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Transcript of 68e4zbp911
Summary of Lecture Notes - ACTSC 232, Winter 2010
Part 2 - Life Benefits
2.1 Life insurances on (x)
A life insurance on (x) is a contract or policy issued by an insurer to a life currently aged
x. The insurer will pay benefits to the beneficiaries of (x) in the future. The payment
times of the benefits are contingent on the death time of (x). Such benefits are called
death benefits or life benefits.
Generally speaking, a life insurance on (x) is called a continuous life insurance if benefits
are payable at the moment of the death of (x). A life insurance is called a discrete life
insurance if benefits are payable at the end of the death year of (x).
Review of the expectations of the functions of Tx and Kx: For a function g,
E[g(Tx)] =
∫ ∞0
g(t)fx(t)dt =
∫ ∞0
g(t) tpx µx+tdt
and
E[g(Kx)] =∞∑k=0
g(k) Pr{Kx = k} =∞∑k=0
g(k) kpx qx+k.
Review of present values: Let vt denote present value (PV) at time 0 of 1 (dollar or unit)
to be paid at time t and vt is called discount function. If the force of interest δt = δ is a
constant, then vt = vt = e−δt = ( 11+i
)t, where v = 11+i
= e−δ and 1 + i = eδ.
Unless stated otherwise, we assume that δt = δ is a constant or vt = vt = e−δt.
(a) A general continuous life insurance on (x) pays death benefits at the death time of
(x). Denote the benefit by bt if Tx = t or (x) dies at time t, t > 0.
Let Z denote the present value at time 0 or at age x of the benefits to be paid by
the insurance. Then Z = bTx vTx = bTx e
−δTx and Z is a random variable, where Tx
is the death time of (x).
The expectation or mean of the present value Z is
E[Z] = E[bTx e
−δTx]
=
∫ ∞0
bt e−δt fx(t)dt =
∫ ∞0
bt e−δt
tpx µx+tdt
1
which is called the actuarial present value (APV) of the insurance, or the expected
present value (EPV) of the insurance, or the pure premium of the insurance, or the
net premium of the insurance, or the single benefit premium of the insurance.
The second moment of the present value Z is
E[Z2] = E[b2Tx
e−2δTx]
=
∫ ∞0
b2t e−2δt fx(t)dt =
∫ ∞0
b2t e−2δt
tpx µx+tdt
and V ar[Z] = E[Z2]− (E[Z])2.
The distribution function of Z is denoted by FZ(z) = Pr{Z ≤ z}. The distribution
function may be continuous, or discrete, or mixed.
(b) A general discrete life insurances on (x) pays death benefits at the end of the death
year of (x). Denote the death benefit by bk+1 if Kx = k or (x) dies in year k + 1,
k = 0, 1, 2, ....
Let Z denote the present value at time 0 or at age x of the benefits. Then,
Z = bKx+1 vKx+1 = bKx+1 e
−δ(Kx+1)
The APV or EPV of the insurance is
E[Z] = E[bKx+1 vKx+1] =
∞∑k=0
bk+1 vk+1 Pr{Kx = k}
=∞∑k=0
bk+1 vk+1
kpx qx+k =∞∑k=0
bk+1 e−(k+1)δ
kpx qx+k.
This expectation is also called the pure premium of the insurance, or the net pre-
mium of the insurance, or the single benefit premium of the insurance.
The second moment of the present value is given by
E[Z2] = E[b2Kx+1 v2(Kx+1)] =
∞∑k=0
b2k+1 v2(k+1)
kpx qx+k =∞∑k=0
b2k+1 e−2(k+1)δ
kpx qx+k.
2.2 Level Benefit Life Insurances
A life insurance on (x) is called a level benefit life insurance if benefits are constant and
independent of the payment times of the benefits.
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(a) A continuous whole life insurance of 1 on (x) pays 1 at the moment of death of (x).
The PV of the benefit is Z = vTx and the APV of the insurance is denoted by
Ax = E[vTx ] =
∫ ∞0
vtfx(t)dt =
∫ ∞0
e−δttpx µx+tdt.
The second moment of Z is denoted by
2Ax = E[v2Tx ] =
∫ ∞0
v2tfx(t)dt =
∫ ∞0
e−2δttpx µx+tdt
and V ar[Z] = 2Ax − (Ax)2.
If the mortality force of (x) follows the constant force law or µx = µ for all x > 0,
then fx(t) = tpx µx+t = µe−µ t for 0 < t <∞ and
Ax =µ
µ+ δand 2Ax =
µ
µ+ 2δ.
If the mortality force of (x) follows De Moivre’s law with the limiting age ω, then
fx(t) = tpx µx+t = 1ω−x , 0 < t < ω − x and
Ax =
∫ ω−x
0
e−δt1
ω − xdt
and
2Ax =
∫ ω−x
0
e−2δt 1
ω − xdt.
A discrete whole life insurance of 1 on (x) pays 1 at the end of the death year of (x).
The PV of the benefit is Z = vKx+1 and the APV of the insurance is denoted by
Ax = E[vKx+1] =∞∑k=0
vk+1kpx qx+k =
∞∑k=0
e−δ(k+1)kpx qx+k.
The second moment of Z is denoted by
2Ax = E[v2(Kx+1)] =∞∑k=0
v2(k+1)kpx qx+k =
∞∑k=0
e−2δ(k+1)kpx qx+k
and V ar[Z] = 2Ax − (Ax)2.
(b) A continuous n-year term life insurance of 1 on (x) pays 1 at the moment of death
if (x) dies during the n-year term and no any payments after the n-year term. The
PV of the benefit is
Z =
{vTx , Tx ≤ n,
0, Tx > n.= vTx I(Tx ≤ n),
3
where I(A) is an indicator function and I(A) = 1 if A holds and 0 otherwise. Note
that (I(A))2 = I(A).
The APV of the insurance is denoted by
A1x:n = E[vTx I(Tx ≤ n)] =
∫ n
0
vtfx(t)dt =
∫ n
0
e−δttpx µx+tdt.
The second moment of Z is denoted by
2A1x:n = E[v2TxI(Tx ≤ n)] =
∫ n
0
v2tfx(t)dt =
∫ n
0
e−2δttpx µx+tdt
and V ar[Z] = 2A1x:n − (A1
x:n)2.
Recursion formulas for Ax and A1x:n:
Ax = A1x:n + vn npx Ax+n
A1x:n = A1
x:1+ v px A
1x+1:n−1
A discrete n-year term life insurance of 1 on (x) pays 1 at the end of the year of
death if (x) dies during the n-year term and nothing after the n-year term. The PV
of the benefit is
Z =
{vKx+1, Kx ≤ n− 1,
0, Kx ≥ n.= vKx+1 I(Kx ≤ n− 1).
The APV of the insurance is denoted by
A1x:n = E[vKx+1 I(Kx ≤ n− 1)] =
n−1∑k=0
vk+1kpx qx+k.
The second moment of Z is denoted by
2A1x:n = E[v2(Kx+1)I(Kx ≤ n− 1)] =
n−1∑k=0
v2(k+1)kpx qx+k =
n−1∑k=0
e−2δ(k+1)kpx qx+k
and V ar[Z] = 2A1x:n − (A1
x:n)2.
Recursion formulas for Ax and A1x:n:
Ax = A1x:n + vn npxAx+n
A1x:n = vqx + v pxA
1x+1:n−1
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(c) An n-year pure endowment of 1 on (x) pays 1 at time n only if (x) is still alive at
the end of the n-year term and nothing if (x) dies during the n-year term. The PV
of the benefit is
Z =
{0, Tx ≤ n,
vn, Tx > n.= vn I(Tx > n).
The APV of the insurance is denoted by
nEx = A 1x:n = E[vn I(Tx > n)] = vn npx = e−δn npx.
The second moment of Z is denoted by
2A 1x:n = E[v2n I(Tx > n)] = v2n
npx = e−2δnnpx
and
V ar[Z] = 2A1
x:n − (A 1x:n)2 = v2n
npx nqx = e−2δnnpx nqx.
Note that
tEx = vt tpx = e−δt tpx, t ≥ 0
is also called actuarial discount function and satisfies for any t > 0 and s > 0,
t+sEx = tEx sEx+t.
(d) A continuous n-year endowment insurance of 1 on (x) pays 1 at the moment of death
if (x) dies during the n-year term and 1 at time n if (x) is still alive at the end of
n-year term. The PV of the benefits is
Z =
{vTx , Tx ≤ n
vn, Tx > n= Z1 + Z2,
where Z1 =
{vTx , Tx ≤ n
0, Tx > n= vTx I(Tx ≤ n) and Z2 =
{0, Tx ≤ n
vn, Tx > n= vn I(Tx >
n) are the present values at time 0 of the continuous n-year term life insurance of
1 on (x) and the n-year pure endowment of 1 on (x), respectively. The APV of the
insurance is denoted by
Ax:n = A1x:n + A 1
x:n =
∫ n
0
vttpx µx+tdt+ vn npx.
5
Note that Z1Z2 = 0.
The second moment of Z is denoted by
2Ax:n = E[Z2] = 2A1x:n + 2A
1x:n =
∫ n
0
v2ttpx µx+tdt+ v2n
npx.
and V ar[Z] = 2Ax:n − (Ax:n)2.
Also,
V ar[Z] = V ar[Z1] + V ar[Z2] + 2Cov[Z1, Z2]
= 2A1x:n − (A1
x:n)2 + 2A1
x:n − (A 1x:n)2 − 2 A1
x:n A1
x:n.
A discrete n-year endowment insurance of 1 on (x) pays 1 at the end of the year of
death if (x) dies during the n-year term and 1 at time n if (x) is still alive at the
end of n-year term. The PV of the benefit is
Z =
{vKx+1, Kx ≤ n− 1
vn, Kx ≥ n= Z1 + Z2,
where Z1 =
{vKx+1, Kx ≤ n− 1
0, Kx ≥ n= vKx+1I(Kx ≤ n−1) and Z2 =
{0, Kx ≤ n− 1
vn, Kx ≥ n=
vnI(Kx ≥ n) are the present values at time 0 of the discrete n-year term life insur-
ance of 1 on (x) and the n-year pure endowment of 1 on (x), respectively.
The APV of the insurance is denoted by
Ax:n = A1x:n + A 1
x:n =n−1∑k=0
vk+1kpx qx+k + vn npx.
Note that Z1Z2 = 0.
The second moment of Z is denoted by
2Ax:n = 2A1x:n + 2A
1x:n =
n−1∑k=0
v2(k+1)kpx qx+k + v2n
npx
and V ar[Z] = 2Ax:n − (Ax:n)2.
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Also,
V ar[Z] = V ar[Z1] + V ar[Z2] + 2Cov[Z1, Z2]
= 2A1x:n − (A1
x:n)2 + 2A1
x:n − (A 1x:n)2 − 2A1
x:n A1
x:n.
(e) A continuous n-year deferred life insurance of 1 on (x) pays 1 at the moment of death
if (x) dies after n years and 0 if (x) dies during the n-year deferred period. The PV
of the benefit is
Z =
{0, Tx ≤ n,
vTx , Tx > n.= vTx I(Tx > n).
The APV of the insurance is denoted by
n|Ax = E[vT I(Tx > n)] =
∫ ∞n
vtfx(t)dt =
∫ ∞n
e−δttpx µx+tdt.
The second moment of Z is denoted by
2n|Ax = E[v2TxI(Tx > n)] =
∫ ∞n
v2tfx(t)dt =
∫ ∞n
e−2δttpx µx+tdt
and V ar[Z] = 2n|Ax
− (n|Ax)2.
The relationship among n|Ax, Ax, and A1x:n:
n|Ax = Ax − A1x:n = vnnpx Ax+n.
A discrete n-year deferred life insurance of 1 on (x) pays 1 at the end of the year of
death if (x) dies after n years and 0 if (x) dies during the n-year deferred period.
The PV of the benefit is
Z =
{0, Kx ≤ n− 1,
vKx+1, Kx ≥ n.= vKx+1I(Kx ≥ n).
The APV of the insurance is denoted by
n|Ax =∞∑k=n
vk+1kpx qx+k =
∞∑k=n
e−δ(k+1)kpx qx+k.
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The second moment of Z is denoted by
2n|Ax =
∞∑k=n
v2(k+1)kpx qx+k =
∞∑k=n
e−2δ(k+1)kpx qx+k
and V ar[Z] = 2n|Ax
− (n|Ax)2.
The relationship among n|Ax, Ax, and Ax:n:
n|Ax = Ax − A1x:n = vnnpxAx+n.
2.3 Variable Benefit Life Insurances:
(a) An annually increasing continuous whole life insurance on (x) pays n at the moment
of death if (x) dies in year n, n = 1, 2, .... The APV of this insurance is denoted by
(IA)x =∞∑n=1
∫ n
n−1
nvt tpx µx+tdt and (IA)x =∞∑n=0
n|Ax,
where 0|Ax = Ax.
An annually increasing discrete whole life insurance on (x) pays n at the end of the
year of death if (x) dies in year n, n = 1, 2, .... The PV of the benefits is Z =
(Kx + 1)vKx+1 and the APV of this insurance is denoted by
(IA)x = E[(Kx + 1)vKx+1] =∞∑k=0
(k + 1) vk+1kpx qx+k and (IA)x =
∞∑k=0
k|Ax,
where 0|Ax = Ax.
The review of the calculations of annuities:
an =n−1∑k=0
vk+1 =1− vn
i,
an =n−1∑k=0
vk =1− vn
d,
(Ia)n =n−1∑k=0
(k + 1)vk+1 =an − nvn
i.
(b) A continuously increasing whole life insurance on (x) pays t if (x) dies at time t > 0.
The PV of the benefits is Z = Tx vTx . The APV of this insurance is denoted by
(IA)x = E[Tx vTx ] =
∫ ∞0
tvt tpx µx+tdt and (IA)x =
∫ ∞0
t|Ax dt.
8
2.4 Relationships between continuous and discrete insurances under the UDD
assumption
Ax =i
δAx
A1x:n =
i
δA1x:n
Ax:n =i
δA1x:n + A 1
x:n
(IA)x =i
δ(IA)x
(IA)x =i
δ
[(IA)x −
(1
d− 1
δ
)Ax
].
2.5 Normal approximation to the sum of present values: Let Zi be the present value
random variable for the ith policy holder in an insurance policy, i = 1, ..., n. Then the
sum S = Z1 + · · · + Zn is the total of the n present values. Assume that the n policy
holders have independent lives. For a large n, the distribution of S can be approximated
by the normal distribution
S − E[S]√V ar[S]
∼ N(0, 1),
or equivalently
S ∼ N(E[S], V ar[S]),
where E[S] = E[Z1] + · · ·+ E[Zn] and V ar[S] = V ar[Z1] + · · ·+ V ar[Zn].
9