>[!abstract] >A modern version of the [[Wheat-and-chessboard problem|wheat-and-chessboard problem]] which asks the following question: would you rather receive $1 million today, or a penny that doubles consecutively every day for 30 days? The option of $1 million upfront is intuitively appealing, especially as the money can be invested to generate interest right away. However, the penny doubling progresses as follows: - Day 1: $\$0.01 \times 2^{0} = \$0.01$ - Day 2: $\$0.01 \times 2^{1} = \$0.02$ - Day 3: $\$0.01 \times 2^{2} = \$0.04$ - Day 4: $\$0.01 \times 2^{3} = \$0.08$ - ... - Day 27: $\$0.01 \times 2^{26} = \$671,088.64$ - Day 28: $\$0.01 \times 2^{27} = \$1,342,177.28$ - Day 29: $\$0.01 \times 2^{28} = \$2,684,354.56$ - Day 30: $\$0.01 \times 2^{29} = \$5,368,709.12$ Generalizing, the money $M$ obtained on day $n$ from doubling pennies can be expressed as follows: $M = 0.01 \times 2^{n-1}$ The progression above shows that it is *more than five times* more lucrative to opt for the penny doubling than for the $1 million upfront payment. For the latter to be more advantageous, one would have to find an investment vehicle that returned *more* than 400% in a month (4,800% APY). What is interesting is that the penny doubling option overtakes the upfront payment option *only* on day 27 out of 30, i.e., *very late* in the month. This shows the subtle nature of geometric growth, which starts deceptively slow until it hits an inflection point and then takes off. ## Related - [[Exponential growth bias]] - [[Wheat-and-chessboard problem]]