Math 3631 Spring 2014
Homework Assignment 1
Due Date: Tuesday, February 4, 2014
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
Your answers should con
Suggested solutions to DHW textbook exercises
Exercise 6.4
Denote by G the required annual premium.
First, consider the actuarial present value (APV) of the benets (note that there is no mention
of any type of approximation, and hence, the APV has to be c
Suggested solutions to DHW textbook exercises
Exercise 7.2
(a) Let G be the annual gross premium. The APV of future gross premiums is therefore
APV(FP) = Ga[35]
and the APV of future benets and expenses is
APV(FB) + APV(FE) = 100000A[35] + 0.35G + 0.05Ga[
Suggested solutions to DHW textbook exercises
Exercise 7.3
(a) With a starting policy value of 0 V = 0, we use the recursive formula to calculate the next
years policy value:
( 0 V + P )(1 + i) = 1 V + q[50] (1000 + 1 V 1 V ).
This simplies to
1V
= P (1 +
Suggested solutions to DHW textbook exercises
Exercise 7.4
(a) Let P denote the annual premium rate payable continuously. The present value of future
benets at issue can be expressed as
T30 P v T30 ,
for T30 10
T30
for 10 < T30 30
PVFB0 = 10P v ,
30 (12)
Math 3631 Spring 2015
Homework Assignment 1
Due Date: Tuesday, February 3, 2015
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
Your answers should con
Math 3631 Spring 2014
Homework Assignment 3
Due Date: Friday, February 28, 2014 at noon
(moved one day because of 3621 exam)
General Notes:
Please hand in your homework either in class before the due date or to my oce (404
MSB) by noon on the due date. I
Math 3631 Spring 2015
Homework Assignment 2
Due Date: Tuesday, February 17, 2015
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
For Part II, you shoul
Math 3631 Spring 2015
Homework Assignment 3
Due Date: Tuesday, March 3, 2015
General Notes:
Please hand in your homework in class on the due date. In order to get full credit, you
must show ALL work.
For Part II, you should use a spreadsheet.
Email the
Math 3631 Spring 2014
Homework Assignment 2
Due Date: Tuesday, February 18, 2014
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
For Part II, you shoul
Math 3631 Spring 2015
Homework Assignment 4
Due Date: Tuesday, March 24, 2015
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
For Part II, you should u
Suggested solutions to DHW textbook exercises
Exercise 7.1
(a) Let P be the required annual benet premium and by the equivalence principle, we have
P = 200000
1
A [41]: 3
a[41]: 3
,
where
a[41]: 3
= 1 + vp[41] + v 2 2p[41] = 1 +
1 99502
1 99689
+
= 2.829
Suggested solutions to DHW textbook exercises
Exercise 6.15
(a) Let P be the required annual premium. By the equivalence principle, we have
P = 250000
A[40]: 20
a[40]: 20
where
= a[40] 20 E[40] a60 = a[40] v 20
a[40]: 20
60
a60
[40]
= 18.45956 (1.05)20
96
Suggested solutions to DHW textbook exercises
Exercise 6.5
Let G the required gross monthly premium.
The APV of the 20-year deferred annuity benets with an initial annual payment of 50,000
increasing by 2% thereafter is given by
50000(1.02)k20 v k kp[40]
Suggested solutions to DHW textbook exercises
Exercise 6.6
Let G the required gross annual premium.
With premiums payable only up to 10 years, we have the APV of gross premiums given by
APV(premiums) = G a[40]: 10 ,
where, based on the Standard Select Sur
Suggested solutions to DHW textbook exercises
Exercise 6.7
Denote by P the single benet premium for the annuity.
(a) The APV of the benets is the single benet premium:
1
P = 20000 30 E[35] a65 + P A[35]: 30
Solving for P , we get
P=
=
20000 30 E[35] a65
1
Suggested solutions to DHW textbook exercises
Exercise 6.8
Let P be the annual premium determined according to the equivalence principle. Then
P=
Ax
ax
and
L0 = v K +1 P a K +1 =
1+
P
d
v K +1
P
.
d
Let P be the annual premium determined according to E[L
Suggested solutions to DHW textbook exercises
Exercise 6.9
First consider the net annual premium. Denote this by P . Assuming UDD between integral
ages (not clearly specied in the problem), the APV of the benets is
APV(benets) = 10000 10A[40] 8 20 E[40] A
Suggested solutions to DHW textbook exercises
Exercise 6.10
Let G be the monthly gross premium.
The APV of the death benets is
(12)
APV(benets) = 100000 A
1
[50]: 10
The APV of the expenses is
(12)
(12)
(12)
APV(expenses) = 100 + 0.15G + 0.15(12G) a[50]:
Suggested solutions to DHW textbook exercises
Exercise 6.11
Let G be the initial annual gross premium. The APV of premiums then is
1
APV(premiums) = G a[55] 2 G 10 E[55] a65 .
The APV of the death benets is
APV(benets) = 100000 A[55] 50000 10 E[55] A65 =
Suggested solutions to DHW textbook exercises
Exercise 6.12
For a whole life insurance of 150000 to (x), the net future loss random variable is
Ln = 150000v K +1 P a K +1
0
=
150000 +
P
d
P
,
d
v K +1
where, based on the equivalence principle,
P = 150000
Suggested solutions to DHW textbook exercises
Exercise 6.13
With the extra risk, we have
t
t
tpx
x+s ds = exp
= exp
0
t
x+s ds et = tpx et
x+s + ds = exp
0
0
so that we have
Ax =
v t tpx x+t dt
0
v t tpx et x+t + dt
=
0
e(+)t tpx x+t dt +
=
0
e(+)t tp
Suggested solutions to DHW textbook exercises
Exercise 6.14
(a) Let K = K[30] be the curtate future lifetime of a select age 30 and P be the gross annual
premium. The loss-at-issue random variable is
L0 = PVFB0 + PVFE0 PVFP0 ,
where
PVFB0 = 250000
v K +1
Math 3631 Spring 2014
Homework Assignment 4
Due Date: Tuesday, March 11, 2014
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
For Part II, you should u
Math 3631 Spring 2015
Homework Assignment 5
Due Date: Tuesday, March 31, 2015
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
Your answers should consi
Math 3631 Spring 2014
Homework Assignment 3
Due Date: Friday, February 28, 2014 at noon
(moved one day because of 3621 exam)
General Notes:
Please hand in your homework either in class before the due date or to my oce (404
MSB) by noon on the due date. I
Math 3631
Spring 2015
Exam 2
02 Apr. 2015
Time Limit: 75 Minutes
Name:
This exam contains 5 pages (including this cover page)
and 3 problems. Check to see if any pages are missing.
You are required to show your work on each problem on
this exam.
Points
1
Math 3631
Spring 2015
Exam 1
19 Feb. 2015
Time Limit: 75 Minutes
Name:
This exam contains 7 pages (including this cover page)
and 4 problems. Check to see if any pages are missing.
You are required to show your work on each problem on
this exam.
Points
1
Math 3631 Spring 2014
Homework Assignment 2
Due Date: Tuesday, February 18, 2014
General Notes:
Please hand in your homework on paper, at the beginning of class on the due date. In
order to get full credit, you must show ALL work.
For Part II, you shoul