Answer :
Let's break down the given problem:
Caroline took out an 8-month loan for [tex]$900 with an APR (Annual Percentage Rate) of 21.6%, compounded monthly. The loan offers no payments for the first 2 months. Therefore, the effective period for monthly payments will be the remaining 6 months. To solve this, we need to adjust the present value (PV) of the loan to account for the interest accrued during the first two months when no payments are made. 1. Interest Calculation for No Payment Period: - The original loan amount is \(\$[/tex] 900\).
- The APR is 21.6%, compounded monthly, which means the monthly interest rate is [tex]\(\frac{21.6 \%}{12} = 1.8\%\)[/tex].
Over the first 2 months, the loan amount [tex]\(\$ 900\)[/tex] will accrue interest. The formula for the compound interest is:
[tex]\[ \text{PV adjusted} = PV \times \left(1 + \frac{I\%}{100 \times \text{C/Y}}\right)^{\text{number of months}} \][/tex]
Plugging in the values:
[tex]\[ \text{PV adjusted} = -900 \times \left(1 + \frac{1.8}{100}\right)^2 \][/tex]
Simplifying within the parentheses first:
[tex]\[ 1 + \frac{1.8}{100} = 1 + 0.018 = 1.018 \][/tex]
Raising this to the power of 2 months:
[tex]\[ (1.018)^2 \approx 1.036324 \][/tex]
Multiplying this result by the loan amount:
[tex]\[ \text{PV adjusted} = -900 \times 1.036324 \approx -932.6916 \][/tex]
So, the adjusted present value considering the interest accrued due to no payments for 2 months is approximately [tex]\(-932.69\)[/tex] dollars.
2. TVM Solver Variables:
- Caroline now has to pay this amount over the remaining 6 months.
- Number of payments ([tex]\(N\)[/tex]) = 6 (the remaining months)
- Interest rate per year ([tex]\(\%\)[/tex]) = 21.6% (APR)
- Adjusted Present Value ([tex]\(PV\)[/tex]) = [tex]\(-932.69\)[/tex]
- Future Value ([tex]\(FV\)[/tex]) = 0 (loan fully paid off)
- Payments per Year ([tex]\(P/Y\)[/tex]) = 12
- Compounding periods per Year ([tex]\(C/Y\)[/tex]) = 12
3. Correct Option:
Given the options:
- Option A: [tex]\( N = 0.5; I \%= 21.6; PV = -900; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
- Option B: [tex]\( N = 6; I \%= 21.6; PV = -900; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
- Option C: [tex]\( N = 6; I \%= 21.6; PV = -932.69; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
- Option D: [tex]\( N = 0.5; I \%= 21.6; PV = -932.69; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
We need to find the group of values that reflect the correct payment schedule and interest calculation. As calculated, the correctly adjusted present value for the loan is [tex]\(-932.69\)[/tex] to be paid over 6 months.
Therefore, Option C is the correct choice:
[tex]\[ N=6 ; I \%=21.6 ; PV=-932.69 ; FV=0 ; P/Y=12 ; C/Y=12 \][/tex]
Caroline took out an 8-month loan for [tex]$900 with an APR (Annual Percentage Rate) of 21.6%, compounded monthly. The loan offers no payments for the first 2 months. Therefore, the effective period for monthly payments will be the remaining 6 months. To solve this, we need to adjust the present value (PV) of the loan to account for the interest accrued during the first two months when no payments are made. 1. Interest Calculation for No Payment Period: - The original loan amount is \(\$[/tex] 900\).
- The APR is 21.6%, compounded monthly, which means the monthly interest rate is [tex]\(\frac{21.6 \%}{12} = 1.8\%\)[/tex].
Over the first 2 months, the loan amount [tex]\(\$ 900\)[/tex] will accrue interest. The formula for the compound interest is:
[tex]\[ \text{PV adjusted} = PV \times \left(1 + \frac{I\%}{100 \times \text{C/Y}}\right)^{\text{number of months}} \][/tex]
Plugging in the values:
[tex]\[ \text{PV adjusted} = -900 \times \left(1 + \frac{1.8}{100}\right)^2 \][/tex]
Simplifying within the parentheses first:
[tex]\[ 1 + \frac{1.8}{100} = 1 + 0.018 = 1.018 \][/tex]
Raising this to the power of 2 months:
[tex]\[ (1.018)^2 \approx 1.036324 \][/tex]
Multiplying this result by the loan amount:
[tex]\[ \text{PV adjusted} = -900 \times 1.036324 \approx -932.6916 \][/tex]
So, the adjusted present value considering the interest accrued due to no payments for 2 months is approximately [tex]\(-932.69\)[/tex] dollars.
2. TVM Solver Variables:
- Caroline now has to pay this amount over the remaining 6 months.
- Number of payments ([tex]\(N\)[/tex]) = 6 (the remaining months)
- Interest rate per year ([tex]\(\%\)[/tex]) = 21.6% (APR)
- Adjusted Present Value ([tex]\(PV\)[/tex]) = [tex]\(-932.69\)[/tex]
- Future Value ([tex]\(FV\)[/tex]) = 0 (loan fully paid off)
- Payments per Year ([tex]\(P/Y\)[/tex]) = 12
- Compounding periods per Year ([tex]\(C/Y\)[/tex]) = 12
3. Correct Option:
Given the options:
- Option A: [tex]\( N = 0.5; I \%= 21.6; PV = -900; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
- Option B: [tex]\( N = 6; I \%= 21.6; PV = -900; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
- Option C: [tex]\( N = 6; I \%= 21.6; PV = -932.69; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
- Option D: [tex]\( N = 0.5; I \%= 21.6; PV = -932.69; FV = 0; P/Y = 12; C/Y = 12 \)[/tex]
We need to find the group of values that reflect the correct payment schedule and interest calculation. As calculated, the correctly adjusted present value for the loan is [tex]\(-932.69\)[/tex] to be paid over 6 months.
Therefore, Option C is the correct choice:
[tex]\[ N=6 ; I \%=21.6 ; PV=-932.69 ; FV=0 ; P/Y=12 ; C/Y=12 \][/tex]
