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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.


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),

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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.


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).


A1x:n = E[vKx+1 I(Kx ≤ n− 1)] =

n−1∑k=0

vk+1kpx qx+k.


2A1x:n = E[v2(Kx+1)I(Kx ≤ n− 1)] =

n−1∑k=0

v2(k+1)kpx qx+k =

n−1∑k=0


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).


nEx = A 1x:n = E[vn I(Tx > n)] = vn npx = e−δn npx.


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.

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Note that Z1Z2 = 0.


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.


Ax:n = A1x:n + A 1

x:n =n−1∑k=0

vk+1kpx qx+k + vn npx.

Note that Z1Z2 = 0.


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).


n|Ax = E[vT I(Tx > n)] =

∫ ∞n

vtfx(t)dt =

∫ ∞n

e−δttpx µx+tdt.


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).


n|Ax =∞∑k=n

vk+1kpx qx+k =

∞∑k=n

e−δ(k+1)kpx qx+k.

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2n|Ax =

∞∑k=n

v2(k+1)kpx qx+k =

∞∑k=n


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.

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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].

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