ACT455H1S - TEST 1 - FEBRUARY 6, 2007

Transcription

1 ACT455H1S - TEST 1 - FEBRUARY 6, 2007 Write name and student number on each page. Write your solution for each question in the space provided. For the multiple decrement questions, it is always assumed that decrements are independent of one another unless indicated otherwise. 1 A three decrement model has forces of decrement Ð"Ñ Ð#Ñ Ð$Ñ. Ð>Ñ œ Þ" ß. Ð>Ñ œ Þ# ß. Ð>Ñ œ Þ$ for >. B B B 2 (a) Find 0 Ð#Ñ œ ;. N B Ð#Ñ 2 (b) Find T ÒX Ÿ "ln œ #Ó. Ð#Ñ 2 (c) Find "l" ; B

2 2. An insurance company estimates that 25% of new policies issued are canceled (lapse) in the 1st policy year, 15% of policies that survive the first year are canceled in the second policy year, and 10% of policies that survive the first two years are canceled in the third policy year (these are multiple decrement table ; Ð6+:=/Ñ probabilities). The insurance company models policy survival with a 2-decrement model, mortality and policy cancellation. The insurer assumes mortality decrement probability of.05 for each year of age (these are multiple decrement table ; Ð79<>+63>CÑ probabilities). 3 (a) Find the probability that a newly issued policy will be canceled while the policyholder is still alive within in the next three years. ; 3 (b) Find "l" B Ð79<>+63>CÑ.

3 3. You are given three separate two-decrement models. Ð"Ñ Ð#Ñ 2 I.. B Ð>Ñ œ + ß. B Ð>Ñ œ,, where + and, are constant. Ð"Ñ 3 II.. Ð>Ñ œ " ß Ð#Ñ ". Ð>Ñ œ, for > &. B & > B & > Ð"Ñ Ð#Ñ " B B & > 3 III.. Ð>Ñ œ + ß. Ð>Ñ œ, for > &, where + is constant. Determine which of the models have marginal distributions of XÐBÑand N that are independent

4 4. A 3-decrement model has the following one year absolute rates of decrement at age B: wð"ñ wð#ñ wð$ñ ; œ Þ# ß ; œ Þ# ß and ; œ Þ% B B B Ð"Ñ Ð#Ñ Ð$Ñ ; B ; B ; B Find, and under each of the following assumptions. 2 (a) Constant force of decrement. 2 (b) UDD in the associated single tables. 2 (c) UDD in the associated single tables for decrements 1 and 2, and decrement 3 is discrete and occurs at age B Þ&.

5 6 5. A company offers an ordinary fully continuous life insurance policy with face amount 1 based on a constant force of mortality of. at all ages, and force of interest $. The fully continuous premium is TÐEBÑ, payable for life. The company also offers a "double indemnity" fully continuous insurance policy with has a death benefit of 2 if death is due to accidental causes and 1 if death is not due to accidental causes. The decrement model used for the double indemnity policy has constant force of decrement for accidental death equal to -. and constant force of decrement for death due to all other causes is Ð" -Ñ.. The company finds that the fully continuous premium (payable until death) for the double indemnity policy is 20% larger than the original policy. Determine -.

6 6. A fully discrete 3-year term insurance based on a single decrement model has death benefits of 1000 if death occurs in the first year, 2000 if death occurs in the second year and 3000 if death occurs in the 3rd year. The mortality probabilities are ; B œþ#ß; B " œþ$, ; B # œþ&, and interest is at an annual effective rate of 25%. Premiums are scheduled to be paid in the first and second years only (it is a "two-payment" policy). There is a 25% premium expense each year while premiums are paid. There is also a level expense of 50 in the first year and 20 in the second and third years. 2+2 (a) Formulate the equation for finding the expense-loaded premium and verify that a premium of satisfies the equation. 2+2 (b) Find the expense-augmented reserve at the end of the first and second years.

7 Ð7 Ñ Þ'> Ð#Ñ Þ'> Þ# " > B B ' Þ' $ 1.(a) : œ / p ; œ / ÐÞ#Ñ.> œ œ. " B Ð#Ñ N ' " Þ'= Þ' (b) T ÒX Ÿ "ln œ #Ó œ ; Î0 Ð#Ñ œ Ò / ÐÞ#Ñ.=ÓÎÐ"Î$Ñ œ " / œ Þ%&". Ð#Ñ Ð7 Ñ Ð#Ñ Þ'Ð"Ñ " Þ'= (c) "l" B " B " ' B " Þ' Þ' ÐÞ#ÑÐ" / Ñ œ / Þ' œ Þ)#&. ; œ : ; œ / / ÐÞ#Ñ.= Ð6+:=/Ñ Ð6Ñ Ð7Ñ Ð6Ñ Ð7Ñ Ð6Ñ 2.(a) $ ; B œ ; B : B ; B " #: B ; B # œ Þ#& Ð" Þ#& Þ&ÑÐÞ"&Ñ Ð" Þ#& Þ&ÑÐ" Þ"& Þ&ÑÐÞ"Ñ œ Þ%"". Ð79<>+63>CÑ Ð7Ñ Ð79<>Ñ "l" B B B " (b) ; œ : ; œ Ð" Þ#& Þ&ÑÐÞ&Ñ œ Þ$&. Ð7Ñ Ð+,Ñ> Ð+,Ñ> Ð+,Ñ> B Ð 7 Ñ 3. I.. œ+,ß > : œ/ ß 0XßNÐ>ß"Ñœ/ +ß0XßNÐ>ß#Ñœ/, Ð+,Ñ> Ð"Ñ + Ð#Ñ, 0XÐBÑÐ>Ñ œ / Ð+,Ñ, 0NÐ"Ñ œ ; B œ +, ß 0NÐ#Ñ œ ; B œ +, p 0XßNÐ>ß "Ñ œ 0XÐBÑÐ>Ñ 0NÐ"Ñ ß 0XßNÐ>ß #Ñ œ 0XÐBÑ Ð>Ñ 0NÐ#Ñ. X ß N are independent.. Ð 7 Ñ # & > # & > & > > B Ð 7 Ñ & XßN #& XßN & > X N ' & & > " "#& #& # N II. œ ß : œ Ð Ñ, 0 Ð>ß "Ñ œ œ 0 Ð>ß #Ñ ß 0 Ð>Ñ œ ß 0 Ð"Ñ œ.> œ œ 0 Ð#Ñß p 0XßNÐ>ß "Ñ œ 0 Ð>Ñ 0NÐ"Ñ ß 0XßNÐ>ß #Ñ œ 0 Ð>Ñ 0NÐ#Ñ X ß N XÐBÑ XÐBÑ. are independent. Ð7Ñ " & > +> & > +> / B Ð 7 Ñ III.. œ+ & > ß > : œð & Ñ/ ß0XßNÐ>ß"ќР& Ñ/ +ß0XßNÐ>ß#Ñœ & ß +> &+ &+ & > +> / " / " / 0XÐ>Ñ œ Ð & Ñ/ + & ß 0NÐ#Ñ œ &+ ß 0NÐ"Ñ œ " &+ Þ +> +> &+ / & > +> / " / 0 Ð>ß #Ñ œ Á ÒÐ Ñ/ + Ó œ 0 Ð>Ñ 0 Ð#Ñ. Not independent. XßN & & & &+ X N +> B Ð7Ñ 4. : œ ÐÞ)ÑÐÞ)ÑÐÞ'Ñ œ Þ$)%. Ð"Ñ 68 : Ð7Ñ 68 Þ) Ð#Ñ B B B 68 : 68 Þ$)% B Ð7Ñ 68 : B wð$ñ Ð7Ñ 68 Þ' B 68 : 7 68 Þ$)% (a) ; œ ; œ ÐÞ'"'Ñ œ Þ"%$' and ; œ Þ"%$' also. Ð$Ñ B B wð"ñ ; œ ; œ ÐÞ'"'Ñ œ Þ$#)) B Ð Ñ Ð"Ñ wð"ñ " wð#ñ wð$ñ " wð#ñ wð$ñ B B # B B $ B B " " # $ and B Ð#Ñ Ð$Ñ wð$ñ " wð"ñ wð#ñ " wð"ñ wð#ñ B B # B B $ B B " " œ ÐÞ%ÑÒ" # ÐÞ# Þ#Ñ $ ÐÞ#ÑÐÞ#ÑÓ œ Þ$#&$. (b) ; œ ; Ò" Ð; ; Ñ ; ; Ó œ ÐÞ#ÑÒ" ÐÞ# Þ%Ñ ÐÞ#ÑÐÞ%ÑÓ œ Þ"%&$ ; œ Þ"%&$ ; œ ; Ò" Ð; ; Ñ ; ; Ó also.

8 wð$ñ (c) > : B œ œ " Ÿ> Þ& Þ' Þ& Ÿ > Ÿ " Ð"Ñ " Ð7Ñ Ð"Ñ " wð"ñ wð#ñ wð#ñ Ð"Ñ ; B œ ' > : B. B Ð>Ñ.> œ ' > : B > : B > : B. B Ð>Ñ.> wð"ñ wð#ñ wð#ñ œ; B Ð' Þ& Ò" > ; B Ó.> ' " Þ& Ò" > ; B ÓÐÞ'Ñ.>Ñ œ ÐÞ#ÑÒÞ%(& ÐÞ%#&ÑÐÞ'ÑÓ œ Þ"%'.. 5. For the single decrement ordinary whole life policy, E œ and " B $. + B œ $. and TÐEB Ñ œ.. For the double indemnity policy, > B Ð 7 Ñ. > : œ /, and the APV of the death benefits are: accident: #' > / $ > /. #- Ð-. Ñ.> œ. $., and non-accident: ' > > Ð" -Ñ / $ /. Ð" -Ñ..> œ. $.. #-. Ð" -Ñ Ð" -Ñ Total APV of benefit is. œ.. $. $. $. Continuous whole life annuity APV is still Ð" -Ñ.. - œ Þ#. " + B œ $., so fully continuous premium is 6. (a) The equation is # $ # "@; B #@ "l; B $@ #l; B BÑ & #Ð@: #: BÑ B Ó. This can be written as # $ # "@; B #@ ; B $@ ; B & #Ð@: #: BÑ œ BÑ Using the given values, # $ # "@; B #@ "l; B $@ #l; B & #Ð@: #: BÑ œ *'(Þ#& and Ñ œ *'(Þ#&. "l #l. B (b) Prospective form of reserves: # " Z/ œ #@; B " $@ "l ; B " Þ#&Ð()'Þ$)Ñ B " Ñ ()'Þ$) œ &*$Þ%# Z œ $@; # œ "##. # / B # Alternatively, we can use the recursive relationship with Z/ œ. Ò ÐÞ(&ÑÐ()'Þ$)Ñ &ÓÐ"Þ#&Ñ "ÐÞ#Ñ œ ÐÞ)Ñ" Z/ p " Z/ œ &*$Þ%", Ò&*$Þ%" ÐÞ(&ÑÐ()'Þ$)Ñ #ÓÐ"Þ#&Ñ #ÐÞ$Ñ œ ÐÞ(Ñ Z p Z œ "## # / # /.