ACTL3002 Life Insurance and Superannuation ModelsModel Solutions to Tutorial 6 - held in Week 7

See Dickson, et al. Solutions Manual for assigned problems.

In class exercises and presentations.

1. The only work required here is writing out the APV of future expenses which is given by

APVFE = e0 + (e1 + e2 Px) ax + e3 Ax.

By the principle of equivalence, we have

APVFP = APVFB + APVFE

G ax = Ax + e0 + (e1 + e2 Px) ax + e3 Ax= e0 + (e1 + e2 Px) ax + (1 + e3) Ax

which implies the expense-loaded premium is

G =e0 + (e1 + e2 Px) ax + (1 + e3) Ax

ax= e0 (Px + d) + (e1 + e2 Px) + (1 + e3) Px= (1 + e0 + e2 + e3) Px + (e1 + de0)

so that clearly, a = (1 + e0 + e2 + e3) and c = (e1 + de0).

2. It is not stated in the problem whether this is fully-discrete, fully continuous, or otherwise. So, wemake the assumption that this is a fully-discrete 10-year endowment policy. Note that for issue agex = 40, we have under De Moivres law, that

kp40 =`40+k`40

=60 k

60.

(a) Let G be the expense-loaded (or gross) annual premium. Thus, we have, by equating APVFPwith APVFB+APVFE,

G a40:10 = 1000A40:10 + 50where

a40:10 =

9k=0

vk k p40 = 160

9k=0

vk (60 k)

= a10 1

60(Ia)9 = 7.848055

andA40:10 = 1 da40:10 = 0.6981517

so that

G =1000A40:10 + 50

a40:10=

1000 (0.6981517) + 50

7.848055= 95.329576.

1

For year k = 1, 2, ..., 9, the gross premium reserve is given by

kV = 1000A40+k:10k G a40+k:10k= 1000A40+k:10k 95.329576 a40+k:10k= 1000

(1 da40+k:10k

) 95.329576 a40+k:10k= 1000 (1000(0.04/1.04) + 95.329576) a40+k:10k= 1000 133.791114a40+k:10k

= 1000 133.791114(9ks=0

vs s p40+k)

= 1000 133.791114(9ks=0

vs (

1 s60 k

))

= 1000 133.791114(a10k

1

60 k (Ia)9k).

The values are summarized below:

k 1 2 3 4 5 6 7 8 9

kV 30.99 116.40 206.52 301.67 402.20 508.50 620.96 740.04 866.21

(b) If we let a be the acquisition expense, in general, the gross annual premium will be

Ga =1000A40:10 + a

a40:10=

1000 (0.6981517) + a

7.848055= 88.9585702 +

a

a40:10

and the gross premium reserve will be

kV = 1000A40+k:10k Ga a40+k:10k= 1000A40+k:10k 88.9585702 a40+k:10k a

(a40+k:10ka40:10

)= 1000

(1 da40+k:10k

) 88.9585702 a40+k:10k a( a40+k:10ka40:10)

= 1000 (1000(0.04/1.04) + 88.9585702) a40+k:10k a(a40+k:10ka40:10

)= 1000 127.420109a40+k:10k a

(a40+k:10ka40:10

).

Reserves will be negative if it ever hits negative at the end of the first year. This is because reserveis increasing obviously (because of the nature of the endowment insurance - not all reserves doincrease over time) with time and if it ever hits negative, it must be in the first year. Therefore,we must have

1000 127.420109a41:9 a(a41:9a40:10

)> 0

or equivalently,

a the prospective reserve at 4% = the retrospective reserveat 4%> the retrospective reserve at 3%

- End of Tutorial 6 model solutions -

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