Example scenario
Loan amount – 500K, loan period – 20 yrs
Interest rate original / hike – 1% / 5%
Monthly payment original / increase – 2.3K / 3.3K
That is a 30 percent increase in monthly loan payments for a 4 percent rate increase.
Why?
Calculated using online calculator
In: 0
Since this 5yo seems to know percentages, the amount you actually end up paying over the 20 years, times the loaned amount is f = r*(1+r)^240 / ((1+r)^240- 1), where r is 5% / 12 (the monthly rate).
(ELI>5) This can be found with some algebra, but ultimately follows from the formula for a “geometric series.” One intuition is to imagine how much each payment saves you in interest: the first month’s payment effectively saves you payment * (1+r)^239 over the lifetime of the loan, the second month’s payment effectively saves you payment * (1+r)^238 over the lifetime of the loan, and the last payment, when you will just owe p, will save you p*(1+r). When the amount your payments “save you” equals the amount you would pay otherwise, you can solve for the payment.
Anyway, plugging into the formula, you get that 5% leads to a 43.502% higher payment than 1%, and indeed 2300*1.43502 = 3300 and change.
[Source](https://www.wolframalpha.com/input?i=Solve+f%28.05%2F12%29%2Ff%28.01%2F12%29+for+f%28r%29+%3D+r*%281%2Br%29%5EM+%2F+%28%281%2Br%29%5EM+-+1%29+where+M+%3D+240)
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