An interesting example of the power of compound interest is the sale of Manhattan Island for $$24$ in $1626$ to Peter Minuit. Based on New York real estate prices today, it seems that Minuit got a great deal. But consider the future value of that $$24$ if it had been invested for $395$ years $(2021$ minus $1626)$ at an interest rate of $8$ percent per year:

$$24\backslash$times FVIF$(\mathrm{.}08,395)=$$$24\backslash$times $(1\mathrm{.}08)395=$$$382,467,607,417,429$

or about $382\mathrm{.}5.$ Perhaps the deal wasn$$t as good as it looked. The total value of land in Manhattan today is only a fraction of $$382\mathrm{.}5$ trillion.

Though entertaining, this analysis is actually somewhat misleading. First, the $8$ percent interest rate we$$ve used to compute future values is quite high by historical standards. At a $3\mathrm{.}5$ percent interest rate, which is more consistent with historical experience, the future value of the $$24$ would be dramatically lower, only

$$24\backslash$times $1\mathrm{.}035395=$ $$19,127,112!$

Second, we have understated the returns to Mr$.$ Minuit and his successors: we have ignored the rental income that the island$$s land has generated over the last three or four centuries.

All things considered, if we had been around in $1626,$ we would have gladly paid $$24$ for the island.

Spreadsheet Questions

$1,$ What was the future value of the $$24$ paid for Manhattan Island by $1726,$ that is$,$ after $100$ years of compounding? What about by $1826?$ What was the growth in value during the second $100$ years? What was the growth in value in the last $95$ years of the $$investment$?$

$2,$Why is the growth of value so much greater in the later years than in the earlier years?

$3,$Suppose that Peter Minuit did not become the first real estate tycoon in New York, but instead had invested his $$24$ at a $5$ percent interest rate in New Amsterdam Savings Bank. What would have been the balance in his account after $5$ years? $50$ years?

$4,$Start$-$up Enterprises had sales last year of only $$0\mathrm{.}5$ million. However, a stock market analyst is bullish on the company and predicts that sales will double each year for four$$years$.$ What are projected sales at the end of this period?