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2. ANNUITIES: COMPOUND FINANCIAL OPERATIONS

2.1. ANNUITY

2.1.1. Concepts and elements

2.1.2. Financial Value of an Annuity

2.2. COMPOUND FINANCIAL OPERATION

2.2.1. Definition

2.2.2. Financial Equivalence Principle

2.3. PERIODIC ANNUITIES

2.3.1. Concept and Elements

2.3.2. Classification

2.3.3. Financial Values of specific annuities

2.1. ANNUITY

2.1.1. CONCEPT AND ELEMENTS

Annuity refers to multiple Financial capitals (cash flows) associated with different intervals or time periods

nnss tCtCtCtC ,,,,,,,,, 2211

Ci: Terms of the annuity. They are payments (PMTS) or charges made in the

financial Operation.

n: number of PMTS

2.1. ANNUITY

2.1.2. Financial Value of an annuity

THE FINANCIAL VALUE OF AN ANNUITY IS the sum of all cash flows

or PMTS , valued at the same point in time (Compounded or

discounted) based on a particular Financial Law.

n

k

t

kkiCV

1

)1(


2.2.1. Definition

A compound financial Operation is a financial operation in which

more than two financial capitals are exchanged (not simultaneously)

For example: a general case is when two annuities are exchanged.

(P)

(CP)

2.2.2. FINANCE EQUIVALENCE PRINCIPLEIn every financial operation annuities on both sides must be financially equivalent

based on a financial Law. That ´s to say that the financial values of both parts must

be financially equivalent in accordance with the aforementioned Law


In every financial operation the financial value of one part must coincide with the

financial value of the other part in the same point, based on a financial Law.

( , ) ~( , )f

V V V V

2.3. FINANCIAL PERIODIC ANNUITIES

2.1.1. The concept:

We consider a financial annuity to be periodic when the time period that exists

between its terms or cash- flows is always the same.

Therefore, every cash-flow is associated with an interval which lasts the same amount of time.

EXAMPLES OF PERIODIC ANNUITIES

NOTE: The maturity of the annuity ´s cash flows can be any period of time. For instance at

the beginning, middle or the end of the interval.

2.2.2. ELEMENTS OF A PERIODIC ANNUITY

• ORIGIN: inferior end of the interval associated with the expiration of the first cash flow.

• END: superior end of the interval associated with the expiration of the last cash flow.

• DURATION: time period elapsed between the beginning and the end of the annuity.

• C: Annuity´s terms (cash flows) .

• n: number of cash flows of an annuity.

• i: type of interest rate. (Always Compound Interest Law)

2.2.3. CLASSIFICATION OF PERIODIC ANNUITIES

2.2.3. CLASSIFICATION

2.2.3.1. According to the maturity of the cash flows

2.2.3.2. According to the quantity of the cash flows

2.2.3.3. According to the duration of the annuity

2.2.3.4. According to the moment of valuation

2.2.3.1. CLASSIFICATION OF PERIODIC ANNUITIES

1 Classification: According to the maturity of the of cash flows

ORDINARY ANNUITY

The maturity of the quantities is at the end of the period The beginning of the annuity: the period before the maturity of the first PMT

The end of annuity : coincides with the maturity of the last PMT

ANNIUITY DUE:

The maturity of the cash flows is at the beginning of every periodThe beginning of annuity: coincides with the maturity of the first PMT

The end of an annuity is (located )one period after the maturity of the last PMT


2 Classification: According to the Quantity of the PMTS

CONSTANT ANNUITIES: Are those in which the quantity of all cash –flows is the same

GEOMETRIC ANNUITIES (Variable Cash-Flows)

Are those in which the C-FS vary geometrically (the PMTS increase or decrease at a

percentage related to the previous percentage)


3 Classification: According to the Duration of the Annuity

TEMPORARY ANNUITIES.

The number of the terms is finite

PERPETUAL ANNUITIES.

The number of the terms tends toward infinity


4 Classification:According to the moment of annuity valuation

The point of valuation is found between the origin and the end of annuity

The point of valuation is previous to

the origin of the annuity

The point of valuation is after the end of

the annuity

FINANCIAL VALUES OF SOME PARTICULAR ANNUITIES

Some Ideas

•The Financial Law applied to Annuities is Compund Interest Law at a

Constant Interest Rate.

• PRESENT VALUE OR DISCOUNTED VALUE (valuation of the annuity at

origin)

We are calculating the finacial value of the annuity at the beginning. t = 0

Future Value ( Valuation of the annuity at the end)

We are calculating the financial value of the annuity at the end. t=n

(1 )n

n oV V i 0 (1 ) n

nV V i

1º Classification: According to

the maturity of the PMTS

ORDINARY ANNUITY

ANNUITY DUE

FINANCIAL VALUES OF SOME PARTICULAR ANNUITIES


the quantity of the PMTS

CONSTANT

GEOMETRIC

ANNUITIES


the duration of the annuity

TEMPORARY

PERPETUAL


the moment of valuation

OVERVIEW OF AN ORDINARY ANNUITY- FV

1 2 3(1 ) (1 ) (1 )

n n nV C i C i C i Cn

1 (1 )

1 2 3(1 ) (1 ) (1 ) 1

r i

n n nV C i i in

1 1

1(1 ) (1 )

1 (1 )

ni i

V Cni

1

1

(1 ) (1 ) 1

1 (1 )

ni i

Ci

1

(1 ) 1

(1 ) 1 (1 )

ni

V Cni i

(1 ) 1(1 ) 1

ni

Ci

(1 ) 1n

iV Cn i

n iC S

nV

FUTURE VALUE

(1 ) 1n

iC

i

1 .

1

na a rS

r

OVERVIEW OF AN ORDINARY ANNUITY- PV

0

1 2 3(1 ) (1 ) (1 ) (1 )

nV C i C i C i C i

0 (1 )n

V V in

0V

PRESENT VALUE

0(1 ) 1

(1 )n

ni

V C ii

01 (1 )

ni

V Ci

01 (1 )

ni

V Ci

n iC a

(1 )n i n i

nS a i

Property

OVERVIEW OF AN ANNUITY DUE - PV

0 0(1 )V i V

01 (1 )

(1 )n

iV C i

i

PRESENT VALUE

1 20 (1 ) (1 ) (1 )

nV C C i C i C i

Equivalent Ordinary Annuity

n iC a

01 (1 )

(1 )n

iV C i

i

(1 )

(1 )

n i n i

n i n i

nS a i

a a i

Properties

0V

OVERVIEW OF AN ANNUITY DUE - FV

(1 )V i Vn n

(1 ) 1(1 )

ni

V C in i

nV

FUTURE VALUE

1 2(1 ) (1 ) (1 ) (1 )

n n nV C i C i C i C in

Equivalent ordinary annuity

n iC S

(1 ) 1(1 )

ni

V C in i

(1 ) n i n iS S i

Property

FINANCIAL VALUE AFTER THE END

(1 )n i

V C S in

nV

(1 )V V in n

(1 ) 1(1 )

ni

V C in i

(1 )h

V V in nh Financial Value after the end

(1 ) 1(1 )

ni

V C in i

FINACIAL VALUE BEFORE THE BEGINNING

0(1 )

hV V i

h

(1 )n iaV C i

V

(1 )h

V V in nh Financial Value before the beginning

0(1 )V V i

1 (1 )(1 ) (1 )

ni

V C i ii

1 (1 )(1 ) (1 )

ni

V C i ii

OVERVIEW OF A PERPETUAL ORDINARY ANNUITY

0

1 (1 )lim

n

n

iV C

i

0 limn in

V C a

0

1 2 3(1 ) (1 ) (1 ) V C i C i C i

0V

01 (1 )i

V Ci

01V Ci

iC a

PRESENT VALUE

01V Ci

OVERVIEW OF A PERPETUAL ANNUITY DUE

01(1 )V C ii

01 (1 )

lim (1 )n

n

iV C i

i

0 limn in

C aV

0V

01 (1 )

(1 )i

C iVi

01(1 )V C ii

iC a

PRESENT VALUE

01 2 3

(1 ) (1 ) (1 ) V C C i C i C i

Property

(1 )i i

a i a

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