On Abby Ellen's graduation from law school, Abby's uncle, Bull Brady, promised her a gift of $24,000 or $2,400 every quarter for the next 4 years after graduating from law school. If the money could be invested at 6% compounded quarterly, which offer should Abby choose?
i = .06/4 = .015
n= 4(4) = 16
Present value of payment option
= 2400(1 - 1.015)^-16)/.015
= 33915.03
Well, what do you think ?
I have done this problem and need help because the answer I got was wrong and way off the answer in the book
Option one: $24,000
F = P(1+r)^t
F = 24,000(1+0.06)^16
16 because the rate is per quarter of a year and there are 16 quarters in 4 years
Option 2: 2400 every quarter for 4 years (so 16 of these 2400 gifts in total)
1st $2400 gift will be in for the full 16 quarters
2nd $2400 gift will be in for 15 quarters
3rd $2400 gift will be in for 14 quaters etc
So, adding these together you get
2400(1+0.06)^16 + 2400(1+0.06)^15 + ... + 2400(1+0.06)^1 = Total
2400[(1.06)^1 + (1.06)^2 + ... + (1.06)^16] = Total
(I just rewrote it in reverse order. Makes the next part easier)
The you use the sum of a geometric series formula where a = 1.06 and r = 1.06 and n = 16 and then multiply your answer by 2400.
Compare the two answers then.
mmmmhhh , been on this earth over 75 years, and guess what, each of those years had 4 seasons (4 quarters) , so 4 years will have 16 quarters.
notice that the interest rate is compounded quarterly, so we need the rate per quarter.
That is i = .06/4 = .015
n = 16
I have no clue where your formula comes from
Trust me!
It is 6% per quarter though. You don't have to divide by 4. It's not 6% per annum. Even if it was you can't just divide by 4.
You have to use
P(1+r)^t = P(1+R)^T
r = first rate
t = 1 (usually)
R = second rate
T = amount of these time periods in t
E.g. 6% AER but I want it quarterly instead
(1+0.06)^1 = (1+R)^4
Power of 1 as it's 1 year
Power of 4 as there are four quarters in a year
R = (1.06)^1/4 - 1 = 1.467% quaterly rate
You can check this by multiplying
1.01467*1.01467*1.01467*1.01467 = 1.06
To determine which offer Abby should choose, we need to calculate the future value of each option and compare them.
Option 1: A lump sum of $24,000
Option 2: Quarterly payments of $2,400 for 4 years
Let's first calculate the future value of option 1. We'll use the formula for compound interest:
Future Value (FV) = P(1 + r/n)^(nt)
Where:
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
For option 1:
P = $24,000
r = 6% = 0.06 (6% converted to decimal form)
n = 4 (quarterly compounding)
t = 4 years
FV = $24,000(1 + 0.06/4)^(4*4)
FV ≈ $31,503.61
The future value of option 1 (the lump sum) is approximately $31,503.61.
Now let's calculate the future value of option 2. Since we have quarterly payments, we can use the formula for the future value of an ordinary annuity:
Future Value (FV) = PMT * [(1 + r/n)^(nt) - 1] / (r/n)
For option 2:
PMT = $2,400 (quarterly payment amount)
r = 6% = 0.06 (6% converted to decimal form)
n = 4 (quarterly compounding)
t = 4 years
FV = $2,400 * [(1 + 0.06/4)^(4*4) - 1] / (0.06/4)
FV ≈ $30,992.16
The future value of option 2 (quarterly payments) is approximately $30,992.16.
Comparing the two options, the future value of option 1 ($31,503.61) is slightly higher than the future value of option 2 ($30,992.16). Therefore, Abby should choose the lump sum of $24,000, as it will result in a higher return on her investment.
Reiny, there are 3 months in a quarter of a year so I'm unsure why you divided by 4 unless you were trying to get the monthly rate but that's not how you get the monthly rate and there aren't 16 months in 4 years.
Present value of 1st 2400 is 2400
Present value of 2nd 2400 is 2400/(1+0.06)
Present value of 3rd 2400 is 2400/(1+0.06)^2 etc
You still get a geometric series
with a = 1/1.06 and r = 1/1.06