Annuity Formulas
Content
Contingent Annuity Models
Lecture: Weeks 6-8
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
1 / 26
Chapter summary
Chapter summary
Life annuities
series of benefits paid contingent upon survival of a given life
single life considered
actuarial present values (APV)
actuarial symbols and notation
Forms of annuities
discrete - due or immediate
payable more frequently than once a year
continuous
“Current payment techniques” APV formulas
Chapter 6 of MQR (Cunningham, et al.)
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
2 / 26
(Discrete) whole life annuity-due
(Discrete) whole life annuity-due
Pays a benefit of a unit $1 at the beginning of each year that the
annuitant (x) survives.
The present value random variable is
Y =a
¨ K+1
where K, in short for Kx , is the curtate future lifetime of (x).
The actuarial present value of the annuity:
∞
X
a
¨ k+1 P (K = k)
a
¨x = E (Y ) = E a
¨ K+1 =
k=0
=
∞
X
k=0
Lecture: Weeks 6-8 (Math 3630)
a
¨ k+1 · k|qx =
∞
X
a
¨ k+1 · kpx qx+k
k=0
Contingent Annuity Models
Fall 2009 - Valdez
3 / 26
(Discrete) whole life annuity-due
Current payment technique
Current payment technique
By using summation by parts, one can show that
a
¨x =
∞
X
k=0
v k kpx =
∞
X
k Ex =
k=0
∞
X
Ax: 1k .
k=0
This is called current payment technique formula for computing life
annuities.
Summation by parts is the discrete analogue of integration by parts
formula; but this formula can be proved differently (to be shown in
lecture!)
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
4 / 26
Relationship to whole life insurance
(Discrete) whole life annuity-due
Relationship to whole life insurance
By recalling from interest theory that a
¨ K+1 = (1 − v K+1 )/d, we
have the useful relationship:
a
¨x = (1 − Ax ) /d.
Alternatively, we write: Ax = 1 − d¨
ax .
It allows us to directly compute the variance formula:
Var(Y ) = Var v K+1 /d2 = 2Ax − (Ax )2 /d2 .
We can also use it to derive recursive relationships such as:
a
¨x = 1 + vE a
¨ K +1 |Kx ≥ 1 P (Kx ≥ 1)
x
= 1 + vpx a
¨x+1 .
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
5 / 26
(Discrete) whole life annuity-due
Example wk6.1
Example wk6.1
This problem is somewhat a variation of Examples 6.1 and 6.3 of
Cunnigham et al.
Suppose you are interested in valuing a whole life annuity-due issued to
(95). You are given:
i = 0.5%; and
the following extract from a life table:
x
95 96 97 98
`x 100 70 40 20
99
4
100
0
1
Express the present value random variable for a whole life annuity-due
to (95).
2
Calculate the expected value of this random variable.
3
Calculate the variance of this random variable.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
6 / 26
Temporary life annuity-due
(Discrete) Temporary life annuity-due
Pays a benefit of a unit $1 at the beginning of each year so long as the
annuitant (x) survives, for up to a total of n years, or n payments.
The present value random variable is
(
a
¨ K+1 , K < n
.
Y =
a
¨n ,
K≥n
The actuarial present value of the annuity:
a
¨x: n = E (Y ) =
n−1
X
a
¨ k+1 kpx qx+k + a
¨ n npx .
k=0
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
7 / 26
Temporary life annuity-due
- continued
- continued
The current payment technique formula for an n-year temporary life
annuity-due is given by:
a
¨x: n =
n−1
X
v k kpx .
k=0
Recursive formula:
¨x+1: y−x−1 .
a
¨x: y−x = 1 + vpx a
The actuarial accumulated value at the end of the term of an n-year
temporary life annuity-due is
s¨x: n =
n−1
X
a
¨x: n
1
=
,
n Ex
n−k Ex+k
k=0
which is analogous to formulas for s¨ n .
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
8 / 26
Temporary life annuity-due
Relationship to endowment life insurance
Relationship to endowment life insurance
Note that Y = (1 − Z) /d, where
K+1
v
, 0≤K<n
Z=
n
v ,
K≥n
is the PV r.v. for a unit of endowment insurance, payable at EOY or
maturity.
Thus, it follows that
a
¨x: n = 1 − Ax: n /d
and the variance is
Var (Y ) = Var (Z) /d2 =
Lecture: Weeks 6-8 (Math 3630)
2
Ax: n − Ax: n
Contingent Annuity Models
2
/d2 .
Fall 2009 - Valdez
9 / 26
Deferred whole life annuity-due
Deferred whole life annuity-due
Pays a benefit of a unit $1 at the beginning of each year while the
annuitant (x) survives from x + n onward.
(
0,
0≤K<n
.
The PV r.v. is Y =
,
K
≥n
a
¨
n| K+1−n
The APV of the annuity is
¨x
n|a
= E (Y ) = n Ex a
¨x+n = a
¨x − a
¨x: n =
∞
X
v k kpx .
k=n
The variance of Y is given by
Var (Y ) =
Lecture: Weeks 6-8 (Math 3630)
2
2 2n
v npx a
¨x+n − 2a
¨x+n + n|2a
¨x − n|a
¨x .
d
Contingent Annuity Models
Fall 2009 - Valdez
10 / 26
Life annuity-due with guaranteed payments
Life annuity-due with guaranteed payments
Consider a life annuity-due with n-year certain. Its PV r.v. is
(
a
¨n ,
0≤K<n
Y =
.
a
¨ K+1 , K ≥ n
APV of the annuity:
a
¨x: n
=
E (Y ) = a
¨ n · n qx +
∞
X
a
¨ K+1 kpx qx+k
k=n
= a
¨n +
∞
X
v k kpx = a
¨n + a
¨x − a
¨x: n
k=n
Guaranteed annuity plus a deferred life annuity!
Exercise: try to find expression for the variance of the PV r.v.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
11 / 26
Whole life annuity-immediate
Whoe life annuity-immediate
Procedures above for annuity-due can be adapted for
annuity-immediate.
Consider the whole life annuity-immediate, the PV r.v. is clearly
Y = a K so that APV is given by
ax = E (Y ) =
∞
X
a K · kpx qx+k =
k=0
∞
X
v k kpx .
k=1
Relationship to life insurance:
Y = 1 − v K /i = 1 − (1 + i)v K+1 /i
leads to 1 = iax + (1 + i)Ax .
Interpretation of this equation - to be discussed in class.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
12 / 26
Example wk6.2
Example 6.2
Find formulas for the:
expectation; and
variance of the present value random variable for a temporary life
annuity-immediate.
See pages also for details of the derivations.
Details to be discussed in lecture.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
13 / 26
Life annuities with m-thly payments
Life annuities with m-thly payments
In practice, life annuities are often payable on a monthly, quarterly, or
semi-annual basis.
Consider a life annuity-due with payments made on an m-thly basis:
PV r.v. is
Y =
mK+J
X
j=0
1 − v K+(J+1)/m
1 j/m
(m)
v
=a
¨
=
K+(J+1)/m
m
d(m)
where J refers to the number of complete m-ths of a year lived in the
year of death, and is given by J = b(T − K) mc.
For example, if the observed T = 45.86 in a life annuity with
quarterly payments, then the observed K = 45 and J = 3.
Here, m = 12 for monthly, m = 4 for quarterly, and m = 2 for
semi-annual.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
14 / 26
Life annuities with m-thly payments
- continued
- continued
The APV of this annuity is
∞
E (Y ) = a
¨(m)
=
x
(m)
1 X h/m
1 − Ax
v
· h/mpx =
m
d(m)
h=0
.
Variance is
2 (m)
(m) 2
A x − Ax
Var v K+(J+1)/m
Var (Y ) =
=
.
2
2
d(m)
d(m)
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
15 / 26
Life annuities with m-thly payments
Important relationships
Important relationships
Here we list some important relationships regarding the life
annuity-due with m-thly payments:
(m)
(m)
¨x + Ax
1 = d¨
ax + Ax = d(m) a
d
1 (m)
(m)
(m)
(m)
(m)
¨1 a
a
¨x = (m) a
¨x − (m) Ax − Ax = a
¨x − a
¨ ∞ Ax − Ax
d
d
(m)
(m)
a
¨x
=
1 − Ax
d(m)
(m)
(m)
(m)
=a
¨∞ − a
¨ ∞ Ax
We discuss in lecture and give interpretations on these formulas.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
16 / 26
Life annuities with m-thly payments
UDD assumption
UDD assumption
If deaths are assumed to have uniform distribution in each year of
age, S is uniform on (0, 1) so that J is uniform also on the integers
{0, 1, ..., m − 1}.
Then we have the following relationship:
(m)
s
−
1
(m)
a
¨(m)
=a
¨1 a
¨x − 1 (m) Ax
x
d
Interpretation can be given on this formula.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
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Alternative expressions
Life annuities with m-thly payments
Alternative expressions
Other expressions for the life annuity-due payable m-thly:
(m)
(m) (m)
s1 a
¨1 a
¨x
s1
−1
(m)
a
¨x
=
(m)
a
¨x
= α (m) a
¨x − β (m) where
−
d(m)
(m) (m)
α (m)
= s1 a
¨1
β (m)
=
(m)
(m)
=a
¨1 a
¨x − β (m) Ax
(m)
=a
¨x −
a
¨x
a
¨x
s1
=
i
d
·
i(m) d(m)
=
i − i(m)
i(m) d(m)
−1
d(m)
(m)
m−1
[very widely used]
2m
Use Taylor’s series expansion to derive some of these formulas.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
18 / 26
(Continuous) whole life annuity
(Continuous) whole life annuity
A life annuity payable continuously at the rate of one unit per year.
One can think of it as life annuity payable m-thly per year, with
m → ∞.
The PV r.v. is Y = a
¯ T where T is the future lifetime of (x).
The APV of the annuity:
a
¯x =
E (Y ) = E a
¯T
Z ∞
Z
t
=
v tpx dt =
0
Lecture: Weeks 6-8 (Math 3630)
Z
∞
=
0
∞
a
¯ T · tpx µx+t dt
t Ex dt
0
Contingent Annuity Models
Fall 2009 - Valdez
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(Continuous) whole life annuity
- continued
- continued
One can also write expressions for the cdf and pdf of Y in terms of
the cdf and pdf of T .
Recursive relation: a
¯x = a
¯x: 1 + vpx a
¯x+1
h
2 i 2
Variance expression: Var a
¯ T = 2A¯x − A¯x
/δ .
Relationship to whole life insurance: A¯x = 1 − δ¯
ax .
Check out Example 5.2.1 where we have constant force of mortality
and constant force of interest.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
20 / 26
Temporary life annuity
Temporary life annuity
A (continuous) n-year temporary life annuity pays 1 per year
continuously while (x) survives during the next n years.
(
a
¯T , 0 ≤ T < n
.
The PV random variable is Y =
a
¯n , T ≥ n
The APV of the annuity:
Z
a
¯x: n = E (Y ) =
0
n
Z
a
¯ t · tpx µx+t dt =
n
v t tpx dt.
0
Recursive formula: a
¯x: y−x = a
¯x: 1 + vpx a
¯x+1: y−x−1 .
To derive variance, one way to get explicit form is to note that
Y = (1 − Z) /δ where Z is the PV r.v. for an n-year endowment ins.
[details to be provided in class.]
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
21 / 26
Deferred whole life annuity
Deferred whole life annuity
Pays a benefit of a unit $1 each year continuously while the annuitant
(x) survives from x + n onward.
The PV r.v. is
(
(
0,
0≤T <n
0,
0≤T <n
Y =
=
.
n
v a
¯ T −n , T ≥ n
a
¯ T −n − a
¯n , T ≥ n
The APV [expected value of Y ] of the annuity is
Z ∞
¯ x = n Ex a
¯x = a
¯x − a
¯x: n =
v t tpx dt.
n| a
n
The variance of Y is given by
2
2
Var (Y ) = v 2n npx a
¯x+n − 2a
¯x+n − n|a
¯x .
δ
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
22 / 26
Special mortality laws
Special mortality laws
Just as in the case of life insurance valuation, we can derive nice
explicit forms for “life annuity” formulas in the case where mortality
follows:
constant force (or Exponential distribution); or
De Moivre’s law (or Uniform distribution).
Try deriving some of these formulas. You can approach them in a
couple of ways:
Know the results for the “life insurance” case, and then use the
relationships between annuities and insurances.
You can always derive it from first principles, usually working with the
current payment technique.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
23 / 26
Whole life annuity with guaranteed payments
Whole life annuity with guaranteed payments
Consider a (continuous) whole life annuity with a guarantee of
payments for the first n years. Its PV r.v. is
(
a
¯n , 0 ≤ T ≤ n
Y =
.
a
¯T , T > n
APV of the annuity:
a
¯x: n
Z ∞
= a
¯ n · n qx +
a
¯ t · tpx µx+t dt
n
Z ∞
= a
¯n +
v t tpx dt = a
¯n + a
¯x − a
¯x: n .
n
Guaranteed annuity plus a deferred life annuity!
Exercise: try to find expression for the variance of the PV r.v.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
24 / 26
Life annuities with varying benefits
Life annuities with varying benefits
Some of these are discussed in details in Section 6.5 of MQR.
You may try to remember to special symbols used, especially if the
variation is a fixed unit of $1 (either increasing or decreasing).
The most important thing to remember is to apply similar concept of
“discounting with life” taught in the life insurance case (note: this
works only for valuing actuarial present values):
work with drawing the benefit payments as a function of time; and
use then your intuition to derive the desired results.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
25 / 26
Miscellaneous Examples
Section 6.6 - miscellaneous examples
It is advisable to study the following illustrative examples also given in the
MQR textbook by Cunningham, et al.
Example 6.12;
Example 6.13;
Example 6.14; and
Example 6.15.
Time permitting, we will discuss some of these in class.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
26 / 26
Lecture: Weeks 6-8
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
1 / 26
Chapter summary
Chapter summary
Life annuities
series of benefits paid contingent upon survival of a given life
single life considered
actuarial present values (APV)
actuarial symbols and notation
Forms of annuities
discrete - due or immediate
payable more frequently than once a year
continuous
“Current payment techniques” APV formulas
Chapter 6 of MQR (Cunningham, et al.)
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
2 / 26
(Discrete) whole life annuity-due
(Discrete) whole life annuity-due
Pays a benefit of a unit $1 at the beginning of each year that the
annuitant (x) survives.
The present value random variable is
Y =a
¨ K+1
where K, in short for Kx , is the curtate future lifetime of (x).
The actuarial present value of the annuity:
∞
X
a
¨ k+1 P (K = k)
a
¨x = E (Y ) = E a
¨ K+1 =
k=0
=
∞
X
k=0
Lecture: Weeks 6-8 (Math 3630)
a
¨ k+1 · k|qx =
∞
X
a
¨ k+1 · kpx qx+k
k=0
Contingent Annuity Models
Fall 2009 - Valdez
3 / 26
(Discrete) whole life annuity-due
Current payment technique
Current payment technique
By using summation by parts, one can show that
a
¨x =
∞
X
k=0
v k kpx =
∞
X
k Ex =
k=0
∞
X
Ax: 1k .
k=0
This is called current payment technique formula for computing life
annuities.
Summation by parts is the discrete analogue of integration by parts
formula; but this formula can be proved differently (to be shown in
lecture!)
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
4 / 26
Relationship to whole life insurance
(Discrete) whole life annuity-due
Relationship to whole life insurance
By recalling from interest theory that a
¨ K+1 = (1 − v K+1 )/d, we
have the useful relationship:
a
¨x = (1 − Ax ) /d.
Alternatively, we write: Ax = 1 − d¨
ax .
It allows us to directly compute the variance formula:
Var(Y ) = Var v K+1 /d2 = 2Ax − (Ax )2 /d2 .
We can also use it to derive recursive relationships such as:
a
¨x = 1 + vE a
¨ K +1 |Kx ≥ 1 P (Kx ≥ 1)
x
= 1 + vpx a
¨x+1 .
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
5 / 26
(Discrete) whole life annuity-due
Example wk6.1
Example wk6.1
This problem is somewhat a variation of Examples 6.1 and 6.3 of
Cunnigham et al.
Suppose you are interested in valuing a whole life annuity-due issued to
(95). You are given:
i = 0.5%; and
the following extract from a life table:
x
95 96 97 98
`x 100 70 40 20
99
4
100
0
1
Express the present value random variable for a whole life annuity-due
to (95).
2
Calculate the expected value of this random variable.
3
Calculate the variance of this random variable.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
6 / 26
Temporary life annuity-due
(Discrete) Temporary life annuity-due
Pays a benefit of a unit $1 at the beginning of each year so long as the
annuitant (x) survives, for up to a total of n years, or n payments.
The present value random variable is
(
a
¨ K+1 , K < n
.
Y =
a
¨n ,
K≥n
The actuarial present value of the annuity:
a
¨x: n = E (Y ) =
n−1
X
a
¨ k+1 kpx qx+k + a
¨ n npx .
k=0
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
7 / 26
Temporary life annuity-due
- continued
- continued
The current payment technique formula for an n-year temporary life
annuity-due is given by:
a
¨x: n =
n−1
X
v k kpx .
k=0
Recursive formula:
¨x+1: y−x−1 .
a
¨x: y−x = 1 + vpx a
The actuarial accumulated value at the end of the term of an n-year
temporary life annuity-due is
s¨x: n =
n−1
X
a
¨x: n
1
=
,
n Ex
n−k Ex+k
k=0
which is analogous to formulas for s¨ n .
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
8 / 26
Temporary life annuity-due
Relationship to endowment life insurance
Relationship to endowment life insurance
Note that Y = (1 − Z) /d, where
K+1
v
, 0≤K<n
Z=
n
v ,
K≥n
is the PV r.v. for a unit of endowment insurance, payable at EOY or
maturity.
Thus, it follows that
a
¨x: n = 1 − Ax: n /d
and the variance is
Var (Y ) = Var (Z) /d2 =
Lecture: Weeks 6-8 (Math 3630)
2
Ax: n − Ax: n
Contingent Annuity Models
2
/d2 .
Fall 2009 - Valdez
9 / 26
Deferred whole life annuity-due
Deferred whole life annuity-due
Pays a benefit of a unit $1 at the beginning of each year while the
annuitant (x) survives from x + n onward.
(
0,
0≤K<n
.
The PV r.v. is Y =
,
K
≥n
a
¨
n| K+1−n
The APV of the annuity is
¨x
n|a
= E (Y ) = n Ex a
¨x+n = a
¨x − a
¨x: n =
∞
X
v k kpx .
k=n
The variance of Y is given by
Var (Y ) =
Lecture: Weeks 6-8 (Math 3630)
2
2 2n
v npx a
¨x+n − 2a
¨x+n + n|2a
¨x − n|a
¨x .
d
Contingent Annuity Models
Fall 2009 - Valdez
10 / 26
Life annuity-due with guaranteed payments
Life annuity-due with guaranteed payments
Consider a life annuity-due with n-year certain. Its PV r.v. is
(
a
¨n ,
0≤K<n
Y =
.
a
¨ K+1 , K ≥ n
APV of the annuity:
a
¨x: n
=
E (Y ) = a
¨ n · n qx +
∞
X
a
¨ K+1 kpx qx+k
k=n
= a
¨n +
∞
X
v k kpx = a
¨n + a
¨x − a
¨x: n
k=n
Guaranteed annuity plus a deferred life annuity!
Exercise: try to find expression for the variance of the PV r.v.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
11 / 26
Whole life annuity-immediate
Whoe life annuity-immediate
Procedures above for annuity-due can be adapted for
annuity-immediate.
Consider the whole life annuity-immediate, the PV r.v. is clearly
Y = a K so that APV is given by
ax = E (Y ) =
∞
X
a K · kpx qx+k =
k=0
∞
X
v k kpx .
k=1
Relationship to life insurance:
Y = 1 − v K /i = 1 − (1 + i)v K+1 /i
leads to 1 = iax + (1 + i)Ax .
Interpretation of this equation - to be discussed in class.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
12 / 26
Example wk6.2
Example 6.2
Find formulas for the:
expectation; and
variance of the present value random variable for a temporary life
annuity-immediate.
See pages also for details of the derivations.
Details to be discussed in lecture.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
13 / 26
Life annuities with m-thly payments
Life annuities with m-thly payments
In practice, life annuities are often payable on a monthly, quarterly, or
semi-annual basis.
Consider a life annuity-due with payments made on an m-thly basis:
PV r.v. is
Y =
mK+J
X
j=0
1 − v K+(J+1)/m
1 j/m
(m)
v
=a
¨
=
K+(J+1)/m
m
d(m)
where J refers to the number of complete m-ths of a year lived in the
year of death, and is given by J = b(T − K) mc.
For example, if the observed T = 45.86 in a life annuity with
quarterly payments, then the observed K = 45 and J = 3.
Here, m = 12 for monthly, m = 4 for quarterly, and m = 2 for
semi-annual.
Lecture: Weeks 6-8 (Math 3630)
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Life annuities with m-thly payments
- continued
- continued
The APV of this annuity is
∞
E (Y ) = a
¨(m)
=
x
(m)
1 X h/m
1 − Ax
v
· h/mpx =
m
d(m)
h=0
.
Variance is
2 (m)
(m) 2
A x − Ax
Var v K+(J+1)/m
Var (Y ) =
=
.
2
2
d(m)
d(m)
Lecture: Weeks 6-8 (Math 3630)
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15 / 26
Life annuities with m-thly payments
Important relationships
Important relationships
Here we list some important relationships regarding the life
annuity-due with m-thly payments:
(m)
(m)
¨x + Ax
1 = d¨
ax + Ax = d(m) a
d
1 (m)
(m)
(m)
(m)
(m)
¨1 a
a
¨x = (m) a
¨x − (m) Ax − Ax = a
¨x − a
¨ ∞ Ax − Ax
d
d
(m)
(m)
a
¨x
=
1 − Ax
d(m)
(m)
(m)
(m)
=a
¨∞ − a
¨ ∞ Ax
We discuss in lecture and give interpretations on these formulas.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
16 / 26
Life annuities with m-thly payments
UDD assumption
UDD assumption
If deaths are assumed to have uniform distribution in each year of
age, S is uniform on (0, 1) so that J is uniform also on the integers
{0, 1, ..., m − 1}.
Then we have the following relationship:
(m)
s
−
1
(m)
a
¨(m)
=a
¨1 a
¨x − 1 (m) Ax
x
d
Interpretation can be given on this formula.
Lecture: Weeks 6-8 (Math 3630)
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Alternative expressions
Life annuities with m-thly payments
Alternative expressions
Other expressions for the life annuity-due payable m-thly:
(m)
(m) (m)
s1 a
¨1 a
¨x
s1
−1
(m)
a
¨x
=
(m)
a
¨x
= α (m) a
¨x − β (m) where
−
d(m)
(m) (m)
α (m)
= s1 a
¨1
β (m)
=
(m)
(m)
=a
¨1 a
¨x − β (m) Ax
(m)
=a
¨x −
a
¨x
a
¨x
s1
=
i
d
·
i(m) d(m)
=
i − i(m)
i(m) d(m)
−1
d(m)
(m)
m−1
[very widely used]
2m
Use Taylor’s series expansion to derive some of these formulas.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
18 / 26
(Continuous) whole life annuity
(Continuous) whole life annuity
A life annuity payable continuously at the rate of one unit per year.
One can think of it as life annuity payable m-thly per year, with
m → ∞.
The PV r.v. is Y = a
¯ T where T is the future lifetime of (x).
The APV of the annuity:
a
¯x =
E (Y ) = E a
¯T
Z ∞
Z
t
=
v tpx dt =
0
Lecture: Weeks 6-8 (Math 3630)
Z
∞
=
0
∞
a
¯ T · tpx µx+t dt
t Ex dt
0
Contingent Annuity Models
Fall 2009 - Valdez
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(Continuous) whole life annuity
- continued
- continued
One can also write expressions for the cdf and pdf of Y in terms of
the cdf and pdf of T .
Recursive relation: a
¯x = a
¯x: 1 + vpx a
¯x+1
h
2 i 2
Variance expression: Var a
¯ T = 2A¯x − A¯x
/δ .
Relationship to whole life insurance: A¯x = 1 − δ¯
ax .
Check out Example 5.2.1 where we have constant force of mortality
and constant force of interest.
Lecture: Weeks 6-8 (Math 3630)
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Temporary life annuity
Temporary life annuity
A (continuous) n-year temporary life annuity pays 1 per year
continuously while (x) survives during the next n years.
(
a
¯T , 0 ≤ T < n
.
The PV random variable is Y =
a
¯n , T ≥ n
The APV of the annuity:
Z
a
¯x: n = E (Y ) =
0
n
Z
a
¯ t · tpx µx+t dt =
n
v t tpx dt.
0
Recursive formula: a
¯x: y−x = a
¯x: 1 + vpx a
¯x+1: y−x−1 .
To derive variance, one way to get explicit form is to note that
Y = (1 − Z) /δ where Z is the PV r.v. for an n-year endowment ins.
[details to be provided in class.]
Lecture: Weeks 6-8 (Math 3630)
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21 / 26
Deferred whole life annuity
Deferred whole life annuity
Pays a benefit of a unit $1 each year continuously while the annuitant
(x) survives from x + n onward.
The PV r.v. is
(
(
0,
0≤T <n
0,
0≤T <n
Y =
=
.
n
v a
¯ T −n , T ≥ n
a
¯ T −n − a
¯n , T ≥ n
The APV [expected value of Y ] of the annuity is
Z ∞
¯ x = n Ex a
¯x = a
¯x − a
¯x: n =
v t tpx dt.
n| a
n
The variance of Y is given by
2
2
Var (Y ) = v 2n npx a
¯x+n − 2a
¯x+n − n|a
¯x .
δ
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
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Special mortality laws
Special mortality laws
Just as in the case of life insurance valuation, we can derive nice
explicit forms for “life annuity” formulas in the case where mortality
follows:
constant force (or Exponential distribution); or
De Moivre’s law (or Uniform distribution).
Try deriving some of these formulas. You can approach them in a
couple of ways:
Know the results for the “life insurance” case, and then use the
relationships between annuities and insurances.
You can always derive it from first principles, usually working with the
current payment technique.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
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23 / 26
Whole life annuity with guaranteed payments
Whole life annuity with guaranteed payments
Consider a (continuous) whole life annuity with a guarantee of
payments for the first n years. Its PV r.v. is
(
a
¯n , 0 ≤ T ≤ n
Y =
.
a
¯T , T > n
APV of the annuity:
a
¯x: n
Z ∞
= a
¯ n · n qx +
a
¯ t · tpx µx+t dt
n
Z ∞
= a
¯n +
v t tpx dt = a
¯n + a
¯x − a
¯x: n .
n
Guaranteed annuity plus a deferred life annuity!
Exercise: try to find expression for the variance of the PV r.v.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
24 / 26
Life annuities with varying benefits
Life annuities with varying benefits
Some of these are discussed in details in Section 6.5 of MQR.
You may try to remember to special symbols used, especially if the
variation is a fixed unit of $1 (either increasing or decreasing).
The most important thing to remember is to apply similar concept of
“discounting with life” taught in the life insurance case (note: this
works only for valuing actuarial present values):
work with drawing the benefit payments as a function of time; and
use then your intuition to derive the desired results.
Lecture: Weeks 6-8 (Math 3630)
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Miscellaneous Examples
Section 6.6 - miscellaneous examples
It is advisable to study the following illustrative examples also given in the
MQR textbook by Cunningham, et al.
Example 6.12;
Example 6.13;
Example 6.14; and
Example 6.15.
Time permitting, we will discuss some of these in class.
Lecture: Weeks 6-8 (Math 3630)
Contingent Annuity Models
Fall 2009 - Valdez
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