ACTU 304- Notes on Life Insurance

Perpetual Saah Andam

University of Ghana

psandam@ug.edu.gh

1st May, 2020

Endowments Payable at the End

of Year of Death

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Endowments

Definition: An endowment insurance is the combination of a

temporary insurance with a pure endowment payable at the end
of the term of the insurance.

The insurance is taken out at the age of x. A death benefit of

GH ¢1 is payable if the insured dies between the ages of x + m
and x + m + n, and a benefit of GH ¢1 is paid at the age of
x + m + n on survival. The number m is a nonnegative integer
and n is a positive integer.

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Depending on the timing of the payment of the death benefit,
we can distinguish between two types of endowments. One of
them pays the death benefit at the end of the year of death and
the other one pays at the moment of death.

This means that, we have the combination of n-year term

insurance deferred for m years and a pure endowment
insurance payable at the age of x + m + n

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Present Value of the Insurance

If the death benefit is payable at the end of the year of death,

the expected value of the endowment is denoted by m |Ax:n .
The present value of the cash flow can be expressed as h(Kx ),
where,


 0, if k < m





h(k ) = v k +1 , if m ≤ k < m + n






 m+n
v , if m + n ≤ k

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From the definition, we have endowment to be the combination
of n-year term insurance deferred for m years and a pure
endowment insurance payable at the age of x + m + n that is;

1 1
m |Ax:n = m |Ax:n + Ax: m+n
Mx+m − Mx+m+n Dx+m+n
= +
Dx Dx
Mx+m − Mx+m+n + Dx+m+n
=
Dx

Then the variance is,

2
= 2m |Ax:n −


V m |Ax:n m |Ax:n

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If m = 0, then we have the notation Ax:n then it implies that,

1 1
Ax:n = Ax:n + Ax: n
Mx − Mx+n Dx+n
= +
Dx Dx
Mx − Mx+n + Dx+n
=
Dx

Then the variance is,

2
V Ax:n = 2 Ax:n − Ax:n

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If n → ∞ then x + m + n will be so large then the endowment
insurance coincides with a whole life insurance since the
insured will definitely die before reaching the age of x + m + n.
That is, as n → ∞ we have

lim Ax:n = Ax
n→∞

and

lim m |Ax:n = m |Ax .

n→∞

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Endowments Payable at the

Moment of Death

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Endowments Payable at the Moment of Death

If the death benefit is payable at the moment of death, the

expected value of the endowment is denoted by m | Āx:n . Then,
the present value of the cash flow is g(Tx ), where


 0, if t < m





g(t) = v t , if m ≤ t < m + n






 m+n
v , if m + n ≤ t

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From the definition, we have endowment to be the combination
of n-year term insurance payable at the moment of death and a
pure endowment insurance payable at the age of x + m + n that
is;

1 1
m | Āx:n = m | Āx:n + Ax: m+n
M̄x+m − M̄x+m+n Dx+m+n
= +
Dx Dx
M̄x+m − M̄x+m+n + Dx+m+n
=
Dx

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Approximation of Endowment Plan

The approximation of endowment can be obtained as;

1 1
1
m | Āx:n ≈ (1 + i) 2 m |Ax:n + Ax: m+n
1
(1 + i) 2 (Mx+m − Mx+m+n ) Dx+m+n
= +
Dx Dx
1
(1 + i) 2 (Mx+m − Mx+m+n ) + Dx+m+n
=
Dx

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It is important to note that m | Āx:n cannot be approximated by
1

(1 + i) 2 m |Ax:n . For the variance we have;

2
= 2m | Āx:n −


V m | Āx:n m | Āx:n

If m = 0, then we have the notation Āx:n then it implies that,

1 1
Āx:n = Āx:n + Ax: n
M̄x − M̄x+n Dx+n
= +
Dx Dx
M̄x − M̄x+n + Dx+n
=
Dx

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Whose approximation can be obtained as;

1 1
1
Āx:n ≈ (1 + i) 2 Ax:n + Ax: n
1
(1 + i) 2 (Mx − Mx+n ) Dx+n
= +
Dx Dx
1
(1 + i) 2 (Mx − Mx+n ) + Dx+n
=
Dx

For the variance we have;

2
V Āx:n = 2 Āx:n − Āx:n

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If n → ∞ then x + m + n will be so large so we have

lim Āx:n = Āx

n→∞

and

lim m | Āx:n = m |Āx .

n→∞

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