Equivalent interest rates

Equivalent interest rates are those that, when applied to the same capital, in the same period, produce equal values.
A principal (P) applied at a rate (i₁) for a period (n) produces an amount (S) . If the same capital applied to a different rate (i₂) to produce the same amount, we say that the rates are equivalents.
See, how to get the generic formula.
By the definition of equivalent rates, as above, we have:
S = S₁.
r = i / 100 (Interest rate in decimal form)
S = P (1 + r_a) (annual rate)
S₁ = P (1 + r_m) ¹² (monthly rate)
As we know, by definition, S = S₁
P (1 + r_a) = P (1 + r_m) ¹²,
1 + r_a = (1 + r_m) ¹²
r_a = (1 + r_m) ¹² -1
Example:
Be the monthly interest rate of 2%, what is the equivalent annual rate?
r_a = (1 + r_m) ¹² - 1
r_a = (1 +0.02) ¹² -1
r_a = (1.02) ¹² -1
r_a = 1.2682 -1
r_a = 0.2682 or 26.82%
Finally, the generic formula for calculating the equivalent rates:
r₁ = {(1+r₂) ^(n₁/n₂)} -1
r₁ = requested interest rate;
r₂ = known interest rate;
n₁ = period corresponding to the desired rate;
n₂ = period corresponding to the known rate.
How to calculateEquivalent interest rates.

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