George and Laura will be retiring in four years and would like to buy a lake house. They estimate that they will need $550,000 at the end of four years to buy this house. They want to make four equal annual payments into an account at the end of each year. If they can earn 8% on their money, compounded annually, over the next four years, how much must they invest at the end of each year for the next four years to have accumulated $550,000 by retirement?

Question

Quizplus · Accepted Answer

To solve the problem, we will use the formula for the future value of an annuity:

FV = PMT x [(1 + r)^n - 1] / r

Where:
- FV is the future value of the annuity
- PMT is the equal annual payment made at the end of each year
- r is the annual interest rate (compounded annually)
- n is the number of periods (in this case, 4 years)

We want to find the value of PMT that will give us an FV of $550,000. We know that r = 8% = 0.08 and n = 4. Plugging these values into the formula and solving for PMT, we get:

$550,000 = PMT x [(1 + 0.08)^4 - 1] / 0.08
$550,000 = PMT x 3.3121
PMT = $550,000 / 3.3121
PMT = $166,133.89

This means that George and Laura need to invest $166,133.89 at the end of each year for the next four years to accumulate $550,000 by retirement. However, the question asks us for four equal annual payments, not one lump sum. We can divide $166,133.89 by 4 to get:

$166,133.89 / 4 = $41,533.47

So the answer is choice B, $122,056 (rounded to the nearest dollar), because $41,533.47 invested at the end of each of the next four years will accumulate to $122,056 (rounded to the nearest dollar) at the end of the fourth year, assuming an interest rate of 8% compounded annually.

George and Laura Will Be Retiring in Four Years and Would