Heriot-Watt University. M.Sc. in Actuarial Science. Life Insurance Mathematics I. Tutorial 2 Solutions

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1 Heriot-Watt University M.Sc. in Actuarial Science Life Insurance Mathematics I Tutorial 2 Solutions. (a) The recursive relationship between pure enowment policy values is: To prove this, write: (V (t) + P )( + i) = p x+t V (t + ). an note that: an: (V (t) + P ) = A x+t:n t P (ä x+t:n t ) A x+t:n t = q x+t v 0 + p x+t v A x+t+:n t ä x+t:n t = + p x+t v ä x+t+:n t so: (Ax+t:n t P (ä x+t:n t ))( + i) = p x+t Ax+t+:n t x+t ä x+t+:n t = p x+t V (t + ). (b) The policy value for a level annuity-ue payable for life is just ä x+t. recursive relationship between policy values is therefore: The (ä x+t )( + i) = p x+t ä x+t+. The statement stans as its own proof. 2. An Excel workbook showing how this example may be lai out can be accesse through the moule webpage. 3. (a) Let (DA) x+r:n r enote the EPV of a ecreasing term assurance with initial sum assure of n r. Define V (t) as the policy value just before the payment of the t th premium an P as the net premium. The policy value at time t, together with the net premium pai at time t, when accumulate at the rate of interest i shoul prouce a value sufficient to provie a sum assure of n t at the en

2 2 of year to the proportion q x+t expecte to ie uring the year, an also to set up a reserve of V (t + ) for the proportion p x+t expecte to survive. Therefore: Proof: (V (t) + P )( + i) = q x+t (n t) + p x+t V (t + ). We note that an V (t) + P = (DA) x+t:n t P ä x+t:n t + P = (DA) x+t:n t P (ä x+t:n t ). () (DA) x+t:n t = q x+t v (n t) + p x+t v (DA) x+t+:n t, ä x+t:n t = a x+t:n t = v p x+t ä x+t+:n t. Substituting these into equation (), we can write: ( (V (t) + P )( + i) = q x+t (n t) + p x+t (DA) x+t+:n t P ä ) x+t+:n t = q x+t (n t) + p x+t V (t + ) as require. (b) The reserve can be foun irectly, or by using the relationship in (a) twice from the bounary conition V (n) = 0. Either way we nee to fin the net premium, P = (DA) x:n ä x:n. We have x = 30, n = 30, mortality is accoring to the A ultimate table an interest is 4% p.a. (DA) 30:30 = 3A 30:30 (IA) 30:30 = 3(M 30 M 60 ) (R 30 R 60 30M 60 ) ä 30:30 ä 30:30 N 30 N 60 = 30M 30 R 3 + R 6 30( ) = N 30 N = Using the recursive relationship for the first time with q 59 = , we have (V (29) )(.04) = so that V (29) = Using the recursive relation for the secon time with q 58 = , we have (V (28) )(.04) = ( )( ) so that V (28) =

3 3 (c) The problem is that the reserve is negative. A level premium has been use to pay for a ecreasing risk. This coul cause problems if the policyholer lapses the policy since it will leave the insurer with a loss. One way to eal with this problem is to charge a higher level premium payable for a term shorter than the term of the policy so that negative reserves o not arise. 4. (a) 6.5V 35:25 ( + i) 0.5 = 0.5 q p 4.5 7V 35:25. But 7 V 35:25 = A 42:8 P 35:25 ä 42:8 = (2.969) = Assuming a constant force of mortality between exact ages 4 an 42, then ( 0.5 p 4.5 ) 2 = p 4 giving 0.5 p 4.5 = Therefore 6.5V 35:25 = ( ) (0.908) (.04) 0.5 =

4 4 (b) ( 6V 35:25 + P ) 35:25 ( + i) 0.25 = 0.25 q p V 35:25 an 6V 35:25 = A5:9 P 35:25 ä 5:9 = A5:9 A 35:25 = D D D 35 (7.625) = D ä 35:25 ä 5:9 Assuming a constant force of mortality between exact ages 5 an 52, ( 0.25 p 5 ) 4 = p 5 giving 0.25 p 5 = Therefore ( )(.04) V 35:25 = = (c) We note that the premiums are pai quarterly an the sum assure is pai at the en of year of eath or on survival to maturity. But ( ) 6.75V (4) (4) P 35:25 35:25 ( + i) 0.25 = 0.25 q p V (4). 35:25 P (4) 35:25 = A 35:25 ä (4) 35:25 = ( D 60 8 D 35 ) = V (4) 35:25 = A 52:8 (0.0244)ä (4) 52:8 = (0.0244)( ( D 60 D 52 )) = Assuming a constant force of mortality between exact ages 5 an 52, then 0.25p 5.75 = , giving Also 6.75V (4) 35:25 = ( ) ( ) (.04) (0.0244) = ( ) 6.5V (4) (4) P 35:25 35:25 ( + i) 0.25 = 0.25 q 5.5 v p V (4) 35:25 such that 6.5V (4) 35:25 = ( )(.04) (0.5570) (.04) (0.0244) =

5 5 5. We have: (V (t)+79.3)(.04) = q x+t (, 000+V (t+))+p x+t V (t+) =, 000 q x+t +V (t+). Solving backwars V (4) = , V (3) = an V (2) = (a) Thiele s equation is: t V (t) = V (t) δ + µ x+t V (t). Intuitive reasoning: uring time t, interest of V (t) t is earne an annuity benefit of t is pai out. If the annuitant ies uring time t, which happens with probability µ x+t t there is no sum assure to pay but the life office keeps the reserve V (t). For proof, note that the policy value is V (t) = ā x+t so that Āx+t = δ V (t) an Thiele s equation above is thus equivalent to Question 2(b) of Tutorial. The appropriate bounary conition is V (n) = 0. (b) Thiele s equation is: t V (t) = V (t) δ + µ x+t V (t). Intuitive reasoning: uring time t, interest of V (t) t is earne an no benefit is pai out. If the annuitant ies uring time t, which happens with probability µ x+t t there is no sum assure to pay but the life office keeps the reserve V (t). For proof, note that the policy value is V (t) = A x+t:n = e δ(n t) n tp x+t. Hence: t V (t) = δ e δ(n t) n tp x+t + e δ(n t) t ( n tp x+t ) = δ V (t) + e δ(n t) ( ) np x t tp x = V (t) δ + e δ(n t) np x ( tp 2 t p x µ x+t ) x ( ) = V (t) δ + e δ(n t) np x µ x+t tp x = V (t) δ + µ x+t V (t). The appropriate bounary conition is V (n) =. (c) Thiele s equation is of the form in (a) above uring payment of the annuity, an of the form in (b) above uring eferral. The appropriate bounary conition is V (ω x) = 0, where ω is the limiting age of the life table.

6 6 7. We have: t V (t) = V (t) δ + P µ x+t (0, V (t) V (t)) = V (t) δ + P 0, 000 µ x+t. The appropriate bounry conition is V (n) = 0, 000.