Question

Your bank account pays interest with an EAR of 4 %. What is the APR quote for this account based on semiannual​ compounding? What is the APR with monthly​ compounding?​ (Note: Be careful not to round any intermediate steps less than six decimal​ places.)

Answer 1

Answer:

APR would be 3.9607805% compounded semiannually

APR would be 3.9284877% compounded monthly

Explanation:

We calcualte consider these rates should be inancially equivalent to an equivalent rate of 4% annually.

The semiannual compounding will capitalize two times:

$(1+APR/2)^2 = 1+EAR\\(\sqrt{1.04} -1) \times 2 = 0.039607805$

The monthly will capitalize 12 times per year:

$(1+APR/12)^12 = 1+EAR\\(\sqrt[12]{1.04} -1) \times 12 = 0.039284877$

Answer 2

Following are the solution to the given points:

for point a:

When semi-annual compounded is used, then APR will be calculated.

EAR (effective annual rate)= 4%

$\to EAR = [ 1 + \frac{APR}{M}]^{MN - 1}$

M = number of compoundings in a year = 2 semi-annual compoundings

The number of years (N)= 1

$\to 4\% = [ 1 + \frac{APR}{2}]^{2\times 1 - 1}\\\\ \to 4\% = [ 1 + \frac{APR}{2}]^{2 - 1}\\\\\to 4\% +1= [ 1 + \frac{APR}{2}]^{2}\\\\\to 1.04^{\frac{1}{2}} = [ 1 + \frac{APR}{2} ]\\\\\to [1.019803902 - 1 ] \times 2 = APR$

APR = Annual percentage rate if compounded semi-annually $\to APR = 3.9607 \%$

for point b:

Calculate your APR if Monthly COMPOUNDING

EAR (Effective annual rate) = 4%

 $\to EAR = [ 1 + \frac{APR}{M} ]^{MN - 1}$

M is the number of compoundings in a year, which is equal to 12 months.

Number of years (N) = 1

$\to 4\% = [ 1 + \frac{APR}{12}]^{12 \times 1 - 1} \\\\\to 4\% + 1 = [ 1 + \frac{APR}{12} ] ^{12 \times 1} \\\\ \to 1.04^{\frac{1}{12}} = [ 1 + \frac{APR}{12}]\\\\\to [1.00327373978 - 1 ] \times 12 = APR$  

APR = Annual percentage rate increased semi-annually

$\to APR = 3.928487\%$

Learn more:

brainly.com/question/23309989