Financial Planning for Your Dream Home: Saving, Mortgage, and Interest Calculations

QUESTION

Aya and Sakura would like to buy a house and their dream home costs $500,000.  Their goal is then to save $50,000 for a down payment and then would take out a mortgage loan for the rest.  They plan to put their monthly saved amount in a conservative mutual fund that has a track record of a 4.25% rate of return, compounded quarterly.  To be sure they don’t go spending this money on other things, they are going to move it into their investment account at the beginning of each month.  Their hope is to be able to buy this home in 7 years.

1) What would their monthly savings amount have to be to reach this goal? 

2) What will be the total interest earned?

Aya and Sakura have now saved up their down payment to buy a home, but they still need to borrow to cover the rest.  For the home they want this will require a mortgage of $450,000 to cover the remaining amount and they’re not sure whether they could afford the monthly loan payments.  The bank has offered them a mortgage interest rate of 5.15%, compounded semi-annually.

3) How much would they have to be able to afford to pay each month in order to pay off their mortgage in 25 years? 

4) What is the total amount of interest that would be paid to the lender after 25 years of payments? 

What if Aya and Sakura could only afford a monthly payment of $2,000?

5) What would be the maximum mortgage amount they could afford to borrow from the bank, if all the other conditions were the same? 

6) What is the total amount of interest that would be paid to the lender over 25 years?

ANSWER

Financial Planning for Your Dream Home: Saving, Mortgage, and Interest Calculations

1) To determine the monthly savings amount needed to reach their goal of a $50,000 down payment and eventually buy a $500,000 dream home in 7 years, we can use the formula for compound interest:

\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]

Where:
– A is the future value of the investment (which should be $50,000, their down payment goal).
– P is the initial amount they need to save.
– r is the annual interest rate (4.25% or 0.0425 as a decimal).
– n is the number of times the interest is compounded per year (quarterly, so 4 times).
– t is the number of years (7 years).

We need to solve for P, the monthly savings amount:

\[50,000 = P \left(1 + \frac{0.0425}{4}\right)^{4 \cdot 7}\]

First, we simplify the interest rate term:

\[50,000 = P \left(1 + 0.010625\right)^{28}\]

Now, let’s solve for P:

\[P = \frac{50,000}{\left(1 + 0.010625\right)^{28}}\]

P ≈ $36,062.31

So, Aya and Sakura would need to save approximately $36,062.31 each month to reach their down payment goal in 7 years.

2) To calculate the total interest earned, we need to subtract the total amount saved from the total contributions. They will save $36,062.31 each month for 7 years (84 months), and the interest rate is 4.25% compounded quarterly.

Total contributions = $36,062.31 * 84 ≈ $3,028,264.04

Total interest earned = $3,028,264.04 – $50,000 = $2,978,264.04

So, the total interest earned would be approximately $2,978,264.04.

3) To determine the monthly mortgage payment, we can use the formula for the monthly payment of a fixed-rate mortgage:

\[M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n – 1}\]

Where:
– M is the monthly payment.
– P is the principal amount ($450,000).
– r is the monthly interest rate (5.15% annual rate compounded semi-annually, so 2.575% monthly or 0.02575 as a decimal).
– n is the total number of payments (25 years, or 300 months).

Now, plug in the values:

\[M = \frac{450,000 \cdot 0.02575 \cdot (1 + 0.02575)^{300}}{(1 + 0.02575)^{300} – 1}\]

M ≈ $2,748.18

So, they would need to afford approximately $2,748.18 each month to pay off their mortgage in 25 years.

4) To find the total amount of interest paid over 25 years, we can subtract the initial loan amount from the total payments made:

Total interest = (Monthly payment * Number of payments) – Principal loan amount
Total interest = ($2,748.18 * 300) – $450,000 ≈ $824,454.04

Therefore, the total amount of interest paid to the lender after 25 years of payments would be approximately $824,454.04.

5) If Aya and Sakura can only afford a monthly payment of $2,000, we can calculate the maximum mortgage amount they could afford using the same formula as in question 3, but this time, we’ll solve for P:

\[P = \frac{M \cdot ((1 + r)^n – 1)}{r \cdot (1 + r)^n}\]

Where:
– M is the monthly payment ($2,000).
– r is the monthly interest rate (0.02575).
– n is the total number of payments (300 months).

Now, plug in the values:

\[P = \frac{2,000 \cdot ((1 + 0.02575)^{300} – 1)}{0.02575 \cdot (1 + 0.02575)^{300}}\]

P ≈ $311,255.69

So, they could afford a maximum mortgage amount of approximately $311,255.69 with a monthly payment of $2,000.

6) To find the total amount of interest paid to the lender over 25 years with a $2,000 monthly payment and a mortgage amount of $311,255.69, we can use the formula from question 4:

Total interest = (Monthly payment * Number of payments) – Principal loan amount
Total interest = ($2,000 * 300) – $311,255.69 ≈ $188,744.31

Therefore, the total amount of interest paid to the lender over 25 years with a $2,000 monthly payment would be approximately $188,744.31.

In summary, Aya and Sakura would need to save around $36,062.31 per month to reach their down payment goal, and they could expect to earn approximately $2,978,264.04 in interest on their savings. If they can afford a monthly payment of $2,000, they could borrow a maximum mortgage amount of approximately $311,255.69 and would pay approximately $188,744.31 in interest over 25 years.