annuity formulas
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Contingent Annuity Models
Lecture: Weeks 6-8
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 1 / 26
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Chapter summary
Chapter summary
Life annuities
series of benefits paid contingent upon survival of a given lifesingle life considered
actuarial present values (APV)
actuarial symbols and notation
Forms of annuitiesdiscrete - 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
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(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 = aK+1
where K, in short for Kx, is the curtate future lifetime of(x).
The actuarial present value of the annuity:
ax = E (Y) =E aK+1
=
k=0
ak+1
P(K=k)
=
k=0
ak+1
qk| x =
k=0
ak+1
pk xqx+k
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 3 / 26
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(Discrete) whole life annuity-due Current payment technique
Current payment technique
By using summation by parts, one can show that
ax =
k=0
vk pk x =
k=0
Ek x =
k=0
A 1x: k
.
This is called current payment technique formula for computing life
annuities.
Summation by parts is the discrete analogue of integration by partsformula; but this formula can be proved differently (to be shown inlecture!)
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 4 / 26
(Di ) h l lif i d R l i hi h l lif i
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(Discrete) whole life annuity-due Relationship to whole life insurance
Relationship to whole life insurance
By recalling from interest theory that aK+1
= (1 vK+1)/d, we
have the useful relationship:
ax = (1 Ax) /d.
Alternatively, we write: Ax = 1 dax.
It allows us to directly compute the variance formula:
Var(Y) =Var
vK+1
/d2 =
A2 x(Ax)2
/d2.
We can also use it to derive recursive relationships such as:
ax = 1 + vE
aKx+1
|Kx1
P(Kx1)
= 1 + vpxax+1.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 5 / 26
(Di t ) h l lif it d E l k6 1
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(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 99 100
x 100 70 40 20 4 0
1 Express the present value random variable for a whole life annuity-dueto (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
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Temporary life annuity-due
(Discrete) Temporary life annuity-due
Pays a benefit of a unit $1at the beginning of each year so long as the
annuitant (x) survives, for up to a total ofn years, or n payments.The present value random variable is
Y = a
K+1, K < n
an
, Kn .
The actuarial present value of the annuity:
ax: n = E (Y) =n1
k=0
ak+1
pk xqx+k+ a n pn x.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 7 / 26
Temporary life annuity due continued
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Temporary life annuity-due - continued
- continued
The current payment technique formula for an n-year temporary life
annuity-due is given by:
ax: n =n1k=0
vk pk x.
Recursive formula:a
x: yx = 1 + vpx ax+1: yx1 .
The actuarial accumulated value at the end of the term of an n-yeartemporary life annuity-due is
sx: n =ax: n
En x=
n1k=0
1
Enk x+k,
which is analogous to formulas for sn
.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 8 / 26
Temporary life annuity-due Relationship to endowment life insurance
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Temporary life annuity-due Relationship to endowment life insurance
Relationship to endowment life insurance
Note that Y = (1 Z) /d, where
Z=
vK+1, 0K < nvn, Kn
is the PV r.v. for a unit of endowment insurance, payable at EOY ormaturity.
Thus, it follows that
ax: n =
1Ax: n
/d
and the variance is
Var (Y) =Var (Z) /d2 =
A2 x: n
Ax: n2
/d2.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 9 / 26
Deferred whole life annuity-due
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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.
The PV r.v. is Y =
0, 0K < n
an| K+1n
, Kn .
The APV of the annuity is
an| x = E (Y) = En x ax+n= ax ax: n =
k=n
vk pk x.
The variance ofY is given by
Var (Y) = 2
dv2n pn x
ax+n a
2x+n
+ a2n| x
an| x
2.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 10 / 26
Life annuity-due with guaranteed payments
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y g p y
Life annuity-due with guaranteed payments
Consider a life annuity-due with n-year certain. Its PV r.v. is
Y =
a n , 0K < na
K+1, Kn .
APV of the annuity:
ax: n
= E (Y) = a n qn x+
k=n
aK+1
pk xqx+k
= a n +
k=n
vk pk x= a n + ax ax: 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
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y
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 clearlyY =a
K so that APV is given by
ax= E (Y) =
k=0
a K pk xqx+k =
k=1
vk pk x.
Relationship to life insurance:
Y =
1 vK
/i=
1(1 + i)vK+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
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Example 6.2
Find formulas for the:
expectation; andvariance of the present value random variable for a temporary lifeannuity-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
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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+Jj=0
1
m vj/m
= a
(m)
K+(J+1)/m =
1 vK+(J+1)/m
d(m)
where Jrefers to the number of complete m-ths of a year lived in theyear of death, and is given by J=(T K) m.
For example, if the observed T = 45.86 in a life annuity withquarterly payments, then the observed K= 45 and J= 3.
Here, m= 12 for monthly, m= 4 for quarterly, and m= 2 forsemi-annual.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 14 / 26
Life annuities with m-thly payments - continued
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- continued
The APV of this annuity is
E (Y) = a(m)x = 1
m
h=0
vh/m ph/m x=1 A(m)x
d(m) .
Variance is
Var (Y) =Var
vK+(J+1)/m
d(m)2 = A
2 (m)x
A
(m)x
2
d(m)2 .
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 15 / 26
Life annuities with m-thly payments Important relationships
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Important relationships
Here we list some important relationships regarding the lifeannuity-due withm-thly payments:
1 =dax+ Ax =d(m)a
(m)x + A
(m)x
a(m)x =
d
d(m)ax
1
d(m)
A
(m)x Ax
= a
(m)
1 axa
(m)
A
(m)x Ax
a(m)x =1 A
(m)
x
d(m) = a(m)
a(m)
A(m)x
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
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UDD assumption
If deaths are assumed to have uniform distribution in each year of
age, S is uniform on (0, 1) so that Jis uniform also on the integers{0, 1,...,m 1}.
Then we have the following relationship:
a(m)x = a(m)1 axs(m)
1
1
d(m)
Ax
Interpretation can be given on this formula.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 17 / 26
Life annuities with m-thly payments Alternative expressions
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Alternative expressions
Other expressions for the life annuity-due payable m-thly:
a(m)x =s
(m)
1 a
(m)
1 ax s
(m)
1 1d(m)
a(m)x = (m) ax (m) where
(m) = s
(m)
1 a
(m)
1 =
i
i(m)
d
d(m)
(m) =s(m)
1 1
d(m) =
i i(m)
i(m)d(m)
a(m)x = a
(m)
1 ax (m) Ax
a(m)x = ax
m 1
2m [very widely used]
Use Taylors 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
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(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, withm .
The PV r.v. is Y = aT
where T is the future lifetime of(x).
The APV of the annuity:
ax = E (Y) = E
aT
=
0
aT
pt xx+tdt
=
0
vt pt xdt=
0
Et x dt
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 19 / 26
(Continuous) whole life annuity - continued
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- continued
One can also write expressions for the cdf and pdf ofY in terms of
the cdf and pdf ofT.Recursive relation: ax = ax: 1 + vpxax+1
Variance expression: Var
aT
=
A2 x
Ax
2
/2.
Relationship to whole life insurance: Ax= 1 ax.Check out Example 5.2.1 where we have constant force of mortalityand constant force of interest.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 20 / 26
Temporary life annuity
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Temporary life annuity
A (continuous) n-year temporary life annuity pays 1 per year
continuously while (x) survives during the next n years.
The PV random variable is Y =
a
T, 0T < n
a n , T n .
The APV of the annuity:
ax: n = E (Y) =
n0
at pt xx+tdt=
n0
vt pt xdt.
Recursive formula: ax: yx
= ax: 1
+ vpx ax+1: yx1 .
To derive variance, one way to get explicit form is to note thatY = (1 Z) /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
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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
Y =
0, 0T < nvna
Tn, T n =
0, 0 T < na
Tn a n , T n
.
The APV [expected value ofY] of the annuity is
an| x = En x ax = ax ax: n =
nvt pt xdt.
The variance ofY is given by
Var (Y) =2
v2n pn x
ax+n a
2x+n
an| x
2.
Lecture: Weeks 6-8 (Math 3630) Contingent Annuity Models Fall 2009 - Valdez 22 / 26
Special mortality laws
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Special mortality laws
Just as in the case of life insurance valuation, we can derive niceexplicit forms for life annuity formulas in the case where mortality
follows:
constant force (or Exponential distribution); or
De Moivres 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 therelationships 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
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Whole life annuity with guaranteed payments
Consider a (continuous) whole life annuity with a guarantee ofpayments for the first n years. Its PV r.v. is
Y =
a n , 0T na
T, T > n
.
APV of the annuity:
ax: n
= a n qn x+
n
at pt xx+tdt
= a n +
n
vt pt xdt= a n + ax ax: 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
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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 thevariation 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: thisworks 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
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