(4) Elwood wishes to purchase a home. She has saved up $13,200 for a down payment. Based on her earnings, she qualifies for a thirty-year mortgage with level monthly payments of $820 including escrow and a nominal interest rate convertible monthly of 5.85%. Her payments are due at the end of each month. From each payment, $240 will be put aside in an escrow account for the payment of taxes and homeowners insurance. What is the most expensive house Elwood can buy if her aunt has promised to give her the money needed for loan applications, inspections, and all other required buyer's closing costs

$111,514.93

Explanation:

Looking at this question one can be forgiven for not seeing it as an Annuity question at first but it is.

It's requires us to calculate the Present value of an Annuity so we shall start by stating the formula,

PV of an Annuity = PMT x [ (1 – (1+i)^-n) / i ]

Where,

PMT is the payment per period

i is the rate of interest

n is the frequency of payments

Calculating PMT

Payment is per month and is $820 including the escrow. We would need to remove that Escrow and Home insurance money to calculate this so we will subtract $240 from the $820.

= 820 - 240

= $580 is the PMT per month

Calculating i

i is on a monthly basis and is stated as an annual figure so we need to convert it to months

= 0.0585/12

= 0.004875 is the monthly interest

Calculating n

Following the i calculation we also convert this to months,

= 30 years x 12

= 360 months.

Time to calculate.

PV of an Annuity = PMT x [ (1 – (1+i) ^-n) / i ]

= 580 [ (1- (1+ 0.004875)^-360)/0.004875]

= 98314.9272065

= $98,314.93

$98,314.93 is the present value of the Annuity and now we have to add the down payment she had already made to account for the true cost of the house.

= 98,314.93 + 13,200

= $111,514.93

The most expensive house Elwood can buy if her aunt has promised to give her the money needed for loan applications, inspections, and all other required buyer's closing costs is $111,514.93

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