Calculation:
Alternative 1: Continue to rent the duplex home.
Determine the payment for year 1.
P1=(House rent+Basic utilities cost)×N ...... (I)
Here, N is the number of months per year, P1 is the payment.
Substitute $450 for house rent, $139 for basic utilities cost and 12 months for N.
P1=($450+$139)×12=589×12=$7,068
Calculate the equivalent year 0 payment in year 0 dollars.
P0=P1(1+f)−n ...... (II)
Here, f is the inflation rate.
Substitute $7,068 for P1, 5% for f and 1 year for n in Equation (II).
P0=$7,068(1+0.05)−1=$7,068(1.05)−1=$7,068(0.9523)=$6,731.42
Calculate the equivalent interest rate.
i=(i'+f+i'×f) ...... (III)
Here, i' is the nominal interest rate and i is the equivalent interest rate.
Substitute 15.5% for i' and 5% for f.
i=(0.155+0.05+0.155×0.07)=(0.205+0.01085)=(0.21585)×100=21.58%
Calculate the present worth of house rent after 10 years.
PW10=A(PA,i,n) ...... (IV)
Substitute $6,731.42 for A, 21.58% for i and 10 years for n.
PW10=$6,731.42(PA,21.58%,10)=$6,731.42((1+0.2158)10−10.2158(1+0.2158)10)=$6,731.42(3.9768)=$26,769.74
Thus, the present worth of the house rent after 10 years is $26,769.74.
Alternative 1: Buying a house.
Calculate the mortgage interest rate per month.
im=Mortgage interest rate12
Substitute 0.08 for Mortgage interest rate.
im=0.0812=0.66%
Calculate the number of payments.
n=N×12 ...... (V)
Substitute 30 years for N in Equation (V).
n=30×12=360 payments
Calculate the down payment.
Down payment=House cost×inflation rate ...... (VI)
Substitute $75,000 for house cost and 5% for inflation rate.
Down payment=$75,000×0.05=$3,750
Calculate the closing cost in constant dollars.
Closing cost in constant dollars=[House cost×interest rate of closing cost] ...... (VII)
Substitute $75,000 for house cost and 1% for interest rate at closing cost in Equation (VII).
Closing cost in constant dollars=$75,000×0.01=$750
Write the formula to Calculate the monthly payment.
A=(House cost−Down payment)(AP,im,n) ...... (VIII)
Here, A is the monthly payment.
Substitute $75,000 for house cost, $3750 for down payment, 0.66% for im and 360 for n in Equation (VIII).
A=($75,000−$3750)(AP,0.66%,360)=($75,000−$3750)(0.00667(1+0.00667)360(1+0.00667)360−1)=$71,250(0.00734)=$523
Calculate the mortgage balance after 10 year comparison period.
A'=A(PA,im,n) ...... (IX)
Substitute $523 for A, 0.66% for im and 240 for n in Equation (IX).
A'=$523(PA,0.66%,240)=$523((1+0.0067)240−10.0067(1+0.0067)240)=$523(119.51)=$62,504
Calculate the total amount paid in payments.
n=A×N×12 ...... (X)
Here, N is number of years.
Substitute $523 for A and 10 years for N in Equation (X).
n=$523×10×12=$62,760
Calculate the principal repayments.
Principal repayments=(Home cost−Down payment)−A' ...... (XI)
Substitute $75,000 for home cost, $3750 for down payment and $62,504 for A' in Equation (XI).
Principal repayments=($75,000−$3750)−$62,504=$8,746
Calculate the interest payments.
Interest payments=n−Principal repayments ...... (XII)
Substitute $62,760 for n and $8,746 for Principal repayments in Equation (XII).
Interest payments=$62,760−$8,746=$54,014
Calculate the monthly tax saving.
Monthly tax saving=(A×Average interest rate on loan payment×Marginal income tax) ...... (XIII)
Substitute $523 for A, 0.8772 for Average interest rate on loan payment, 0.3 for Marginal tax rate in Equation (XIII).
Monthly tax saving=(523×0.877×0.3)=$138
Calculate the cost of mortgage after tax.
Cost of mortgage after the tax=A−Monthly tax saving ...... (XIV)
Substitute $523 for A and $138 for monthly tax saving in equation (XIV).
Cost of mortgage after the tax=523−138=$385
Calculate the sale amount of the property after 10 years.
Sale amount of property=Home cost(FP,i,n) ...... (XV)
Substitute $75,000 for home cost and 6% for i in Equation (XV).
Sale amount of property=$75,000(FP,6%,10)=$75,000(1+0.06)10=$134,314
Calculate the net income from the sale.
Net income from the sale=(Sale amount of the property−(Commission×sale amount of the property)−A') ...... (XVI)
Substitute $134,314 for sale amount and 5% for Commission, $62,504 for A' in Equation (XVI).
Net income from the sale=($134,314−(0.05×$134,314)−$62,504)=$134,314−6716−62504=$65,094
Calculate the present worth of home cost of owning house for year 1.
PW1=[(Down payment+Closing cost)+(Cost of mortgage×12)(PA,i',n)+(Constant dollar utilities×12)(PA,i,n)+(Constant dollar insurance and maintenance×12)(PA,i,n)−Net income from the sale(PF,i',n)] ...... (XVII)
Substitute $3,750 for down payment, $750 for closing costs, $385 for Cost of mortgage, 0.155 for i', 10 years for n, 0.10 for i and $65,094 for Net income from the sale in Equation (XVII).
PW1=[(3750+750)+(385×12)((1+0.155)10−10.155(1+0.155)10)+(160×12)((1+0.10)10−10.10(1+0.10)10)+(50×12)((1+0.10)10−10.10(1+0.10)10)−65,094(1(1+0.155)10)]=[4500+4620(4.925)+(1920)(6.144)+(600)(6.144)−65,094(0.2366)]=$27,335.14
Thus, the present worth of cost of owning the house is $27,335.14.
Conclusion:
Thus, buying the house is an affective alternative.