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 %.
To understand why a savings and loan association that pays a nominal rate of 4.5% interest actually pays an effective rate of 4.6%, we need to consider the concept of compounding.
Compounding refers to the process of earning interest on both the initial deposit and any previously accumulated interest. In simple terms, it means that as time passes, interest is added to the original amount, and subsequent interest is calculated based on the new, increased balance.
In the case of the savings and loan association, the nominal rate of 4.5% indicates the annual interest rate. However, it is compounded daily, meaning that interest is calculated and added to the account balance each day.
To calculate the effective rate, we need to consider the compounding frequency. In this scenario, since interest is compounded daily, we need to convert the nominal rate into an effective annual rate. The formula to convert the nominal rate to an effective rate is:
Effective rate = (1 + (r/n))^n - 1
Where:
- r is the nominal rate expressed as a decimal (4.5% = 0.045)
- n is the number of compounding periods in one year
In this case, the compounding period is daily, so n = 365 (since there are 365 days in a year).
Plugging in the values into the formula:
Effective rate = (1 + (0.045/365))^365 - 1
Calculating this expression, we find that the effective rate is approximately 0.046, or 4.6%.
Therefore, even though the nominal rate is 4.5%, due to the daily compounding, the savings and loan association actually pays an effective rate of 4.6%. This is because compounding allows for the accumulation of additional interest on a more frequent basis, resulting in a slightly higher effective rate.