There’s simple interest rate (which is compounded one time) and then there’s compounded interest rate. Typically the “interest rate” number you hear quoted or talked about is the annual percentage rate (APR) or the figure to determine the yearly cost of the loan.
Simple interest rate: There is no additional cost (or interest) that accumulates on accumulated interest of the loan. For example, say you borrow $100 for one year at an APR rate of 4%. One payment option may be to repay the full $104 one time at the end of the 12th month. Another payment option may be to pay $52 every 6 months. Another option may be $26 every 4 months.
Compound interest rate: Interest is also paid on accumulated interest. Therefore, interest (amount extra you pay) is also compounded at at set intervals on top of what has accumulated, with a rate determined from the APR rate. For example, say that $100 you borrow for one year at APR of 4% is compounded every 3 months. Every time the interest is calculated / compounded it is calculated using APR/4 on top of the original loan amount and the additional interest that has accumulated. The APR/4 rate is the periodic rate used to compound the accumulated amount owed.
One year loan at 4% interest compounded every quarter of the year. APR/4 = 0.04/4
End of 3th month owed = $100 x (1 + 0.04/4) = $101
End of 6th month owed = $101x (1 + 0.04/4) = $102.01
End of 9th month owed = $102.01 x (1 + 0.04/4) = $103.03
End of 12th month owed = $103.03 x (1 + 0.04/4 ) = $104.06
You see here that you’re really paying $4.06 expense and not the $4 because the interest rate is compounded more frequently. The lender may set you up to pay the whole thing off in one year with one payment of $104.06 at the end of the year.
You could also have a $100 loan taken out for 2 years with APR of 4%. If it is compounded every single month then the periodic rate for compounding is (1 + 0.04/12) for 12 times a year for a total of 2 years. Say the bank tells you to pay $4.34 every month, this is how it works:
End of 1st month owed = $100 x (1 + 0.04/12) = $100.33 minus $4.34 payment = $95.99 owed
End of 2nd month owed = $95.99 x (1 + 0.04/12) = $96.31 minus $4.34 payment = $91.97 owed
..
End of 12th month owed = $55.16 x (1 + 0.04/12) = $55.34 minus $4.34 payment = $51 owed
..
End of 24th month owed = $4.33 x (1 + 0.04/12) = $4.34 minus $4.34 payment = $0 owed
There are compact formulas to calculate all this. I broke everything down so you can see how it works step by step. Go to Wikipedia or investopedia and search for compound interest rate calculation formulas.
Interest rates on bonds are different. On a $100 US treasury bond if the rate is 4% then that means a 4% return per year. If they make coupon payments to you two times a year then you will get $2 after 6 months and another $2 at end of 12 months. You get these coupon payments periodically until the bond matures and at the end you get your $100 back.