Objectoriented modeling#
Multiple objects of similar types tend to have common definitions of logic and data. Modeling these objects manually one by one is not a good idea, because you would end up having multiple copies of the same definitions, which are hard to maintain and error prone.
modelx supports an inheritance mechanism, which enables you to define the parts common to the multiple objects only once as part of a base object, and model each object by inheriting from the base object and defining only the parts unique to the object. By making full use of inheritance, you can organize the multiple objects sharing similar features into inheritance trees, minimizing duplicated formulas and keeping your model organized and transparent while maintaining the model’s integrity.
Inheritance in objectoriented programming#
You may have heard about objectoriented programming (OOP). OOP is a programming paradigm, and most modern programming languages, such as Python and C++, support OOP. Such languages include powerful mechanisms, such as inheritance, for elegantly modeling complex objects. Inheritance in the OOP languages greatly enhances code reusability and extensibility.
modelx is inspired by OOP, and implements an inheritance mechanism similar to that of OOP. However, while most popular objectoriented programming languages use classbased inheritance, modelx uses prototypebases inheritance.
Python for example uses classbased inheritance. Python lets you define classes and objects are instances of the classes. Inheritance relationships in Python are defined in terms of classes.
In modelx, there is no class equivalent, and inheritance relationships are defined between Space objects. A space object inherits from another space object, in order to use the other object as a prototype.
How Inheritance works in modelx#
An inheritance relationship is established when you define a space(let’s name it A
)
as a base space of another space(let’s name it B
).
In this case, A
is called a base space of B
, and
B
is called a sub space of A
.
When B
inherits from A
,
copies of all the cells, references and spaces contained in A
are automatically created in B
.
This automatic copying is called deriving.
For example, let A
have a child cells foo
and a child reference bar
.
As the figure shows, another set of foo
and bar
are derived in B
.
The lines between A
and B
with a hollow triangle arrowhead on the side of A
indicates that B
inherits from A
.
Initially, the formula of foo
and the value of bar
in B
are copied from those in A
, but you can update foo
’s formula and bar
’s value
in B
. This act of updating derived objects are called overriding.
You can also add a new child object in B
, for example a new cells named baz
.
Adjustablemortgage Example#
Let’s learn how to implement inheritance by modeling a simple financial product.
Earlier in this tutorial, a simple fixedrate mortgage loan is modeled as the Fixed
space.
Let’s say we also want to model an adjustablerate mortgage loan.
For simplicity, we assume the adjustablerate mortgage has the same loan term and principal as the fixedrate mortgage’s.
Let the loan term be 10, and the principal be 100,000 in this example.
During the first 5 years, the interest rate of the adjustable mortgage is fixed at 2%, but from the 6th year the interest rate is updated every year till the end of the loan period. Let’s assume the interest rate is expected as follows:
Year 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

Interest Rate 
2% 
2% 
2% 
2% 
2% 
4% 
5% 
6% 
5% 
4% 
Note that the interest rate applicable after the first 5 years is not known at the inception of the loan, because the rate is not fixed in advance. So the interest rate table above is an assumption or a scenario if we are modeling a loan to be paid off at a future point in time.
We want to model the adjustable mortgage as the Adjustable
space.
Since Fixed
and Adjustable
are both mortgages and expected to have some shared
definitions of formulas and values, we can make use of inheritance.
we create a base space, BaseMortgage
, and define cells and references
common to Fixed
and Adjustable
in BaseMortgage
.
Fixed
and Adjustable
inherit from BaseMortgage
, and we override
some of the derived objects inherited from BaseMortgage
to reflect their own features. The diagram below depicts the relationship of the spaces.
To identify the commonality between the two types of mortgages, let’s review the contents of Fixed
from the earlier example one by one,
and think about whether and how they should be updated.
Term
is an integer representing the length of the loan term in years. We’ve assumed above that the fixed and adjustable mortgages have the same term.Principal
represents the initial loan balance and is given as an input. We’ve assumed above that the principals of the fixed and adjustable mortgages are the same amount.Rate
is a constant interest rate that applies through the lifetime of the fixed mortgage. Since the interest rate onAdjustable
is adjusted periodically,Rate
forAdjustable
should have a different definition from that ofFixed
. The adjustable interest rate can be defined as adict
indexed with loan duration.Payment
represents the amount of a payment to be made regularly to repay the loan. In the case ofFixed
,Payment
is defined as the constant amount calculated fromPrincipal
,Term
, andRate
.Payment
forAdjustable
needs to be timedependent, because it is recalculated periodically in response to changes in the interest rate. We will redefine the formula ofPayment
to make it timedependent and applicable to bothFixed
andAdjustable
.Instead of directly referring to
Rate
fromPayment
, it’s better to refer toRate
fromPayment
indirectly through a new cells with a time index, because the fixed and adjustable rate can be referenced with the time index in the same fashion. Let’s name the cellsIntRate
.Balance
is indexed with the time indext
, and represents the remaining balance of the loan at timet
. The formula ofBalance
calculates the loan balance at timet
recursively from the previous balance. The initial balance is input fromPrincipal
andRate
is referenced for interest accretion. By replacingRate
withIntRate(t)
, the formula becomes common betweenFixed
andAdjustable
.
The tables below summarizes how the contents of each space should be defined.
Contents 





10 
Inherited from 
Inherited from 

100000 
Inherited from 
Inherited from 

To be defined 

a 

Shared formula 
Inherited from 
Inherited from 

To be defined 
Unique formula 
Unique formula 

Shared formula 
Inherited from 
Inherited from 
You may have noticed that instead of creating BaseMortgage
,
it is possible to model Adjustable
by inheriting from Fixed
.
Although it’s technically possible, it’s not a good design, because
the adjustable mortgage is not a special form of the fixed mortgage.
Good practice is to make sure that an inheritance relationship should
always represent “is a” relationship.
Modeling Inheritance#
We start from the Mortgage
model from the earlier example, but you may also start from scratch if you prefer:
>>> import modelx as mx
>>> model = mx.read_model("Mortgage")
Let’s use the Fixed
space as the base space. Rename it BaseMortgage
:
>>> model.Fixed.rename('BaseMortgage')
>>> model.BaseMortgage
<UserSpace Mortgage.BaseMortgage>
Now set 10 to Term
, which is a constant shared between the sub spaces:
>>> model.BaseMortgage.Term = 10
Now let’s create Fixed
under the model by inheriting from BaseMortgage
.
You can do so by passing BaseMortgage
to the bases
parameter of the model’s new_space
method:
>>> model.new_space('Fixed', bases=model.BaseMortgage)
In the same way, create Adjustable
by inheriting from BaseMortgage
:
>>> model.new_space('Adjustable', bases=model.BaseMortgage)
You can also define an inheritance relationship between existing spaces
using the add_bases
method. Alternatively to calling the new_space
with the bases
parameter, you could also create Fixed
and Adjustable
by calling new_space
without bases
, and later calling
add_bases
on Fixed
and Adjustable
to set BaseMortgage
as their
base space:
>>> model.new_space('Fixed')
>>> model.new_space('Adjustable')
>>> model.Fixed.add_bases(model.Mortgage)
>>> model.Adjustable.add_bases(model.Mortgage)
Next, we set the interest rates by duration for Adjustable
as a dict
.
Note that the index starts from 0, so the key for the Nth rate is (N1):
>>> model.Adjustable.Rate = {
... 0: 0.02,
... 1: 0.02,
... 2: 0.02,
... 3: 0.02,
... 4: 0.02,
... 5: 0.04,
... 6: 0.05,
... 7: 0.06,
... 8: 0.05,
... 9: 0.04
... }
You may also assign 0.03
to Rate
in Fixed
, although the value is inherited:
>>> model.Fixed.Rate = 0.03
To refer to Rate
in the same manner in both Fixed
and Adjustable
,
we create a cells IntRate
indexed with t
.
First we create IntRate
in BaseMortgage
and define its formula to raise a NotImplementedError to indicate that it needs to be defined in the sub spaces.
There are a few ways to define the formula of IntRate
.
Here we define it by first defining a Python function and then assigning it
to InRate
’s formula:
>>> IntRate = model.BaseMortgage.new_cells('IntRate')
>>> def temp(t): # the name of the function can be anything.
raise NoteImplementedError
>>> IntRate.formula = temp
>>> IntRate.formula
def IntRate(t):
raise NoteImplementedError
Then override IntRate
in Fixed
and Adjustable
to refer to their own Rate
:
>>> model.Fixed.IntRate.formula = lambda t: Rate
>>> model.Adjustable.IntRate.formula = lambda t: Rate[t]
>>> model.Adjustable.IntRate[5]
0.04
Next, we are going to define Payment
in BaseMortgage
so that the definition of Payment
in the base space can be inherited and used both in Fixed
and Adjustable
without change.
The formula before update should look like below in BaseMortgage
because
we developed it from Fixed
form the earlier example:
def Payment():
return Principal * Rate * (1+Rate)**Term / ((1+Rate)**Term  1)
The formula above exactly represents the math expression below, which is a known formula to calculate the amount of level annual payments to pay off in Term `years a debt with interest accruing at ``Rate` a year.
To make the formula applicable to Adjustable
, we need to apply the following changes.
Parameterize
Payment
witht
Replace
Rate
withIntRate(t1)
Replace
Principal
withBalance(t1)
Replace
Term
withTerm  t + 1
The expression now looks like below:
Balance(t1) * IntRate(t1) * (1 + IntRate(t1))** (Term  t + 1) / ((1 + IntRate(t1))** (Term  t + 1)  1)
The corresponding math expression is as follows:
You may wonder why Payment(t)
refer to Balance(t1)
and IntRate(t1)
,
instead of Balance(t)
and IntRate(t)
.
You may also wonder why the remaining period is not Term  t
but Term  t + 1
.
The figure below illustrates how Payment(6)
is calculated.
Payment(6)
is calculated at t=5
such that paying the amount for the rest of
the loan term (5 years) would pays off Balance(5)
with interest accruing at IntRate(5)
,
assuming that IntRate(5)
would apply for the rest of the loan period.
In reality, the interest rate is updated annually, so one year later at t=6
,
the IntRate(6)
may be different from IntRate(5)
. In that case, Payment(7)
is updated
such that the updated amount would pays off Balance(6)
with interest
accruing at IntRate(6)
for the rest of the loan term.
Note the Payment
formula above is also valid for Fixed
, because
the formula Payment
returns the same value for t
during the loan period if
the interest rate does not change. So we define Payment
in BaseMortgage
.
The code below update Payment
in BaseMortgage
.
r
and u
are defined to make the expression compact:
>>> def temp(t):
... r = IntRate(t1)
... u = Term  t + 1
... return Balance(t1) * r * (1 + r)**u / ((1 + r)**u  1)
>>> model.BaseMort.Payment.formula = temp
We need to update one more cells. Balance
is defined in BaseMortgage
as follows:
>>> model.Mortgage.Balance.formula
def Balance(t):
if t > 0:
return Balance(t1) * (1+Rate)  Payment
else:
return Principal
The formula should refer to IntRate(t1)
and Payment(t)
instead of Rate
and Payment
respectively:
>>> def temp(t):
... if t > 0:
... return Balance(t1) * (1 + IntRate(t1))  Payment(t)
... else:
... return Principal
>>> model.BaseMortgage.Balance.formula = temp
Checking the results#
Now that we have completed making all the necessary changes,
let’s check the results.
Below the adjustable payments are output as a dict
.
As expected, the payments increase after the first 5 years
because the interest rate at t=5
is higher than before.
The payments then vary every year, reflecting the changes in the interest rate:
>>> {t: model.Adjustable.Payment(t) for t in range(1 ,11)}
{1: 11132.652786531637,
2: 11132.65278653164,
3: 11132.652786531638,
4: 11132.652786531644,
5: 11132.65278653164,
6: 11786.927741021387,
7: 12065.96444749335,
8: 12292.72989621633,
9: 12120.72411143264,
10: 12005.288643704713}
>>> model.Adjustable.Payment.series.plot()
To compare against the adjustable payments,
let’s also output and plot the fixed payments.
As you see below, the fixed payments are constant
throughout the loan period, even though the payments
are recalculated every year by the formula shared with Adjustable
:
>>> {t: model.Fixed.Payment(t) for t in range(1 ,11)}
{1: 11723.050660515952,
2: 11723.050660515952,
3: 11723.050660515953,
4: 11723.050660515959,
5: 11723.05066051596,
6: 11723.050660515968,
7: 11723.05066051596,
8: 11723.050660515977,
9: 11723.05066051599,
10: 11723.05066051596}
>>> model.Fixed.Payment.series.plot()
Below is the output of Adjustable.Balance
.
You can see that the balance is actually paid off at t=0
:
>>> {t: model.Adjustable.Balance(t) for t in range(0 ,11)}
{0: 100000,
1: 90867.34721346837,
2: 81552.0413712061,
3: 72050.42941209857,
4: 62358.78521380889,
5: 52473.30813155344,
6: 42785.31271579419,
7: 32858.613904090555,
8: 22537.40084211966,
9: 11543.546772793003,
10: 1.0913936421275139e11}
>>> model.Adjustable.Balance.series.plot()