Question

Step 1: Choose the price of the house. Then calculate 20% (which will be your down payment). Write down the price and 20% of the price.

Step 2: You don't have 20% now, so you will use an annuity to save up until you have the 20%. Choose a time in the future (2 years, 3 years, 4 years, 5, 10?) that you will purchase this house. Choose an APR that the bank will give you. Calculate how much you need to deposit every month in order to have the 20% down payment down the road. Write down the numbers of years, the interest rate, the formula with all the numbers plugged in, and the monthly deposits you will need to make.


Step 3: Now you take out a mortgage on the remaining 80%. Choose an APR that the bank will charge you (to be realistic, more than the APR in step 2) and the time you will take to pay off the loan. Write down the formula with all the numbers plugged in, and write down the minimum monthly payments.


Please show me proper work and a step by step explanation on how you got your answers. Anyone who attempts to answer just to steal points will be reported. Btw this is due midnight tonight so I could really use some help with this

Answer 1

9514 1404 393

Answer:

1. $250,000 house price. $50,000 down payment
2. 2 years, 3% from the bank, monthly: $2024.06
3. 5% APR, 30 years, monthly: 1073.64

Explanation:

1. House prices vary considerably. In January, 2021, the median US house price was about $269,000, growing at the rate of about 3.2% per year. For the purpose of this problem, we have chosen a slightly lower price of ...

$250,000 . . . selected house price

20% of this price is ...

0.20 × $250,000 = $50,000 . . . amount of down payment

__

2. House prices are growing faster than the interest rate we can get at the bank, so we want to minimize the amount of time we save for a down payment. At the same time, we recognize that saving this amount quickly will put a significant strain on the budget. We choose a period of 2 years, and assume a bank rate on savings of 3%. (US rates in mid-2021 average about 0.04%.) This annuity formula gives the future value of a series of payments:

A = P((1+r/12)^(12t)-1)(12/r) . . . . monthly payment P at annual rate r for t years

Solving for P, we have ...

P = A(r/12)/((1 +r/12)^(12t) -1)

Filling in the chosen numbers, we find we need to save ...

P = $50,000(0.03/12)/(1 +0.03/12)^(12·2) -1) = $50,000(0.0025)/0.06175704

P = $2024.06

$2024.06 needs to be deposited every month for 2 years at 3%.

__

3. The mortgage will be for ...

$250,000 -50,000 = $200,000

We assume we can get an APR of 5% on a 30-year loan. (US rates in mid-2021 are around 3.2%.) The formula for the payment amount is ...

A = P(r/12)/(1 -(1 +r/12)^(-12t)) . . . principal P at rate r for t years

Filling in the chosen numbers, we find the monthly payment to be ...

A = $200,000(0.05/12)/(1 -(1 +0.05/12)^(-12·30))

= $200,000(0.0041666667)/0.77617340 = $1073.64

The monthly mortgage payment will be $1073.64.