Document Sample

```					                             MATH20962 Tutorial 1

Life expectancy

1. Find values for the following using your table books (and write down the correct
actuarial notation):

(a) The expected future lifetime of a newborn (also known as the complete future

(b) The expected complete future lifetime of a person aged exactly 55 assuming
ELT15(Female) mortality.

(c) The expected age at death (including fractional years) of a person aged exactly
55 assuming ELT15(Female) mortality.

Bonus question: explain why this is higher than the expected age at death of a
newborn.

(d) The expected curtate future lifetime of a person aged exactly 55 assuming AM92
mortality.

2. Is a person aged 60 assumed to live longer if he follows PMA92C20 or she follows
0
PFA92C20? Lookup e x for both tables and state a common conclusion for male
versus female pensioner life expectancy.

Looking up

3. Lookup:

(a) The expected present value of a 25 year endowment assurance of a life aged
exactly 40, using an interest rate of 6% and AM92 ultimate mortality. Write down the
correct actuarial notation.

(b) The expected present value of a whole life assurance payable at the end of year
of death of a life aged exactly 40, using an interest rate of 6% and AM92 mortality.

(c) The expected present value of a whole life assurance payable at the end of year
of death of a life aged exactly 40, using an interest rate of 12.36% and AM92 select
mortality. [Hint: 1:06^2 = 1:1236]

Tx, Kx

4. A person dies aged 82 yrs and 142 days. What are the values of T25 and K25?
Expected value and variance

5. April 2011 (subject CT5) Institute exam Q10:

Calculate the expected present value and variance of the present value of an
endowment assurance of 1 payable at the end of the year of death for a life aged 40
exact, with a term of 15 years.

Basis:
Mortality       AM92 Select
Rate of interest 4% per annum

l55    9557.8179
[You may use:        15   p[ 40]                      0.969913 ]
l[ 40] 9854.3036

General relationships between expected present value

6. State with reason whether the following statements are true or false. Where
necessary assume 5%pa interest.
(i)     A25:3|  A35:3|
(ii)     A35:10|  0.6
(iii)    A1         0.6
35:10|

7. Evaluate A70:2| using AM92 ultimate mortality and 4%pa interest.
(You may assume             2   p 70 = l 72 / l 70  7637 .6208 / 8054 .0544  0.94830 )

8. Correct the following formulas
n| Ax  v Ax  n
n
(i)
(ii)     Ax:n|  Ax:t|  v t t p x . Axt:nt|

9. State which has the highest value:
2
A[55 ]             A55           A25:5|                  A55
2
The suffix indicates a modified interest rate of i’ where 1+i’ = (1+i)2
Derivation of key formula

10. April 2005 (subject CT5) Institute exam Q12:

(i) By considering a term assurance policy as a series of one year deferred term
assurance policies, show that:

i
A1             A1
x:n|           x:n|

(ii) Calculate the expected present value and variance of the present value of a term
assurance of 1 payable immediately on death for a life aged 40 exact, if death occurs
within 30 years.

Basis:
Interest 4% per annum
Mortality AM92 Select
l70    8054.0544
You may assume            30   p[ 40]                      0.8173134
l[ 40] 9854.3036

Derivation of formula for the variance of the present value of a deferred
assurance

11. April 2007 (subject CT5) Institute exam Q10:

Let X be a random variable representing the present value of the benefits of a whole
of life assurance, and Y be a random variable representing the present value of the
benefits of a temporary assurance with a term of n years. Both assurances have a
sum assured of 1 payable at the end of the year of death and were issued to the
same life aged x.

(i) Describe the benefits provided by the contract which has a present value
represented by the random variable X - Y.                                  [1 mark]

(ii) Show that

cov( X , Y ) 2 A1  Ax . A1
x:n|         x:n|

and hence or otherwise that

var (X – Y) = 2 Ax ( n| Ax ) 2  2 A1
x:n|

where the functions A are determined using an interest rate of i, and functions 2 A
are determined using an interest rate of i 2  2i .

January 2012

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 11 posted: 8/29/2012 language: English pages: 3