In the Futurama episode “A Fistful of Dollars” Fry finds out that after 1000 years of being frozen, the $0.93 in his bank account has turned into $4.3 billion, with an interest rate of 2.25%. Is this accurate? How much would be in the account after 1000 years?
This depends on how interest is calculated. The most basic type of interest is simple interest. This type lets you gain interest on the money invested, but not gain interest on the interest. So it doesn’t compound. As an example, if you invest $1000 at a 2% interest, you will get $20 each year (2% of $1000). For this type of interest, we find the amount in the account by using the formula P + Prt, where P is the initial amount, r is the rate, and t is the time in years. Using this approach, Fry would have 0.93 + 0.93*.0225*1000, or $21.86. This isn’t close to the $4.3 billion the show claims, but this is also only the most basic type of interest.
Compound interest allows you to gain interest on the money invested, and gain interest on the interest. Let’s suppose you invested $1000, and got $20 in interest after the first year. In the second year, you will gain interest on the new balance, $1020. This will compound the interest, causing the amount to increase every year. How often this type of interest is compounded will change the amount of money in the account. The amount in the account is calculated using P(1 + r/n)^nt, where P is the initial amount invested, r is the rate, t is the time in years, and n is the number of times a year the interest is compounded.
Let’s suppose the interest is compounded yearly. The amount in the account after 1000 years would then be 0.93*(1 + .0225/1)^(1*1000), or $4,283,508,449.71. This is much closer to the $4.3 billion the show claimed. What if instead, the interest is compounded quarterly, or four times a year? In this case, we would have n = 4, and the amount in the account would be 0.93*(1 + .0225/4)^(4*1000), or $5,160,938,941.06, approximately $5.2 billion. This is significantly more, but isn’t even the most you could gain.
The highest account balance would occur if the interest is compounded continuously. In this case, the amount in the account would be Pe^(rt), where P is the original amount, r is the rate, t is the time in years, and e is Euler’s number, approximately 2.718. In this case, the amount in the account would be 0.93e^(.0225*1000), or $5,496,785,518.61, approximately $5.5 billion!