Question
What is the formula to calculate the total amount needed to save for a down payment on a house if the target down payment is 20% of the purchase price, the expected annual interest rate is 5%, and the savings period is 10 years, compounded annually?
Answers
GPT-4o mini
To calculate the total amount needed to save for a down payment on a house, we can follow these steps:
1. **Determine the purchase price (P)**: Let \( P \) be the purchase price of the house. The down payment will be \( 0.20 \times P \).
2. **Use the future value of a lump sum formula**: If you plan to save a certain amount (let's denote this amount as \( A \)) every year and want to find out how much you will have in the future, you can use the formula for the future value of an annuity compounded annually:
\[
FV = A \times \frac{(1 + r)^n - 1}{r}
\]
where:
- \( FV \) is the future value of the annuity (the total amount you will have saved for the down payment),
- \( A \) is the annual savings,
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of years.
3. **Calculate the amount needed for down payment**: Since we want the future value to equal the down payment amount, we set:
\[
0.20 \times P = A \times \frac{(1 + r)^n - 1}{r}
\]
4. **Rearranging the formula to solve for \( A \)**: To calculate the total amount needed \( A \):
\[
A = \frac{0.20 \times P \times r}{(1 + r)^n - 1}
\]
5. **Substituting in the given values**:
- The target down payment is 20%.
- The expected annual interest rate, \( r = 0.05 \).
- The savings period \( n = 10 \) years.
Now, if we substitute these values into the equation to find out how much you need to save annually to reach the down payment amount, it results in:
\[
A = \frac{0.20 \times P \times 0.05}{(1 + 0.05)^{10} - 1}
\]
The expression can be simplified further knowing that \( (1 + 0.05)^{10} \) calculates to approximately 1.62889:
\[
(1 + 0.05)^{10} - 1 \approx 1.62889 - 1 = 0.62889
\]
Thus:
\[
A = \frac{0.20 \times P \times 0.05}{0.62889} \approx 0.0159 \times P
\]
So, the total amount needed to save annually is approximately \( 1.59\% \) of the purchase price each year for 10 years at an interest rate of 5% to achieve a down payment of 20% of the house price.
This means over 10 years, you will accumulate enough to meet your 20% down payment target.
1. **Determine the purchase price (P)**: Let \( P \) be the purchase price of the house. The down payment will be \( 0.20 \times P \).
2. **Use the future value of a lump sum formula**: If you plan to save a certain amount (let's denote this amount as \( A \)) every year and want to find out how much you will have in the future, you can use the formula for the future value of an annuity compounded annually:
\[
FV = A \times \frac{(1 + r)^n - 1}{r}
\]
where:
- \( FV \) is the future value of the annuity (the total amount you will have saved for the down payment),
- \( A \) is the annual savings,
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of years.
3. **Calculate the amount needed for down payment**: Since we want the future value to equal the down payment amount, we set:
\[
0.20 \times P = A \times \frac{(1 + r)^n - 1}{r}
\]
4. **Rearranging the formula to solve for \( A \)**: To calculate the total amount needed \( A \):
\[
A = \frac{0.20 \times P \times r}{(1 + r)^n - 1}
\]
5. **Substituting in the given values**:
- The target down payment is 20%.
- The expected annual interest rate, \( r = 0.05 \).
- The savings period \( n = 10 \) years.
Now, if we substitute these values into the equation to find out how much you need to save annually to reach the down payment amount, it results in:
\[
A = \frac{0.20 \times P \times 0.05}{(1 + 0.05)^{10} - 1}
\]
The expression can be simplified further knowing that \( (1 + 0.05)^{10} \) calculates to approximately 1.62889:
\[
(1 + 0.05)^{10} - 1 \approx 1.62889 - 1 = 0.62889
\]
Thus:
\[
A = \frac{0.20 \times P \times 0.05}{0.62889} \approx 0.0159 \times P
\]
So, the total amount needed to save annually is approximately \( 1.59\% \) of the purchase price each year for 10 years at an interest rate of 5% to achieve a down payment of 20% of the house price.
This means over 10 years, you will accumulate enough to meet your 20% down payment target.