You plan to buy the house of your dreams in 14 years. You have estimated that the price of the house will be $81,742 at that time. You are able to make equal deposits every month at the end of the month into a savings account at an annual rate of 11.44 percent, compounded monthly. How much money should you place in this savings account every month in order to accumulate the required amount to buy the house of your dreams?
Round the answer to two decimal places.
To achieve the goal of buying your dream house in 14 years, which is estimated to cost $81,742 at that time, you can start saving by making regular monthly deposits into a savings account. In this scenario, we will assume an annual interest rate of 11.44 percent, compounded monthly. To determine how much money you should deposit each month, we can use the concept of future value of annuities.
The formula for calculating the future value of an annuity is:
\[FV = PMT \times \left(\frac{(1 + r)^n – 1}{r}\right)\]
Where:
– FV represents the future value (the cost of your dream house in this case).
– PMT is the monthly payment you need to determine.
– r is the monthly interest rate (annual rate divided by 12 and expressed as a decimal).
– n is the total number of compounding periods (the number of months in this case).
In your case, the future value (FV) is $81,742, the annual interest rate (r) is 11.44 percent or 0.1144 as a decimal, and the total number of compounding periods (n) is 14 years multiplied by 12 months, which is 168 months.
Now, let’s plug these values into the formula:
\[81,742 = PMT \times \left(\frac{(1 + 0.1144/12)^{168} – 1}{0.1144/12}\right)\]
Now, let’s calculate the expression inside the parentheses first:
\[0.1144/12 = 0.0095333\]
\[(1 + 0.0095333)^{168} = 5.5560665\]
Now, we can rewrite the equation:
\[81,742 = PMT \times \left(\frac{5.5560665 – 1}{0.0095333}\right)\]
Next, simplify the expression inside the parentheses:
\[\frac{5.5560665 – 1}{0.0095333} \approx 461.82\]
Now, the equation becomes:
\[81,742 = PMT \times 461.82\]
To isolate PMT (the monthly deposit), divide both sides of the equation by 461.82:
\[PMT \approx \frac{81,742}{461.82}\]
Now, calculate PMT:
\[PMT \approx 176.89\]
So, to accumulate enough money to buy your dream house in 14 years, you should place approximately $176.89 in this savings account every month. Rounded to two decimal places, the monthly deposit should be $176.89.
By consistently saving this amount each month and earning an annual interest rate of 11.44 percent compounded monthly, you will be well on your way to achieving your goal of purchasing your dream house in 14 years.
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