Final answer:
To find the effective annual interest rate, use the formula (1+(Nominal Interest Rate/Number of Compounding Periods))^Number of Compounding Periods - 1. For monthly, weekly, and daily compounding, the effective annual interest rates are 6.1681%, 6.1837%, and 6.1846% respectively. For continuous compounding, it is 6.1837%.
Step-by-step explanation:
To find the effective annual interest rate corresponding to each compounding period, we need to use the formula:
Effective Annual Interest Rate = (1 + (Nominal Interest Rate / Number of Compounding Periods))Number of Compounding Periods - 1
Using the given nominal interest rate of 6% per year:
a. For monthly compounding, substitute the values into the formula:
Effective Annual Interest Rate = (1 + (0.06 / 12))12 - 1 = 6.1681%
b. For weekly compounding, substitute the values into the formula:
Effective Annual Interest Rate = (1 + (0.06 / 52))52 - 1 = 6.1837%
c. For daily compounding, substitute the values into the formula:
Effective Annual Interest Rate = (1 + (0.06 / 365))365 - 1 = 6.1846%
d. For continuous compounding, we can use the formula:
Effective Annual Interest Rate = eNominal Interest Rate - 1, where e is Euler's number (approximately 2.7183):
Effective Annual Interest Rate = e0.06 - 1 = 6.1837%
The effective annual interest rate depends on the compounding frequency and is calculated differently for monthly, weekly, daily, and continuous compounding using specific formulas. The result is expressed as a percentage rounded to four decimal places.
The calculation for the effective annual interest rate for different compounding periods involves using the formula (1 + nominal rate/n)^(n*t) - 1, where 'n' is the number of compounding periods per year and 't' is the time in years. Since the time is one year for the effective annual rate, 't' will be 1.
a. For monthly compounding (n = 12), the effective annual interest rate is ((1 + 0.06/12)^(12*1) - 1), which should be calculated and then expressed as a percentage, rounded to four decimal places.
b. For weekly compounding (n = 52), the effective rate is ((1 + 0.06/52)^(52*1) - 1), again expressed as a percentage to four decimal places.
c. For daily compounding (n = 365), the formula is ((1 + 0.06/365)^(365*1) - 1) and the resulting percentage should be rounded to four decimal places.
d. Continuous compounding is calculated using the formula e^(r * t) - 1, where 'e' is the mathematical constant approximately equal to 2.71828, 'r' is the nominal rate, and 't' is the time in years. For continuous compounding with a 6% rate, this is e^(0.06*1) - 1, and the result should be expressed as a percentage and rounded to four decimal places.