International Journal of Experimental Algorithms (IJEA) Volume: 2 Issue: 1

INTERNATIONAL JOURNAL OF
EXPERIMENTAL ALGORITHMS (IJEA)







VOLUME 2, ISSUE 2, 2011

EDITED BY
DR. NABEEL TAHIR








ISSN (Online): 2180-1282
International Journal of Experimental Algorithms (IJEA) is published both in traditional paper form
and in Internet. This journal is published at the website http://www.cscjournals.org, maintained by
Computer Science Journals (CSC Journals), Malaysia.


IJEA Journal is a part of CSC Publishers
Computer Science Journals
http://www.cscjournals.org

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INTERNATIONAL JOURNAL OF EXPERIMENTAL ALGORITHMS
(IJEA)
Book: Volume 2, Issue 1, May 2011
Publishing Date: 31-05-2011
ISSN (Online): 1985-4129

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CSC Publishers, 2011

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EDITORIAL BOARD


ASSOCIATE EDITORS (AEiCs)


Associate Professor Dursun Delen
Oklahoma State University
United States of America

Professor Nizamettin Aydin
Yildiz Technical University
Turkey

EDITORIAL BOARD MEMBERS (EBMs)


Dr. Doga Gursoy
Graz University of Technology (Austria )

Dr. Kenneth Revett
British University in Egypt (Egypt)

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TABLE OF CONTENTS




Volume 2, Issue 2, May 2011



Pages
27 - 41
Mathematical Derivation of Annuity Interest Rate
and its Application
Karam A. Fayed












International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (2) : 2011

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Karam A. Fayed
Mathematical Derivation of Annuity Interest Rate
and its Application


K.A.Fayed



karamfayed_1@hotmail.com
Ph.D.From Dept. of applied Mathematics and Computing,
Cranfield University, UK.
Faculty of commerce/Dept. of applied Statistics and Computing,
Port Said University, Port Fouad, Egypt.

Abstract

A fundamental task in business for investor or borrower is to know the interest rate of an
annuity. In this type of problem, the size of each periodic payment(R), the term(n), and the
amount(Sn) or the present value of the annuity(An) are usually given.However, a direct
equation representing the Annuity Interest Rate(i) is not available, since an approximate value
of the Annuity Interest Rate is obtained by interpolation methodbased on table showing
(Sn/R) values. This paper emphasizes the real time computational problem for Annuity
interest rate. It has therefore been important to derive an equation for computing the Annuity
Interest rate. The evaluation of error analysis has been discussed. The new algorithm saved
computational energy by approximately 99.9%than that of the tabulated one.

Keywords: Investment Mathematics, Statistical Toolbox, MATLAB Programming.


1. INTRODUCTION
There are many situations in which both businesses and individuals would be faced with
either receiving or paying a constant amount for a length of period. When a firm faces a
stream of constant payments on a bank loan for a period of time, we call that stream of cash
flows an annuity.

An annuity is a series of periodic payments, usually made in equal amounts. The payments
are computed by the compound interest method[1] and are made at equal intervals of time.
Individual investors may make constant payments on their home or car loans, or invest a fixed
amount year after year to save for their retirement. Any financial contract that calls for equally
spaced and level cash flows over a finite number of periods is called an annuity. If the cash
flow payments continue forever, the contract is called perpetuity. Constant cash flows that
occur at the end of each period are called ordinary annuities.

2. THE AMOUNT OF AN ANNUITY
In Business, the amount of an annuity is the final value at the end of the term of the annuity.
To derive the formula for the amount of an ordinary annuity,
let:

R is the size of each regular payment.

i is the interest rate per conversion period.

n is the number of payments during the term of an annuity.

Sn is the amount of an ordinary annuity.
Then:

The amount of an ordinary annuity is given by:









International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (2) : 2011

27

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Karam A. Fayed

Term
Amount of
0
1
2
---------------
(n-2)
(n-1)
n
annuity(Sn)





R R +




R
R(1+i) +



R

R(1+i)2 +
Payment


---------------


--------- +

R


R(1+i)n-2+
R



R(1+i)n-1
i annuity interest rate per conversion


Multiply both sides by (1+i), we have:

Subtracting Eq.(2) from Eq.(1), we get:




3. THE PRESENT VALUE OF AN ANNUITY
The present value of an annuity is the value at the beginning of the term of the annuity. The
present value of an annuity can be derived by the same way to get the following formula:

Where:

An is the present value of an ordinary annuity.

4. ANNUITY INTEREST RATE PER CONVERSION(i)
The annuity equation (Eq.3 or Eq.4)can also be used to the find the interest rate or discount
rate for an annuity.To determine an accurate valueof the Annuity interest rate instead of using
a trial-and-error approach, we need to solve the equation for the unknown value i as follow:

a) When the Amount is Known (Sn)
(1) Two_Term Simplification


To find the annuity interest rate when the amount is known, use the Eq.(3) as
follow:


From eq.(5), the term
can be simplified using binomial theorem, since it can
obtain the binomial series which is valid for any real number
as follow:

The term
can be rewritten in the following form:
By replacing
, we have:

From eq.(5) & the two ith term expansion of eq.(7), We have:




Dividing both sides by i, we get:

International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (2) : 2011

28

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Karam A. Fayed


Therefore:
Equation(8) represents the annuity interest rate equation for computingiafter the two ithterm
expansion.

(2) Three_Term Simplification
From eq.(5) & the threeith term expansion of eq.(7), We have:





Dividing both sides by i, we get:






Solving the above quadratic equation for i, we get:


Simplifying the above equation, we get:


Therefore, equation(9) represents the annuity interest rate equation for computing i after the
threeith term expansion.

b) When the Present Value is Known (An)
(1) Two_Term Simplification:
Using Eq.(4), we can get the following formulae:

From eq.(10) & the two ith term expansion of eq.(7), We have:



International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (2) : 2011

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Karam A. Fayed

Dividing both sides by i, we get:




Therefore, equation(11) represents the annuity interest rate equation for computing i after the
two ith term expansion.

(2) Three_Term Simplification
From eq.(10) & the three ith term expansion of eq.(7), We have:





Dividing both sides by i, we get:





Solving the above quadratic equation for i, we get:


Simplifying the above equation, we get:


Therefore, equation(12) represents the annuity interest rate equation for computing i after the
three ith term expansion.

5. CALCULATION OF ANNUITY INTEREST RATE

a) Tabulated Annuity Interest Rate
(1) Known Amount:

Table_1 includes selection of annuity interest rate used in the investment market. The
ratio
in Table_1 has been computed for given values of conversion period(n) and the
corresponding annuity interest rate. This ratio is used back to extract the annuity interest
rate(i_tabulated) from Tables given in [1].

International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (2) : 2011

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Karam A. Fayed
Time period (n)
i% exact
n=10
n=20
i_tabulated
Sn/R
i_tabulated
Sn/R
0.2
0.249
10.0904816840387
0.248
20.3845990093093
0.4
0.416
10.1819335047275
0.416
20.7785540890338
0.6
0.624
10.2743656882306
0.584
21.1821069182341
0.8
0.872
10.3677885591048
0.868
21.5955054350601
1
1.000
10.4622125411205
1.000
22.0190039947967
1.2
1.247
10.5576481580867
1.133
22.4528635317327
1.4
1.493
10.6541060346834
1.268
22.8973517249426
1.6
1.623
10.7515968972984
1.515
23.3527431680687
1.8
1.868
10.8501315748704
1.758
23.8193195431968
2
2.000
10.9497209997379
2.000
24.2973697989177
2.2
2.244
11.0503762084931
2.238
24.7871903326693
2.4
2.488
11.1521083428429
2.475
25.2890851774580
2.6
2.511
11.2549286504744
2.525
25.8033661930578
2.8
2.756
11.3588484859271
2.764
26.3303532617892
3
3.000
11.4638793114707
3.000
26.8703744889805
3.2
3.242
11.5700326979890
3.233
27.4237664082190
3.4
3.483
11.6773203258690
3.464
27.9908741914986
3.6
3.516
11.7857539858976
3.536
28.5720518643747
3.8
3.758
11.8953455801620
3.769
29.1676625262402
4
4.000
12.0061071229586
4.000
29.7780785758355
4.2
4.037
12.1180507417060
4.084
30.4036819421117
4.4
4.479
12.2311886778653
4.453
31.0448643205664
4.6
4.520
12.3455332878658
4.547
31.7020274151745
4.8
4.953
12.4610970440374
4.895
32.3755831860388
5
5.000
12.5778925355488
5.000
33.0659541028884

TABLE 1: Computing the ratio
and tabulated annuity interest rate

(2) Known Present Value
Similarly, Table_2 computes the ratio
for given values of conversion period(n) and the
corresponding annuity interest rate. This ratio is used back to extract the annuity interest
rate(i_tabulated) from Tables given in [1].























International Journal of Experimental Algorithms (IJEA), Volume (2) : Issue (2) : 2011

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Karam A. Fayed
Time period (n)
i% exact

n=10
n=20
i_tabulated
An/R
i_tabulated
An/R
0.2
0.251
9.89087431187258
0.251
19.5860898344387
0.4
0.332
9.78347474743335
0.331
19.1840839823320
0.6
0.626
9.67776811620015
0.627
18.7935810581347
0.8
0.748
9.57372195913692
0.746
18.4141947010670
1
1.000
9.47130453070169
1.000
18.0455529662705
1.2
1.253
9.37048478137687
1.256
17.6872977422976
1.4
1.373
9.27123234066807
1.372
17.3390841937310
1.6
1.764
9.17351750055746
1.776
17.0005802277864
1.8
1.745
9.07731119939899
1.742
16.6714659838048
2
2.000
8.98258500624224
2.000
16.3514333445971
2.2
2.256
8.88931110557294
2.261
16.0401854686493
2.4
2.513
8.79746228245785
2.524
15.7374363422453
2.6
2.487
8.70701190808258
2.477
15.4429103506104
2.8
2.743
8.61793392567109
2.737
15.1563418672197
3
3.000
8.53020283677584
3.000
14.8774748604555
3.2
3.258
8.44379368792813
3.265
14.6060625168388
3.4
3.518
8.35868205763838
3.532
14.3418668800934
3.6
3.778
8.27484404373630
3.801
14.0846585053389
3.8
4.040
8.19225625104152
4.072
13.8342161277393
4
4.000
8.11089577935504
4.000
13.5903263449677
4.2
3.961
8.03074021176281
3.930
13.3527833128750
4.4
4.522
7.95176760324237
4.539
13.1213884537818
4.6
4.478
7.87395646956413
4.461
12.8959501768371
4.8
5.049
7.79728577647907
5.086
12.6762836099142
5
5.000
7.72173492918482
5.000
12.4622103425400

TABLE 2: Computing the ratio
and tabulated annuity interest rate

b) Simplified Annuity Interest Rate

(1) Known Amount
Table_3 computes annuity interest rate derived in Eq.(9), Eq.(10), Eq.(11), and Eq.(12)
respectively. These Calculations have been computed for given values of
(Table_3_a)
or
(Table_3_b) in addition to different conversion period(n).


Figure_1 shows the tabulated annuity interest rate, the exact annuity interest and the
simplified one against different annuity interest rate. This figure indicates that the simplified
annuity interest rate moves smoothly without any abrupt change or fluctuations. On the other
hand, the tabulated annuity interest rate moves irregularly along with different interest rate.
This variation reverses a wide range of errors associated with the tabulated calculation of
annuity interest rate.














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