Suggested solutions to DHW textbook exercises
Exercise 7.5
(a) Let P denote the annual premium rate payable continuously. The present value of future
benefits at issue can be expressed as


for T30 ≤ 10
T P v T30 ,

 30
T30
for 10 < T30 ≤ 30
PVFB0 = 10P v ,

(12)
30

, for T30 > 30
v ä (12)
K30 +(1/12)−30
The present value of future premiums at issue can be expressed as
(
P āT , for T30 ≤ 10
30
PVFP0 =
P ā10 , for T30 > 10
The future loss random variable L0 is the difference between

T30 P v T30 − P āT ,


30

T
L0 = PVFB0 − PVFP0 = 10P v 30 − P ā10 ,


v 30 ä(12)
− P ā10 ,
(12)
the two:
for T30 ≤ 10
for 10 < T30 ≤ 30
for T30 > 30
K30 +(1/12)−30
(b) The actuarial present value of future premiums at issue is
APV(FP) = P ā30: 10 .
The actuarial present value of future benefits at issue is
1
(12)
1
APV(FB) = P I¯Ā 30:
+ 10P 10 E30 Ā40:
+ 30 E30 ä60 .
10
20
Equating the two APV’s by the equivalence principle, we get
P =
ā30: 10 −
(12)
30 E30 ä60
1
1
I¯Ā 30:
− 1010 E30 Ā40:
10
20
.
(c) For a policy still in force at duration 5, the present value of
can be expressed as

(5 + T35 )P v T35 − P āT ,


35

T
L5 = PVFB5 − PVFP5 = 10P v 35 − P ā5 ,


v 25 ä(12)
− P ā5 ,
(12)
future loss random variable
for T35 ≤ 5
for 5 < T35 ≤ 25
for T35 > 25
K35 +(1/12)−25
(d) For a policy still in force at duration 5, the policy value can be derived using
APV(FP5 ) = P ā35: 5
and
1
(12)
1
1
APV(FB5 ) = 5Ā35:
+ P I¯Ā 35:
+ 10P 5 E35 Ā40:
+ 25 E35 ä60 .
5
5
20
The policy value at duration 5 is therefore
(12)
1
1
¯ 1
5 V = 5Ā35: 5 + P I Ā 35: 5 + 10P 5 E35 Ā40: 20 + 25 E35 ä60 − P ā35: 5 .
Prepared by E.A. Valdez
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