14. Assume Julian has a choice between two deposit accounts. Account A has an annual percentage rate of 7.55 percent but with interest compounded monthly. Account B has an annual percentage rate of 7.45 percent with interest compounded continuously. Which account provides the highest effective annual return? (Please calculate the arithmetic solution and show your work)

first account:

i = .0755/12 = .006291666..
amount of $1
= 1(1.006291666)^12 = 1.078168

second account:
amount of $1
= 1 e^(.0745) = 1.07734

What do you think?

To determine which account provides the highest effective annual return, we can calculate the effective annual rate (EAR) for each account.

For Account A, with interest compounded monthly, the formula for EAR is:

EAR = (1 + (APR / m))^m - 1

Where APR is the annual percentage rate and m is the number of compounding periods per year.

APR for Account A is 7.55%, and it compounds monthly, so m = 12.

EAR(A) = (1 + (0.0755 / 12))^12 - 1

Calculating this, we get:

EAR(A) ≈ (1 + 0.00629166667)^12 - 1
EAR(A) ≈ 0.0766 or 7.66%

For Account B, with continuous compounding, the formula for EAR is:

EAR = e^(APR) - 1

Where e is the mathematical constant approximately equal to 2.71828.

APR for Account B is 7.45%.

EAR(B) = e^(0.0745) - 1

Calculating this, we get:

EAR(B) ≈ 2.71828^(0.0745) - 1
EAR(B) ≈ 0.0772 or 7.72%

Comparing the two effective annual rates, we find that Account B with continuous compounding provides the highest effective annual return at approximately 7.72%.

To determine which account provides the highest effective annual return, we need to compare the effective annual rates (EAR) of both accounts.

Let's start with Account A:
The annual percentage rate (APR) of Account A is 7.55 percent, but the interest is compounded monthly. To find the effective annual rate, we can use the formula:
EAR = (1 + APR/n)^n - 1,
where n is the number of compounding periods per year.

In this case, since interest is compounded monthly, n = 12 (number of months in a year).

EAR_A = (1 + 7.55%/12)^12 - 1
EAR_A = (1 + 0.062916)^12 - 1
EAR_A ≈ 0.076313 or 7.6313%

Moving on to Account B:
The APR of Account B is 7.45 percent, and interest is compounded continuously. To calculate the EAR for continuous compounding, we use the formula:
EAR = e^APR - 1,
where e is the mathematical constant approximately equal to 2.71828.

EAR_B = e^(7.45%) - 1
EAR_B = e^(0.0745) - 1
EAR_B ≈ 0.077661 or 7.7661%

Comparing the two EARs, we find that Account B (with continuous compounding) has a higher effective annual return of approximately 7.7661%, while Account A (with monthly compounding) has an effective annual return of approximately 7.6313%.

Therefore, Account B provides the higher effective annual return.

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