Math Help - Effective Compounded Daily Rate

I am stumped on the below - hopefully someone better than me at math can explain.

The effective annual interest rate for a nominal interest rate of 10% compounded monthly is equal to the below: = (1 + (10% / 12)) ^ 12 - 1
= 10.47%

So, assuming a $100 single capital contribution, total annual accrued interest is $10.47 in year 1 assuming no payments. How do you back into the daily compounded rate that will produce total annual interest of 10.47% or the same total pref amount of $10.47 off the same contribution?

Comments (12)

Jul 10, 2019

General Formula: [1 + (i / n)] ^ n -1
Where: i = annual rate, n = number of compounding periods (i.e. months, days, etc.)

Given total effective annual rate of 10.47%:

0.1047 = (1 + i / 365) ^ 365 - 1

1.1047 = (1 + i / 365) ^ 365

1.1047 ^ (1 / 365) = 1 + i / 365

1.1047 ^ (1 / 365) - 1 = i / 365

[1.1047 ^ (1 / 365) - 1] * 365 = i = 0.099587

Jul 10, 2019

Thanks for the reply. Let me further clarify, I need the rate below (i) that would result in $10.47 of total interest after Day 365.

Day 0 - $100 contribution
Day 1 - $100 * (1+i) = a
Day 2 - a * (1+i) = b
Day 3 - b * (1+i) = c
etc...

Jul 10, 2019
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