Discount function

Discounting is a common calculation in the actuarial cash flow models. The discount() function available in the package supports the vectorized approach to discounting.

from cashflower import discount

@variable(array=True)
def present_value():
    return discount(cash_flows=cash_flow(), discount_rates=discount_rate())

The discount() function takes two mandatory parameters, both of which must be NumPy arrays containing float values:

  • cash_flows - an array representing the cash flows to be discounted,

  • discount_rates- an array of forward discount rates corresponding to each period.

The discount() function returns an array, so you can specify @variable(array=True) for the variable that will hold the result.


Scalar equivalent

The discount() function’s calculation is equivalent to the following recursive formula:

from settings import settings

@variable()
def present_value(t):
    if t == settings["T_MAX_CALCULATION"]:
        return cash_flow(t)
    else:
        return cash_flow(t) + present_value(t+1) * discount_rate(t+1)

Using the discount() function significantly improves the calculation time.


Calculation example

Here’s an example of how the discount() function works:

import numpy as np
from cashflower import discount

cash_flow = np.array([90.00, 120.00, 100.00])
discount_rate = np.array([1, 0.8, 0.9])
print(discount(cash_flow, discount_rate))
# [258. 210. 100.]

In this example, the present values of future cash flows for three periods are calculated as follows:

  • 258.0 is the present value of all three cash flows at the beginning of projection 90.0 + 120.0 * 0.8 + 100.0 * 0.8 * 0.9.

  • 210.0 is the present value of two cashflows after the first period 120.0 + 100.0 * 0.9.

  • 100.0 represents the present value of the last cash flow after two periods 100.0.

Note that the discount rate at time t represents the value of 1 at time t-1.