Assume that your parents wanted to have $180,000 saved for college by your eighteenth birthday and they started saving on your first birthday (deposited the first amount on your first birthday). They saved the same amount each year on your birthday and earned 8% per year on their investments.
How much would they have to save each year to reach their goal?
How would your answer change if they decided to save a certain amount each month? (APR is still 8%)
If they think you will take five years instead of four to graduate and decide to have $220,000 saved just in case, how much more would they have to save each year to reach their new goal?
1)
Formula for future value of annuity can be used to compute annual cash deposit as:
FV = P x [(1+r) n -1]/r
P = FV/[(1+r) n -1]/r
FV = Future value of deposits = $ 180,000
P = Periodic cash deposit
r = Rate of return = 0.08
n = Number of periods = 18
P = $ 180,000/ [(1+0.08) 18 -1]/0.08
= $ 180,000/ [(1.08) 18 -1]/0.08
= $ 180,000/ [(3.99601949918493-1)/0.08]
= $ 180,000/ [(2.99601949918493/0.08)
= $ 180,000/ 37.4502437398117
= $ 4,806.37726287079 or $ 4,806.38
Parents need to save $ 4,806.38 annually to reach the goal.
2)
r = 0.08/12 = 0.006666667 p.m.
n = 18 years x 12 months = 216 periods
P = $ 180,000/ [(1+0.006666667) 216 -1]/ 0.006666667
= $ 180,000/ [(1.006666667) 216 -1]/ 0.006666667
= $ 180,000/ [(3.2005744878448-1)/ 0.006666667]
= $ 180,000/ [(4.2005744878448/0.006666667)
= $ 180,000/ 480.08614917241
= $ 374.93270803644 or $ 374.93
Parents need to save $ 374.93 monthly to reach the goal.
3)
FV = $ 220,000
P = $ 220,000/ [(1+0.08) 18 -1]/0.08
= $ 220,000/ [(1.08) 18 -1]/0.08
= $ 220,000/ [(3.99601949918493-1)/0.08]
= $ 220,000/ [(2.99601949918493/0.08)
= $ 220,000/ 37.4502437398117
= $ 5,874.4610990643 or $ 5,874.46
Parents need to save $ 5,874.46 annually to reach the new goal.
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