if Mark wants to retire in 25 years with $1,011,000.00 and can save $9,680.00 annually for 25 years with his first savings contribution made in 1 year, then what annual return does he need to earn? (Round the value to 2 decimal places and Please enter the value only without converting it to a decimal format. If the answer is 8.55%, enter 8.55)
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To find the annual return Mark needs to earn, we can use the future value of an annuity formula:
FV = P * [(1 + r)^n - 1] / r
where:
FV = future value of the annuity
P = annual savings contribution
r = annual interest rate
n = number of years
Plugging in the given values, we have:
$1,011,000 = $9,680 * [(1 + r)^25 - 1] / r
Now we need to solve for r. This can be done through trial and error or using a financial calculator or software. Let's use Newton's method:
1. Assume a value for r (e.g. 0.08).
2. Calculate the left-hand side (LHS) and right-hand side (RHS) of the equation.
3. If the LHS is greater than the RHS, increase the value of r; if the LHS is less than the RHS, decrease the value of r.
4. Repeat steps 2 and 3 until the LHS is very close to the RHS.
Using this method, we find that an annual return of approximately 4.11% is needed.
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