SOLUTION: Assume you are planning to invest 5,000 each year for 6 yrs and will earn 10% per yr. Determine the future value of the annuity if your first 5,000 is invested at the end of the f

Question 1043353: Assume you are planning to invest 5,000 each year for 6 yrs and will earn 10% per yr. Determine the future value of the annuity if your first 5,000 is invested at the end of the first year.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

First investment: you deposit $5,000 in the account and it will compound interest for 5 years (not 6 since you deposited at the end of the year). So

P = 5000
r = 0.10
n = 1 (assuming annual compounding)
t = 5

Use the compound interest formula to get
A = P*(1+r/n)^(n*t)
A = 5000*(1+0.10/1)^(1*5)
A = 8052.55

So at the end, $5000 would turn into $8052.55

This value of 8052.55 will be used later (see below)

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Second investment: you invest another $5000 at the same time of year (I'm assuming) so you will have it sit in the account and accumulate interest for 4 years

P = 5000
r = 0.10
n = 1 (assuming annual compounding)
t = 4

Use the compound interest formula to get
A = P*(1+r/n)^(n*t)
A = 5000*(1+0.10/1)^(1*4)
A = 7320.50

So at the end, $5000 would turn into $7320.50

This value of 7320.50 will be used later (see below)

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Third investment: you invest another $5000 at the same time of year (I'm assuming) so you will have it sit in the account and accumulate interest for 3 years

P = 5000
r = 0.10
n = 1 (assuming annual compounding)
t = 3

Use the compound interest formula to get
A = P*(1+r/n)^(n*t)
A = 5000*(1+0.10/1)^(1*3)
A = 6655

So at the end, $5000 would turn into $6655

This value of 6655 will be used later (see below)

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Fourth investment: you invest another $5000 at the same time of year (I'm assuming) so you will have it sit in the account and accumulate interest for 2 years

P = 5000
r = 0.10
n = 1 (assuming annual compounding)
t = 2

Use the compound interest formula to get
A = P*(1+r/n)^(n*t)
A = 5000*(1+0.10/1)^(1*2)
A = 6050

So at the end, $5000 would turn into $6050

This value of 6050 will be used later (see below)

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Fifth investment: you invest another $5000 at the same time of year (I'm assuming) so you will have it sit in the account and accumulate interest for 1 year

P = 5000
r = 0.10
n = 1 (assuming annual compounding)
t = 1

Use the compound interest formula to get
A = P*(1+r/n)^(n*t)
A = 5000*(1+0.10/1)^(1*1)
A = 5500

So at the end, $5000 would turn into $5500

This value of 5500 will be used later (see below)

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Sixth investment: you invest another $5000 at the same time of year (I'm assuming) so you will have it sit in the account and accumulate interest for 0 years (essentially a few months and not enough to be considered a whole year)

P = 5000
r = 0.10
n = 1 (assuming annual compounding)
t = 0

Use the compound interest formula to get
A = P*(1+r/n)^(n*t)
A = 5000*(1+0.10/1)^(1*0)
A = 5000

So the last $5000 stays at $5000

This value of 5000 will be used later (see below)

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Remember all those values that I mentioned were going to be used later? Well we're going to use them now. Specifically, we're going to add them all up

8052.55+7320.50+6655+6050+5500+5000 = 38578.05

So over the 6 years, the six investments ($5000 each) will grow to $38578.05

Not bad considering that 6*5000 = 30000 dollars would be in your possession if you just hang onto the money and put it under your mattress. To make matters worse, inflation would eat into those savings.

We can use the future value of annuity formula to get

FV = P * [ (1+r)^n-1 ]/r
FV = 5000 * [ (1+0.10)^6-1 ]/0.10
FV = 38578.05

and we get the same answer. So this formula is a nice shortcut to compute annuity problems.

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