5.4
Fully Discrete Net Premium Reserve
5.4.1 Net Future Loss of Fully Discrete Insurance
Net Future Loss Random Variable at time k, 𝒌𝑳
The insurer's future loss random variable at time k (or at age x+k).
In whole life insurance:
𝑘𝐿 =
𝐵𝑉 (𝐾𝑥+𝑘)+1 − 𝑃𝑥 . 𝑎̈ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(𝐾𝑥 +𝑘)+1|
for k = 0, 1, 2, …,∞.
In n-year endowment insurance:
𝑘𝐿
= 𝐵𝑉 𝑚𝑖𝑛(𝐾𝑥+𝑘+1,
𝑛−𝑘)
− 𝑃𝑥:𝑛|
̅̅̅ . 𝑎̈ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑚𝑖𝑛(𝐾𝑥 +𝑘+1, 𝑛−𝑘)|
for k = 0, 1, 2, …, n. Loss is zero for k > n.
Variance of Net Future Loss Random Variable, 𝑽𝒂𝒓[ 𝑘𝐿]
In whole life insurance:
𝑘𝐿
= 𝐵𝑉 (𝐾𝑥 +𝑘)+1 − 𝑃𝑥 . 𝑎̈ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(𝐾𝑥 +𝑘)+1|
= 𝐵𝑉
(𝐾𝑥 +𝑘)+1
= (𝐵 +
1 − 𝑉 (𝐾𝑥 +𝑘)+1
1 − 𝑉𝑚
− 𝑃𝑥 (
) , 𝑤ℎ𝑒𝑟𝑒 𝑎̈ 𝑚|
̅̅̅̅ =
𝑑
𝑑
𝑃𝑥 (𝐾 +𝑘)+1 𝑃𝑥
)𝑉 𝑥
−
𝑑
𝑑
𝑽𝒂𝒓[ 𝑘𝐿] = 𝑽𝒂𝒓 [(𝐵 +
= (𝐵 +
𝑃𝑥 (𝐾 +𝑘)+1 𝑃𝑥
)𝑉 𝑥
− ]
𝑑
𝑑
𝑃𝑥 2
) 𝑽𝒂𝒓[𝑉 (𝐾𝑥 +𝑘)+1 ] − 0
𝑑
𝑃𝑥 2 2
= (𝐵 + ) [ 𝐴𝑥+𝑘 − (𝐴𝑥+𝑘 )2 ]
𝑑
𝐵. 𝐴𝑥+𝑘 2 2
) [ 𝐴𝑥+𝑘 − (𝐴𝑥+𝑘 )2 ]
= (𝐵 +
𝑑. 𝑎̈ 𝑥+𝑘
,
𝒘𝒉𝒆𝒓𝒆 𝑃𝑥 =
𝐵. 𝐴𝑥+𝑘
𝑎̈ 𝑥+𝑘
𝐴𝑥+𝑘 2 2
) [ 𝐴𝑥+𝑘 − (𝐴𝑥+𝑘 )2 ] ,
= 𝐵. (1 +
1 − 𝐴𝑥+𝑘
= 𝐵. (
= 𝐵.
𝒘𝒉𝒆𝒓𝒆 𝑑. 𝑎̈ 𝑥+𝑘 = 1 − 𝐴𝑥+𝑘
2
1
) [ 2𝐴𝑥+𝑘 − (𝐴𝑥+𝑘 )2 ]
1 − 𝐴𝑥+𝑘
[ 2𝐴𝑥+𝑘 − (𝐴𝑥+𝑘 )2 ]
(1 − 𝐴𝑥+𝑘 )2
In n-year endowment insurance:
𝑘𝐿
= 𝐵𝑉 𝑚𝑖𝑛(𝐾𝑥 +𝑘+1,
= (𝐵 +
𝑃𝑥:𝑛|
̅̅̅
𝑑
𝑛−𝑘)
− 𝑃𝑥:𝑛|
̅̅̅ . 𝑎̈ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑚𝑖𝑛(𝐾𝑥 +𝑘+1, 𝑛−𝑘)|
) 𝑉 𝑚𝑖𝑛(𝐾𝑥 +𝑘+1,
𝑉𝑎𝑟[ 𝑘𝐿] = 𝐵. (1 +
𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅
𝑑. 𝑎̈ 𝑥+𝑘:𝑛−𝑘
̅̅̅̅̅̅ |
𝑛−𝑘)
−
𝑃𝑥:𝑛|
̅̅̅
𝑑
2
2
) [ 2𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ − (𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ ) ] , 𝑤ℎ𝑒𝑟𝑒 𝑃𝑥:𝑛|
̅̅̅ =
𝐵. 𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅
𝑎̈ 𝑥+𝑘:𝑛−𝑘
̅̅̅̅̅̅ |
2
= 𝐵.
[ 2𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ − (𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ ) ]
(1 − 𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ )
2
,
𝑤ℎ𝑒𝑟𝑒 𝑑. 𝑎̈ 𝑥+𝑘:𝑛−𝑘
̅̅̅̅̅̅̅
̅̅̅̅̅̅ | = 1 − 𝐴𝑥+𝑘:𝑛−𝑘|
5.4.2 General Formula in Fully Discrete Net Premium Reserve
Value of 𝒂̈ 𝒙 , 𝒏𝑬𝒙 , 𝑨𝒙 can be obtained from Illustrative Life Table at interest i .
𝑎̈ 𝑥:𝑛|
̅̅̅ = 𝑎̈ 𝑥 − 𝑛𝐸𝑥 . 𝑎̈ 𝑥+𝑛
𝐴1𝑥:𝑛|
̅̅̅ = 𝐴𝑥 − 𝑛𝐸𝑥 . 𝐴𝑥+𝑛
1
𝐴𝑥:𝑛|
̅̅̅ = 𝑛𝐸𝑥
1
1
𝐴𝑋:𝑛|
̅̅̅ = 𝐴𝑥:𝑛|
̅̅̅ − 𝐴𝑥:𝑛|
̅̅̅
5.4.3 Formula of Fully Discrete Net Premium Reserve in Difference Insurance
Consider the difference case of a fully discrete insurance issued to a life (x) where premium
of P is paid at the beginning of each year and benefit of B1$ is paid at the end of year of death.
Whole Life Insurance
The Net Future Loss Random Variable at time t, 𝐿𝑡
𝑘𝐿
= 𝑽(𝑲𝒙+𝒌)+𝟏 − 𝑃𝑥 . 𝑎̈ ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(𝑲𝒙 +𝑘)+1|
̅
The Fully Continuous Net Premium Reserve, 𝒕𝑽
𝑘 𝑉𝑥
= 𝑬[ 𝑘𝐿] =
𝐴𝑥+𝑘 − 𝑃𝑥 𝑎̈ 𝑥+𝑘
If a whole life insurance is funded under the equivalence principle:
𝑬[ 0𝐿] = 𝟎
𝐴𝑥 − 𝑃𝑥 𝑎̈ 𝑥 = 0
Then the benefit premium is:
𝑃𝑥 =
𝐴𝑥
𝑎̈ 𝑥
n-year term life insurance
The Net Future Loss Random Variable at time t, 𝐿𝑡
𝑘𝐿 = 𝑽
(𝑲𝒙 +𝒌)+𝟏
𝑰(𝑲𝒙 < 𝒏) − 𝑃𝑥 {𝑎̈ 𝑘|
̅̅̅ 𝑰(𝑲𝒙 < 𝒏) + 𝑎̈ 𝑛|
̅̅̅ 𝑰(𝑲𝒙 = 𝒏)}
̅
The Fully Continuous Net Premium Reserve, 𝒕𝑽
1
̅̅̅
𝑘 𝑉𝑥:𝑛|
= 𝑬[ 𝑘 𝐿 ] = {
1
1
𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ − 𝑃𝑥:𝑛|
̅ . 𝑎̈ 𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ ,
0,
𝑘=𝑛
𝑘<𝑛
If a n-year term life insurance is funded under the equivalence principle:
𝑬[ 0𝐿] = 𝟎
1
1
𝐴𝑥:𝑛|
̅̅̅ − 𝑃𝑥:𝑛|
̅̅̅ . 𝑎̈ 𝑥:𝑛|
̅̅̅ = 0
Then the benefit premium is:
1
𝑃𝑋:𝑛|
̅̅̅
=
𝐴1𝑋:𝑛|
̅̅̅
𝑎̈ 𝑋:𝑛|
̅̅̅
n-year endowment insurance
The Net Future Loss Random Variable at time t, 𝐿𝑡
𝑘𝐿
= 𝑽(𝑲𝒙 +𝒌)+𝟏 𝑰(𝑲𝒙 < 𝒏) + 𝑽𝒏−𝒌 𝑰(𝑲𝒙 ≥ 𝒏) − 𝑃𝑥 {𝑎̈ ̅̅̅
𝑘| 𝑰(𝑲𝒙 < 𝒏) + 𝑎̈ ̅̅̅
𝑛| 𝑰(𝑲𝒙 = 𝒏)}
̅
The Fully Continuous Net Premium Reserve, 𝒕𝑽
̅̅̅
𝑘 𝑉𝑥:𝑛|
=
𝑬[ 𝑘 𝐿 ]
𝐴 ̅̅̅̅̅̅̅ − 𝑃𝑥:𝑛|
̅ . 𝑎̈ 𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ ,
= { 𝑥+𝑘:𝑛−𝑘|
1,
𝑘=𝑛
𝑘<𝑛
If a n-year endowment insurance is funded under the equivalence principle:
𝑬[ 0𝐿] = 𝟎
𝐴𝑥:𝑛|
̅̅̅ − 𝑃𝑥:𝑛|
̅̅̅ . 𝑎̈ 𝑥:𝑛|
̅̅̅ = 0
Then the benefit premium is:
𝑃𝑥:𝑛|
̅̅̅ =
𝐴𝑥:𝑛|
̅̅̅
𝑎̈ 𝑥:𝑛|
̅̅̅
n-year pure endowment insurance
The Net Future Loss Random Variable at time t, 𝐿𝑡
𝑘𝐿
= 𝑽𝒏−𝒌 𝑰(𝑲𝒙 ≥ 𝒏) − 𝑃𝑥 {𝑎̈ ̅̅̅
𝑘| 𝑰(𝑲𝒙 < 𝒏) + 𝑎̈ ̅̅̅
𝑛| 𝑰(𝑲𝒙 = 𝒏)}
̅
The Fully Continuous Net Premium Reserve, 𝒕𝑽
1
̅̅̅
𝑘 𝑉𝑥:𝑛|
1
1
𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ ,
̅̅̅̅̅̅̅ − 𝑃𝑥:𝑛|
̅ . 𝑎̈ 𝑥+𝑘:𝑛−𝑘|
= 𝑬[ 𝑘 𝐿 ] = {
1,
𝑡=𝑛
𝑡<𝑛
If a n-year pure endowment insurance is funded under the equivalence principle:
𝑬[ 0𝐿] = 𝟎
1
1
𝐴𝑥:𝑛|
̅̅̅ = 0
̅̅̅ − 𝑃𝑥:𝑛|
̅̅̅ . 𝑎̈ 𝑥:𝑛|
Then the benefit premium is:
1
𝑃𝑋:𝑛|
̅̅̅
=
1
𝐴𝑥:𝑛|
̅̅̅
𝑎̈ 𝑥:𝑛|
̅̅̅
h-payment year of whole life insurance
The Net Future Loss Random Variable at time t, 𝐿𝑡
𝑘𝐿
= 𝑽(𝑲𝒙 +𝒌)+𝟏 − 𝑃𝑥 {𝑎̈ ̅̅̅
̅̅̅ 𝑰(𝑲𝒙 ≥ 𝒉)}
𝑘| 𝑰(𝑲𝒙 < 𝒏) + 𝑎̈ ℎ|
̅
The Fully Continuous Net Premium Reserve, 𝒕𝑽
ℎ
𝑘 𝑉𝑥
=
𝑬[ 𝑘 𝐿 ]
𝐴𝑥+𝑘 − ℎ𝑃𝑥 𝑎̈ 𝑥+𝑘:ℎ−𝑘|
̅̅̅̅̅̅̅ , 𝑘 < ℎ
= {
𝐴𝑥+𝑘 , 𝑘 ≥ ℎ
If a h-payment year of whole life insurance is funded under the equivalence principle:
𝑬[ 0𝐿] = 𝟎
𝐴𝑥 − ℎ𝑃𝑥 𝑎̈ 𝑥:ℎ|
̅̅̅ = 0
Then the benefit premium is:
ℎ𝑃𝑥
=
𝐴𝑥
𝑎̈ 𝑥:ℎ|
̅̅̅
h-payment year of n-year endowment insurance
The Net Future Loss Random Variable at time t, 𝐿𝑡
𝑘𝐿
= 𝑽(𝑲𝒙 +𝒌)+𝟏 𝑰(𝑲𝒙 < 𝒏) + 𝑽𝒏−𝒌 𝑰(𝑲𝒙 ≥ 𝒏) − 𝑃𝑥 {𝑎̈ ̅̅̅
̅̅̅ 𝑰(𝑲𝒙 ≥ 𝒉)}
𝑘| 𝑰(𝑲𝒙 < 𝒏) + 𝑎̈ ℎ|
̅
The Fully Continuous Net Premium Reserve, 𝒕𝑽
𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅ − 𝑃𝑥:𝑛|
̅ . 𝑎̈ 𝑥+𝑘:ℎ−𝑘|
̅̅̅̅̅̅̅ ,
ℎ
̅
𝑘 𝑉𝑥:𝑛|
=
𝑬[ 𝑘 𝐿 ]
= { 𝐴𝑥+𝑘:𝑛−𝑘|
̅̅̅̅̅̅̅
1
ℎ
,
,
𝑘<ℎ<𝑛
ℎ≤𝑘<𝑛
𝑘=𝑛
If a h-payment year of n-year endowment insurance is funded under the equivalence principle:
𝑬[ 0𝐿] = 𝟎
𝐴𝑥:𝑛|
̅̅̅ − ℎ𝑃𝑥:𝑛|
̅̅̅ . 𝑎̈ 𝑥:ℎ|
̅̅̅ = 0
Then the benefit premium is:
̅̅̅ =
ℎ𝑃𝑥:𝑛|
𝐴𝑥:𝑛|
̅̅̅
𝑎̈ 𝑥:ℎ|
̅̅̅