Math / Algebra

QuestionFirst National Bank charges 14.1 percent compounded monthly on its business loans. First United Bank charges 14.4 percent compounded semiannually. Calculate the EAR for First National Bank and First United Bank (Do not round intermediate calculations and enter your answers as a percent rounded to 2 decimal places, e.g., 32.16.) \begin{tabular}{|l|l|l|} \hline First National & & $\%$ \\ \hline First United & & $\%$ \\ \hline \end{tabular}
As a potential borrower, to which bank would you go for a new loan? First National Bank First United Bank

Studdy Solution

STEP 1

1. The Effective Annual Rate (EAR) is calculated to compare the annual interest rates of loans with different compounding periods.
2. The formula for the EAR is given by: $ \text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1 \] where $ r $ is the nominal annual interest rate and $ n $ is the number of compounding periods per year.
3. The problem requires finding the EAR for both banks and comparing them to determine which bank offers a more favorable loan in terms of lower effective annual interest.

STEP 2

1. Calculate the EAR for First National Bank.
2. Calculate the EAR for First United Bank.
3. Compare the EARs to determine which bank offers a lower effective annual rate.

STEP 3

Identify the nominal annual interest rate and the number of compounding periods for First National Bank. The nominal rate $r$ is 14.1%, and it is compounded monthly ( $n = 12$ ).
$r = 0.141, \quad n = 12$

STEP 4

Use the formula for EAR to calculate the effective annual rate for First National Bank.
$\text{EAR}_{\text{First National}} = \left(1 + \frac{0.141}{12}\right)^{12} - 1$

STEP 5

Calculate the intermediate value $\frac{0.141}{12}$ .
$\frac{0.141}{12} = 0.01175$

STEP 6

Calculate $\left(1 + 0.01175\right)^{12}$ .
$\left(1 + 0.01175\right)^{12} \approx 1.149$

STEP 7

Subtract 1 to find the EAR for First National Bank.
$\text{EAR}_{\text{First National}} = 1.149 - 1 = 0.149$

STEP 8

Convert the EAR to a percentage.
$\text{EAR}_{\text{First National}} = 0.149 \times 100\% = 14.90\%$

STEP 9

Identify the nominal annual interest rate and the number of compounding periods for First United Bank. The nominal rate $r$ is 14.4%, and it is compounded semiannually ( $n = 2$ ).
$r = 0.144, \quad n = 2$

STEP 10

Use the formula for EAR to calculate the effective annual rate for First United Bank.
$\text{EAR}_{\text{First United}} = \left(1 + \frac{0.144}{2}\right)^{2} - 1$

STEP 11

Calculate the intermediate value $\frac{0.144}{2}$ .
$\frac{0.144}{2} = 0.072$

STEP 12

Calculate $\left(1 + 0.072\right)^{2}$ .
$\left(1 + 0.072\right)^{2} \approx 1.148$

STEP 13

Subtract 1 to find the EAR for First United Bank.
$\text{EAR}_{\text{First United}} = 1.148 - 1 = 0.148$

STEP 14

Convert the EAR to a percentage.
$\text{EAR}_{\text{First United}} = 0.148 \times 100\% = 14.80\%$

STEP 15

Compare the EARs of both banks to determine which bank offers a lower effective annual rate.
$\text{EAR}_{\text{First National}} = 14.90\%, \quad \text{EAR}_{\text{First United}} = 14.80\%$
Since 14.80% is less than 14.90%, First United Bank offers a lower effective annual rate.
Solution: \begin{tabular}{|l|l|l|} \hline First National & 14.90 & $\%$ \\ \hline First United & 14.80 & $\%$ \\ \hline \end{tabular}
As a potential borrower, you would go to First United Bank for a new loan.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the EAR for First National Bank.
2. Calculate the EAR for First United Bank.
3. Compare the EARs to determine which bank offers a lower effective annual rate.

STEP 3

Identify the nominal annual interest rate and the number of compounding periods for First National Bank. The nominal rate $r$ is 14.1%, and it is compounded monthly ( $n = 12$ ).
$r = 0.141, \quad n = 12$

STEP 4

Use the formula for EAR to calculate the effective annual rate for First National Bank.
$\text{EAR}_{\text{First National}} = \left(1 + \frac{0.141}{12}\right)^{12} - 1$

STEP 5

Calculate the intermediate value $\frac{0.141}{12}$ .
$\frac{0.141}{12} = 0.01175$

STEP 6

Calculate $\left(1 + 0.01175\right)^{12}$ .
$\left(1 + 0.01175\right)^{12} \approx 1.149$

STEP 7

Subtract 1 to find the EAR for First National Bank.
$\text{EAR}_{\text{First National}} = 1.149 - 1 = 0.149$

STEP 8

Convert the EAR to a percentage.
$\text{EAR}_{\text{First National}} = 0.149 \times 100\% = 14.90\%$

STEP 9

Identify the nominal annual interest rate and the number of compounding periods for First United Bank. The nominal rate $r$ is 14.4%, and it is compounded semiannually ( $n = 2$ ).
$r = 0.144, \quad n = 2$

STEP 10

Use the formula for EAR to calculate the effective annual rate for First United Bank.
$\text{EAR}_{\text{First United}} = \left(1 + \frac{0.144}{2}\right)^{2} - 1$

STEP 11

Calculate the intermediate value $\frac{0.144}{2}$ .
$\frac{0.144}{2} = 0.072$

STEP 12

Calculate $\left(1 + 0.072\right)^{2}$ .
$\left(1 + 0.072\right)^{2} \approx 1.148$

STEP 13

Subtract 1 to find the EAR for First United Bank.
$\text{EAR}_{\text{First United}} = 1.148 - 1 = 0.148$

STEP 14

Convert the EAR to a percentage.
$\text{EAR}_{\text{First United}} = 0.148 \times 100\% = 14.80\%$

STEP 15

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Studdy Solution

STEP 1

STEP 2

1. Calculate the EAR for First National Bank.2. Calculate the EAR for First United Bank.3. Compare the EARs to determine which bank offers a lower effective annual rate.

STEP 3

Identify the nominal annual interest rate and the number of compounding periods for First National Bank. The nominal rate r r r is 14.1%, and it is compounded monthly (n=12 n = 12 n=12).r=0.141,n=12r = 0.141, \quad n = 12r=0.141,n=12

STEP 4

Use the formula for EAR to calculate the effective annual rate for First National Bank.EARFirst National=(1+0.14112)12−1\text{EAR}_{\text{First National}} = \left(1 + \frac{0.141}{12}\right)^{12} - 1EARFirst National​=(1+120.141​)12−1

STEP 5

Calculate the intermediate value 0.14112 \frac{0.141}{12} 120.141​.0.14112=0.01175\frac{0.141}{12} = 0.01175120.141​=0.01175

STEP 6

Calculate (1+0.01175)12 \left(1 + 0.01175\right)^{12} (1+0.01175)12.(1+0.01175)12≈1.149\left(1 + 0.01175\right)^{12} \approx 1.149(1+0.01175)12≈1.149

STEP 7

Subtract 1 to find the EAR for First National Bank.EARFirst National=1.149−1=0.149\text{EAR}_{\text{First National}} = 1.149 - 1 = 0.149EARFirst National​=1.149−1=0.149

STEP 8

Convert the EAR to a percentage.EARFirst National=0.149×100%=14.90%\text{EAR}_{\text{First National}} = 0.149 \times 100\% = 14.90\%EARFirst National​=0.149×100%=14.90%

STEP 9

Identify the nominal annual interest rate and the number of compounding periods for First United Bank. The nominal rate r r r is 14.4%, and it is compounded semiannually (n=2 n = 2 n=2).r=0.144,n=2r = 0.144, \quad n = 2r=0.144,n=2

STEP 10

Use the formula for EAR to calculate the effective annual rate for First United Bank.EARFirst United=(1+0.1442)2−1\text{EAR}_{\text{First United}} = \left(1 + \frac{0.144}{2}\right)^{2} - 1EARFirst United​=(1+20.144​)2−1

STEP 11

Calculate the intermediate value 0.1442 \frac{0.144}{2} 20.144​.0.1442=0.072\frac{0.144}{2} = 0.07220.144​=0.072

STEP 12

Calculate (1+0.072)2 \left(1 + 0.072\right)^{2} (1+0.072)2.(1+0.072)2≈1.148\left(1 + 0.072\right)^{2} \approx 1.148(1+0.072)2≈1.148

STEP 13

Subtract 1 to find the EAR for First United Bank.EARFirst United=1.148−1=0.148\text{EAR}_{\text{First United}} = 1.148 - 1 = 0.148EARFirst United​=1.148−1=0.148

STEP 14

Convert the EAR to a percentage.EARFirst United=0.148×100%=14.80%\text{EAR}_{\text{First United}} = 0.148 \times 100\% = 14.80\%EARFirst United​=0.148×100%=14.80%

STEP 15

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1. Calculate the EAR for First National Bank.
2. Calculate the EAR for First United Bank.
3. Compare the EARs to determine which bank offers a lower effective annual rate.

Identify the nominal annual interest rate and the number of compounding periods for First National Bank. The nominal rate $r$ is 14.1%, and it is compounded monthly ( $n = 12$ ).
$r = 0.141, \quad n = 12$

Use the formula for EAR to calculate the effective annual rate for First National Bank.
$\text{EAR}_{\text{First National}} = \left(1 + \frac{0.141}{12}\right)^{12} - 1$

Calculate the intermediate value $\frac{0.141}{12}$ .
$\frac{0.141}{12} = 0.01175$

Calculate $\left(1 + 0.01175\right)^{12}$ .
$\left(1 + 0.01175\right)^{12} \approx 1.149$

Subtract 1 to find the EAR for First National Bank.
$\text{EAR}_{\text{First National}} = 1.149 - 1 = 0.149$

Convert the EAR to a percentage.
$\text{EAR}_{\text{First National}} = 0.149 \times 100\% = 14.90\%$

Identify the nominal annual interest rate and the number of compounding periods for First United Bank. The nominal rate $r$ is 14.4%, and it is compounded semiannually ( $n = 2$ ).
$r = 0.144, \quad n = 2$

Use the formula for EAR to calculate the effective annual rate for First United Bank.
$\text{EAR}_{\text{First United}} = \left(1 + \frac{0.144}{2}\right)^{2} - 1$

Calculate the intermediate value $\frac{0.144}{2}$ .
$\frac{0.144}{2} = 0.072$

Calculate $\left(1 + 0.072\right)^{2}$ .
$\left(1 + 0.072\right)^{2} \approx 1.148$

Subtract 1 to find the EAR for First United Bank.
$\text{EAR}_{\text{First United}} = 1.148 - 1 = 0.148$

Convert the EAR to a percentage.
$\text{EAR}_{\text{First United}} = 0.148 \times 100\% = 14.80\%$