Exercises_-_Chapter_7

Exercises_-_Chapter_7 - ACTSCI 331 Life Contingencies I...

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ACTSCI 331 - Life Contingencies I Exercises (Chapter 7) 1. A fully continuous 20-year payment, 30-year term insurance of 2000 is issued to (35) . We assume that the force of interest is 6% and the force of mortality is given by ° ( x ) = 0 : 00005 ° 1 : 1 x ( x > 0 ) which implies that t p x = exp ° ± 0 : 0005246 ° 1 : 1 x ± 1 : 1 t ± 1 ²³ ; t ² 0 ; x > 0 . Under these assumptions, n A 1 35+ n : 30 ° n j a 35+ n : 20 ° n j 0 0.06873 11.39534 10 0.09710 7.35175 25 0.07952 ° : (a) Calculate the bene°t premium (Answer: 12.063) (b) De°ne the loss r.v.±s j L for j = 10 ; 25 . Plot roughly the curves of j L in function of T for j = 10 ; 25 . (c) Find the value of 10 L if T = 12 : 3 and the one for 25 L if T = 27 : 4 . (Answers: 1716.3 and 1731.8) (d) Compute the bene°t reserves j V = E [ j L j T > j ] for j = 10 and 25 . (Answers: 105.52 and 159.04) (e) Given that T (35) > 10 , °nd the probability that the prospective loss 10 L is less than or equal to 1800. (Answer: 0.99377) 2. A fully continuous 20-year deferred whole life insurance of 4000 is issued to (45). The bene°t premium is payable as long as the insured is alive without exceeding the deferred period. We assume that the force of interest is 6% and the force of mortality is given by ° ( x ) = 0 : 00005 ° 1 : 1 x for x > 0 ; which implies that t p x = exp ° ± 0 : 0005246 ° 1 : 1 x ± 1 : 1 t ± 1 ²³ ; t ² 0 ; x > 0 . Under these assumptions, n 20 ° n j A 45+ n A 45+ n a 45+ n : 20 ° n j 0 0.11630 0.21341 11.01577 10 0.22524 0.33092 7.09581 25 ² 0.56246 ² (a) Calculate the bene°t premium. (Answer: 42.23) (b) De°ne the loss r.v.±s j L for j = 10 ; 25 . Plot roughly the curves of j L in function of T for j = 10 ; 25 . (c) Find the value of 10 L if T = 12 : 3 and the one for 25 L if T = 27 : 4 . (Answers: -90.725 and 3463.6) (d) Compute the bene°t reserves j V for j = 10 ; 25 . (Answers: 601.3 and 2249.8) (e) Given that T (45) > 10 , °nd the probability that the prospective loss 10 L is less than or equal to 0. (Answer: 0.15051) 3. A fully continuous 35-year payment, whole life insurance of $1000 is issued to (30). For calculation purposes, we assume (1) ° ( x ) = 1 100 ° x ; x ² 20 ; (2) ± = 6% ; With the bene°t premium P EP = 19 : 28043 , °nd 10 ° such that Pr ( 10 L > 10 ° j T > 10) = 5% . (Answer: 782.34) 1
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4. A fully continuous 15-year payment, 25-year deferred whole life annuity with an annualized payment of $15000
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