Chapter 4, Problem 2CS

To determine

Total worth of plan B.

Explanation of Solution

Time period is 15 years. Down payment (DP) is 10%. Fixed rate (i) is 5%. Price of house (Pr) is $330,000. Fee (UF) is $3,000.

Total payment (TP) can be calculated as follows:

$\begin{array}{c}\text{Total payment}=\left(\text{Pr}\times \text{DP}\right)+\text{UF}\\ =\left(330,000\times 0.1\right)+3,000\\ =33,000+3,000\\ =36,000\end{array}$

Total payment is $36,000.

Loan amount (LM) can be calculated as follows:

$\begin{array}{c}\text{Loan payment}=\text{Pr}-\left(\text{Pr}\times \text{DP}\right)\\ =330,000-\left(330,000\times 0.1\right)\\ =330,000-33,000\\ =297,000\end{array}$

Loan payment is $297,000.

Tax and insurance (TI) is $500 per month. Interest rate per month (im) is 0.4167% $\left(\frac{5}{12}\right)$. Time period (n) is 180 $\left(15\times 12\right)$ months. Equivalent annual value of monthly payment can be calculated as follows:

$\begin{array}{c}\text{A}=\text{LM}\left(\frac{\text{im}{\left(1+\text{im}\right)}^{\text{n}}}{{\left(1+\text{im}\right)}^{\text{n}}-1}\right)+\text{TI}\\ =\text{297,000}\left(\frac{\text{0}\text{.004167}{\left(1+\text{0}\text{.004167}\right)}^{\text{180}}}{{\left(1+\text{0}\text{.004167}\right)}^{\text{180}}-1}\right)+500\\ =\text{297,000}\left(\frac{\text{0}\text{.004167}\left(2.1138302\right)}{2.1138302}\right)+500\\ =\text{297,000}\left(\frac{0.0088083}{1.1138302}\right)+500\\ =\text{297,000}\left(0.00790081\right)+500\\ =2,346.54+500\\ =2,846.54\end{array}$

Equivalent monthly loan repayment is $2,847 per month.

The remaining money after monthly payment and down payment (ia) is deducted, earns 0.5% interest rate per month. Available amount (AV) is $40,000. Time period (n1) is 120 $\left(10\times 12\right)$ months.  Future worth of available amount (FA) can be calculated as follows:

$\begin{array}{c}\text{FA}=\left(\text{AV}-\text{TP}\right){\left(1+\text{ia}\right)}^{\text{n1}}\\ =\left(40,000-36,000\right){\left(1+0.005\right)}^{\text{120}}\\ =\left(4,000\right)\left(1.819397\right)\\ =7,277.59\end{array}$

Future value of remaining available amount is $7,278.

The person can afford the monthly payment (MP) of $2,850. The future value of remaining monthly available amount (FM) can be calculated as follows:

$\begin{array}{c}\text{FM}=\left(\text{MP}-\text{A}\right){\left(1+\text{ia}\right)}^{\text{n1}}\\ =\left(2,850-2,847\right){\left(1+0.005\right)}^{\text{120}}\\ =\left(3\right)\left(1.819397\right)\\ =5.46\end{array}$

The future value of remaining monthly available amount is $5.

Future value of loan balance (FB) can be calculated as follows:

$\begin{array}{c}\text{FB}=\text{LM}{\left(1+\text{im}\right)}^{\text{n1}}-\text{A}\left(\frac{{\left(1+\text{im}\right)}^{\text{n}}-1}{\text{im}}\right)\\ =297,000{\left(1+\text{0}\text{.004167}\right)}^{\text{120}}-2,847\left(\frac{{\left(1+\text{0}\text{.004167}\right)}^{\text{120}}-1}{0.004167}\right)\\ =297,000\left(1.647075\right)-2,847\left(\frac{1.647075-1}{0.004167}\right)\\ =297,000\left(1.647075\right)-2,847\left(\frac{0.647075}{0.004167}\right)\\ =297,000\left(1.647075\right)-2,847\left(155.285577\right)\\ =489,181.275-442,098.038\\ =47,083.24\end{array}$

Future value of loan balance is $47,083.

Selling price (SP) is $231,000. Increase in the future value of house (FH) can be calculated as follows:

$\begin{array}{c}\text{FH}=\text{SP}-\text{FB}\\ =231,000-47,083\\ =183,917\end{array}$

Future value of house is $183,917.

Total future worth (TFW) of plan B can be calculated as follows:

$\begin{array}{c}\text{TFW}=\text{FA}+\text{FM}+\text{FH}\\ =7,278+5+183,917\\ =191,200\end{array}$

Total future worth of plan B is $191,200.

The value of FA for plan A is 7,278. The value of FM for plan A is $116,354. The value of Loan balance for plan A is 243,386.

Total future worth (TFW) of plan A can be calculated as follows:

$\begin{array}{c}\text{TFW}=\text{FA}+\text{FM}+\text{FH}\\ =7,278+116,354+\left(231,000-243,386\right)\\ =111,246\end{array}$

Total future worth of plan A is $111,246. Total future worth of plan B is greater than plan A. Thus, select plan B.