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.)

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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

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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!)


(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.


(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.


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.


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

.


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.


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.


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.


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.


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.


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.


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 .


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.


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.


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.


(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


(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.


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.]


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.


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.


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.


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.



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Annuity Formulas

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