Question 1107138: Jenna’s parents give her a gift of $15,000 to put in a college savings account for her son. She is deciding between two different plans: One pays 3.5% interest compounded daily, and the other pays 2.5% interest compounded quarterly. Which plan should she choose if she wants to maximize the amount in the account in 14 years?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! compounding daily will give you a higher effective interest rate per year than compounding quarterly, if the annual interest rates are the same.
to find the effective annual interest rate, using the same annual interest rate, you wold do the following.
2.5% = .025.
.025 compounded annually = (1 + .025/1)^1 - 1 = .025
.025 compounded quarterly = (1 + .025/4)^4 - 1 = .025235353
.025 compounded daily = (1 + .025/360)^360 - 1 = .025314231
the higher the compounding rate per year, the higher the effective interest rate, up to the point when you get to continuous compounding.
.025 compounded continuously = 1 * e^(.025*1) - 1 = .025315121
that's the best you can do because you can't compound any more times per year than continuous compounding.
now, you take 3.5% per year and it will give you a greater effective annual intereest rate than all of the above, regardless of how many time you compound per year.
3.5% = .035.
.035 compounded annually = (1 + .035/1)^1 - 1 = .035
.035 compounded quarterly = (1 + .035/4)^4 - 1 = .035462061
.035 compounded daily = (1 + .035/365)^365 - 1 = .035617971
.035 compounded continuously = 1 * e^(.035 * 1) - 1 = .035619709
no contest.
3.5% per year is better than 2.5% per year, regardless of the compounding rate per year.
let's look at 14 years.
the formula for discrete compounding is f = p * (1 + r/c) ^ (n*c)
f is the future value
p is the present value
r is the interest rate per year.
c is the number of compounding periods per year.
n is the number of years.
f = what you want to find
p = 15,000 for both plans.
r = .035 for plan 1 and .025 for plan 2.
c = 365 for plan 1 and 4 for plan 2.
n = 14 years both plans.
for plan 1, you get f = 15,000 * (1 + .035/365)^(365 * 14) = 24,484.16812.
for plan 2, you get f = 15,000 * (1 + .025/4)^(4*14) = 21,262.84083
plan 1 is the winner.
that's the plan with 3.5% compounded daily.
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