An investor can earn 9.1% interest compounded semi-annually or 9% interest compounded monthly. Determine which option he should prefer?
To determine which option the investor should prefer, we need to calculate the effective annual interest rate for each option.
Option 1: 9.1% interest compounded semi-annually
The formula to calculate the effective interest rate for compound interest is:
Effective Interest Rate = (1 + (Nominal Interest Rate / Number of Compounding Periods)) ^ Number of Compounding Periods - 1
For this option, the nominal interest rate is 9.1% and it compounds semi-annually, so the number of compounding periods is 2.
Effective Interest Rate = (1 + (0.091 / 2)) ^ 2 - 1
= (1 + 0.0455) ^ 2 - 1
= (1.0455) ^ 2 - 1
= 1.09245025 - 1
= 0.09245025 or 9.245%
Option 2: 9% interest compounded monthly
Using the same formula, we can calculate the effective interest rate for this option. Since the interest compounds monthly, the number of compounding periods is 12.
Effective Interest Rate = (1 + (0.09 / 12)) ^ 12 - 1
= (1 + 0.0075) ^ 12 - 1
= (1.0075) ^ 12 - 1
= 1.0933578 - 1
= 0.0933578 or 9.336%
Comparing the effective annual interest rates for both options, we can see that Option 2 with 9% interest compounded monthly has a higher effective interest rate (9.336%) compared to Option 1 with 9.1% interest compounded semi-annually (9.245%). Therefore, the investor should prefer Option 2.
To determine which option the investor should prefer, we need to compare the effective annual interest rates.
Option 1: 9.1% interest compounded semi-annually:
The effective annual interest rate for compounding semi-annually can be calculated using the formula:
(1 + r/n)^n - 1
Where:
r = annual interest rate (9.1% or 0.091)
n = number of times interest is compounded per year (2 for semi-annually)
Using the formula, plugging in the values, the effective annual interest rate for compounding semi-annually is:
(1 + 0.091/2)^2 - 1
= (1.0455)^2 - 1
= 1.0924 - 1
= 0.0924 or 9.24%
Option 2: 9% interest compounded monthly:
The effective annual interest rate for compounding monthly can be calculated using the formula:
(1 + r/n)^n - 1
Where:
r = annual interest rate (9% or 0.09)
n = number of times interest is compounded per year (12 for monthly)
Using the formula, plugging in the values, the effective annual interest rate for compounding monthly is:
(1 + 0.09/12)^12 - 1
= (1.0075)^12 - 1
= 1.0937 - 1
= 0.0937 or 9.37%
Comparing the effective annual interest rates, we can see that Option 2 (9% interest compounded monthly) has a higher effective annual interest rate of 9.37%, compared to Option 1 (9.1% interest compounded semi-annually) at 9.24%.
Therefore, the investor should prefer Option 2, which offers 9% interest compounded monthly.