5.4: Savings Plans
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Sometimes it makes better financial sense to put small amounts of money away over time to purchase a large item instead of taking out a loan with a high interest rate. When looking at depositing money into a savings account on a periodic basis we need to use the savings plan formula.
where, F = Future value PMT = Periodic payment r = Annual percentage rate (APR) changed to a decimal t = Number of years n = Number of payments made per year |
Example \(\PageIndex{1}\): Savings Plan—Vacation
Henry decides to save up for a big vacation by depositing $100 every month into an account earning 4% per year. How much money will he have at the end of two years? PMT = $100, r = .04, t = 2, n = 12
Henry will have $2,494.29 for his vacation.
For some problems, you will have to find the payment instead of the future value. In that case, it is helpful to just solve the savings plan formula for PMT. Since PMT is multiplied by a fraction, to solve for PMT, you can just multiply both sides of the formula by that fraction. You should just think of the savings plan formula in two different forms, one solving for future value, F, and one solving for payment, PMT.
Example \(\PageIndex{2}\): Savings Plan—Finding Payment
Joe wants to buy a pop-up trailer that costs $9,000. He wants to pay in cash so he wants to make monthly deposits into an account earning 3.2% APR. How much should his monthly payments be to save up the $9,000 in 3 years?
F = $9,000, r = .032, t = 3, n = 12
Joe has to make monthly payments of $238.52 for 3 years to save up the $9,000.
Example \(\PageIndex{3}\): Savings Plan—Finding Time
Sara has $300 a month she can deposit into an account earning 6.8% APR. How long will it take her to save up the $10,000 she needs?
F = $10,000, PMT = $300, r = 0.068, n = 12
Note: We will use the original savings plan formula which solves for the future value, F to solve this problem.
To solve for time you have to take the logarithm (log) of both sides. You can then use the “Power Rule” of logs, which states, when , as stated in section 4.3.
It will take Sara about 2.6 years to save up the $10,000.
Example \(\PageIndex{4}\): Savings Plan—Two-Part Savings Problem
At the end of each quarter a 50-year old woman puts $1200 in a retirement account that pays 7% interest compounded quarterly. When she reaches age 60, she withdraws the entire amount and places it into a mutual fund that pays 9% interest compounded monthly. From then on she deposits $300 in the mutual fund at the end of each month. How much is in the account when she reaches age 65?
First, she deposits $1200 quarterly at 7% for 10 years.
PMT = $1200, r = 0.07, t = 10, n = 4
Second, she puts this lump sum plus $300 a month for 5 years at 9%. Think of the lump sum and the new monthly deposits as separate things. The lump sum just sits there earning interest so use the compound interest formula. The monthly payments are a new payment plan, so use the savings plan formula again.
Total = (lump sum + interest) + (new deposits + interest)
She will have $130,159.72 when she reaches age 65.