Question

Task 2: Buying a House

Bethany and Samuel are buying a new house. They have $25,000 saved for a down payment and know that they can afford a monthly payment of $1,500 or less. They also know that the best interest rate they can get is 5.1% annually and they want to sign a 30-year mortgage.

If 5.1% is the annual interest rate, what is the monthly interest rate?

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If they sign a 30-year mortgage, how many monthly payments will they make?

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This equation is used to find the monthly payment, m, given the monthly interest rate, i, the principle, P, and the number of interest periods, n, in months:


Using the equation, find the largest possible principle P for their situation. In other words, what is the largest amount of money they can borrow? (Hint: To answer the question, you must rearrange the equation.)

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Since Bethany and Samuel also have $25,000 for a down payment, what is the highest price of a home they can afford?

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

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Suppose the couple buys the house worth $280,000. What would be the monthly payment if they pay $25,000 as a down payment and finance the remainder of the house cost? Use the same monthly interest rate from part a.

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Using this spreadsheet, complete an amortization schedule for the first year of payments. What is the balance of the mortgage after the first 12 payments? Copy the results into the table below.

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Payment Number
Payment
Amount Toward Principle
Amount Toward Interest
Balance
 
 
 
 
$255,000.00
1
$1,384.52
$300.77
$1,083.75
$254,699.23
2




3




4




5




6




7




8




9




10




11




12





Can we use the same compound interest formula to calculate the monthly payments on a car loan? If so, then indicate the values of i, P, and n for a four-year loan of $12,000 at an annual interest rate of 4.2%.

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Resources
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Answer 1

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


`\displaystyle{ P=R[(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=30r=%5.1 = 5.1 /100 = 0.051m=12, because they make the payments monthlyi=r/m=0.051/12=0.00425n=mt = 12*30=360 ------------------------------------------------------------------------------------------------

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[(1-(1+i)^(-n))/(i)]`

substituting the known values:


`\displaystyle{ P=1,500\cdot[(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+ (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=0.00425n=360Applying 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+ (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
-----------------------------------------------------------------------------------------------------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%."areP: $12,000 (principal)t: 4r: 42/100=0.042m: 12i=0.042/12= 0.0035n=mt=12*4=48