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:

Did anyone notice how every drop in the stock market is reported as breaking news but the subsequent recovery is completely ignored?

Later edit, 1 October 2019:

The drop from 158 to 142USD was breaking news. But 9 months later the price is 226USD, thank you very much.

I’m not a big fan of inspirational quotes, but I recently found myself resonating with a few pragmatic perspectives.
The first one comes from Jerry Seinfeld. In a recent interview he says that simply asking yourself ‘what am I really sick off?’ is key to innovation:

It’s very important to know what you don’t like. A big part of innovation is saying, “You know what I’m really sick of?” For me, that was talk shows where music plays, somebody walks out to a desk, shakes hands with the host, and sits down. “How are you?” “You look great.” I’m also sick of people who are really there to sell their show or product.
“What am I really sick of?” is where innovation begins.
— Jerry Seinfeld, An interview by Daniel McGinn

The second perspective comes from Jeff Bezos, the guy who revolutionized the way we shop online. He warns about the inevitable criticism associated with any pragmatic approach:

If you never want to be criticized, for goodness’ sake don’t do anything new.
— Jeff Bezos

Too bad this quote was also used by Trump in a tweet…

The third perspective was triggered by the previous two: I remembered a principle from the book ‘Soft Skills‘, written by John Sonmez. This principle was ‘Any action is better than no action’, and when I looked back in the book, I found this quote:

Any action is often better than no action, especially if you have been stuck in an unhappy situation for a long time. If it is a mistake, at least you learn something, in which case it’s no longer a mistake. If you remain stuck, you learn nothing.
—Eckhart Tolle, The Power of Now

Update: While proof reading this post I remembered about another perspective, about forcing yourself to do things imperfectly. It comes from Sara Mauskopf, the founder of Winnie, who gives a very concrete example:

I have given myself an hour to write this post before I’m on childcare duty. I can publish the post after that hour or I can spend more time later polishing it and making it perfect. I’m forcing myself to publish the piece before the hour is up even though it probably has some typos and maybe could be written more concisely. The extra couple hours I could spend polishing it won’t make a massive difference in the number of people who read and benefit from this post.
Perfectionism is a tough habit to break so you have to set time limits and force yourself to just put things out there even if they aren’t 100% perfect.
— Sara Mauskopf, How to start a company with no free time

Finally, there is no better way to motivate yourself into doing something than saying ‘Challenge accepted‘. It works for me 🙂