RATIO MATHEMATICA 27 (2014) 81-90 ISSN:1592-7415

Some applications of linear difference
equations in finance

with wolfram|alpha and maple

Dana Ř́ıhová, Lenka Viskotová
Department of Statistics and Operation Analysis,

Faculty of Business and Economics,

Mendel University in Brno, Zemědělská 1, 613 00 Brno

dana.rihova@mendelu.cz, lenka.viskotova@mendelu.cz

Abstract

The principle objective of this paper is to show how linear differ-
ence equations can be applied to solve some issues of financial mathe-
matics. We focus on the area of compound interest and annuities. In
both cases we determine appropriate recursive rules, which constitute
the first order linear difference equations with constant coefficients,
and derive formulas required for calculating examples. Finally, we
present possibilities of application of two selected computer algebra
systems Wolfram|Alpha and Maple in this mathematical area.

Key words: linear difference equation, compound interest, future
value of an annuity, periodic payment, computer algebra systems.

2000 AMS: 39A06, 39-04.

1 Introduction

The values of most economic variables are given as a sequence of values
observed at discrete time intervals or periods. These sequences are often
specified by recursion with some initial elements. But it is preferable to
know a rule in the form of an equation for the n-th element to calculate the
values of sequence elements. The recursive rule of a sequence represents a
difference equation and the functional notation for the n-th element can be
obtained by solving this difference equation (see [7]).

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D. Ř́ıhová, L. Viskotová

Many formulas used in financial mathematics can be derived from the
recursive rules between two consecutive elements which constitute difference
equations of the first order. This includes for example simple and compound
interest calculation, the present and future value of an annuity and loan
amortization.

2 Compound Interest

2.1 Derivation of Formula

A sum of money deposited in a bank earns interest which is added to
the principal at regular intervals and the new amount is used for calculating
the interest for the next conversion period. We shall develop a formula for
the total amount of money that is accumulated by a given principal after a
certain number of conversion periods, see also [3], [4].

Let r stand for the annual interest rate and k denote the number of
conversion periods in a year. Let n be equal to the number of conversion
periods in the term of the deposit. Let yn represent the amount on deposit
at the end of n conversion periods and P the initial sum deposited (i.e.
principal). We obtain the following recursive rule

yn+1 = yn +
r

k
yn =

(
1 +

r

k

)
yn, n = 0, 1, 2, . . .

with y0 = P , where the fraction
r
k

stands for the interest rate per conversion
period and r

k
yn is the interest generated during (n+1)st period. The previous

formula represents the first order homogeneous linear difference equation with
constant coefficients

yn+1 −
(

1 +
r

k

)
yn = 0 (1)

with initial condition

y0 = P. (2)

The above problem (1), (2) can be solved by using the properties of a geo-
metric sequence, see [1] and [8]. But our approach will be different due to the
use of a difference equation. The characteristic equation of (1) takes form

z −
(

1 +
r

k

)
= 0

with real root

z = 1 +
r

k
.

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Some applications of linear difference equations in finance

According to [5], the general solution is a geometric sequence

yn = C
(

1 +
r

k

)n
, C ∈ R.

A constant C can be specified from the initial condition (2) for period n = 0,
hence

C = P.

Thus the general solution is given by

yn = P
(

1 +
r

k

)n
(3)

and represents the compound interest formula. This formula gives the amount
yn into which principal P grows when it earns compound interest for n con-
version periods at an interest rate of r

k
per conversion period.

2.2 Illustrative Examples

Example 2.1. An amount of EUR 1, 000 is deposited into a savings account
at an annual interest rate of 2.5%, compounded yearly. What will the value
of the account be worth after 20 years?

To find the amount we use formula (3). We have principal P = 1, 000, annual
interest rate r = 0.025, number of conversion periods per year k = 1 and
total number of conversion periods n = 20. After plugging those figures into
the formula, we get

y20 = 1000

(
1 +

0.025

1

)20
.
= 1638.62

Example 2.2. Find the number of years required for a given sum of money
to double itself if the interest rate is 3%, compounded quarterly.

Substituting yn = 2P in the compound interest formula (3), we have

2P = P
(

1 +
r

k

)n
which implies

2 =
(

1 +
r

k

)n
. (4)

Taking natural logarithms on both sides and using properties of logarithms
gives

n =
ln 2

ln
(
1 + r

k

).
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D. Ř́ıhová, L. Viskotová

To calculate the number of years N we have to divide the total number of
conversion periods n by their number in a year k

N =
1

k

ln 2

ln
(
1 + r

k

). (5)
Setting r = 0.03, k = 4 we get the required number of years

N =
1

4

ln 2

ln
(
1 + 0.03

4

) .= 23.19
3 Future Value of an Annuity

3.1 Derivation of Formula

An annuity is essentially a sequence of periodic payments, usually equal
in amount, payable at equal intervals of time over the course of a fixed time
period. The future value of an annuity is the total value of its periodic
payments enhanced at interest rate for given number of conversion periods. It
is defined as the sum of the amounts of all payments and the total compound
interest earned on these payments to the time of the last payment. See for
example [1], [4].

Suppose the constant sum R is deposited at the end of each conversion
period in a bank which credits interest at the annual rate r. The deposits are
made k times each year over n conversion periods. Let yn denote the total
amount in the account at the end of n conversion periods. We shall find the
total worth of an annuity after n deposits.

The recursive rule for the future value of an annuity can be written as

yn+1 = yn +
r

k
yn + R =

(
1 +

r

k

)
yn + R, n = 0, 1, 2, . . .

with y0 = 0, where
r
k

is the interest rate per conversion period.
This equation constitutes the first order nonhomogeneous linear difference

equation with constant coefficients

yn+1 −
(

1 +
r

k

)
yn = R (6)

with initial condition
y0 = 0. (7)

In financial mathematics, the above problem (6), (7) is solved by us-
ing the properties of a geometric sequence. But we will proceed by means of

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Some applications of linear difference equations in finance

difference equations like in the case of compound interest. To solve nonhomo-
geneous difference equation (6) we consider the corresponding homogeneous
difference equation

yn+1 −
(

1 +
r

k

)
yn = 0 (8)

which is the same as (1) in the case of compound interest. Hence the general
solution ȳn of this homogeneous difference equation is given by

ȳn = C
(

1 +
r

k

)n
, C ∈ R. (9)

The right-hand side of the nonhomogeneous difference equation (6) is a con-
stant R which is a polynomial of degree zero. Thus a particular solution Yn
can be estimated by

Yn = b, b ∈ R.

For more details see [5]. Using the method of undetermined coefficients (see
[2]) we substitute the above estimate into (6). We get

b−
(

1 +
r

k

)
b = R

and solving for b we obtain

b = −R
k

r
.

Therefore the particular solution of (6) takes the form

Yn = −R
k

r
. (10)

Using (9), (10) according to the superposition principle (see [6]), the general
solution of the nonhomogeneous linear difference equation (6) is the sum

yn = Yn + ȳn = −R
k

r
+ C

(
1 +

r

k

)n
, C ∈ R.

A constant C can be specified from the initial condition (7) for period n = 0.
Hence we obtain

C = R
k

r
.

Consequently, the general solution of (6) takes the form

yn = −R
k

r
+ R

k

r

(
1 +

r

k

)n
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D. Ř́ıhová, L. Viskotová

which can be written as

yn = R

(
1 + r

k

)n − 1
r
k

. (11)

The above relation represents the future value of an annuity formula
which gives the amount of an annuity of n payments of R at the compound
rate r

k
per conversion period under the assumption that the payment interval

equals the conversion period. The future value of an annuity formula is used
to calculate what value at a future date would be for a series of periodic
payments.

In financial mathematics, it is common to use the following form of the
formula (11) setting i = r

k
where i represents the interest rate per compound-

ing interval (see [8] and [10])

yn = R
(1 + i)

n − 1
i

. (12)

3.2 Illustrative Examples

Example 3.1. Suppose EUR 500 is deposited at the end of every six-month
period in a bank, whose annual rate is 3.4%, compounded semiannually. How
much will this account be worth after 7 years?

We get the solution using (11), where R = 500, r = 0.034, k = 2, n = 14.
Then we obtain

y14 = 500

(
1 + 0.034

2

)14 − 1
0.034
2

.
= 7828.64

Example 3.2. Find the payment amount that you should deposit at the end
of each month in a bank so that EUR 35, 000 will be available after 10 years
if the interest rate is 1.6%, compounded monthly after each deposit.

Solving for R from the future value of an ordinary annuity formula (11) we
get

R = yn

r
k(

1 + r
k

)n − 1.
In our case we have r = 0.016, k = 12, n = 120, y120 = 35, 000. Hence the
monthly payment is calculated as follows

R = 35, 000
0.016
12(

1 + 0.016
12

)120 − 1 .= 269.149
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Some applications of linear difference equations in finance

4 Solving with Computer Algebra Systems

In mathematics of finance, Excel is commonly used for calculations. In
this paper the quoted calculations of compound interest and annuitity are
completed by computational tool Wolfram|Alpha and mathematical software
Maple, respectively.

4.1 Wolfram|Alpha

We will demonstrate the computation of compound interest (3) and Ex-
ample 2.2 through the free online service Wolfram|Alpha, which is available
via any web browser at http://wolframalpha.com. This tool provides math-
ematical computations based on software Mathematica and accepts com-
pletely free-form input, commands are specified by the name of operation in
English.

To solve the difference equation (1) with the initial condition (2) we type
both equations together separated by comma into an input field writing in-
dexes in parentheses. The provided general solution (3) is shown in Figure 1.

Fig. 1. Compound interest formula

Further, in Figure 2 you can see calculation of number of years from Example
2.2 using derived formula (5). We get the same result by solving equation
(4) and using the command solve and the reserved word for as shown in
Figure 3.

87



D. Ř́ıhová, L. Viskotová

Fig. 2. Calculation of number of years by using derived formula

Fig. 3. Calculation of number of years by solving equation

4.2 Maple

Now we show the computation of the future value of an annuity (11) and
illustrative Example 3.2.

We assign the recurrence relation (6) to the name REq.

> REq:=y(n+1)-y(n)*(1+r/k)=R;

Maple returns the output:

REq := y(n + 1) −y(n)
(

1 +
r

k

)
= R

Then we make the assignment of the initial condition (7) to the name IC.

> IC:=y(0)=0:

To solve the given difference equation we execute the command rsolve which
solves among others the first order linear difference equations. A single recur-
rence relation and a boundary condition are the first argument, the second
argument indicates the function that is solved for. Indexes are written in

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Some applications of linear difference equations in finance

parentheses.

> rsolve({REq,IC},y(n));

Rk
(
k+r
k

)n
r

−
kR

r

The above obtained expression corresponds to the future value of an annuity
(11).

For determining the payment R we type the following command, where
FV equals to the total amount yn in the account upon the last deposit (i.e
the future value of an annuity).

> isolate(%=FV,R):simplify(%);

R =
FV r

k
((

k+r
k

)n − 1)
To make the calculation of Example 3.2 we use command subs.

> subs(k=12,r=0.016,FV=35000,n=120,%);

R = 269.1493510

4.3 Comparison of Used Systems

The professional Maple is very powerful tool which enables to make new
procedures and modules, save and read them or together with other data
store in a library. On the other hand, it requires certain programming skills.

In comparison with Maple, Wolfram|Alpha does not allow to save and
reload the results of computations and make own procedures, also its perfor-
mance is rather slow. But its significant advantage is that it is free online
and very simple to use. Moreover, Wolfram|Alpha provides a variety of com-
putations from other fields, for example from money and finance.

5 Conclusion

This paper has discussed linear difference equations and their applications
in economics (see also [9]). These equations are frequently used especially
in financial mathematics and some of their typical applications have been
presented here.

Our main aim was to show relationship between some formulas of financial
topics and mathematical knowledge which is required for their deriving. We
have focused on derivation of the compound interest and the future value

89



D. Ř́ıhová, L. Viskotová

of an annuity formula by means of solution of difference equations. The
simultaneous application of mathematical software has been demonstrated,
the supplementary computations have been performed through Maple and
Wolfram|Alpha.

Finally, the paper emphasizes the need for mathematics in economic sub-
jects. The presented approach can be used in teaching of mathematics at
economic universities and helps to provide students with the opportunities
to apply their mathematics in relevant economics contexts.

References

[1] J. J. Costello, O. S. Gowdy and M. A. Rash, Mathematics for the Man-
agement, Life, and Social Sciences, New York: Harcourt Brace Jo-
vanovich, Inc., 1982.

[2] S. Elaydi, An Introduction to Difference Equations, New York: Springer
Science+Business Media, Inc., 2005.

[3] G. Fulford, P. Forrester and A. Jones, Modelling with Differential and
Difference Equations, Cambridge: Cambridge University Press, 1997.

[4] S. Goldberg, Introduction to Difference Equations: With Illustrative Ex-
amples from Economics, Psychology and Sociology, New York: Dover
Publications, Inc., 2010.

[5] J. Moučka and P. Rádl, Matematika pro studenty ekonomie, Praha:
Grada Publishing, Inc., 2010.

[6] K. Neusser, Difference Equations for Economists, Bern: Univer-
sity of Bern, 2012, [Online], [Cited 2015-07-15]. Available from:
http://www.neusser.ch/downloads/DifferenceEquations.pdf.

[7] P. Pražák, Diferenčńı rovnice s aplikacemi v ekonomii, Hradec Králové:
Gaudeamus, 2013.

[8] J. Radová, P. Dvořák and J. Málek, Finančńı matematika pro každého,
Praha: Grada Publishing, Inc., 2005.

[9] D. Ř́ıhová and L. Viskotová, Compound Interest and Annuities with
Linear Difference Equations and CAS, MITAV 2015 (proceedings of ab-
stracts), Brno: University of Defence, 2015.

[10] O. Šoba, M. Šir̊uček and R. Ptáček, Finančńı matematika v praxi, Praha:
Grada Publishing, Inc., 2013.

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