Question

A deferred annuity is an annuity which delays its payouts. This means that the payouts do not start until after a certain duration. Notice that a deferred annuity is just a deposit at the start, followed by an annuity. Your task is to define a Higher-order Function that returns a function that takes in a given interest rate and outputs the amount of money that is left in a deferred annuity.

Define a function new_balance(principal, gap, payout, duration) that returns a single-parameter function which takes in a monthly interest rate and outputs the balance in a deferred annuity. gap is the duration in months before the first payment, payout is monthly and duration is just the total number of payouts. Written in Python code.



Test Case:
new_balance(1000, 2, 100, 2)(0.1)
Output:
1121.0

Answer 1

A higher-order function called new_balance can be defined in Python to calculate the balance of a deferred annuity at a given interest rate. The function first computes the future value of principal during the deferred period and then subtracts each payout, compounded monthly, to find the remaining balance.

Step-by-step explanation:

The student is interested in creating a function that calculates the balance of a deferred annuity based on a given interest rate using Python programming. To accomplish this, one can define a higher-order function new_balance that takes the principal, gap before payouts begin, the payout amount, and the duration of payouts. This function then returns another function that, when given a monthly interest rate, calculates and returns the balance of the annuity.

The logic behind the function involves calculating the future value of the initial deposit for the duration of the gap followed by the deduction of each payout adjusted for the remaining period at the given monthly interest rate.

def new_balance(principal, gap, payout, duration):
def balance(rate):
total = principal * ((1 + rate) ** gap)
for i in range(duration):
total = (total - payout) * (1 + rate)
return total
return balance

The test case new_balance(1000, 2, 100, 2)(0.1) demonstrates using the higher-order function by first creating a specific function for the given annuity parameters and then immediately calling it with an interest rate of 0.1 to find the balance.

Answer 2

Step-by-step explanation:

Here is the requested Python code for the new_balance function:

def new_balance(principal, gap, payout, duration):

def balance(interest_rate):

return principal * (1 + interest_rate)**gap + payout * ((1 + interest_rate)**(gap + duration) - (1 + interest_rate)**gap) / interest_rate

return balance

This function defines a single-parameter function balance which takes in a monthly interest rate and calculates the balance in a deferred annuity based on the given parameters. The balance is calculated using the formula for a deferred annuity, which accounts for the gap between the initial deposit and the start of the payouts, as well as the total number of payouts and the monthly payout amount.