A bank says, instant loan at the rate of x% (say, 1.25%) per month, how to calculate rate of interest per year for the loan offered by the bank?
And vice-versa, if the rate of interest of a loan is y% (say, 16%) per year, how to calculate it for per month?
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In the United States, banks normally quote interest as nominal annual rates, compounded monthly, so that that $1.25\%$ per month is $12\cdot1.25=15\%$ annually, and $16\%% annually means $1.3333\%$ per month. This is different from the APR in the truth-in-lending law, which is the effective annual rate.
If you receive $1.25\%$ interest per month, your account is multiplied by $1.0125$ every month. At the end of the year it will be multiplied by $1.0125^{12}\approx 1.16075$ so you have received $16.075\%$ annual interest. Similarly, if the annual interest is $r$, your account will be multiplied by $1+r$ at the end of the year. Each month it will be multiplied by $(1+r)^{1/12}$, so the monthly interest is $(1+r)^{1/12}-1$
