Chapter 7.8, Problem 18PFA

a.

To determine

To calculate: The effective monthly interest rate for the saving account at XYZ savings.

Answer to Problem 18PFA

So, then the effective monthly interest rate is 0.15%.

Explanation of Solution

Given: Sharonda is comparing the saving account that are offered at the two banks.

  

Concept Used:

Use the compound interest formula:

  $A=P{(1+\frac{r}{n})}^{nt}.$

Where P is the initial investment, r is the interest rate , n is the number of times the interest compounds each year ,and t is the number of years

Calculation:

Here substitute $P=1andr=1.8\%=0.018andn=1$ in the compound interest formula.

  $\begin{array}{}$ A=1{(1+\frac{0.018}{1})}^{1.t}.$$ A={(1.018)}^{1.t}.$$ A={({1.018}^{\frac{1}{12}.12})}^{1.t}.$$ A={({1.018}^{\frac{1}{12}})}^{12t}.$$ A\approx {1.0015}^{12t}.$\end{array}$

The effective monthly interest rate is then 1.0015-1=0.0015=0.15%.

So, then the effective monthly interest rate is 0.15%.

b.

To determine

To find: The bank which offer a better deal.

Answer to Problem 18PFA

The bank which offer a better deal is XYZ Savings.

Explanation of Solution

Given: Sharonda is comparing the saving account that are offered at the two banks.

  

Calculation:

It is given that Statewide Financial has a monthly interest rate of 0.12% . From part (a). XYZ savings has an effective monthly interest of 0.15%. XYZ saving then offers a better deal since it has a higher monthly interest rate.

c.

To determine

To calculate: The effective annual interest rate for the saving account at Statewide Financial.

Answer to Problem 18PFA

The effective annual interest rate is 1.4%.

Explanation of Solution

Given: Sharonda is comparing the saving account that are offered at the two banks.

  

Concept Used:

Use the compound interest formula:

  $A=P{(1+\frac{r}{n})}^{nt}.$

Where P is the initial investment, r is the interest rate, n is the number of times the interest compounds each year ,and t is the number of years

Calculation:

Here substitute $P=1andr=0.12\%=0.0012andn=12$ in the compound interest formula.

  $\begin{array}{}$ A=1{(1+\frac{0.0012}{1})}^{12.t}.$$ A={(1.0012)}^{12.t}.$$ A={(1.0012)}^{12.t}.$$ A={(1.0012)}^{12t}.$$ A\approx {1.014}^t.$\end{array}$

The effective annual interest rate is then 1.014-1=0.014=1.4%.

Conclusion:

The effective annual interest rate is 1.4%.

d.

To determine

To identify: The bank out of three given banks Sharonda should choose.

Answer to Problem 18PFA

Capital Street.

Explanation of Solution

Given: Sharonda is comparing the saving account that are offered at the two banks.

  

Use the compound interest formula:

  $A=P{(1+\frac{r}{n})}^{nt}.$

Where P is the initial investment, r is the interest rate , n is the number of times the interest compounds each year ,and t is the number of years

Calculation:

Here, substitute $P=1\text{and}\frac{r}{n}=0.55\%=0.0055\text{and}n=4$ in the compound interest formula.

  $\begin{array}{}$ A=1{(1+\frac{r}{n})}^{4.t}.$$ A={(1+0.0055)}^{4.t}.$$ A={(1.0055)}^{4.t}.$$ A={(1.0055)}^{4t}.$$ A\approx {1.022}^t.$\end{array}$

The effective annual interest rate is then 1.022-1=0.022=2.2%.

Conclusion:

This is higher than the annual interest of 1.8% for XYZ Savings and the effective annual interest rate of 1.4% for Statewide Financial so Capital Street is the bank she should choose.