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I am creating an amortization table using an annual rate (5%) and a monthly-equivalent rate (4.8889%). When you compound the monthly rate 12 times you get the equivalent of compounding the annual rate 1 time i.e. (1+4.8889%/12)^12 = 1.0500 which is the same as (1+5%/1)^1. However, the inverse does not seem to be true - (1-4.8889%/12)^12 = 0.952 whereas (1+5%/1)^1=0.950.

Excel output

How do I convert the 5% into a rate, such that the resulting rate will result in the same end value when compounded negatively?

jerry miller
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    Related to your problem: If you increase a number by $5%$ and then reduce that by $5%$ then you get to $99.75%$ of the number you started from, not $100%$. A $4.7619%$ reduction would have got you closer. – Henry Aug 07 '24 at 01:18
  • Yes, I understand that point. However, I'm wondering how I can convert a annual rate x (payable annually) into a annual rate y (payable monthly) such that (1-x/1)^1 = (1-y/12)^12 – jerry miller Aug 07 '24 at 15:18

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If you are saying $\left(1+\tfrac{0.05}{1}\right)^1 \approx \left(1+\tfrac{0.048889}{12}\right)^{12}$

by using $12\left(\left(1+0.05\right)^{1/12}-1\right) \approx 0.048889$

then $12\left(\left(1-0.05\right)^{1/12}-1\right) \approx -0.051184$

would imply $\left(1-\tfrac{0.05}{1}\right)^1 \approx \left(1-\tfrac{0.051184}{12}\right)^{12}$

so you seem to want about $5.1184\%$ in the decreasing case.

Henry
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