In the Seinfeld episode ‘The Kiss Hello’ (S06E17), Jerry’s Nana remembers that Uncle Leo was supposed to give Jerry’s mom 50 dollars 53 years before. When Morty (Jerry’s father) finds out, he starts to calculate the amount that Uncle Leo would owe his wife, Helen:

MORTY: Do you know what the interest on that fifty dollars comes to over fifty-three years?

HELEN: Oh, Morty, please.

MORTY: Six hundred and sixty-three dollars and forty-five cents. And that’s figuring conservatively at five percent interest, over fifty-three years, compounded quarterly. #

Using the 5% interest rate that Morty mentions, the best case scenario for the $50 over 53 years would be for continuous compounding:

Total = P * e ^ rt, where P=50, r=0.05 and t=53, so Total ~= $707

The value that Morty came up with – $663.45 is extremely close to what he would have received for a yearly compound:

Total = P * ( 1 + r/n) ^ nt, where P=50, r=0.05, t=53 and n=1, so Total ~= $663.75

However he does mention “compounded quarterly“, so the final expected amount should be:

Total = P * ( 1 + r/n) ^ nt, where P=50, r=0.05, t=53 and n=4, so Total ~= $696.17

In conclusion, Morty underestimated the amount that Helen should have received by $32.42. Quarterly compound is better than yearly compound, and the continuous compound is the best. Also, $696.17 in 1995 (when the episode aired) is worth $1,451.14 today.

Intuition about the e number

Imagine an initial investment of $1, earning 100% annual interest. So P=1, r=1, t=1.

• If interest is compounded once per year, you end up with: (1+1)^1 = 2
• If interest is compounded twice per year, you end up with (1 + 1/2) ^2 = 2.25
• Quarterly compounding: (1 + 1/4) ^ 4 = 2.44
• Monthly compounding: (1 + 1/12) ^ 12 = 2.61
• Weekly compounding: (1 + 1/52) ^ 52 = 2.69
• Continuous compounding: e = 2.718

Discrete vs continuous growth

At its core, e measures the idea of continuous growth, which is different from the discrete growth. Discrete growth means that a given population only grows at certain moments (example at the hour mark). Continuous growth is different: instead of waiting until the end of the hour, the population is growing all the time, at every moment, like a plant that grows gradually instead of in sudden jumps.

The discrete growth is modelled by the (1 + r)^t formula, while the continuous growth is modeled by the e^rt formula. The second one results in a slightly higher number because you’re adding a little bit of growth continuously.

Assuming a population of 1000 astrophage, and a growth rate of 5% per hour, here is how it will grow over a 10-hours period: