Mat 462 - Winter 2012
Dr. C. Schmegner
Name:
Test 4 - Due February 22nd at the beginning of class
This exam is to be solved individually without any help from other people. Oering or receiving help
is considered cheating. Nevertheless, if you have trouble
Problem 6-23
You are given:
(i) q70 = 0.04.
(ii) q71 = 0.044.
(iii) Deaths are uniformly distributed over each year of age.
Calculate e70:1.5| .
Recall that under the UDD assumption, lx+t = lx tdx , where x is an integer and
0 t 1.
We know that t px = lx+
Mat 462 - Winter 2012
Dr. C. Schmegner
Name:
Test 3
1. Given e0 = 25, and lx = x, 0 x , calculate V ar(T10 ).
2. Given the UDD assumption and the values q1 = 0.1, q2 = 0.05, q3 = 0.01, nd the value
of e1:1.5 .
1
3. Given the table below,
x
lx
90 100
91 87
PREMIUM CALCULATION
Loss=present value of benets - present value of premium payments
By the equivalence principle, the premiums are such that E(Loss) = 0. In general, the
loss will be denoted by L and the premiums by P .
1. Fully continuous models
1.1. Wh
M “g t "
O l, A _.
1’ l "EC; «
'l l
a; _ ' w '7‘ ’~———-—— V 1 ‘ a ‘
5: PM? ‘(L " “:53 f 4- /2 nalgm
Mime) "—15 [373:1 E Ex— MM
:E[ \i V1]; if. LQO‘JC w-
“SA 1 h? ' »SU(x+b) Cu: ‘7‘0' 3—
agé 76:. (F53
_ 2. )6
a '79 moi“; 7'“; 70 {
Mat 462 - Winter 2016
Dr. C. Schmegner
Name:
Test 1
1. a. Mortality follows de Moivres Law and e32 = 30. Calculate V ar(T80 ).
b. Mortality follows an exponential distribution and e50 = 20. Calculate V ar(T50 ).
1
2. Given
1
, for x 0,
1+x
nd expressions
Mat 462 - Winter 2016
Dr. C. Schmegner
Name:
Test 3 - Due February 18th at the beginning of class
Please write your solutions clearly and nicely and IN ORDER from 1 on .
I will not grade the test if your solutions are not in order
1. Z is the present valu
Mat 362/462 — Winter 2012 Name:
Dr. C. Schmegner
Test 5
1. For a constant force model with Mm) = .05, and 6 = .02, calculate (140 and Varmﬂ).
40
—-—I
lo NEXF (.US') =>T+O NExf (J75)
— 4,. ’ 275—:
"5L >2 t a W. 2‘1 (5‘40 “1+5— 9747” A”)
v.— no
mam]; W‘Eo
CONTINGENT ANNUITIES
1. Continuous case
1.1. Whole Life Annuity.
1 eTx
1 Zx
1 Tx
=
=
Yx = aTx =
ax = E(Yx ) =
x
t Ex
0
ax =
ax =
1E(Zx )
2 ax =
12x
A
2
x
0
at fx (t)dt =
x 1et
fx (t)dt
0
x
0
= . =
t t px dt
dt
1E( Tx )
=
=
1Ax
Ax = 1 x
a
2 Ax = 1 2
Mat 462 - Winter 2013
Dr. C. Schmegner
Name:
Test 2
1. Given that e0 = 25 and lx = x, for 0 x , nd the value of Var(T10 ).
2. Given that x = k + e2x , for all x > 0, and .4 p0 = .5, nd the value of k .
1
3. If the density function of the lifetime T0 of a
SURVIVAL MODELS
Continuous Survival Models
T0 = age-at-death.
Concepts and Notations.
T0 =lifetime random variable = age-at-death
Survival Model: the distribution of T0 .
Examples: DeMoivre (uniform), constant force model (exponential)
F0 (t) = P (T0
LIFE TABLES
Tabular Survival Models
Concepts and Notations. Consider a cohort of newborns.
l0 = number of newborns in the cohort
lx = E (number of people, out of the initial cohort, surviving age x)
lx = l0 S0 (x)
t dx = E ( number of people, out of th
INSURANCE MODELS
Continuous case: benefit payable at death
Whole Life.
Zx = Tx = eTx
Ax = E (Zx ) =
2 Ax =
x t
e fx (t)dt
0
x 2t
e
fx (t)dt
0
V ar(Zx ) = V ar( Tx ) = 2 Ax A2
x
T0 Uniform(0, ) Ax =
1e(x)
( x)
T0 Exponential() Ax =
=
, 2Ax
+
a x
Insurance Models Review
1. Whole Life Insurance
(a) Continuous: Know what it is and how to calculate Zx , Ax , 2Ax , V (Zx ) in general,
for any survival model. Calculate all these for X Uniform(0, ) and for X
Exponential() for a given . Know how to go e
Mat 462 - Winter 2013
Dr. C. Schmegner
Name:
Test 1
This test is to be solved individually without oering or receiving help. Feel free to address
to me any questions you may have.
2
1. (10p) If t = 100t for 0 t < 100, nd the survival function, distributio