Hello thanks for helping me!
Here's the problem:
You are told that if you invest $11,000 per year for 23 years (all payments made at the end of each year), you will have accumulated $366,000 at the end of the period. What annual rate of return is the investment offering?
This is how far I've come with the problem:
366000/11000 = ((1+r)^23 - 1)/r
I don't know what to do next.
Thank you!
This is actually a geometric series and r can not be solved for manually (easily).
As one of the formulas on the Wikipedia page I provided a link to just above implies, the sum of the terms in a geometric series of the form below can be calculated by
t^(N-1) + t^(N-2) + ... + t^0 = (1 - t^N)/(1-t).
First, let's first generalize the problem you have shared slightly to make use of the formula above. Suppose one invests P amount of money at the end of every year for N years and every year's investment earns an interest rate of r annually. Then, the total Q to be accumulated at the end of the year N would be
Q = P(1+r)^(N-1) + P(1+r)^(N-2) + ... + P(1+r)^(0)
= P[(1+r)^(N-1) + (1+r)^(N-2) + ... + 1]
Q/P = [(1+r)^(N-1) + (1+r)^(N-2) + ... + 1]
= [1 - (1+r)^N]/(1 - (1+r)) (using the geometric series summation formula above assuming t = 1+r)
= [(1+r)^N - 1]/r. (after rearranging the terms and a little simplification)
We can easily adapt this formula to the problem you have posted by assuming Q = $366,000, P = $11,000, N = 23 and r, of course, is the annual rate of return. Then
366,000 / 11,000 = [(1+r)^23 - 1]/r
which is the same as the equation you came up with.
This equation can not be simplified meaningfully any further in order to find r directly. It is a root-finding problem for a polynomial of the 23rd degree. You need to use a numerical method such as the Newton-Rhapson method to solve it (which is how it is usually done) or you can go by trial-end-error manually in a systematic fashion. Another tool that can be used is the Solver add-in in Excel.
Of course, another way to go about finding r is to use a financial calculator which would have a built-in function for such problems and all you would need to do is to provide the Q, P, and N to the calculator. Yet another way is to use the RATE() function in Excel in a similar fashion to the financial calculator. Note that both the financial calculators and the RATE() function in Excel actually define the problem as a polynomial root-finding problem and use a numerical method such as the Newton-Rhapson behind-the-scenes.
Having said all that, the answer to your question is r = 3.21%.
Editing notice
Formulas and equations are revised to account for the fact that money is invested at the end of every year. Thx for the warning @base64.
You can try guesses for r until you get close enough to zero, i.e.
(11000 ((1 + r)^23 - 1))/r - 366000 = 0
Results plotted for r = 0.02, 0.025, 0.03, 0.035, 0.04
with d = 11000
n = 23
s = 366000
Zero error at r = 0.032
Or you can find a solver, e.g. https://www.wolframalpha.com
If you want a simple calculation, you have 253000 of principal, which gives the interest as 366000-253000 = 113000, which is 44.66% of the principal. That's an arithmetic average of 1.94% per year. We can make further adjustments to that. For instance, each dollar of principal is earning interest, on average, for only half the time period, so that gives 3.88%. Taking compounding into account gives a lower amount, but 3.88% is a good start for Newton's method. Obviously it's not good enough for an exam question, but for a lot of real-world needs, it's a good estimate.
In addition to all the suggestions in other answers, Wolfram Alpha should give you the answer.
