This is a finance question.  "An ad read, 'Pay us $100 per year for 10
years and we will pay you $100 per year thereafter in perpetuity.'  Is
this a fair deal?  What is the rate of interest?"

Hi there,

Based on receiving a single interest payment at the end of each year,
with all interest re-invested, the rate of interest is 6.83%. Any
higher than that, and the company providing the deal would profit from
it, any less than that and you would profit from it, from a purely
non-real world, hypothetical point of view.

Here is the math:

End of year 1: $100 + 6.83% = $106.83
Start of year 2: $106.83 + $100 = $206.83

End of year 2: $206.83 + 6.83% = $220.95
Start of year 3: $220.95 + $100 = $320.95

End of year 3: $320.95 + 6.83% = $342.87
Start of year 4: $342.87 + $100 = $442.87

End of year 4: $442.87 + 6.83% = $473.13
Start of year 5: $473.13 + $100 = $573.13

End of year 5: $573.13 + 6.83% = $612.27
Start of year 6: $612.27 + $100 = $712.27

End of year 6: $712.27 + 6.83% = $760.92
Start of year 7: $760.92 + $100 = $860.92

End of year 7: $860.92 + 6.83% = $919.72
Start of year 8: $919.72 + $100 = $1019.72

End of year 8: $1019.72 + 6.83% = $1089.37
Start of year 9: $1089.37 + $100 = $1189.37

End of year 9: $1189.37 + 6.83% = $1270.60
Start of year 10: $1270.60 + $100 = $1370.60

End of year 10: $1370.60 + 6.83% = $1464.21

From then onwards, $1464.21 at 6.83% returns $100.00 per year.



Real world reason for it to be a fair deal
------------------------------------------

If the rate of interest of the deal (after deducting the $1000 from
the final interest sum) works out better than the average rate of
interest from banks for the life of the deal.

Real world reasons for it to be a bad deal
------------------------------------------

They might go bankrupt before you see any return.
You might not live long enough to see any return.



Math strategy: I'm not a mathematician, but I know how to use a
calculator. I worked it out by trying 10% and 5%, then narrowing it
down.


Best wishes,
robertskelton-ga

Robert:

Thank you very much for the excellent answer.

However, you didn't explain that the value you used to compare the
future value of the annuity to during the iterative process was the
present value of the perpetuity 10 years hence, which is given by the
equation PVp = C/r, where C is the perpetuity payment and r is the
interest rate.