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Question 163714: I am trying to solve the following problem:
r=(S/P)1/n-1 The numbers I have are:
S=24,780 (Value after 5 years)
P=10,000 (Initial investment)
nth power=5
Trying to find the 5-year average annual return.
I hope this makes sense; I can't make the problem look like it should.
Thank you!
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! S = 24780 = value after 5 years.
P = 10,000 = initial investment.
n = 5 years.
not sure what r is, but i think you mean annual rate of return.
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so i assume you want to know what the annual rate of return is.
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if it's simple interest (no compounding), then you take the take S and subtract P from it and divide by 5.
you get (S-P)/5 = (24780-10000)/5 = 2956 increase per year.
2956 / initial investment of 10,000 = an interest rate of .2956 per year.
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assuming that's true, then simple interest (no compounding) is all on the original investment, and 10000 * .2956 = 2956 and 5 * 2956 = 14780 which, when added to the initial investment, equals 24780.
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if it's compound interest, then you get a different equation.
compound interest means you apply the interest rate to the principal plus whatever interest has been earned previously. it is assumed you are reinvesting the interest so it becomes part of the principal for the next year.
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as an example, if your initial investment is 10,000 and your interest rate is 10% per year, then you would calculate what you earn each year as follows:
10000 * 10% = 1000 interest
10000 + 1000 = 11000 invested after year 1.
11000 * 10% = 1100
11000 + 1100 = 12100 invested after year 2.
12100 * 10% = 1210
12100 + 1210 = 13310 invested after year 3.
etc.....
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the formula for compound interest is F = P * (1+i)^n
in the example above, if we assume P = 10000 and i = .10, and we want to know the value in the 3d year, we would make the equation as
F = future value = P * (1+i)^n
which becomes
F = 10000 * (1.1)^3
which becomes
F = 10000 * 1.331
which becomes
F = 13310.
if you look at the example above, you'll see that the amount invested after 3 years is 13310 which is the same as F.
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in your problem
you have P = 10000
you have S = 24780
you have n = 5
you want to solve for r.
the equation for compound interest in your problem is
S = P * (1+r)^n
substituting known values, we get
24780 = 10000 * (1+r)^5
dividing both sides by 10000 gets
24780/10000 = (1+r)^5
which becomes
2.478 = (1+r)^5
taking the fifth root of both sides of the equation and it becomes
(2.478)^(1/5) = 1+r
using the calculator to solve for fifth root makes the equation become
1.199002974 = 1+r
subtracting 1 from both sides of the equation and it becomes
.199002974 = r
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r = .199002974 which is the rate you are earning at per year assuming compound interest.
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to prove the answer is correct, substitute in the original equation.
24780 = 10000 * (1+.199002974)^5
which is the same as
24780 = 10000 * (1.199002974)^5
which becomes
24780 = 10000 * 2.478
which becomes
24780 = 24780
proving the answer is correct.
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so you have two answers.
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if you assume simple interest, your answer is
r = .2956
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if you assume compound interest, your answer is
r = .199002974
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r is the annual interest rate.
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