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QuestionWhich account has the highest annual interest rate: A. 3.28%3.28\%, B. 3.25%3.25\%, C. 3.10%3.10\%, D. 3.15%3.15\%, E. 3.25%3.25\%?

Studdy Solution

STEP 1

Assumptions1. The interest rates given are nominal rates, not effective annual rates. . The compounding frequency is different for each account.
3. We need to find the effective annual rate (EAR) for each account to compare them.

STEP 2

The formula to calculate the effective annual rate (EAR) isEAR=(1+in)n⋅t−1EAR = (1 + \frac{i}{n})^{n \cdot t} -1where- ii is the nominal interest rate, - nn is the number of compounding periods per year, - tt is the number of years.

STEP 3

For Account1, the interest is compounded annually, so n=1n =1 and i=3.28%=0.0328i =3.28\% =0.0328.
EAR1=(1+0.03281)1⋅1−1EAR_{1} = (1 + \frac{0.0328}{1})^{1 \cdot1} -1

STEP 4

Calculate the EAR for Account1.
EAR1=(1+0.0328)1−1=0.0328=3.28%EAR_{1} = (1 +0.0328)^{1} -1 =0.0328 =3.28\%

STEP 5

For Account2, the interest is compounded monthly, so n=12n =12 and i=3.25%=0.0325i =3.25\% =0.0325.
EAR2=(1+0.032512)12⋅1−1EAR_{2} = (1 + \frac{0.0325}{12})^{12 \cdot1} -1

STEP 6

Calculate the EAR for Account2.
EAR2=(1+0.032512)12−1≈0.0331=3.31%EAR_{2} = (1 + \frac{0.0325}{12})^{12} -1 \approx0.0331 =3.31\%

STEP 7

For Account3, the interest is compounded weekly, so n=52n =52 and i=3.10%=0.031i =3.10\% =0.031.
EAR3=(1+0.03152)52⋅1−1EAR_{3} = (1 + \frac{0.031}{52})^{52 \cdot1} -1

STEP 8

Calculate the EAR for Account3.
EAR3=(1+0.03152)52−1≈0.0316=3.16%EAR_{3} = (1 + \frac{0.031}{52})^{52} -1 \approx0.0316 =3.16\%

STEP 9

For Account4, the interest is compounded daily, so n=365n =365 and i=3.15%=.0315i =3.15\% =.0315.
EAR4=(+.0315365)365⋅−EAR_{4} = ( + \frac{.0315}{365})^{365 \cdot} -

STEP 10

Calculate the EAR for Account4.
EAR4=(+0.0315365)365−≈0.0320=3.20%EAR_{4} = ( + \frac{0.0315}{365})^{365} - \approx0.0320 =3.20\%

STEP 11

For Account5, the interest is compounded quarterly, so n=4n =4 and i=3.25%=0.0325i =3.25\% =0.0325.
EAR5=(+0.03254)4⋅−EAR_{5} = ( + \frac{0.0325}{4})^{4 \cdot} -

STEP 12

Calculate the EAR for Account5.
EAR5=(+0.03254)4−≈0.0330=.30%EAR_{5} = ( + \frac{0.0325}{4})^{4} - \approx0.0330 =.30\%

STEP 13

Now that we have the EAR for each account, we can compare them to find the account with the highest annual interest rate.
Account2 has the highest annual interest rate with an EAR of3.31%.
The correct answer is B. Account2.

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