Question

Use the following information to answer the next two questions.

(1) You plan to purchase your dream home in 10 years.  Currently, it would cost you $500,000 to purchase the land and build the house to your specifications, but you believe this cost will grow by 3% per year. You have decided to place equal monthly payments into an investment account that earns 5% annually in order to make a 20% down payment on the home when you purchase it in 10 years. How much must you place into the account each month to make your down payment?

(2) Ten years has passed and you're ready to purchase the home. You will make a 20% down payment and finance the rest over 30 years at a 4% interest rate. Assuming that your mortgage requires monthly payments and that you take 30 years to pay off the loan, how much will you end up paying for the house? Round intermediate steps to four decimals.

Answer 1

(1)

The first step is to calculate the down payment. The house costs $500,000 now but it increases at a rate of 3% per year. To calculate the cost after 10 years, we simply use the time value equation i.e.

$FV=PV(1+r)^{n}$

where

• $FV$ is the future value
• $PV$ is the present value
• $r$ is the growth rate
• $n$ is the number of periods

Substituting the values, we get

$FV=500,000(1+0.03)^{10}$

or FV = $ 671,958.1897

Now, the down payment is 20% of this amount so

$down \:payment = 0.2 (671,958.1897)$

down payment = $ 134,391.6379

Now, we can calculate the monthly payments required to be invested in a portfolio that yields 5% per year for 10 years to give us an amount equal to our down payment. Here we can use the future value annuity formula which is given below:

$FV=P\left [ \frac{(1+r)^{n}-1}{r} \right ]$

where

• $FV$ is going to be our down payment of $ 134,391.6349
• P is the monthly to calculated
• r is the monthly rate which is going to be (5/12%). [Remember we are calculating monthly payments]
• n is the period which is (10*12) or 120 months

Substituting the values, we get

$134,391.6349 =P\left [ \frac{(1+0.0042)^{120}-1}{0.0042} \right ]$

$134,391.6349 =P(155.6150)$

P = $ 863.6162

(2)

First step is to calculate the amount that will be financed which is 80% of the Future value of house calculated in (1).

So, Amount to be financed = 80% of 671,958.1897

Amount to be financed = $ 537,566.5517

Now, in order to calculate the monthly loan payments we can use the annuity formula which is:

$PV=P\left [ \frac{1-(1+r)^{-n}}{r} \right ]$

where

• PV is the amount to be financed $ 537,566.5517
• r is the monthly interest rate which is (4/12%)
• n is the period which 30*12 months
• P is the payment to be calculated

Substituting the values we have

$537,566.5517=P\left [ \frac{1-(1+0.0033)^{-360}}{0.0033} \right ]$

$537,566.5517=P(210.4848)$

or P = $ 2,553.9448

To know the total payment for the house, we take these monthly mortgage and multiply by 360 (the number of payments) and we take the monthly investment for down payment and multiply that by 120 and sum the two.

$Total\: payment=2553.9448(360) + 863.6162(120)$

$Total\: payment=1,023,054.072$