Question 118 explain why the discount function for

Question 11.8
Explain why the discount function for age 65 is
10
v rather than
10½
v .
Using commutation functions
The calculation of pension EPVs can be aided by the use of commutation functions.
These are not included in the standard international actuarial notation, so any symbols
we use must be defined as we go along. The idea will become clearer from some
examples.
So, we will take the EPV of the lump sum we have just described, and define some
suitable commutation functions that we can use. The main step in this process is to
multiply the numerator and denominator of the EPV formula by
x
v , where x is the
current age of the member (the “valuation age”). So, in our case, we multiply and
divide through by
55
v to obtain:
55½ 56½ 64½ 65
55 56 64 65
55
55
10,000
 
    
 
 EPV v r v r v r v r
v l
We then define symbols to represent all the elements of the expression.

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While we could use any symbols for these (provided we define them!), we will adopt
the same notation as used in the Pension Scheme Tables. So for the denominator we
define:

x
x x
D v l
and for the numerator define:
½
65
65
65
65
 







x
x
r
x
v r x
C
v r x
Note that we have to define the function differently for the two age ranges to
accommodate the retirements occurring on the exact 65th birthday.
So the EPV now looks like this:
10
55
55
0
10,000




r
t
t
EPV C
D
We now tidy this up further by defining:
65
0





x
r r
x x t
t
M C
and so:
55
55
10,000 
r
M
EPV
D
Question 11.9
Calculate the expected present value of a lump sum benefit of £50,000 paid on normal
age retirement, for a scheme member aged exactly 52, assuming that this benefit is paid:
(a) regardless of the actual age at retirement
(b) only if the member retires after the 64th birthday.
Use the assumptions underlying the Pension Scheme Tables, and 4% pa interest.

CT5-11: Pension funds Page 15
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From this question note that, in general:
 an
j
x
M function will include all decrements by cause j occurring at exact age
x and in all future years after age x
 a
j
x
C function will include decrements by cause j occurring at age x and in the
single year of age beginning at age x only.
Dividing by
y
D (where y is the valuation age) will ensure that the correct discounting
and survival probabilities are included.
3.2 Lump sums paid on other types of decrement
The other possible decrements are withdrawal, death and ill-health retirement.
Assuming, as before, that all remaining active members retire “normally” on the 65th
birthday, then we know that:
65 65 65
0    w d i
Other than this, the formulae used are of identical construction to before. So, for
example, the EPV of a lump sum of £100,000 payable immediately on death in service
of a member currently aged 44 would be:
44
44
100,000 
d
M
EPV
D
where:
64
0
½








x
x x
x
d d
x x t
t
d x
x x
D v l
M C
C v d
noting that the
d
x
M function sums to 64   t x (not 65  x ), because there is no
65
d
C
function to include.