You want to have $75,000 in your savings account 12 years from now, and you’re prepared to make equal annual deposits into the account at the end of each year. If the account pays 6.8% interest, what amount must you deposit each year?
To find out the amount you need to deposit each year, you can use the formula for the future value of an annuity. The future value (FV) of an annuity is given by the formula:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value of the annuity (desired savings amount)
P = Annual deposit amount
r = interest rate per period
n = number of periods
In this case, the future value (FV) is $75,000, the interest rate (r) is 6.8% or 0.068, and the number of periods (n) is 12 years.
Now, we need to rearrange the formula to solve for the deposit amount (P). Here's how:
1. Divide both sides of the formula by [(1 + r)^n - 1] to isolate P:
(P * [(1 + r)^n - 1] / r) / [(1 + r)^n - 1] = FV / [(1 + r)^n - 1]
2. Simplify the right-hand side of the equation:
P = (FV * r) / [(1 + r)^n - 1]
Now, let's substitute the given values into the formula and solve for P:
P = ($75,000 * 0.068) / [(1 + 0.068)^12 - 1]
P = $5,100 / [1.886610 - 1]
P = $5,100 / 0.886610
P ≈ $5,756.98
Therefore, you would need to deposit approximately $5,756.98 each year to accumulate $75,000 in your savings account after 12 years with a 6.8% interest rate.
To calculate the annual deposit needed to reach $75,000 in 12 years with 6.8% interest, you can use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value ($75,000 in this case)
P = Annual deposit
r = Interest rate per period (6.8% or 0.068)
n = Number of periods (12 years)
Plugging in the given values into the formula:
$75,000 = P * ((1 + 0.068)^12 - 1) / 0.068
Now we can solve for P:
$75,000 * 0.068 = P * ((1 + 0.068)^12 - 1)
$5,100 = P * ((1.068)^12 - 1)
$5,100 = P * (1.949358 - 1)
$5,100 = P * 0.949358
P = $5,100 / 0.949358
P ≈ $5,367.42
Therefore, you would need to deposit approximately $5,367.42 each year to reach a savings balance of $75,000 in 12 years with a 6.8% interest rate.