Chapter 6, Problem 77P

To determine

(a)

The monthly leasing cost for $1$ apartment.

Answer to Problem 77P

The monthly leasing cost for $1$ apartment is $\$58853.13$.

Explanation of Solution

Given:

The cost of land is $\$3.2\text{ M}$.

The cost of building is $\$4.8\text{ M}$.

The annual operating and maintenance cost is $\$850000$.

The annual property tax and insurance is $6\%$ of initial value of building.

The vacancy rate is $12\%$ of initial value of building.

The owner wants rate of return of $15\%$.

Concept used:

Write the expression for initial value of building.

$V_I=C_L+C_B$ ...... (I)

Here, the initial value of building is $V_1$, cost of land is $C_L$ and cost of building is $C_B$.

Write the expression for salvage value at the end of $20$ years.

$S=C_L+\left(40\%\times C_B\right)$ ...... (II)

Here, the salvage value is S.

Write the expression for annual expenses.

$E=C_{OM}+P$ ...... (III)

Here, operating cost is $C_{OM}$ and property tax and insurance is $P$.

Write the expression for principle.

$P=\text{Annual investment}+E$ ...... (IV)

Write the expression to calculate monthly leasing total amount.

$V_1=\left(A-P\right)\left[\frac{{\left(1+i\right)}^n-1}{i{\left(1+i\right)}^n}\right]+S\left[\frac{1}{{\left(1+i\right)}^n}\right]$ ...... (V)

Here, monthly leasing total amount is $A$.

Write the expression for monthly leasing cost for $1$ apartment.

$M=\frac{A}{N}$ ...... (VI)

Here number of rented apartment is $N$.

Calculations:

Calculate the initial value of building.

Substitute $\$3.2\text{ M}$ for $C_L$ and $\$4.8\text{ M}$ for $C_B$ in Equation (I).

$\begin{array}{c}{V}_I=\$3.2\text{ M}+\$4.8\text{ M}\\ =\$8\text{ M}\\ =\$8000000\end{array}$

Calculate salvage value at the end of $35$ years.

Substitute $\$3.2\text{ M}$ for $C_L$ and $\$4.8\text{ M}$ for $C_B$ in Equation (II).

$\begin{array}{c}S=\$\text{3}\text{.2 M}+\left(\frac{10}{100}\times \$4.8\text{ M}\right)\\ =\$3.2\text{ M}+\$0.48\text{ M}\\ =\$3.68\text{ M}\\ \text{=\$3680000}\end{array}$

Calculate the annual investment.

$\begin{array}{c}\text{Annual investment}=\frac{\left(\$8\text{ M}-\$3.68\text{ M}\right)}{35}\\ =\frac{\$4.32\text{ M}}{35}\\ =\$123428.57\end{array}$

Calculate annual expenses.

Substitute $\$850000$ for $C_{OM}$ and $6\%\times \$8\text{ M}$ for $P$ in Equation (III).

$\begin{array}{c}E=\$850000+\left(6\%\times \$8\text{ M}\right)\\ =\$850000+\$480000\\ =\$1330000\end{array}$

Calculate the principle.

Substitute $\$1330000$ for $E$ and $\$123428.57$ for annual investment in Equation (IV).

$\begin{array}{c}P=\$123428.57+\$1330000\\ =\$1453428.57\end{array}$

Calculate the total amount for leasing 30 units.

Substitute $\$1453428.57$ for $P$, $\$8000000$ for $V_I$, $\text{\$3680000}$ for $S$, $\frac{0.15}{12}=0.0125$ for $i$ and $12\times 35=420$ for $n$ in Equation (V).

$\begin{array}{l}\$8000000=\left[\begin{array}{l}\left(A-\$1453428.57\right)\left[\frac{{\left(1+0.0125\right)}^{420}-1}{0.0125{\left(1+0.0125\right)}^{420}}\right]\\ +\text{\$3680000}\left[\frac{1}{{\left(1+0.0125\right)}^{420}}\right]\end{array}\right]\\ \$8000000=\left(A-\$1453428.57\right)\left(79.5663\right)+\text{\$3680000}\left(0.0054\right)\\ \$8000000=A\left(79.5663\right)-\$115643933.6+\text{\$}19949.61\\ A=\$1553722.67\end{array}$

Calculate monthly leasing cost for $1$ apartment.

Substitute $\$1553722.67$ for $A$ and $88\%\times 30$ for $N$ in Equation (VI).

$\begin{array}{c}\text{Monthly cost}=\frac{\$1553722.67}{88\%\times 30}\\ =\frac{\$1553722.67}{26.4}\\ =\$58853.13\end{array}$

The monthly leasing cost for $1$ apartment is $\$58853.13$.

Conclusion:

The monthly leasing cost for $1$ apartment is $\$58853.13$.

To determine

(b)

If combining $2$ units into exercise facility is a good decision or not.

Answer to Problem 77P

Combining $2$ units into exercise facility does not seem to be a good decision as it fetches less per-unit lease rentals.

Explanation of Solution

Given:

$2$ units are turned into exercise facility and the vacancy rate is reduced by $5\%$.

By turning 2 units into exercise facility total available apartments are $28$.

The vacancy rate is reduced by $5\%$ so occupancy rate is $95\%$.

Calculate monthly leasing cost for $1$ apartment.

Substitute $\$1553722.67$ for $A$ and $95\%\times 28$ for $N$ in Equation (VI).

$\begin{array}{c}\text{Monthly cost}=\frac{\$1553722.67}{95\%\times 28}\\ =\frac{\$1553722.67}{26.6}\\ =\$58410.63\end{array}$

The monthly leasing cost for $1$ apartment by turning $2$ units into exercise facility is $\$58410.63$.

Combining $2$ units into exercise facility does not seem to be a good decision as it fetches less per-unit lease rentals.

Conclusion:

The monthly leasing cost for $1$ apartment by turning $2$ units into exercise facility is $\$58410.63$.

Combining $2$ units into exercise facility does not seem to be a good decision as it fetches less per-unit lease rentals.