explain why a savings and loan association that pays a nominal rate of 4.5% interest, compounded daily, actually pays an effective rate of 4.6%.
Because of daily compounding, the accumulated interest at the end of a year is 4.602%, not 4.5%.
Actual APR = (1+.045/365)^365 = 1.04602
To understand why a savings and loan association that pays a nominal rate of 4.5% interest, compounded daily, actually pays an effective rate of 4.6%, we need to understand the concept of nominal interest rate and effective interest rate, as well as how compounding frequency affects the overall rate.
The nominal interest rate is the stated interest rate, which in this case is 4.5%. It represents the interest rate before considering the effects of compounding.
On the other hand, the effective interest rate takes into account the compounding frequency and provides a more accurate measure of the interest earned over a specific time period.
In the given scenario, the savings and loan association compounds the interest daily. This means that the interest is recalculated and added to the account balance every day.
To determine the effective interest rate, we can use the following formula:
Effective interest rate = (1 + (nominal interest rate / number of compounding periods))^number of compounding periods - 1
Here, the nominal interest rate is 4.5%, but since compounding is done daily, the number of compounding periods is 365 (assuming it is not a leap year).
Let's calculate the effective interest rate:
(1 + (0.045 / 365))^365 - 1 = 0.4608
The effective interest rate, rounded to the nearest hundredth, is approximately 4.6%.
Therefore, even though the nominal interest rate is 4.5%, the compounding effect over daily periods leads to an effective interest rate of 4.6%. This shows that the interest earned on the savings and loan association is slightly higher due to the compounding effect.