Chapter 4, Problem 44P

(a):

To determine

Calculate the equivalent monthly value.

Explanation of Solution

The borrowing (B) is $80,000. The interest rate (i1) is 6% per year and it is compounded semiannually for two payments. The effective interest rate 1 (Ei1) is 0.5% $\left(\frac{6}{12}\right)$. The interest rate (i2) is 4.2% per year and it is compounded semiannually for third payments. The effective interest rate 2 (Ei2) is 0.35% $\left(\frac{4.2}{12}\right)$. The time period (n) is 60 months $\left(12\times 5\right)$.

The equivalent monthly value (A) can be calculated as follows:

$\begin{array}{c}\text{A}=\text{B}\left(\frac{\text{Ei1}{\left(1+\text{Ei1}\right)}^{\text{n}}}{{\left(1+\text{Ei1}\right)}^{\text{n}}-1}\right)\\ =\text{80,000}\left(\frac{\text{0}\text{.005}{\left(1+\text{0}\text{.005}\right)}^{\text{60}}}{{\left(1+\text{0}\text{.005}\right)}^{\text{60}}-1}\right)\\ =\text{80,000}\left(\frac{\text{0}\text{.005}\left(1.3488502\right)}{1.3488502-1}\right)\\ =\text{80,000}\left(\frac{0.0067443}{0.3488502}\right)\\ =\text{80,000}\left(0.01933\right)\\ =1,546.4\end{array}$

The equivalent monthly value is $1,546.4.

(b):

To determine

Calculate the current balance.

Explanation of Solution

The interest payment in the first year can be calculated as follows:

$\begin{array}{c}\text{Interest payment 1}=\text{B}\times \text{Ei1}\\ =80,000\times 0.005\\ =400\end{array}$

The interest payment for the first year is $400.

The interest payment in the second year can be calculated as follows:

$\begin{array}{c}\text{Interest payment 2}=\left(\text{B}-\left(\text{A}-\text{Interest payment 1}\right)\right)\times \text{Ei}\\ =\left(80,000-\left(1,546.40-400\right)\right)\times 0.005\\ =\left(80,000-1,146.40\right)\times 0.005\\ =78,853360\times 0.005\\ =394.27\end{array}$

The interest payment for the second year is $394.27.

The current principal payment (CP) can be calculated as follows:

$\begin{array}{c}\text{CP}=\text{B}-\left(\left(\text{A}\times \text{2}\right)-\left(\begin{array}{l}\text{Interest payment 1}\\ -\text{Interest payment 2}\end{array}\right)\right)\\ =80,000-\left(\left(\text{1,546}\text{.4}\times \text{2}\right)-400-394.27\right)\\ =80,000-\left(3,092.8-400-394.27\right)\\ =80,000-2,298.53\\ =77,701.47\end{array}$

The current principal payment is $77,701.47.

(c):

To determine

Calculate the total interest payment.

Explanation of Solution

The total interest payment for the first two years can be calculated as follows:

$\begin{array}{c}\text{Total interest payment}=\left(\begin{array}{l}\text{Interest payment 1}\\ +\text{Interest payment 2}\end{array}\right)\\ =400+394.27\\ =794.27\end{array}$

The total interest payment for the first two years is $794.27.

(d):

To determine

Calculate the equivalent monthly value.

Explanation of Solution

The borrowing (B) is $80,000. The interest rate (i1) is 6% per year and it is compounded semiannually for two payments. The effective interest rate 1 (Ei1) is 0.5% $\left(\frac{6}{12}\right)$. The interest rate (i2) is 4.2% per year and it is compounded semiannually for third payment. The effective interest rate 2 (Ei2) is 0.35% $\left(\frac{4.2}{12}\right)$. The time period (n) is 60 months $\left(12\times 5\right)$.

The equivalent monthly value (A) for second loan can be calculated as follows:

$\begin{array}{c}\text{A}=\text{Current principal payment}\left(\frac{\text{Ei2}{\left(1+\text{Ei2}\right)}^{\text{n}}}{{\left(1+\text{Ei2}\right)}^{\text{n}}-1}\right)\\ =\text{77,701}\text{.47}\left(\frac{\text{0}\text{.0035}{\left(1+\text{0}\text{.0035}\right)}^{\text{60}}}{{\left(1+\text{0}\text{.0035}\right)}^{\text{60}}-1}\right)\\ =\text{77,701}\text{.47}\left(\frac{\text{0}\text{.0035}\left(1.233226\right)}{1.233226-1}\right)\\ =\text{77,701}\text{.47}\left(\frac{0.004316}{0.233226}\right)\\ =\text{77,701}\text{.47}\left(0.018506\right)\\ =1,437.94\end{array}$

The equivalent monthly value is $1,437.94.