Contact: Roberto Osorno Hinojosa, rosorno@iteso.mx
Víctor Germán Ledesma García, vigelega@hotmail.com
ITESO University
Periférico Sur Manuel Gómez Morín 8585,
Tlaquepaque, Jalisco, México. C.P. 45090
Telephone: (052+33)36693434 ext.3099
Version: AGB October 2006
Abstract:
Content is focused on the creation of goal oriented present value and future value models, using system dynamics
methodologies, allowing the value of money free transit over the timeline, and configuring the composition of sub
periods ranging from discrete to continuous events.
Keywords:
System Dynamics Application, Simulation, Compound Interest, Present Value, Future Value.

System Dynamics applied to the value of money. 2 de 11
Introduction
The present is an updated recapitulation from a thesis research (Ledesma, 2003) at ITESO. This material
can be thought of as a support tool to empower educational field knowledge on the process of compound
interest, and its inverse process.
The process of money value over the timeline is explained systematically by flow actions which add or
substract from the amount, through the feedback loop; the experimental model allows iterating through
different scenarios and can be goal oriented. The explanation of processes using this model structure
permits learning inheritance as time goes.
In this research the problem is methodologically separated in seven steps and takes on the mathematical-
temporal interpretation in a synthetic way, it also offers a new formulation of an effective interest rate
subject to the time step; also of the reversible process through discount, of which little is said.
Background
Historically, financial calculations have been subject to numerous applications and improvement. We
have found that financial analysis can be understood not only as the study of a determined state in a
moment, but as a continuous, subject to changes, adjustments, and environment. This way we can find
more exact financial behavior in systemic models. The financial behavior of an enterprise obeys the
interaction of different agents, which add dynamic complexity to the system: we find not only causal
relations, but accumulation, delays and feedback cycles (Sterman 2000).
The financial behavior vision as a function of flows and levels is not entirely new; nevertheless modeling
and application for decision making and learning are fields that are yet to be explored. The challenge of
finding practical applications within organizations is approached by this work.
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System Dynamics applied to the value of money. 3 de 11
Present and future value cases.
Step 1. Formulation of the problem.
Spreadsheets and financial calculators are common tools for money operations, but these instruments only
operate with fixed interest rates and cannot incorporate sub-composition periods, eliminating the
possibility of fluctuation or manipulation within the range of total computation; they calculate by means
of a multiplicative factor for capitalization. The following formulas are widely used:
Rate Fixed Fixed fraction
mn
i (3)
Vf n = Vp(1 + i ) (1)
n
Interest Vf n = Vp 1 +
m
− mn
i (4)
Vp = Vf n (1 + i ) (2)
−n
Discount Vp = Vf n 1 +
m
Due to this kind of limitation, the objective is to obtain a model through system dynamics which would
allows to understand the process of interest accumulation and substraction through discount; both in
discrete time steps and in continuous time. We want to verify if there exists an adaptative feedback
process in contrast with the traditional geometric progress capitalization through time by means of a
factor.
First challenge is to identify all the elements of the system and their mathematical counterpart.
Known premises: Developed formulas: Terminology:
Vf 0 = Vp (P1) m Vf Future value
i (5)
e i = lim 1 + Vp Present value
m→ +∞ m
1 (P2) −m h Time step frequency
h= d
e −d = lim 1 + (6)
dt m → +∞ m m Number of sub-periods
(capitalization or de-capitalization
1 (P3)
m= periods)
∆t
Vf k = Vf k −1 (1 + ie ) (P4) Vf n = Vp(1 + ie ) (7)
n i Nominal interest rate
redit = Vf k −1ie (P5) Vf n = Vp e in (8) k Element between 0 to n
Vf k −1 = Vf k (1 − d e ) (P6) Vp = Vf n (1 − d e ) (9)
n d Nominal discount rate
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System Dynamics applied to the value of money. 4 de 11
Discount = Vf k d e (P7) Vp = Vf n e − dn (10) n Period number
Vf (t + ∆t ) − Vf (t ) (P8) t Time
(Vf )
n
Discrete gain = Sec(t ) = (11)
+
Vf n = Vp + j −1 e ji
∆t t Time increment
j =1
Vf n = Vp∏ (1 + iej ) (12)
n
j =1
Vf (t ) − Vf (t − ∆t ) (P9) dt Time differential
(Vf d e j ) (13)
1
Discrete discount = Sec(t ) =
−
Vp = Vf n − j
∆t j =n
1 d ej
Vp = Vfn ∏ 1 − (14)
j =n 1 + d ej
Vf (t +dt ) − Vf (t ) i
m ie Effective interest
Continuous gain = Vf ′(t ) + = lim+ (P10) ie = 1 + − 1 (15)
dt →0 dt m
Vf (t ) − Vf (t − dt ) d
−m de Effective discount
Continuous discount = Vf ′(t ) − = lim− de = 1 − 1 + (16)
dt →0 dt m
(P11)
Step 2. Causal Loop Diagram.
Initial amount Initial amount
(Vp) (Vf)
+ + + +
+ +
+ -
Revenue + Final
Discount +
(Vp)(ie) or
(liquid value)(ie)
+
+ Liquid value
(Vp+Redit)
amount
(Vf)(de) or
(liquid
-- Liquid value
(Vf-Discount)
Final
amount
(Vf)
value)(de) (Vp)
+ m + m
+ + + +
Effective Effective discount
interest rate (ie) + i rate (de) + d
Fig. 1. Future value causal diagram. Fig. 2. Present value causal diagram.
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System Dynamics applied to the value of money. 5 de 11
Step 3. Computer simulation prototype.
Revenue collect Vf Vp Discount withdraw
Vf0 Vfn
Must give Must receive Must give Must receive
i ie de d
m Vp Vf m
Fig. 3. Vf schema Fig. 4. Vp schema
“Revenue collect” as revenue flow is constituted by (P5) and effective interest rate (15), and “Discount
withdraw” is composed by (P7) and effective discount (16).
Step 4. Running, verification and, validation of the model.
Table I. m’s Table III. Present and future value with different composition frequencies.
10% ie de Time Vf (m=0.5) Vp (m=0.5) Vf (m=1) Vp (m=1) Vf (m=2) Vp (m=2) Vf (m + ) Vp(m + )
m=0.5 9.544% 8.712% 0.0 4.0 10.000 12.000 10.000 14.641 10.000 14.774 10.000 14.918
m=1.0 10.000% 9.090% 0.5 3.5 10.500 14.071 10.512 14.190
m=2.0 10.250% 9.297% 1.0 3.0 11.000 13.310 11.025 13.400 11.051 13.498
m + 10.517% 9.516% 1.5 2.5 11.576 12.762 11.618 12.840
2.0 2.0 10.954 10.954 12.100 12.100 12.155 12.155 12.214 12.214
Table II. h’s 2.5 1.5 12.762 11.576 12.840 11.618
m=1 Vf(t=1) Vp(t=1)
3.0 1.0 13.310 11.000 13.400 11.025 13.498 11.051
h=1 11.000 11.000
3.5 0.5 14.071 10.500 14.190 10.512
h=2 10.976 10.973
4.0 0.0 12.000 10.000 14.641 10.000 14.774 10.000 14.918 10.000
h=4 10.964 10.960
Within table I are collected run results when testing i cases every two years capitalization (m=2), annual
(m=1), biannual (m=0.5) and continuous (m + ), and compared to the result of formula (15).
When experimenting with different h´s (simulator time steps), as shown in table II and figure 5, we find
new effective interest rates for the different value curves, even when m remains unchanged.
Table III and figure 6 show how future value increases either due to capitalization frequency m, or
simulation time step h, adding importance to the fraction of intervals which compose a sub-period of time
subject to m.
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System Dynamics applied to the value of money. 6 de 11
m=0 m=1/10 m=1/5
m=1/2 m=1 m=2
h = 10 h=5 h=2
m=4 m=360 m=INF
h=1 h = 1/2 h-->0
17.5% 28
16.5%
15.5% V 26
14.5% 24
F 22
13.5%
I 12.5% 20
11.5% 18
e 10.5%
(
16
9.5% $ 14
8.5% 12
7.5%
)
10
6.5% 0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Time (years)
h`s (years)
Fig. 5. Ie for different m' and h'
s s Fig. 6. VF at 10% with different h'
s
While solving for a single m with different h´s we obtain different results for the same liquid
value in time. This shows that there are different effective values for interest (ie) and discount (de)
subject to m (capitalization or de-capitalization frequency), as well as effective values subject to m-h
(frequency of composition and time step) for interest (ieh) and discount (deh).
Step 5. More complete model formulation
It follows that the schemas from figures 3 and 4 are valid only if m=h, and work correctly for constant
interest and discount rates. Under this limitation it is of benefit to add a loop working as a change policy,
in order to allow the incorporation of floating rates, it would also be of advantage to extend the
capabilities of the model by adding goal seeking.
Revenue + +
+ (Vp)(ieh) or (liquid ieh i
+
value)(ieh) +
Goal +
+ + +
+ + h
Initial amount + Liquid value + Final
(Vp) o (Vf) Discrepancy (Vp + Revenue) amount
(Vf - Discount) (Vf) o (Vp)
+
- -
-
+ m
+
+ Discount +
(Vf)(deh) or (liquid deh d
value)(deh) + +
Fig. 7. Causal diagram for Present value /Future value
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System Dynamics applied to the value of money. 7 de 11
In addition it could incorporate flow integration and disintegration capability in order to achieve a more
generic and minimal model. And for a final improvement, by means of a visual interface layer, selectors
could be installed to the model layer to choose the event type (continuous or discrete) and the final
amount (VF or VP).
For discrete events:
m
Revenue collect Final amount Discount withdraw i h
ie h = h 1+ −1 (17)
m
−m
d h
deh = h 1− 1+ (18)
m
ieh deh
i d
h m For continuous events:
i
Initial amount
VF Continue
ie h = h e − 1
h
(19)
Stop
i
Goal difference VP Discrete −
deh = h 1 − e h
(20)
Fig. 8. Generic schema
Formula 17 has its root in the combination of composition and time step frequencies, to obtain the
m h
effective interest rate, from ie = 1 + i ieh
−1 = 1 + − 1 solving for ieh. The findings on formulas 18, 19
m h
and 20 are now incorporated within the generic schema; the model has the extra capability to stop time
advance when its goal is reached.
Step 6. Experimentation scenarios, applying goals and policies.
Supposing an initial amount of $10, to find the time needed to reach a final amount of $5,000,000; an
error margin of one day is required. Annualized interest rates with three monthly capitalizations require a
minimal amount, and follow this policy: (10%, $10), (12%, $2,000), (15%, $40,000), (20%, $1,000,000).
To solve this we need to adjust the parameters as follows: m=4, initial amount=10, Goal=5000000,
h=360, and condition i with respect to liquid value; we can also add the next maximal annual search
period in the visual interface layer, which defaults to 120 years.
As a result to this scenario, the goal is reached within 109 years and 32 days.
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System Dynamics applied to the value of money. 8 de 11
Step 7. Learning in the time horizon.
Based on the model behavior we can conclude in essence that interest capitalization is a process by which
the initial amount is accumulating revenue; de-capitalization by discounts is the process by which the
final amount is substracted discounts.
Time is present to effect influence on interest and discount rates, for effective value through capitalization
frequency and in time step.
When h=m formula for ieh and deh are simplified as those from step 1. The finding of h influenced
formula is relevant because it allows us to obtain the value of a simple fraction within a time interval with
no repercussion in the formation of a new sub-composition period.
The improved model has the possibility to be used in scenarios where nominal interest or discount rates
fluctuates through time and can simplify elaborated calculations.
Conclusions
In order to understand the process of money value through time it is known that we start with an initial
amount which goes through a liquid value to reach a final amount, in this time it can be added revenue, or
substracted discount which marks this as an addition or substraction task which gives little relevance to
the origin of revenue and the destination of discount.
The mathematical part is fundamental in the elaboration of the prototype model, it bears mention that with
revenue and discount for discrete events (P8, P9) we obtain secant lines, and when working with
continuous events we get tangent lines (P10, P11), in both cases we use the left line for discount, and the
right one for revenue.
The operations for future Value and present Value from formulas (7, 9), when the time interval is
infinitely small, as pointed by (5, 6) we get (8, 10). Being this the way to understand how to elevate cases
from discrete events to continuous. In this context (11, 13) express the existence of recursive processes
motivated by feedback loops, in contrast with (12, 14) which only calculate by multiplication factors.
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System Dynamics applied to the value of money. 9 de 11
The importance of revenue and discount flow actions as leverage instruments to achieve the system goal
is manifest.
We can visualize the relevancy of feedback loops to operate with change policies in variables which in
this particular case are allowed by a floating interest rate.
During experimentation it is obligatory to consider the simulator timestep (h) and the period composition
frequency (m) in order to obtain effective values as time step, this experience can be useful as a procedure
to avoid lineal interpolation techniques for values within a composition sub-period.
As for step 6, experimenting with an scenario where period composition is done in a three month basis we
require to monitor discrepancies on a daily basis, this gives more relevance to the composition
frequencies and time step from formulas (17,18), and to the feedback to interact with floating (or variable)
interest rates.
System dynamics is again successful as a methodology because it enhances knowledge communication by
using experimental models which, as a didactic support instrument improves the reasoning of formulas,
and is a resource which can potentially be used to teach financial mathematics.
Final observations.
This interdisciplinary investigation was developed using mathematic constructivism, and by managing the
project with the spiral method; system dynamics methodology consisted in seven steps, in this work it
applied a procedure subject to open proposal as recommended in (Sterman, 2000) and (Cavana, 2002).
Actually, this represents the fourth round of evolutionary refinement since September 2001.
With system dynamics methodology we get answers for any time fraction, so we can analyze step by step
any event in time; this opens the way to other interest derivative processes and very probably to financial
analysis and interpretation instruments.
In these models we needed to make clear that valves are labeled with actions derived from object flow, in
this cases they were not labeled only as “ Revenue” or “ Discount” because we wanted to differentiate flow
valves from simple change indexes (rates) to actions dependent on time change; we think this facilitates
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System Dynamics applied to the value of money. 10 de 11
language use when leveraging models, because it allows to express the transition between a subject and
an action, leading to a chronologic flow of actions.
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About the authors
MIA. Roberto Osorno Hinojosa is coordinator of the “ Maestría en Informática Aplicada” (Applied
informatics masters degree) at ITESO. His interests are System dynamics and Information Technology
based strategies.
ISC. Víctor Germán Ledesma Garcia is a former student at ITESO. His interests are System dynamics
and its application in business.
Instituto Tecnológico y de Estudios Superiores de Occidente, 2006