The minimum yearly repayment (MYR) is calculated using the formula:
MYR = P * r / (1 - (1 + r) ^ -n)

Where:
P = $50,000
r = 0.0877
n = 7

Calculation:
MYR = 50,000 * 0.0877 / (1 - (1 + 0.0877) ^ -7)
MYR = 4,385 / (1 - (1.0877) ^ -7)
MYR = 4,385 / (1 - 0.54845479...)
MYR = 4,385 / 0.45154521...
MYR = 9,711.099...

Rounded to the nearest cent, the minimum yearly repayment is **$9,711.10**.

The controlling authority for the Division 7A minimum yearly repayment rule is the **Australian Taxation Office (ATO)**.