ifs/Eiiciiigan State University
STT 455 ~ Actsariai lviodels E
Class Test 2
R'iondayg is November 2913
Total Mimics: 390 points
Please Write your name and student number at the spaces provided:
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Name: . gait EETED M55 L UT? (3 A; Section No.:
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lK/licliigan State cfw_Juiversity
$TT 4:553 - Actuariai lvlodeis Li
Final Examination
Illday it December 283% 5:45 w 27:45 PM:
Total Sabre: lil points
Name: SUljCcigi gaihllmi SectionZ
0 There are ten (10) multi- Question Worth Score
ple choice qu
h/l'icitigan State University
STT 455 - Actuarial iviodels 35
Class Test 2
ltlonday, 17 November 2-2913
Totai Eviarks: 190 points
Please write your name and student number at the spaces provided:
Mi.
Name: zgf-cfw_LEEWEM l2 SDQMTEQNS
0 There are ve (5) mu
Niieiiigan State University
STT 455 Actuarial Mlodels E
"Ciass Test 1, _
- lVIonday, 20 cfw_)Ctober 2014
Total fMarks: 300 points
Please- write your name at the space provided:
Name: EM, glALJZEZ
0 There are ve (5) multiple Choice (MC) and one (1) written
Michigan State University
STT 455 - Actuarial Models I
Final Examination
Tuesday, 10 December 2013 5:45 - 7:45 PM
Total Score: 100 points
Section 2
Name:
There are ten (10) multiple choice questions here and
you are to answer all questions asked. Each qu
Insurance Benefits
Lecture: Weeks 6-8
Lecture: Weeks 6-8 (STT 455)
Insurance Benefits
Fall 2014 - Valdez
1 / 36
An introduction
An introduction
Central theme: to quantify the value today of a (random) amount to
be paid at a random time in the future.
main
Suggested solutions to DHW textbook exercises
Exercise 5.2
(a) Y is the present value random variable associated with an annuity that pays continuously
at the rate of $1 per year where the payment starts at the moment of death of (x)
continuing until the
Suggested solutions to DHW textbook exercises
Exercise 2.9
To verify the formula, we need the Leibnitz rule for differentiating an integral:
d
dz
Z
b(z)
Z
b(z)
f (x, z)dx =
a(z)
a(z)
f
b(z)
a(z)
dx + f (b(z), z)
f (a(z), z)
z
z
z
Therefore, we have
Z x+
Suggested solutions to DHW textbook exercises
Exercise 4.1
All answers below did not match textbook answers in the 1st edition. I am glad to see they
corrected this in the 2nd edition - now they match!
(a) 5 E35 = v 5 5p35 = (1.06)5
98485.58
= 0.7359423
1
Michigan State University
STT 455 - Actuarial Models I
Class Test 2
Monday, 11 November 2013
Total Marks: 100 points
Please write your name and student number at the spaces provided:
Name:
Section No.:
There are five (5) multiple choice (MC) and one (1)
Suggested solutions to DHW textbook exercises
Exercise 2.10
For Gompertz law, we have x = Bcx so that
0.000344
172
50
=
=
= c20 .
30
0.000130
65
This gives us c = (172/65)1/20 and thus, we have
Z 10
40+s ds
10p40 = exp
0
Z 10
s
40
c ds
= exp Bc
0
B 40 1
Annuities
Lecture: Weeks 9-11
Lecture: Weeks 9-11 (STT 455)
Annuities
Fall 2014 - Valdez
1 / 43
What are annuities?
What are annuities?
An annuity is a series of payments that could vary according to:
timing of payment
beginning of year (annuity-due)
1 1
Premium Calculation
Lecture: Weeks 12-14
Lecture: Weeks 12-14 (STT 455)
Premium Calculation
Fall 2014 - Valdez
1 / 31
Preliminaries
Preliminaries
An insurance policy (life insurance or life annuity) is funded by contract
premiums:
once (single premium) ma
Suggested solutions to DHW textbook exercises
Exercise 5.4
We simply apply recursion formulas. Starting with a60 = vp60 (1 + a61 ), we get
vp60 =
10.996
a60
=
.
1 + a61
11.756
Extending the recursion to two years, we have a60 = vp60 + v 2 2p60 (1 + a62 )
Michigan State University
STT 455 - Actuarial Models I
Class Test 2
Monday, 17 November 2013
Total Marks: 100 points
Please write your name and student number at the spaces provided:
Name:
There are five (5) multiple choice (MC) and one (1) written-answe
Suggested solutions to DHW textbook exercises
Exercise 5.17
(a) For a whole life annuity-due on (65), we can write the present value random variable of
the benefits as
Y =a
K+1 ,
where K is the curtate future lifetime of (65) with probability mass
Pr[K =
Suggested solutions to DHW textbook exercises
Exercise 3.3
(a) The probability that a life currently aged 75 who has just been selected will survive to
reach age 85 is
`[75]+10
`85
10542
= 0.6617702.
=
=
10p[75] =
`[75]
`[75]
15930
(b) The probability tha
Michigan State University
STT 455 - Actuarial Models I
Class Test 1
Monday, 20 October 2014
Total Marks: 100 points
Please write your name at the space provided:
Name:
There are five (5) multiple choice (MC) and one (1) written-answer questions
here and
Michigan State University
STT 455 - Actuarial Models I
Sample Test 1
Total Marks: 100 points
Please write your name and student number at the spaces provided:
Name:
Section No.:
There are five (5) multiple choice (MC) and one (1) written-answer questions
Suggested solutions to DHW textbook exercises
Exercise 5.15
Recall from basic interest theory the following relationships:
i = e 1, i(m) = m e/m 1 ,
d = 1 e ,
and d(m) = m 1 e/m .
(a) Write (m) as a function of :
(m) =
i
d
i(m) d(m)
1 e
1
e 1
= 2 e[(1/m)1
Suggested solutions to DHW textbook exercises
Exercise 6.19
(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
(
v K+1 (1.025)K ,
PVF
STT 455 Review Session for Class Test 2
November 13, 2014
1. For a special whole life insurance on (45), you are given:
Death benefit is payable at the end of the year of death.
Death benefit is $10,000 during the first 10 years, and $20,000 thereafter.
Suggested solutions to DHW textbook exercises
Exercise 6.9
Let G the required gross monthly premium.
The APV of the 20-year deferred annuity benefits with an initial annual payment of 50,000
increasing by 2% thereafter is given by
APV(benefits) =
X
50000(
Suggested solutions to DHW textbook exercises
Exercise 4.7
(a) For an n-year increasing term insurance, the benefit starts at $1 and increases by $1 each
year thereafter, provided death occurs within the first n years.
(IA)x:1 n
=
n1
X
(k + 1)v k+1 k|qx
k
Suggested solutions to DHW textbook exercises
Exercise 6.3
(a) Let B be the amount of death benefit, payable at the end of year of death and K = K[41] ,
the curtate future lifetime of select age 41. Then the loss-at-issue random variable can
be expressed
Suggested solutions to DHW textbook exercises
Exercise 4.4
Although not clearly stated in the problem, we assume that benefits are payable at the end of
the year of death. With the reversionary bonus, we find that the benefit payment is
bK+1 = 100000(1.03
Suggested solutions to DHW textbook exercises
Exercise 6.8
Denote by G the required annual premium.
First, consider the actuarial present value (APV) of the benefits (note that there is no mention
of any type of approximation, and hence, the APV has to be