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Gross Premium Reserves, Variance of Future Loss
Gross loss at issue:
𝑔𝑟𝑜𝑠𝑠 𝑓𝑢𝑡𝑢𝑟𝑒 𝑙𝑜𝑠𝑠 = 𝑃𝑉(𝑓
...
𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠)
𝐴𝑃𝑉(𝑓𝑢𝑡𝑢𝑟𝑒 𝑝𝑟𝑒𝑚𝑖𝑢𝑚) = 𝐴𝑃𝑉(𝑓𝑢𝑡𝑢𝑟𝑒 𝑏𝑒𝑛𝑒𝑓𝑖𝑡𝑠)
• Notations:
o G: gross premium
o 𝜋: premium
o 𝑒: the level renewal expense
o 𝑒𝑓 : the first year expense
o 𝐸: settlement expense
o 𝑏: the face amount
1
...

𝐴30: 20 = 0
...

𝐴30: 1 = 0
...


Expenses are in the following table:

Per premium
First year
35%
Renewal
3%
IV
...

V
...
96
VI
...

VII
...

Calculate 𝐸[ 0𝐿 ]
...
45
0
...
00
2
...
75

𝐸[ 0𝐿] = 100,000𝐴130: 20 + (0
...
03𝑎̈ 30: 20 )𝐺 + 8 + 2𝑎̈ 30: 20 − 13
...
45 − 0
...
32 + 13
...
03)400 + 8 + 2 × 13
...
75𝐺
= −171
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Variance – discrete (2 approaches to calculate: formula & first principles)
a
...

o For fully discrete whole life insurance, the gross future loss is
𝐾𝑥+1 + (𝑒 − 𝑒) − (𝐺 − 𝑒)𝑎̈
𝑓
0𝐿 = (𝑏 + 𝐸)𝑣
𝐾𝑥+1
𝑉𝑎𝑟( 0𝐿) = ( 2𝐴𝑥 − 𝐴2𝑥 ) (𝑏 + 𝐸 +
o

𝐺 −𝑒 2
)
𝑑

Notes:
1) If G is determined by the equivalence principle, expenses do not differ between first year
and renewal, and there are no settlement expenses (E), then 𝐺 − 𝑒 = 𝑛𝑒𝑡 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
and the formula reduces to the formula for the variance of the future net loss
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2
𝜋 2
𝑉𝑎𝑟( 0𝐿) = ( 2𝐴𝑥: 𝑛 − (𝐴𝑥: 𝑛 ) ) (𝑏 + )
𝑑
2

𝐴𝑥: 𝑛 −(𝐴𝑥: 𝑛 )

2

𝑉𝑎𝑟( 0𝐿) =

𝑏2 [

𝑉𝑎𝑟( 0𝐿) =

𝑞(1−𝑞)
for whole life with equivalence principle and constant rate of mortality only
𝑞+ 2𝑖

(1−𝐴𝑥: 𝑛 )

2

] if equivalence principle premium is used

3) For fully discrete endowment insurance with face amount b, settlement expenses E, level
2

renewal expenses e, 𝑉𝑎𝑟( 0𝐿) = ( 2𝐴𝑥: 𝑛 − (𝐴𝑥: 𝑛 ) ) (𝑏 + 𝐸 +
b
...

Expenses are as follows:
Percent of premium
First year
20%
renewal
5%

𝐺−𝑒 2
)
𝑑

Per policy
25
5

II
...

III
...
05
IV
...
05; 1|𝑞85 = 0
...

Let 𝑒𝑘 = 𝑡ℎ𝑒 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑡𝑜 𝑦𝑒𝑎𝑟 𝑘
𝑒1 = 400(0
...
95)−5

𝑒3 = 652
...
14

1
...
95)−5
+ 1
...
28

Then the present value of the loss given death in year k is
1000
1
...
053

1000
− 295
1
...
3810 in year 1,

− 652
...
28 = −128
...

Then:
𝐸[ 0𝐿] = 0
...
3810) + 0
...
8866) + 0
...
4413) = −50
...
05((657
...
1(254
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85(128
...
8
2

𝑉𝑎𝑟( 0𝐿) = 𝐸[ 0𝐿2 ] − 𝐸[ 0𝐿] = 42126
...
81742 = 39544
3
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Probabilities and percentiles – 2 examples on the slides
Gross premium reserve (the expected value of the gross future loss)
Gross premium reserve is calculated using a set of mortality and interest assumptions — reserve basis; the set of
assumptions used for calculating the gross premium is called premium basis
...
(The gross premium reserve is often negative in early durations, because expenses are higher in the 1st year
than in renewal years

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