Aldo took out a loan for $7500 that charges an annual interest rate of 9.3%, compounded monthly. Answer each part. If necessary, refer to the
list of financial formulas.
(a) Find the amount owed after one year, assuming no payments are made.
Do not round any intermediate computations, and round your answer to the nearest cent.
$
(b) Find the effective annual interest rate, expressed as a percentage.
Do not round any intermediate computations, and round your answer to the nearest hundredth of a percent.
(a) Using the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = amount owed after one year
P = principal (initial amount borrowed) = $7500
r = annual interest rate = 9.3% = 0.093
n = number of times interest is compounded per year = 12 (monthly)
t = time in years = 1
A = 7500(1 + 0.093/12)^(12*1) = $8227.51
Therefore, the amount owed after one year is $8227.51
(b) Using the formula for effective annual interest rate:
EAR = (1 + r/n)^n - 1
where:
EAR = effective annual interest rate
r = annual interest rate = 9.3% = 0.093
n = number of times interest is compounded per year = 12 (monthly)
EAR = (1 + 0.093/12)^12 - 1 = 0.0973 or 9.73%
Therefore, the effective annual interest rate is 9.73%.
amount owed is actually 8,228.01
Thank you for letting me know. If the actual amount owed after one year is $8228.01, then we can adjust the answer to (a) accordingly:
(a) Using the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = amount owed after one year
P = principal (initial amount borrowed) = $7500
r = annual interest rate = 9.3% = 0.093
n = number of times interest is compounded per year = 12 (monthly)
t = time in years = 1
A = 7500(1 + 0.093/12)^(12*1) = $8228.01 (rounded to the nearest cent)
Therefore, the amount owed after one year is $8228.01.
Then what is the effective annual interest rate?
To find the effective annual interest rate (EAR) when the loan is compounded monthly with periodic interest rate r, we can use the formula:
EAR = (1 + r/n)^n - 1
where r is the annual interest rate, and n is the number of times the interest is compounded in a year.
In this case, r = 9.3%, and n = 12 (monthly compounding). Substituting these values into the formula, we get:
EAR = (1 + 0.093/12)^12 - 1 = 0.0973 or 9.73% (rounded to two decimal places)
Therefore, the effective annual interest rate charged on the loan is 9.73%.
To find the amount owed after one year, assuming no payments are made, we can use the compound interest formula:
A = P * (1 + r/n)^(n*t)
Where:
A = Final amount after one year
P = Principal (initial loan amount)
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
Given:
P = $7500
r = 9.3% or 0.093 (as a decimal)
n = 12 (compounded monthly)
t = 1 year
Let's plug in the values into the formula:
A = 7500 * (1 + 0.093/12)^(12*1)
A = 7500 * (1 + 0.00775)^(12)
A = 7500 * (1.00775)^(12)
A ≈ $8220.29
Therefore, the amount owed after one year, assuming no payments are made, is approximately $8220.29.
To find the effective annual interest rate, we can use the following formula:
Effective Annual Interest Rate = (1 + r/n)^n - 1
Where:
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
Given:
r = 9.3% or 0.093 (as a decimal)
n = 12 (compounded monthly)
Let's plug in the values into the formula:
Effective Annual Interest Rate = (1 + 0.093/12)^12 - 1
Effective Annual Interest Rate ≈ 0.0973 or 9.73%
Therefore, the effective annual interest rate is approximately 9.73%.
To find the amount owed after one year, assuming no payments are made, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount owed after one year
P = the principal amount (initial loan amount)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
In this case:
P = $7500
r = 9.3% = 0.093 (as a decimal)
n = 12 (compounded monthly)
t = 1 (one year)
Plugging these values into the formula:
A = $7500(1 + 0.093/12)^(12*1)
Calculating this expression, we get:
A ≈ $7500(1 + 0.00775)^12
A ≈ $7500(1.00775)^12
A ≈ $7500(1.101386169)
A ≈ $8260.89
Therefore, the amount owed after one year, assuming no payments are made, is approximately $8260.89.
To find the effective annual interest rate, we need to find the rate that would yield the same amount of interest if compounded annually. We can use the formula for effective annual interest rate:
EAR = (1 + r/n)^n - 1
Where:
EAR = effective annual interest rate (as a decimal)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
In this case, we have already calculated the value of A for one year and we know the principal amount (P = $7500). We can rearrange the formula to solve for r:
EAR = (1 + r/n)^n - 1
(1 + r/n)^n = EAR + 1
1 + r/n = (EAR + 1)^(1/n)
r/n = (EAR + 1)^(1/n) - 1
r = n((EAR + 1)^(1/n) - 1)
We can plug in the values we know:
EAR = ?
r = 0.093 (as a decimal)
n = 12 (compounded monthly)
r = 12((EAR + 1)^(1/12) - 1)
To solve for EAR, we can use trial and error or iterative methods. However, depending on the level of accuracy required, we can estimate the value of EAR using a financial calculator, spreadsheet, or online calculator.
Using a financial calculator or a spreadsheet, we can input the formula:
0.093 = 12((EAR + 1)^(1/12) - 1)
Solving this equation, we find:
EAR ≈ 0.0968 or 9.68% (rounded to the nearest hundredth of a percent)
Therefore, the effective annual interest rate, expressed as a percentage, is approximately 9.68%.