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Amount Functions
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.. meta::
   :description: An amount function specifies how money grows for an arbitrary growth pattern.
   :keywords: amount function, interest, financial mathematics, actuarial, python, package, simple interest, compound interest, money growth.

Although we have introduced the familiar cases of simple and compound interest, not all growth patterns are linear or geometric. Sometimes a growth pattern might be geometric, cubic, or some arbitrary user-defined pattern.

To accommodate these new patterns, we can define an :term:`amount function`, which specifies how money grows for an arbitrary growth pattern:

.. math::
   A_K(t)

Where :math:`K` specifies the amount of principal, :math:`t` specifies the amount of time, and :math:`A_K(t)` returns the value at time :math:`t` of :math:`K` invested at time 0.

Examples
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Suppose money exhibits a quadratic growth pattern, specified by the amount function:

.. math::
   A_K(t) = K(.05t^2 + .05t + 1)

If we invest :math:`K=5` at time 0, how much does it grow to at time 5?

TmVal's :class:`.Amount` class allows us to model this behavior. To solve the above problem, simply call the class and supply the growth function and principal. First, define the growth function as a Python function that takes the time and principal as arguments:

.. ipython:: python

   from tmval import Amount

   def f(t, k):
       return k * (.05 * (t **2) + .05 * t + 1)

Now supply the growth function to the :class:`.Amount` class, and call :code:`my_amt.val(5)` to get the answer:

.. ipython:: python

   my_amt = Amount(gr=f, k=5)

    print(my_amt.val(5))