We’ve inspected exponential growth and decay. Here we consider applications of their integrals.

The exponential growth integral is

\[\int_0^T P_0e^{kt}dt = \frac{P_0}{k}(e^{kt} - 1)\]

The exponential decay integral is

\[\int_0^T P_0e^{-kt}dt = \frac{P_0}{k}(1 - e^{-kt})\]

Accumulated Future Value

In financial applications, you can substitute \(P_0\) with \(R(t)\) (a function representing the rate, per year, of a continuous income stream). \(k\) then becomes the interest rate, compounded continuously, and \(T\) becomes the number of years for which the income stream is invested. The accumulated future value of a continuous income stream is then

\[A = \int_0^T R(t)e^{kt}dt = \frac{R(t)}{k}(e^{kT} - 1)\]

Future Value Example

Suppose you receive a continuous income stream of $225,000 per year for 20 years, and you invest that money at 3.2% compounded continuously in a fund. What is the accumulated future value of the income stream?

from math import e

def afv(r, k, t):
  k = k / 100
  return (r / k) * (e**(k*t) - 1)
  
afv(225000, 3.2, 20)
## 6303381.182612939

Accumulated Present Value

Using the same logic, we can go from an expected accumulated future value back to an accumulated present value of a continuous income stream.

\[B = \int_0^T R(t)e^{-kt}dt = \frac{R(t)}{k}(1 - e^{-kT})\]

Present Value Example

Given a constant income stream of $275,000 for 8 years, what is the accumulated present value?

def apv(r, k, t):
  k = k / 100
  return (r / k) * (1 - e**(-k*t))

apv(275000, 5, 8)
## 1813239.7468039838