QuestionFind the account with the highest effective annual interest rate among four options with different compounding frequencies and rates.

Studdy Solution

STEP 1

Assumptions1. Account1 Interest is compounded quarterly at an annual rate of4.20%. . Account Interest is compounded monthly at an annual rate of4.15%.
3. Account3 Interest is compounded semiannually at an annual rate of4.10%.
4. Account4 Interest is compounded annually at a rate of4.25%.

STEP 2

We need to calculate the effective annual rate (EAR) for each account. The formula for EAR is $EAR = (1 + i/n)^{n*t} -1$ where- i is the nominal interest rate- n is the number of compounding periods per year- t is the number of yearsIn this case, t =1 year for all accounts.

STEP 3

First, let's calculate the EAR for Account1. The nominal interest rate i =.20% and n = (quarterly compounding).
$EAR_{1} = (1 +0.0420/)^{*1} -1$

STEP 4

Calculate the EAR for Account1.
$EAR_{1} = (1 +0.010)^{4} -1$

STEP 5

Calculate the EAR for Account1.
$EAR_{1} =1.0420744 -1 =0.0420744$

STEP 6

Convert the EAR for Account1 back to a percentage.
$EAR_{1} =0.0420744 \times100\% =4.20744\%$

STEP 7

Now, let's calculate the EAR for Account2. The nominal interest rate i =4.15% and n =12 (monthly compounding).
$EAR_{2} = (1 +0.0415/12)^{12*1} -1$

STEP 8

Calculate the EAR for Account2.
$EAR_{2} = (1 +0.00345833)^{12} -1$

STEP 9

Calculate the EAR for Account2.
$EAR_{2} =.042601 - =.042601$

STEP 10

Convert the EAR for Account2 back to a percentage.
$EAR_{2} =0.042601 \times100\% =4.260\%$

STEP 11

Next, let's calculate the EAR for Account3. The nominal interest rate i =4.10% and n = (semiannual compounding).
$EAR_{3} = ( +0.0410/)^{*} -$

STEP 12

Calculate the EAR for Account.
$EAR_{} = ( +0.0205)^{2} -$

STEP 13

Calculate the EAR for Account3.
$EAR_{3} =.0412025 - =0.0412025$

STEP 14

Convert the EAR for Account3 back to a percentage.
$EAR_{3} =0.041202 \times100\% =4.12025\%$

STEP 15

Finally, let's calculate the EAR for Account4. The nominal interest rate i =4.25% and n = (annual compounding).
$EAR_{4} = ( +0.0425/)^{*} -$

STEP 16

Calculate the EAR for Account4.
$EAR_{4} = ( +0.0425)^{} -$

STEP 17

Calculate the EAR for Account4.
$EAR_{4} =.0425 - =0.0425$

STEP 18

Convert the EAR for Account4 back to a percentage.
$EAR_{4} =0.0425 \times100\% =4.25\%$

STEP 19

Now that we have the EAR for all accounts, we can compare them to find the account with the highest EAR.
The EAR for each account is- Account14.20744% - Account4.26011% - Account34.12025% - Account44.25%
Account has the highest effective annual interest rate.

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