Question

Task 2: Buying a House

Answer 1

The main formula used in this problem is the Present Value of an annuity Formula:

$\displaystyle{ P=R[\frac{1-(1+i)^{-n}}{i}]$

where:

P: present value (the mortgage value)

R: periodic payment

t: time (in years)

r: annual interest rate

m: number of compoundings per year

i=r/m      (rate per period)

n=mt      the total number of payments

The maximal monthly payment that Bethany and Samuel can afford is $1,500.

The best interest rate is %5.1 annually, and they want to sign a 30-year mortgage.
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thus

R=1,500

t=30

r=%5.1 = 5.1 /100 = 0.051

m=12, because they make the payments monthly

i=r/m=0.051/12=0.00425

n=mt = 12*30=360

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That is, 

i) they will make 360 monthly payments, 

ii) the largest amount of money they can borrow can be found using the formula:

$\displaystyle{ P=R[\frac{1-(1+i)^{-n}}{i}]$

substituting the known values:

$\displaystyle{ P=1,500\cdot[\frac{1-(1+0.00425)^{-360}}{0.00425}]=276,268.65$         dollars

the calculation can be performed using powerful calculating engines like wolframalpha or a scientific calculator.


iii) Since Bethany and Samuel also have $25,000 for a down payment, they can buy a house with a maximum cost of:

276,268.65+25,000=301,268.65 $, which we can round to 301,300$ for the sake of common sense.

iv)

If the house they want to buy has a price of $280,000, they can afford it.


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For the second part of the problem, we use the 

Compound  interest  formula:

$A=P(1+ \frac{r}{m} )^{mt}=P(1+ i)^{n}$

where

A= Future value
P = the Principal (the initial amount of money) 

If the couple buys a house of value  $280,000  and they own 

$25,000 (the down payment)  then the mortgage is :

$280,000-$25,000=$255,000

the monthly interest rate is the same, that is:

i=0.00425

n=360

Applying the formula we have:

$A=P(1+ i)^{n} \\\\A=255,000\cdot(1+ 0.00425)^{360}=1,173,820$

this is the total value that has to be payed in 360 monthly payments,

thus every month the couple will pay  $1,173,820/360= $3,260.61

after the firs 12 payments, t=1, the future value becomes:

$A=P(1+ \frac{r}{m} )^{mt}\\\\A=255,000(1+0.00425 )^{12}=268,313$

thus the value is 268,313, so the balance of the mortage is 

268,313-12*3,260.61=229,185.68   dollars

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Last part: YES

"the values of i, P, and n for a four-year loan of $12,000 at an annual interest rate of 4.2%."

are

P: $12,000  (principal)

t: 4

r: 42/100=0.042

m: 12

i=0.042/12= 0.0035

n=mt=12*4=48