General benet reserves
Maths of Life Contingencies 2
Analysis of benet reserves
Edward Furman
Department of Mathematics and Statistics
York University
April 14, 2015
Edward Furman
Maths of Life contingenices MATH 3281
1 / 39
General benet reserves
Loss at
Mathematics of Life Contingencies. Math 3281 3.00 W
Instructor: Edward Furman
Homework 1
Unless otherwise indicated, all lives in the following questions are
subject to the same law of mortality and their times until death are
independent random variables
Mathematics of Life Contingencies 2, MATH 3281 3.00 W
http:/math.yorku.ca/~ efurman/MATH3281_2011
Lecture time and location
13:00-14:30 MW TBA
Tutorial time and location
15:30-16:30 F
TBA
Instructor name and contacts Ed Furman
[email protected]
ht
Economics of insurance
Maths of Life Contingencies 2
Economics of insurance
Edward Furman
Department of Mathematics and Statistics
York University
January 11, 2012
Edward Furman
Maths of Life Contingencies 2
1/9
Economics of insurance
Denition 1.1 (see, B
Mathematics of Life Contingencies. Math 3281 3.00 W
Instructor: Edward Furman
Homework 1
Unless otherwise indicated, all lives in the following questions are
subject to the same law of mortality and their times until death are
independent random variables
Final Test
MATH 3281 3.00
April, 4th, 2012
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This exam is a closed book exam, duration 180 minutes.
3. Only non-programmable calculators are permitted.
4. The
Exercise 1
IL = N HmIfOIO 3 U < n t,andv rey; n r.
_ _ U
IQO?VLL = 1FU - IJ-f =2 v[ P'(]'_5v ) I: Vcfw_f[1 F %;] - %;
Then VarQL) = [1 + Var(vU) = -Var(vU).
Since var U) : (2Xx+r:;f-l _ Zx+tm 2)
22 _ _ E _2
from (4.2.10), then VargL) =
(t5 - alga)
Not
Mathematics of Life Contingencies MATH
3281
Benet premiums
Edward Furman
Department of Mathematics and Statistics
York University
March 16, 2011
Edward Furman
Mathematics of Life Contingencies MATH 3281
1 / 39
Fairness/identity utility.
Recall that the pr
Quiz 2
MATH 3281 3.00
Jan 19, 2015
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This quiz is a closed book test, duration 15 minutes.
3. Only non-programmable calculators are permitted.
4. The text has
Mathematics of Life Contingencies MATH
3281
Benet reserves
Edward Furman
Department of Mathematics and Statistics
York University
April 5, 2011
Edward Furman
Mathematics of Life Contingencies MATH 3281
1 / 47
Urgent:
It is not enough to just determine a p
Quiz 1
MATH 3281 3.00
Jan 12, 2015
Given name and surname:
Student No:
Signature:
INSTRUCTIONS:
1. Please write everything in ink.
2. This quiz is a closed book test, duration 15 minutes.
3. Only non-programmable calculators are permitted.
4. The text has
Actuarial mathematics, MATH 3280 6.00 Y
Instructor: Edward Furman
Homework 13
1. For an n-year unit endowment insurance issued on a fully continuous basis to
(x), dene t L, the prospective loss after duration t. Conrm that
2
V ar(t L|T > t) =
Ax+t:nt (Ax+
Mathematics of Life Contingencies. Math 3281 3.00 W
Instructor: Edward Furman
Homework 7
Unless otherwise indicated, all lives in the following questions are
subject to the same law of mortality and their times until death are
independent random variables
QUIZ 1 _\.-IATH 3281 3.00 JAN 1) 0015
ada-d
(liven name and surname:-_*_
Student No:-_._*_
Signature:_
INSTRUCTIONS:
1. Please write everything in ink.
. This quiz is a. closed book test, duration 15 minutes.
2
3. Only non-programmable calculators are per
Recall.
. (m)
an
:=
mn1
j=0
1 1/m j
1 vn
1 vn
.
=
(v
) =
m
m(1 (1 d )1/m )
d (m)
Example 0.5 (m-thly payable whole life annuity due.)
Let J (m) be an r.v. representing the number of m-thly periods
lived by (x) at the year of death. Then the m-thly payable
Mathematics of Life Contingencies 2, MATH 3281 3.00 W
http:/math.yorku.ca/~ efurman/MATH3281_2014
Lecture time and location
13:00-14:15 MW VH 3006 304
Tutorial time and location
13:30-14:30 F
CLH 110
Instructor name and contacts Ed Furman
[email protected]
Exercises
Assume, unless otherwise stated, that insurances are payable at the moment of
death, and that the force of interest is a constant 8 with i andtd as the equivalent
rates of interest and discount.
126 Section 4.6 Notes and Referenes a. Integrate b
CHAPTER 4
n
Exercise 4.1
If the force of mortality is constant then
Fiat -e -/1 dt
0
A): = fvt ' 1px 'x(t)dt
H
= ,u~ e(MW dt
= #I[_ e(#+6)t]""+°° = . -
,u + 6 6 y + 5
Exercise 4.2
We are given that
y(x) = i, x > 0.
01+x+s Il+x+t
=
Mathematics of Life Contingencies. Math 3281 3.00 W
Instructor: Edward Furman
Homework 2Solution
Unless otherwise indicated, all lives in the following questions are
subject to the same law of mortality and their times until death are
independent random v
Mathmatics of Life Contingencies 2. Math 3281 3.00 W
Instructor: Edward Furman
Homework 1
Let (x) and (y) be simple life statuses denoting two persons of age x 0 and
y 0, respectively, with independent future lifetimes. Also, for n > 0, let (n )
denote t
4.11. The random variable Z is the present-value random variable for a Whole life
insurance of unit-amount payable at the moment of death and issued to (x).
If 8 = 0.05 and ux(t) = 0.01:
a. Display the formula for the p.d.f. of Z.
b. Graph the p.d.f. of Z
Exercise 4.1 1
(a) It is shown in the textbook that the probability density function (PDF) of Z is:
-1114
gm=(al
62
CHAPTER 4 37
@)Nmema -
Therefore
1
A = =_
x y+5 6
2_ 2 u 21
x y+26 u
Exercise 4.12
The random variable under con
Mathematics of Life Contingencies. Math 3281 3.00 W
Instructor: Edward Furman
Homework 2
Unless otherwise indicated, all lives in the following questions are
subject to the same law of mortality and their times until death are
independent random variables
Mathematics of Life Contingencies MATH
3281
Life annuities contracts
Edward Furman
Department of Mathematics and Statistics
York University
February 13, 2012
Edward Furman
Mathematics of Life Contingencies MATH 3281
1 / 23
Denition 0.1 (Life annuity.)
Lif
Actuarial mathematics 2
Life insurance contracts
Edward Furman
Department of Mathematics and Statistics
York University
January 30, 2012
Edward Furman
Actuarial mathematics MATH 3280
1 / 45
Denition 0.1 (Life insurance.)
Life insurance is a contract that
Actuarial mathematics, MATH 3280 6.00 Y
Instructor: Edward Furman
Homework 14
Unless otherwise indicated, all lives in the following questions are subject to the same law of mortality and their times until death are independent random variables.
1. The jo