥) 𝑓𝑦(𝑦)

Law of Total Probability (Discrete) Pr(𝐴) = ∑𝑖 Pr(𝐴 ∩ 𝐵𝑖 ) =∑ 𝑖 Pr(𝐵𝑖 )Pr(𝐴|𝐵𝑖 ) Law of Total Probability (Continuous) Pr(𝐴) = ∫ Pr(𝐴|𝑥) 𝑓(𝑥)𝑑𝑥 Conditional Mean Formula 𝐸𝑋 [𝑋] = 𝐸𝑌 [𝐸𝑋 [𝑋|𝑌]] Double Expectation Formula 𝐸𝑋 [𝑔(𝑋)] = 𝐸𝑌 [𝐸𝑋 [𝑔(𝑋 )|𝑌]] Conditional Variance Formula

Lesson 2 – Survival Distributions: Probability Functions

2.1 𝑆𝑥+𝑡 (𝑢) =

2.2 𝑆𝑥 (𝑡)

𝑆𝑥 (𝑡+𝑢 ) 𝑆𝑥 (𝑡) 𝑆0(𝑥+𝑡 ) = 𝑆 (𝑥) 0 𝐹0(𝑥+𝑡 )−𝐹0 (𝑥)

2.3 𝐹𝑥 (𝑡) =

2.4 2.5

𝑡|𝑢 𝑞𝑥

𝑡|𝑢 𝑞𝑥

1−𝐹0 (𝑥)

= 𝑡 𝑝𝑥 − 𝑡+𝑢 𝑝𝑥

= 𝑡+𝑢 𝑞𝑥 − 𝑡 𝑞𝑥

Life Table Functions 𝑡 𝑝𝑥

𝑡 𝑞𝑥

=

𝑡|𝑢 𝑞𝑥

=

𝑡+𝑢 𝑝𝑥

𝑙𝑥+𝑡

=

𝑙𝑥

𝑡 𝑑𝑥

𝑙𝑥

=

𝑢𝑑𝑥+𝑡

𝑙𝑥

𝑙𝑥 −𝑙𝑥+𝑡

=

𝑙𝑥

𝑙𝑥+𝑡−𝑙𝑥+𝑡+𝑢 𝑙𝑥

= 𝑡 𝑝𝑥 𝑢 𝑝𝑥+𝑡

𝑉𝑎𝑟𝑋 (𝑋) = 𝐸𝑌 [𝑉𝑎𝑟𝑋 (𝑋|𝑌)] + 𝑉𝑎𝑟𝑌 (𝐸𝑋 [𝑋|𝑌])

Distribution Bernoulli Binomial Uniform Exponential

Mean 𝑞 𝑚𝑞 𝑎+𝑏 2

𝜃

Variance 𝑞(1 − 𝑞) 𝑚𝑞(1 − 𝑞)

(𝑏−𝑎) 2 12

𝜃2

Bernoulli Shortcut: If a random variable can only assume two values 𝑎 and 𝑏 with prob 𝑞 and 1 − 𝑞 , then 𝑉𝑎𝑟(𝑋) = 𝑞 (1 − 𝑞 )(𝑏 − 𝑎)2

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

2 3.1 𝜇𝑥+𝑡 =

𝑓𝑥 (𝑡 ) 𝑆𝑥 (𝑡)

Lesson 3 – Survival Distributions: Force of Mortality

3.2 𝜇𝑥+𝑡 =

𝑑 𝑞 𝑑𝑡 𝑡 𝑥

3.3 𝜇𝑥+𝑡 = −

3.4 𝜇𝑥+𝑡 = −

𝑡𝑝𝑥

𝑑 ln(𝑆𝑥 (𝑡 ))

4.2

𝑑𝑡 𝑑 ln( 𝑡𝑝𝑥 ) 𝑑𝑡

Lesson 4 – Survival Distribution: Mortality 4.1 Gompertz’ Law 𝜇𝑥 = 𝐵𝑐 𝑥 𝑐>1

3.5 𝑆𝑥 (𝑡) = exp(− ∫ 𝜇𝑥+𝑠 𝑑𝑠) 0 𝑡

3.7 𝑡 𝑝𝑥 = exp(− ∫𝑥 𝜇𝑠 𝑑𝑠) 3.8 𝑓𝑥 (𝑡) = 𝑆𝑥 (𝑡)𝜇𝑥+𝑡 = 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑡+𝑢 3.9 𝑃(𝑡 < 𝑇𝑥 < 𝑡 + 𝑢) = 𝑡|𝑢 𝑞𝑥 = ∫𝑡 𝑠 𝑝𝑥 𝜇𝑥+𝑠 𝑑𝑠 𝑥+𝑡

𝑡 𝑞𝑥

= ∫0 𝑠 𝑝𝑥 𝜇𝑥+𝑠 𝑑𝑠 −𝑘𝑡 If 𝜇′𝑥+𝑠 = 𝜇𝑥+𝑠 + 𝑘 for 0 ≤ 𝑠 ≤ 𝑡 then 𝑡𝑝𝑥′ = 𝑝 . 𝑡 𝑥𝑒 If 𝜇𝑥+𝑠 = 𝜇 𝑥+𝑠 + 𝜇𝑥+𝑠 for 0 ≤ 𝑠 ≤ 𝑡 then 𝑡 𝑝𝑥 = 𝑡 𝑝𝑥 𝑡 𝑝𝑥 3.10

𝑡

If 𝜇′𝑥+𝑠 = 𝑘𝜇𝑥+𝑠 for 0 ≤ 𝑠 ≤ 𝑡 then 𝑡𝑝𝑥′ = ( 𝑡 𝑝𝑥)

= exp(−

𝑡 𝑝𝑥

= exp(−𝐴𝑡 −

𝐵𝑐 𝑥 (𝑐 𝑡−1) ln(𝑐)

𝑘

4.4

)

𝜇𝑥 = 𝐴 + 𝐵𝑐 𝑥 𝑐>1 A is constant part of force of mortality *Adding A to 𝜇 multiplies 𝑡 𝑝𝑥 by e−μt

4.3 Makeham’s Law

3.6 𝑡 𝑝𝑥 = exp(− ∫0 𝜇𝑥+𝑠 𝑑𝑠) 𝑡

𝑡 𝑝𝑥

𝐵𝑐 𝑥 (𝑐 𝑡−1) ln(𝑐)

)

𝜇𝑥 = 𝑘𝑥 𝑛 𝑘𝑥 (𝑛+1) ) 0𝑝𝑥 = exp(− 𝑛+1 Constant Force of Mortality 4.5 𝜇𝑥 = 𝜇 4.6 𝑡 𝑝𝑥 = e −μt 4.7 (BLANK) Weibull Distribution

Uniform Distribution 1 4.8 𝜇𝑥 = 0≤𝑥 ≤𝜔

4.9

𝑡 𝑝𝑥

𝜔−𝑥 𝜔−𝑥−𝑡

=

𝑡 𝑞𝑥

𝜔−𝑥 𝑡

0≤𝑡≤𝜔−𝑥

= 𝜔−𝑥 0 ≤ 𝑡 ≤ 𝜔 − 𝑥

4.11 𝑡|𝑢 𝑞𝑥 = 0≤𝑡+𝑢 ≤𝜔−𝑥 𝜔−𝑥 4.12 (BLANK)

4.10

𝑢

Beta Distribution 𝛼 4.13 𝜇𝑥 = 𝜔−𝑥 0 ≤ 𝑥 ≤ 𝜔 4.14

𝑡 𝑝𝑥

=(

𝜔−𝑥−𝑡 𝛼 𝜔−𝑥

) 0≤𝑡≤𝜔−𝑥

*The force of mortality is the sum of two uniform forces. 𝑡 𝑝𝑥 is the product of uniform probabilities

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

3 Lesson 5 – Survival Distributions: Moments Complete Future Lifetime ∞ 5.1 𝑒󰇗𝑥 = ∫ 𝑡 𝑡 𝑝𝑥𝜇𝑥+𝑡 𝑑𝑡 0 ∞ 5.2 𝑒󰇗𝑥 = ∫0 𝑡 𝑝𝑥 𝑑𝑡

5.3 𝐸[𝑇𝑥2 ] = 2 ∫0 𝑡 𝑡 𝑝𝑥 𝑑𝑡 ∞ 5.4 𝑉𝑎𝑟(𝑇𝑥 ) = 2 ∫ 𝑡 𝑡 𝑝𝑥 𝑑𝑡 − 𝑒󰇗𝑥2 0 5.5 𝑒󰇗𝑥:𝑛|  = 𝐸[min(𝑇𝑥 , 𝑛 )] 𝑛 𝑒󰇗𝑥:𝑛|  = ∫ 𝑡 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + 𝑛 𝑛 𝑝𝑥 0 𝑛 5.6 𝑒󰇗𝑥:𝑛|  = ∫ 𝑡 𝑝𝑥 𝑑𝑡 0 ∞

5.7 𝐸[min(𝑇𝑥 , 𝑛)2 ] = 2 ∫0 𝑡 𝑡 𝑝𝑥 𝑑𝑡 𝑛

5.20 𝑒𝑥 = ∑𝑘=1 e−μk = e (CFM) 1−e−μ 5.21 𝑒󰇗𝑥 = 𝑒𝑥 + 0.5 (UDD) 5.22 𝑒󰇗𝑥:𝑛|  = 𝑒𝑥:𝑛|  + 0.5 𝑛 𝑞𝑥 (UDD) ∞

−μ

*For those surviving n years, min(𝑇𝑥 , 𝑛) = 𝑛 𝑛 *For those not surviving n years, average future lifetime is 2 , since future lifetime is uniform. * 𝑒𝑥 = 𝐸[min(𝐾𝑥 , 𝑛)] * 𝑒󰇗𝑥 = 𝐸[𝑇𝑥 ] *If curtate,𝑒𝑥+𝑠:𝑎+𝑏| , 𝑏 < 1, 𝑠 < 1, 𝑎𝜖Ƶ is the same as 𝑒𝑥+𝑠:𝑎| 

Special Mortality Laws 𝜔−𝑥 5.8 𝑒󰇗𝑥 = 𝐸[𝑇𝑥 ] = 𝛼+1 (Beta) 𝑒󰇗𝑥 = 𝐸 [𝑇𝑥 ] =

𝑒󰇗𝑥 = 𝐸 [𝑇𝑥 ] =

𝜔−𝑥 1

2

(UDD)

𝜇 𝛼(𝜔−𝑥) 2

(CFM)

5.9 𝑉𝑎𝑟(𝑇𝑥 ) = (𝛼+1)2(𝛼+2) (Beta) 𝑉𝑎𝑟(𝑇𝑥 ) =

𝑉𝑎𝑟(𝑇𝑥 ) =

(𝜔−𝑥 )2 1

𝜇2

12

(UDD)

(CFM)

5.10 𝑒󰇗𝑥:𝑛|  = 𝑛 𝑝𝑥 (𝑛) + 𝑛 𝑞𝑥 ( ) (UDD) 2 𝑛

𝑒󰇗𝑥:𝑛| (𝑛) +  = 𝜔−𝑥 5.11 𝑒󰇗𝑥:1|  = 𝑝𝑥 + 0.5𝑞𝑥 (UDD) 𝜔−𝑥−𝑛

𝑛 𝑛 ( ) 𝜔−𝑥 2

(UDD)

Curtate Future Lifetime 5.12 𝑒𝑥 = ∑ ∞ 𝑘=0 𝑘 𝑘| 𝑞𝑥 𝑛−1 5.13 𝑒𝑥:𝑛|  = ∑ 𝑘=0 𝑘 𝑘| 𝑞𝑥 + 𝑛 𝑛 𝑝𝑥 2 2 5.14 𝐸[𝐾𝑥 ] = ∑ ∞ 𝑘=0 𝑘 𝑘|𝑞𝑥 2 2 2 5.15 𝐸[min(𝐾𝑥 , 𝑛) ] = ∑ 𝑛−1 𝑘=0 𝑘 𝑘| 𝑞𝑥 + 𝑛 𝑛 𝑝𝑥 5.16 𝑒𝑥 = ∑ ∞ 𝑝 𝑘=1 𝑘 𝑥 𝑛 5.17 𝑒𝑥:𝑛|  = ∑ 𝑘=1 𝑘 𝑝𝑥 2 5.18 𝐸[𝐾𝑥 ] = ∑ ∞ 𝑘=1(2𝑘 − 1) 𝑘 𝑝𝑥 5.19 𝐸[min(𝐾𝑥 , 𝑛)2 ] = ∑ 𝑛𝑘=1(2𝑘 − 1) 𝑘 𝑝𝑥

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

4 Lesson 6 – Survival Distributions: Percentiles and Recursions 6.1 𝑒󰇗𝑥 = 𝑒󰇗 𝑥:𝑛|  + 𝑛 𝑝𝑥 𝑒󰇗 𝑥+𝑛 6.2 𝑒𝑥 = 𝑒𝑥:𝑛|  + 𝑛 𝑝𝑥 𝑒𝑥+𝑛 6.3 𝑒𝑥 = 𝑒𝑥:𝑛−1|  + 𝑛 𝑝𝑥 (1 + 𝑒𝑥+𝑛 ) 6.4 𝑒𝑥 = 𝑝𝑥 + 𝑝𝑥 𝑒𝑥+1 = 𝑝𝑥 (1 + 𝑒𝑥+1 )𝑛 = 1 6.5 𝑒𝑥:𝑛|  = 𝑒𝑥:𝑚|  + 𝑚 𝑝𝑥𝑒𝑥+𝑚:𝑛−𝑚|  𝑚 < 𝑛 6.6 𝑒𝑥:𝑛|  = 𝑒𝑥:𝑚−1|  + 𝑚 𝑝𝑥(1 + 𝑒𝑥+𝑚:𝑛−𝑚|  )𝑚 < 𝑛  = 𝑝𝑥 + 𝑝𝑥 𝑒𝑥+1:𝑛−1|  = 𝑝𝑥 (1 + 𝑒𝑥+1:𝑛−1| ) 6.7 𝑒𝑥:𝑛|

Lesson 7 – Survival Distributions: Fractional Ages Uniform Distribution of Deaths 7.1 𝑙𝑥+𝑠 = (1 − 𝑠)𝑙𝑥 + 𝑠𝑙𝑥+1 = 𝑙𝑥 − 𝑠𝑑𝑥 7.2 𝑠 𝑞𝑥 = 𝑠𝑞𝑥 𝑠𝑞 𝑠𝑑𝑥 𝑙 7.3 𝑠 𝑞𝑥+𝑡 = 𝑥 = = 1 − ( 𝑥+𝑠+𝑡 ) , 0 ≤ 𝑠 + 𝑡 ≤ 1 1−𝑡𝑞𝑥

𝑙𝑥 −𝑡𝑑𝑥

7.4 𝑠 𝑝𝑥 𝜇𝑥+𝑠 = 𝑞𝑥 𝑞𝑥 𝑞 7.5 𝜇𝑥+𝑠 = 𝑥 = 1−𝑠𝑞 𝑠 𝑝𝑥

𝑙𝑥+𝑡

𝑥

7.6 𝑒󰇗𝑥 = 𝑒𝑥 + 2 (UDD) Recall: 5.11 (𝑒󰇗𝑥:1|  = 𝑝𝑥 + 0.5𝑞𝑥 ) 7.7 𝑒󰇗𝑥:𝑛|  = 𝑒𝑥:𝑛|  + 0.5 𝑛 𝑞𝑥 1

Constant Force of Mortality 7.8 𝑝𝑥 = 𝑒 −𝜇 7.9 𝜇 = −ln(𝑝𝑥 ) 7.10 𝑠 𝑝𝑥 = 𝑒 −𝜇𝑠 = (𝑝𝑥 ) 𝑠 7.11 𝑠 𝑝𝑥+𝑡 = (𝑝𝑥 ) 𝑠 0 ≤ 𝑡 ≤ 1 − 𝑠

Table 7.1: Summary of Formulas for Fractional Ages UDD CFM 𝑙𝑥+𝑠 𝑙𝑥 − 𝑠𝑑𝑥 𝑙𝑥 𝑝𝑥𝑠 𝑠𝑞 1 − 𝑝𝑥𝑠 𝑥 𝑠 𝑞𝑥 1 − 𝑠𝑞 𝑝𝑥𝑠 𝑝 𝑥 𝑠 𝑥 𝑠𝑞 𝑥 1 − 𝑝𝑥𝑠 𝑠 𝑞𝑥+𝑡 1 − 𝑡𝑞𝑥 𝑞𝑥 − ln(𝑝𝑥 ) 𝜇𝑥+𝑠 1 − 𝑠𝑞𝑥 𝑞𝑥 −𝑝𝑥𝑠 ln(𝑝𝑥 ) 𝑠 𝑝𝑥 𝜇𝑥+𝑠 𝑒󰇗𝑥 𝑒𝑥 + 0.5 𝑒󰇗 𝑥:𝑛|  𝑒𝑥:𝑛|  + 0.5 𝑛 𝑞𝑥 𝑒󰇗 𝑥:1| 𝑝𝑥 + 0.5𝑞𝑥

Function



Monica E. Revadulla EXAM MLC – Models for Life Contingencies

𝑒󰇗 𝑥:𝑡| = 𝑒󰇗𝑥:𝑘|  + 𝑘 𝑝𝑥 𝑒󰇗 𝑥+𝑘:𝑡−𝑘| 

Formula Summary of ASM 2014

5 Lesson 8 – Survival Distributions: Select Mortality  A man whose health was established 5 years ago will have better mortality than a randomly selected man.  A life selected at age 𝑥 can never become a life selected at any higher age. [𝑥] will never become [𝑥 + 1].

Lesson 10 – Insurance: Annual and 1/mthly – Moments ∞ ∞ 10.1 𝐸[𝑍] = ∑𝑘=0 𝑏𝑘 𝑣 𝑘+1 𝑘|𝑞𝑥 = ∑ 𝑘=0 𝑏𝑘 𝑣 𝑘+1 𝑘𝑝𝑥 𝑞𝑥+𝑘 ∞ 2] 2 2 2(𝑘+1) 10.2 𝐸[𝑍 = ∑𝑘=0 𝑏𝑘 𝑣 2(𝑘+1) 𝑘| 𝑞𝑥 = ∑ ∞ 𝑘𝑝𝑥 𝑞𝑥+𝑘 𝑘=0 𝑏𝑘 𝑣

Table 10.1: Actuarial notation for standard types of insurance Name Present Value of RV Symbol Whole life 𝑣 𝐾𝑥 +1 𝐴𝑥 𝐴𝑥 :𝑛| Term life  𝑣 𝐾𝑥 +1 𝐾𝑥 < 𝑛 { 0𝐾𝑥 ≥ 𝑛 0𝐾𝑥 < 𝑛 Deferred life 𝑛| 𝐴𝑥 { 𝐾𝑥 +1 𝑣 𝐾𝑥 ≥ 𝑛 Deferred term 0𝐾𝑥 ≤ 𝑛  𝑛| 𝐴𝑥 :𝑚| {𝑣 𝐾𝑥 +1 𝑛 < 𝐾𝑥 < 𝑛 + 𝑚 𝐴 𝑛|𝑚 𝑥 0𝐾𝑥 ≥ 𝑛 + 𝑚 0𝐾𝑥 < 𝑛 Pure Endowment 𝐴𝑥:𝑛| 󰆷  { 𝑛 𝑣 𝐾𝑥 ≥ 𝑛 𝑣𝑡 𝑝 𝑣 𝐾𝑥 +1 𝐾𝑥 < 𝑛 𝑣 𝑛 𝐾𝑥 ≥ 𝑛 2 − (𝐴𝑥:𝑛| ) {

Endowment

10.3 𝑉𝑎𝑟(𝑍) = 2𝐴𝑥:𝑛|  10.4 2𝑖 = 2𝑖 + 𝑖 2 2 𝑑 = 2𝑑 − 𝑑 2 2 𝑣 = 𝑣2 10.5 𝑛|𝐴𝑥 = 𝑛 𝐸𝑥 𝐴𝑥+𝑛 10.6 𝐴𝑥 :𝑛|  = 𝐴𝑥 − 𝑛|𝐴𝑥 = 𝐴𝑥 − 𝑛 𝐸𝑥 𝐴𝑥+𝑛 10.7 𝐴𝑥:𝑛|  = 𝐴𝑥 :𝑛|  + 𝑛 𝐸𝑥 = 𝐴 𝑥 − 𝑛 𝐸𝑥 𝐴𝑥+𝑛 + 𝑛 𝐸𝑥 𝑞 10.8 𝐴𝑋 = (𝐶𝐹𝑀) 𝑞+𝑖

𝑉𝑎𝑟(𝑍) = 𝑞+2𝑖+𝑖 2 − (

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

𝑞

𝑞

𝑞+𝑖

)

2

𝑡 𝑥

𝐴𝑥:𝑛| 

(CFM)

Formula Summary of ASM 2014

6 Table 10.2: EPV for insurance payable at EOY of death Type of Insurance CFM UDD 𝑞 𝑎 Whole Life 𝜔−𝑥| 𝑞+𝑖 2 𝑞 𝑎  n-year term 𝑛| (1 − (𝑣𝑝)𝑛 ) 𝑞+𝑖 𝜔−𝑥 𝑞 𝑣 𝑛 𝑎𝜔−(𝑥+𝑛)| n-year deferred life   (𝑣𝑝)𝑛 𝑞+𝑖 𝜔−𝑥 (𝑣𝑝)𝑛 n-year pure endowment 𝑣 𝑛 (𝜔 − (𝑥 + 𝑛)) 𝜔−𝑥

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Lesson 11 – Insurance: Continuous – Moments – Part I ∞ 11.1 𝐸(𝑍) = 𝐴𝑥 = ∫0 𝑣 𝑡 𝑓𝑥 (𝑡)𝑑𝑡 ∞ −𝛿𝑡 11.2 𝐴𝑥 = ∫0 𝑒 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 2 11.3 𝑉𝑎𝑟(𝑍) = 𝐸(𝑍 2 ) − (𝐸(𝑍)) = 2 𝐴𝑥 − (𝐴𝑥 )2 𝜇 ( ) −𝑛 𝜇+𝛿 11.4 𝑛|𝐴𝑥 = 𝑒 𝐴𝑥 =

𝜇

𝜇+𝛿

𝜇+𝛿

(𝐶𝐹𝑀)

𝐴𝑥+𝑛 = 𝐴𝑥 (𝐶𝐹𝑀 ) 11.5 𝐴𝑥 :𝑛|  = 𝐴𝑥 (1 − 𝑛 𝐸𝑥 ) = 

11.6

 𝑛| 𝐴𝑥 :𝑚|

11.8

𝑛| 𝐴𝑥

𝜇 (1 𝜇+𝛿

= 𝐴𝑥 ( 𝑛 𝐸𝑥 − 𝑚+𝑛 𝐸𝑥) =

− 𝑒 −𝑛(𝜇+𝛿) ) 𝜇𝑒 −𝑛(𝜇+𝛿) 𝜇+𝛿

𝜇 (1 − 𝑒 −𝑛(𝜇+𝛿) ) + 𝑒 −𝑛(𝜇+𝛿 ) 11.7 𝐴𝑥:𝑛|  = 𝜇+𝛿

(1 − 𝑒 −𝑚(𝜇+𝛿) )

 = 𝑛 𝐸𝑥 𝐴𝑥+𝑛 −𝑛(𝜇+𝛿 ) 𝐴𝑥:𝑛| 󰆷 = 𝑒

Formula Summary of ASM 2014

7 Lesson 12 – Continuous – Moments – Part II

Lesson 13 – Insurance: Probabilities and Percentiles

Table 12.1 EPV for insurance payable at moment of death Type of Insurance CFM UDD 𝜇 𝑎  Whole Life 𝜔−𝑥| 𝜇+𝛿 𝜔−𝑥 𝜇 𝑎𝑛| n-year term (1 − 𝑒 −𝑛(𝜇+𝛿) ) 𝜇+𝛿 𝜔−𝑥 𝜇 n-year deferred life 𝑒 −𝛿𝑛 𝑎𝜔−(𝑥+𝑛)|  𝑒 −𝑛(𝜇+𝛿) 𝜇+𝛿 𝜔−𝑥 n-year pure endowment 𝑒 −𝑛(𝜇+𝛿) 𝑒 −𝛿𝑛 (𝜔 − (𝑥 + 𝑛)) 12.1 12.2 12.3 12.4 12.5

∞ 𝑛! Gamma 𝐸𝑃𝑉 = ∫0 𝑡 𝑛 𝑒 −𝑐𝑡 𝑑𝑡 = 𝑐 𝑛 +1 ∞ 1 If n=1, 𝐸𝑃𝑉 = ∫0 𝑡𝑒 −𝑐𝑡 𝑑𝑡 = 2 𝑐 ∞ 2 If n=2, 𝐸𝑃𝑉 = ∫0 𝑡 2 𝑒 −𝑐𝑡 𝑑𝑡 = 3 𝑐 𝑢 1 ∫0 𝑡𝑒 −𝑐𝑡 𝑑𝑡 = 2 (1 − (1 + 𝑐𝑢)𝑒 −𝑐𝑢 ) 𝑐 𝑢 (𝑎𝑢| −𝑢𝑣 ) (𝐼𝑎 ) 𝑢| = 𝛿

𝜔−𝑥

To calculate Pr(𝑍 ≤ 𝑧) for continuous 𝑍, draw a graph of 𝑍 as a function of 𝑇𝑥 . Identify the parts of the graph that are below the horizontal line 𝑍 = 𝑧, and the corresponding 𝑡’s. Then calculate the probability of 𝑇𝑥 being in the range of those 𝑡’s. For CFM, Pr(𝑍 ≤ 𝑧) = 𝑧 𝛿

𝜇

For discrete 𝑍 , identify 𝑇𝑥 and then identify 𝐾𝑥 + 1 corresponding to that 𝑇𝑥 .

To calculate percentiles of continuous 𝑍 , draw a graph of 𝑍 as a function of 𝑇𝑥 . Identify where the lower parts of the graph are, and how they vary as a function of 𝑇. For example, for whole life, higher 𝑇 leads to lower 𝑍 . For 𝑛year deferred whole life, both 𝑇𝑥 < 𝑛 and higher 𝑇𝑥 lead to lower 𝑍 . Write an equation for the probability 𝑍 is less than 𝑧 in terms of mortality probabilities expressed in terms of 𝑡 . Set it equal to the desired percentile, and solve for 𝑡 or for 𝑒 𝑘𝑡 for any 𝑘. Then solve for 𝑧 (which is often 𝑣 𝑡 )

Variance If 𝑍3 = 𝑍1 + 𝑍2 , 𝑍1 &𝑍2 are mutually exclusive, 𝑉𝑎𝑟(𝑍3 ) = 𝑉𝑎𝑟(𝑍1 ) + 𝑉𝑎𝑟(𝑍2 ) − 2𝐸(𝑍1 )𝐸(𝑍2 )

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

8 Lesson 14 – Insurance: Recursive Formulas, Varying Insurance Recursive Formulas 14.1 𝐴𝑥 = 𝑣𝑞𝑥 + 𝑣𝑝𝑥 𝐴𝑥+1 14.2 𝐴𝑥:𝑛|  = 𝑣𝑞𝑥 + 𝑣𝑝𝑥 𝐴𝑥+1:𝑛−1|  14.3 𝐴𝑥 :𝑛| 󰆷 :𝑛−1|  = 𝑣𝑞𝑥 + 𝑣𝑝𝑥 𝐴𝑥+1  14.4 𝑛| 𝐴𝑥 = 𝑣𝑝𝑥 𝑛−1|𝐴𝑥+1

Applying whole life recursive equation twice: 𝐴𝑥 = 𝑣𝑞𝑥 + 𝑣 2 𝑝𝑥 𝑞𝑥+1 + 𝑣 2 2𝑝𝑥 𝐴𝑥+2   )𝑥 = 𝜇 14.5 (𝐼𝐴 (𝜇+𝛿) 2

14.6 Continuously whole life insurance (CFM) 2𝜇 𝐸(𝑍 2 ) = (𝜇 + 2𝛿)3  )𝑥 :𝑛|  𝐴)𝑥 :𝑛| 14.7 (𝐼𝐴  + (𝐷  = 𝑛𝐴𝑥 :𝑛|   (𝐷𝐴 ) 14.8 (𝐼𝐴)𝑥 :𝑛| +   = (𝑛 + 1)𝐴𝑥 :𝑛|  𝑥 :𝑛| 14.9 (𝐼𝐴)𝑥 :𝑛|  + (𝐷𝐴) 𝑥 :𝑛|  = (𝑛 + 1)𝐴𝑥 :𝑛|  𝑛   (𝐼𝐴)𝑥  :𝑛| = ∑ 𝑘=1 𝑘 𝑘−1|𝐴𝑥 :1|

Recursive Formulas for Increasing and Decreasing Insurance 14.10 (𝐼𝐴)𝑥 :𝑛|  = 𝐴 𝑥 :𝑛|   + 𝑣𝑝𝑥 (𝐼𝐴)𝑥+1  󰆷 :𝑛−1| 14.11 (𝐼𝐴)𝑥 :𝑛|  = 𝐴𝑥 :1|   + 𝑣𝑝𝑥 (𝐼𝐴𝐴)𝑥+1 󰆷 :𝑛−1| 14.12 (𝐷𝐴)𝑥 :𝑛| 󰆷 :𝑛−1|  = 𝑛𝐴𝑥 :1| + 𝑣𝑝𝑥 (𝐷𝐴)𝑥+1  14.13 (𝐷𝐴)𝑥 :𝑛|  = 𝐴𝑥 :𝑛|  + (𝐷𝐴)𝑥 :𝑛−1| 

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Lesson 15 – Insurance: Relationships (𝐴𝑥 , 𝐴𝑥 , 𝐴𝑥 ) Uniform Distribution of Deaths 𝑖 15.1 𝐴𝑥 = ( ) 𝐴𝑥 𝑚

𝛿

𝑖 15.2 𝐴𝑥 :𝑛|  = ( ) 𝐴𝑥 :𝑛|  𝛿

 = ( 𝑖 ) 𝑛|𝐴𝑥 𝛿

𝑛|𝐴𝑥

𝑖 15.4 𝐴𝑥:𝑛|  = ( ) 𝐴𝑥 :𝑛|  + 𝐴𝑥:𝑛| 󰆷 

15.3

15.5 15.6

(𝑚) 𝐴𝑥 2

=

𝑖

𝛿

𝑖 (𝑚)

𝐴𝑥

2𝑖+𝑖 𝐴𝑥 = 2𝛿

2

𝐴𝑥

2

Claims Acceleration Approach

𝐴𝑥 = (1 + 𝑖)0.5 𝐴𝑥 0.5 𝐴𝑥 :𝑛|  = (1 + 𝑖) 𝐴𝑥 :𝑛|  0.5  𝑛| 𝐴𝑥 = (1 + 𝑖) 𝑛| 𝐴𝑥 0.5 𝐴𝑥:𝑛|  = (1 + 𝑖) 𝐴𝑥 :𝑛|  + 𝐴𝑥:𝑛| 󰆷  𝑚−1

𝐴𝑥 = (1 + 𝑖 ) 2𝑚 𝐴𝑥 2  𝐴𝑥 = (1 + 𝑖) 2𝐴𝑥 (𝑚)

Formula Summary of ASM 2014

9  = 𝑎󰇘 𝑥 − 𝑛 𝐸𝑥 𝑎󰇘 𝑥+𝑛 17.12 𝑎󰇘 𝑥:𝑛| 𝑛−1 17.13 𝑎󰇘 𝑥:𝑛|  = ∑𝑘=1 𝑎󰇘 𝑘| 𝑘−1 𝑝𝑥 𝑞𝑥+𝑘−1 + 𝑎󰇘 𝑛|  𝑛−1 𝑝𝑥 𝑛−1 𝑘 ∑ = 17.14 𝑎󰇘 𝑥:𝑛|  𝑘=0 𝑣 𝑘𝑝𝑥 ∞ 𝑘 𝑛| 𝑎󰇘 𝑥 = ∑ 𝑘=𝑛 𝑣 𝑘𝑝𝑥 17.15 Constant Force of Mortality

Lesson 17 – Annuities: Discrete, Expectation Annuities-Due Whole Life Annuities 1−𝐴 17.1 𝑎󰇘 𝑥 = 𝑑 𝑥 17.2 𝐴𝑥 = 1 − 𝑑𝑎󰇘 𝑥

17.3 𝑎󰇘 𝑥:𝑛|  =

𝑎󰇘 𝑥 =

𝑛| 𝑎󰇘 𝑥

1−𝐴𝑥:𝑛| 

Temporary Life Annuities 𝑑

 = 1 − 𝑑 𝑎󰇘 𝑥:𝑛|  17.4 𝐴𝑥:𝑛|

17.7 𝑎󰇘 𝑛|  =

1−𝑣 𝑛

Whole life annuities 1 − 𝑖𝑎𝑥 𝐴𝑥 = 1+𝑖

𝑣 𝑛 −𝑣 𝐾𝑥 +1 𝑑

𝐾𝑥 ≤ 𝑛 − 1

𝐾𝑥 ≥ 𝑛

Temporary life annuities 1 = 𝑖𝑎𝑥:𝑛|  + 𝐴𝑥:𝑛|  + 𝑖𝐴𝑥 :𝑛|  17.16 𝐴𝑥 :𝑛| = 𝑣𝑎󰇘 𝑥:𝑛|  − 𝑎𝑥:𝑛| 

n-year certain-and-life annuity-due 𝑑

17.8 𝑎󰇘 𝐾𝑥 +1| =

1−𝑣 𝐾𝑥 +1 𝑑

𝐾𝑥 ≥ 𝑛

Table 17.2: Actuarial notation for standard types of annuity-due Name Whole Life Temporary Life Deferred Life Deferred Temporary Life Certainand-life

Annual pmt at k 10 ≤ 𝑘 ≤ 𝐾𝑥 10 ≤ 𝑘 ≤ min(𝐾𝑥 , 𝑛 − 1) 0𝑘 > min(𝐾𝑥 , 𝑛 − 1)

00 ≤ 𝑘 < n ork > Kx 1𝑛 ≤ 𝑘 ≤ K x 00 ≤ 𝑘 < n 1𝑛 ≤ 𝑘 < min(𝑛 + 𝑚, K x + 1) 0𝐾𝑥 ≥ min(𝑛 + 𝑚, Kx + 1) 10 ≤ 𝑘 < max(Kx + 1, 𝑛) 0𝑘 ≥ max(Kx + 1, 𝑛)

  + 𝑛| 𝑎󰇘 𝑥 𝑎󰇘 𝑥:𝑛| | = 𝑎󰇘 𝑛|

PV 𝑎󰇘   𝐾𝑥 +1| 𝑎󰇘   𝐾𝑥 < 𝑛 𝐾𝑥 +1| 𝑎󰇘 𝑛| 𝐾𝑥 ≥ 𝑛 0𝐾𝑥 < 𝑛 𝑎󰇘   − 𝑎󰇘 𝑛|𝐾𝑥 ≥ 𝑛 𝐾𝑥 +1| 0𝐾𝑥 < 𝑛 𝑎󰇘   − 𝑎󰇘 𝑛|𝑛 ≤ 𝐾𝑥 ≤ 𝑛 + 𝑚 𝐾𝑥 +1| 𝑎󰇘  − 𝑎󰇘 𝑛|  𝐾𝑥 ≥ 𝑛 + 𝑚 𝑛+𝑚| 𝑎󰇘 𝑛| 𝐾𝑥 < 𝑛 𝑎󰇘   𝐾𝑥 ≥ 𝑛 𝐾𝑥 +1|

𝑛| 𝑎󰇘 𝑥 = 𝑛 𝐸𝑥 𝑎󰇘 𝑥+𝑛 17.11 𝑎󰇘 𝑥 = 𝑎󰇘 𝑥:𝑛|  + 𝑛|𝑎󰇘 𝑥

17.9

17.10

𝑞+𝑖

= 𝑛 𝐸𝑥 𝑎󰇘 𝑥

Annuities-immediate

n-year Deferred Whole Life Annuity 17.5 0𝐾𝑥 ≤ 𝑛 − 1 17.6 𝑎󰇘 𝐾𝑥 𝑛| = +1| − 𝑎󰇘 

1+𝑖

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Symbol 𝑎󰇘 𝑥 𝑎󰇘 𝑥:𝑛| 𝑛| 𝑎󰇘 𝑥

 𝑛| 𝑎󰇘 𝑥:𝑛|

𝑎󰇘𝑥:𝑛|  |

Certain-and-life annuities 17.17 𝑎󰇘 𝑥 = 𝑎𝑥 + 1 17.18 𝑎󰇘 𝑥:𝑛|  = 𝑎𝑥:𝑛−1|  + 1

17.19 𝑎󰇘 𝑥:𝑛|  = 𝑎𝑥:𝑛|  + 1 − 𝑛 𝐸𝑥 17.20 𝑛| 𝑎󰇘 𝑥 = 𝑛| 𝑎𝑥 + 𝑛 𝐸𝑥

1/mthly annuities (17.1)  𝑎󰇘 𝑥

(𝑚)

(17.2)  𝐴𝑥

(𝑚)

17.21 𝑠󰇘𝑥:𝑛|  =

=

(𝑚)

1−𝐴𝑥

𝑑(𝑚)

= 1 − 𝑑(𝑚) 𝑎󰇘 𝑥

𝑎󰇘 𝑥:𝑛|

(𝑚)

𝑛 𝐸𝑥

Official Definition: 𝑎󰇘 𝑥 = ∑∞ 𝑛=1 𝑎󰇘 𝑛 ( 𝑛−1𝑞𝑥 ) Alternative ∞ 𝑎󰇘 𝑥 = ∑𝑛=0 𝑣 𝑛 𝑛 𝑝𝑥

Formula Summary of ASM 2014

10 18.1 𝑎 𝑇𝑥| =

Lesson 19 – Variance

Lesson 18 – Annuities: Continuous, Expectation Name Whole Life Temporary Life Deferred Life Deferred Temporary Life Certainand-life

1−𝑣 𝑇𝑥 𝛿

Annual pmt at k 1𝑡 ≤ 𝑇 1𝑡 ≤ min(𝑇, 𝑛) 0𝑡 > min(𝑇, 𝑛)

PV 𝑎 𝑇|  𝑎 𝑇|  𝑇 ≤ 𝑛 𝑎𝑛| 𝑇 > 𝑛 0𝑇 ≤ 𝑛 𝑎 𝑇| 𝑛|  − 𝑎 𝑇 > 𝑛 0𝑇 ≤ 𝑛 𝑎 𝑇| 𝑛|  − 𝑎 𝑛 < 𝑇 ≤ 𝑛 + 𝑚 𝑛| 𝑎  − 𝑎  𝑇 > 𝑛 + 𝑚 𝑛+𝑚| 𝑎 𝑛| 𝑇 < 𝑛 𝑎  𝑇 ≥ 𝑛 𝐾𝑥 +1|

0𝑡 ≤ nort > T 1𝑛 < 𝑡 ≤ 𝑇 0𝑡 ≤ nort > T 1𝑛 < 𝑡 ≤ 𝑛 + 𝑚𝑜𝑟𝑡 ≤ 𝑇 0𝑇 > 𝑛 + 𝑚 1𝑡 ≤ max(𝑇, 𝑛) 0𝑡 > max(T, 𝑛)

18.2 𝑎𝑥 = 𝛿 𝑥 18.3 𝐴𝑥 = 1 − 𝛿𝑎𝑥 ∞ 18.4 𝑎𝑥 = ∫0 𝑎𝑡| 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 ∞ 18.5 𝑎𝑥 = ∫0 𝑣𝑡 𝑡 𝑝𝑥 𝑑𝑡 1−𝐴

18.6 𝑎𝑥 =

𝜇 1− 𝜇+𝛿

=

𝛿 𝜇+𝛿

𝛿 𝛿 1−𝐴𝑥:𝑛| 

18.7 𝑎𝑥:𝑛|  = 𝛿 18.8 𝐴𝑥:𝑛| 𝑥:𝑛|  = 1 − 𝛿𝑎  18.9

18.10

𝑥 𝑛| 𝑎

=

𝑥 𝑛| 𝑎

1−𝐴𝑥 𝛿

−

=

𝛿

= 𝑛 𝐸𝑥 𝑎𝑥 =

1

=

∞

19.2 𝐸[𝑎2𝑇 ] = ∫0 ( 𝑥|

𝑥 𝑛| 𝑎

19.3 𝑉𝑎𝑟(𝑎 𝑇𝑥 | ) =

𝑎𝑥:𝑛|  |

− 𝑎 19.6 𝑉𝑎𝑟(𝑌) = 𝑥 𝛿 𝑥 − (𝑎𝑥 )2 2 = 1 − (2𝛿) 𝑎𝑥:𝑛| 19.7 2𝐴𝑥:𝑛|  

 𝑥:𝑛| 𝑛| 𝑎

𝜇+𝛿

𝛿

(𝐶𝐹𝑀)

1−𝑒 −𝑛(𝜇+𝛿) 𝜇+𝛿

𝛿2

2 2  𝐴 𝑥:𝑛|  −(𝐴  )

𝑥:𝑛| 19.4 𝑉𝑎𝑟(𝑌) = 𝛿2 2  19.5 𝐴𝑥 = 1 − (2𝛿) 2 𝑎𝑥

2(𝑎

19.8 𝑉𝑎𝑟(𝑌) =

2

)

2 2(𝑎𝑥:𝑛|  − 𝑎  ) 𝑥:𝑛|

− (𝑎𝑥:𝑛|) Note: 𝑎𝑥 is 1st moment at twice FOI 2

𝛿

2

2

𝐴𝑥−(𝐴𝑥 )2 𝑑2

𝐴𝑥:𝑛|  −(𝐴𝑥:𝑛|  )

2

19.10 𝑉𝑎𝑟(𝑌) = 𝑑2 19.11 2𝐴𝑥 = 1 − 2𝑑 2𝑎󰇘 𝑥 = 1 − (2𝑑 − 𝑑2 ) 2𝑎󰇘 𝑥

(𝐶𝐹𝑀)

𝑒 −𝑛(𝜇+𝛿)

2

) 𝑡 𝑝𝑥 µ𝑥+𝑡 𝑑𝑡 𝛿 2  (  )2 𝐴 𝑥− 𝐴 𝑥

∞ 1−𝑣𝑡

19.9 𝑉𝑎𝑟(𝑎󰇘  𝐾𝑥 +1| ) =

19.12 𝑉𝑎𝑟(𝑌) =

𝐴𝑥:𝑛|  −𝐴𝑥

18.11 𝑎𝑥:𝑛| 𝑥 (1 − 𝑛 𝐸𝑥 ) =  = 𝑎

CFM: 𝑎𝑥 = 𝑎𝑥+𝑛 Relationships: 𝑎𝑥 = 𝑎𝑥:𝑛| 𝑥+𝑛  + 𝑛 𝐸𝑥 𝑎

2 ] = ∫0 𝑎𝑇| 19.1 𝐸[𝑎2𝑇  𝑡 𝑝𝑥 µ𝑥+𝑡 𝑑𝑡 𝑥|

Whole Life and Temporary Life

2

𝜇+𝛿

1−𝐴𝑥:𝑛| 

Symbol 𝑎𝑥 𝑎𝑥:𝑛|

(𝐶𝐹𝑀)

2(𝑎󰇘 𝑥 − 2𝑎󰇘 𝑥 ) + 2𝑎󰇘 𝑥 𝑑

Other Annuities 2 19.13 𝐸[𝑌𝑥󰇘 2 ] = ∑ ∞  𝑘=1 𝑎󰇘 𝑘|

𝑘−1|𝑞𝑥

2 󰇘 2  ] = ∑𝑛𝑘=1 𝑎󰇘 𝑘| 19.14 𝐸[𝑌𝑥:𝑛| 

𝑘−1|𝑞𝑥

− (𝑎󰇘 𝑥 )2

2 + 𝑛 𝑝𝑥𝑎󰇘 𝑛|2 = ∑𝑛−1 𝑘| 𝑘=1 𝑎󰇘

𝑘−1|𝑞𝑥

+ 𝑛−1 𝑝𝑥𝑎󰇘 𝑛|2

𝑛| 𝑥 𝑎𝑥:𝑛|   + 𝑛|𝑎  | = 𝑎

Monica E. Revadulla EXAM MLC – Models for Life Contingencies

Formula Summary of ASM 2014

11 Lesson 20 – Annuities: Probabilities and Percentiles

For the continuous whole life annuity PVRV Y, the relationship of 𝐹𝑌 (𝑦)to 𝐹𝑥 (𝑡) as follows: 𝐹𝑌 (𝑦) = Pr(𝑌 ≤ 𝑦) 1 − 𝑣 𝑇𝑥 = Pr( ≤ 𝑦) 𝛿 = Pr(𝑣 𝑇𝑥 ≥ 1 − 𝛿𝑦) = Pr(𝑇𝑥 ln 𝑣 ≥ ln(1 − 𝛿𝑦)) = Pr(−𝑇𝑥 𝛿 ≥ ln(1 − 𝛿𝑦)) ln(1 − 𝛿𝑦) = Pr (𝑇𝑥 ≤ ) 𝛿 ln(1 − 𝛿𝑦) = 𝐹𝑥 (− ( )) 𝛿

To calculate a probability for an annuity, calculate the 𝑡 for which 𝑎𝑡 has the desired property. Then calculate the probability 𝑡 is in that range.

To calculate a percentile of an annuity, calculate the percentile of 𝑇𝑥 , then calculate 𝑎𝑇𝑥 |

Some adjustments may be needed for discrete annuities or non-whole-life annuities, as discussed in the lesson. If forces of mortality and interest are constant, then the probability that the present value of payments on a continuous whole life annuity will be greater than its expected present value is Pr(𝑎 𝑥 ) = ( 𝑇𝑥 | > 𝑎

𝜇

𝜇 𝛿 ) 𝜇+𝛿

Monica E. Revadu...