This notebook is called PeriodicPaymentInvestment.ipynb

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Then open this notebook file. Make sure your Jupyter notebook has SageMath as the kernel:

Kernel ➤ Change Kernel ➤ Select Kernel = SageMath 10.2

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Compute the equivalent interest for the year if we make the same contribution periodically every paycheck.

Let's assume compounding is every pay period. Usually banks do monthly or daily compounding. Thus the compound interest will be a little more.

Compound interest formula for n periods is $(1 + \frac{i}{n})^n$ See https://www.investopedia.com/terms/a/apy.asp

If you deposited $100$ for one year at 5% interest and your deposit was compounded quarterly, at the end of the year you would have $105.09$. If you had been paid simple interest, you would have had $105$. The APY would be $(1+4.05​)4−1=.05095=5.095%$.

def compound_interest( i, n ):
    return (1 + i / n) ^ n

compound_interest( i, 1 )-1

I

print( f"{100.0 * (compound_interest(0.05, 4)-1):8.3f} %")

   5.095 %

def equivalent_interest( i, num_pay_periods=26, yearly_contribution=1000, **kwargs ):
    """What is the interest for an entire year if we periodically contribute the same amount"""
    debug = kwargs.get('debug')
    
    # Define symbolic variables so we can have infinite precision.
    var('k')
    var('gain')
    var('contribution_per_pay_period')
    var('contribution_with_compounding')
    var('yearly_compounded_salary')
    var('yearly_equivalent_interest')
    
    # Use fractions instead of percentages in calculations.
    i /= 100
    
    # Divide up yearly salary into equal paychecks.
    contribution_per_pay_period = yearly_contribution/num_pay_periods
    
    # Apply compound interest to each paycheck in kth pay period to the end of the year.  Then sum them up.
    yearly_compounded_salary = 0
    for k in [1..num_pay_periods]:
        # Let's assume compounding is every pay period.  Usually banks do monthly or daily compounding.  So not exact, but close enough.
        contribution_with_compounding = compound_interest(i,k) * contribution_per_pay_period
        equivalent_interest = (contribution_with_compounding - contribution_per_pay_period)/contribution_per_pay_period
        if debug:
            print( f"k = {k:2d} interest = {100.0*float(i):8.1f}% contribution per pay period = {float(contribution_per_pay_period):8.2f} with compounding = {float(contribution_with_compounding):8.2f} equivalent interest = {100.0*float(equivalent_interest):8.5f}%" )
        yearly_compounded_salary += contribution_with_compounding

    # The equivalent simple interest for the year.
    yearly_equivalent_interest = (yearly_compounded_salary - yearly_contribution)/yearly_contribution

    if debug:
        print(f"\ninterest rate                        = {100.0*float(i):8.1f}%")
        print(f"contribution per pay period          = {float(contribution_per_pay_period):8.2f}\n")
        print(f"yearly contribution                  = {float(yearly_contribution):8.2f}")
        print(f"yearly contribution with compounding = {float(yearly_compounded_salary):8.2f}")
        print(f"Yearly interest rate                 = {100.0*float(i):8.2f} %")
        print(f"Yearly equivalent interest           = {100.0 * float(yearly_equivalent_interest):8.2f} %")
    
    return 100 * float(yearly_equivalent_interest)

equivalent_interest( 10 )

10.437408355371522

# axes='False' centers the labels along the axes. See https://ask.sagemath.org/question/9235/2d-plot-axes-labels/
plot(equivalent_interest,(1,40),xmin=0,xmax=40,ymin=0, ymax=50,axes_labels=['interest','compounded interest'],axes='False',figsize=(5,5))