A man age 55 is a member of his employer’s occupational pension scheme. He
has been a member of the scheme since he joined the employer at age 35, and
he expects to retire at age 65. However, at any time before age 65 he could: (a)
die; (b) leave the employer and hence have to leave the pension scheme; or (c)
be forced to retire early because of ill health.
His salary is £40,000 per annum, and this is not expected to change in future.
It is assumed to be paid continuously.
The pension scheme’s actuary uses the following Markov model to represent the
man’s life history before age 65.

Z
Z
Z
Z
Z
Z
Z
ZZ~
>
State 1
Active
State 3
Withdrawn
State 2
Retired Ill
State 4
Dead
μ12(x)
μ13(x)
μ14(x)
The transition intensities (defined as rates per annum) are as follows, where x
is age:
μ12(x) = 0.0006 + 0.000034 × 1.1015x+10
μ13(x) = 0.02
μ14(x) = 0.0006 + 0.000034 × 1.1015x
.
The scheme rules define the benefits he shall receieve in any event as follows:
• On reaching the planned retirement age of 65, he will receive an annuity
for life, of annual amount 1/60th of his salary at retirement for each year
of service to the employer. It will be paid annually in advance.
• On retiring in illhealth, he will receive a lump sum equal to 1/20th of
his salary at that time for each year of service to the employer. Years of
service will be counted exactly (that is, fractions of a year will count).
• On withdrawing from service, the scheme will transfer 90% of the reserve
then being held to the pension scheme of his new employer.
• On dying before age 65, a lump sum death benefit of £150,000 will be
paid.
The basis for the annuity on retirement at age 65 is the PMA92C20 life table
with interest of 4% per annum effective (see the Yellow Tables).
(a) Assuming that the scheme must immediately set up a reserve sufficient to
meet all future liabilities, calculate that reserve. Interest before age 65 is
assumed to be 6% per annum effective.
(b) Suppose the scheme rules require all employees to pay a contribution into
the scheme (the same thing as premiums paid into an insurance contract).
The contribution rate is set at 5% of salaries, payable continuously. By how
much is the reserve held now reduced?
Instructions
Write a short report describing in your own words:
• any equations you need to solve;
• any numerical methods you use; and
• if you use Excel, an extract from an Excel spreadsheet in which the numer
ical solution is implemented, of not more than one A4 page, illustrating
the main lines of your approach.
Your report should not exceed 1,500 words and should not be longer than 4
A4 pages (excluding any extract from an Excel spreadsheet). The deadline for
submission, in the AM&S coursework postbox outside Room EM 1.24, is 4.00
p.m. on Friday 19 March 2010.
has been a member of the scheme since he joined the employer at age 35, and
he expects to retire at age 65. However, at any time before age 65 he could: (a)
die; (b) leave the employer and hence have to leave the pension scheme; or (c)
be forced to retire early because of ill health.
His salary is £40,000 per annum, and this is not expected to change in future.
It is assumed to be paid continuously.
The pension scheme’s actuary uses the following Markov model to represent the
man’s life history before age 65.

Z
Z
Z
Z
Z
Z
Z
ZZ~
>
State 1
Active
State 3
Withdrawn
State 2
Retired Ill
State 4
Dead
μ12(x)
μ13(x)
μ14(x)
The transition intensities (defined as rates per annum) are as follows, where x
is age:
μ12(x) = 0.0006 + 0.000034 × 1.1015x+10
μ13(x) = 0.02
μ14(x) = 0.0006 + 0.000034 × 1.1015x
.
The scheme rules define the benefits he shall receieve in any event as follows:
• On reaching the planned retirement age of 65, he will receive an annuity
for life, of annual amount 1/60th of his salary at retirement for each year
of service to the employer. It will be paid annually in advance.
• On retiring in illhealth, he will receive a lump sum equal to 1/20th of
his salary at that time for each year of service to the employer. Years of
service will be counted exactly (that is, fractions of a year will count).
• On withdrawing from service, the scheme will transfer 90% of the reserve
then being held to the pension scheme of his new employer.
• On dying before age 65, a lump sum death benefit of £150,000 will be
paid.
The basis for the annuity on retirement at age 65 is the PMA92C20 life table
with interest of 4% per annum effective (see the Yellow Tables).
(a) Assuming that the scheme must immediately set up a reserve sufficient to
meet all future liabilities, calculate that reserve. Interest before age 65 is
assumed to be 6% per annum effective.
(b) Suppose the scheme rules require all employees to pay a contribution into
the scheme (the same thing as premiums paid into an insurance contract).
The contribution rate is set at 5% of salaries, payable continuously. By how
much is the reserve held now reduced?
Instructions
Write a short report describing in your own words:
• any equations you need to solve;
• any numerical methods you use; and
• if you use Excel, an extract from an Excel spreadsheet in which the numer
ical solution is implemented, of not more than one A4 page, illustrating
the main lines of your approach.
Your report should not exceed 1,500 words and should not be longer than 4
A4 pages (excluding any extract from an Excel spreadsheet). The deadline for
submission, in the AM&S coursework postbox outside Room EM 1.24, is 4.00
p.m. on Friday 19 March 2010.
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