Answer to Question #6197 in Finance for O Oliver

Question

So far, John and Daphne have accumulated $15,000 in their college savings account (at t = 0). Their long-run financial plan is to add an additional $5,000 in each of the next 4 years (at t = 1, 2, 3, and 4). Then they plan to make 3 equal annual contributions in each of the following years, t = 5, 6, and 7. They expect their investment account to earn 9%. How large must the annual payments at t = 5, 6, and 7 be to cover Ellen's anticipated college costs?

Answer 1

At first we need to calculate, how much we will get after 4 year:
FV = ((((15000 + 5000)*1,09+5000)*1,09+5000)*1,09+5000)*1,09 = 46 097,3
Next we find& how large must& be the annual payments using annuity formula:
<img src="data:image/png;base64,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" alt="">
Where PV – present value
A – annual payments
r – interest rate
n - number of years
<img src="data:image/png;base64,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" alt="">
So,& annual payments must& be
<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFAAAAARCAIAAADFd7wzAAABPElEQVRYhe1WyRHDMAhUXRSkeqhGzVCM8vDFKZhJJvEk4WexrHaNhN3mj0X7tIB3x9/wt8ff8JyjN0CSa4TQrjBpF+eiPPI5Rz+r+ljJZcAAuSNCidLwoVfCCYEvje4TEsKlYmMSmnxyUad2crycydGN50hYYJgQGiBJWrsRQ65DVMXkasnxwXblGfWs01HYI13SlPv1Bdha/fZCdkPIkVW7RcPHcnaDFdwRkB+f+HWqE8uP/+YX2QAJzZcMiy6s75nFJ+R1w2p0AlxN3bpRklgxrLu1Pj+EECd9co6vXRgFNEXBAZslw8bfyvDoV4YQNCodELWBqDUkA41HpcPqE7McpKxSPkXkks8d7b5b24PVJ+uMpopEqKGfTC1br0dMQM6TyedvR3oKxA7xwL7zr6X/V/Zk3NawdyFeEQ/0la7+4GFJSAAAAABJRU5ErkJggg==" alt="">