In 1626, Peter
Minuit traded trinkets worth $24 to a tribe of
Native Americans for land on Manhattan Island.
Assume that in 1990 the same land was worth $25.2 billion. If the sellers in this transaction had
invested their $24 at 7% annual interest compounded continuously during the entire 364-year period, who would have gotten the better
end of the deal? By how much?
1990 - 1626 = 364 (I see where this is going :)
1.07^364 = 4.9624 *10^10 !!!!!!
times 24 = 1.19 * 10^12
their return at 7% is far better
1.19 * 10^12 - 2.52 * 10^7
LOL, forget it. 25.2 billion is not even in the ball game.
To determine who got the better end of the deal, we need to calculate the future value of the initial investment at 7% annual interest compounded continuously over the 364-year period.
The formula for calculating the future value (FV) of an investment with continuous compounding is:
FV = P * e^(rt)
Where:
- FV is the future value
- P is the principal amount ($24 in this case)
- e is the mathematical constant approximately equal to 2.71828
- r is the interest rate (7% or 0.07 in decimal form)
- t is the time period (364 years)
Using this formula, we can calculate the future value of the initial investment:
FV = 24 * e^(0.07 * 364)
Using a scientific calculator or a calculator with an exponential function, we can evaluate this expression:
FV ≈ 24 * e^(25.48)
≈ 24 * 520807267.67096
≈ 12499344824.104
So, the future value of the initial investment would be approximately $12,499,344,824. The sellers would have gained this much if they had invested their $24 at 7% compounding continuously over the 364-year period.
Comparing this to the value of $25.2 billion, we can see that the sellers would have gotten a much better end of the deal. The difference between the value of the land in 1990 and the future value of the investment is:
$25,200,000,000 - $12,499,344,824 ≈ $12,700,655,176
Therefore, the sellers would have gained approximately $12.7 billion more by investing their $24 at 7% compounded continuously over the 364-year period.