72

‘tis a beautiful number isn’t it, seventy two ?

‘tis not a prime number, and hence it plays ball with everyone, as in, it is divisible by a lot of integers other than one and by itself

72 is divisible by a whole host integers such as 1, 2, 3 … 4, 6, 8, and 9, 12, 18 … 24, 36 and 72. We call 72 an abundant number, because, the sum of its divisors (other than itself) add up to more than the number in itself. In this case, sum of (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36) = 123 (which is greater than 72)

A perfect number is one where the sum of the divisors add up the number perfectly, such as 28, where 1 + 2 + 4 + 7 + 14 = 28

Why are we talking about this ?

Say you have a retirement portfolio, or really, any investment portfolio that is worth $ 10,000 or $ 100,000 today , and if you ever wonder how long it would take - for it to double

Then you need to know about 72

Whatever mix you have in stocks, bonds, crypto, real estate, or the packaged 2050 retirement target date portfolio

- if the average rate of return on them is (x) 7% for every year for the next t years

Then the time it takes to double your portfolio is 72/x, so in this case, it would be 72/7 ~ 10.3 years

So, you’ll double your portfolio in a decade, even if you don’t add a dime to it from where it stands today

That’s about it

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But why 72 ? and why not say, 92 or 22 ? There is a foundational reason for that, I love smokey Bourbon, the smell of bacon grease, and I love deriving equations myself … so we can just derive this from its foundational blocks. There is no audio this week because there are too many equations

Think about this equation

\(Future Value = Current Value * (1 +r)^t\)

Where, r is the interest rate per time period (essentially, your yield), and t is the total number of time periods. t is what we are solving for, we are solving for the value of t at which your

\(Future Value = Current Value * 2\)

So, you can rewrite the above as

\((1 + r)^t = 2\)

We talked about supermodels and natural logarithms (called ln) last week. Typically we solve these types of equations by taking the natural log (ln) on both sides

Why ? I don’t know … it’s kind of instinctual, a 300-pound linebacker is comin’ at your head, just take the knee and move out the way - you know, instinctual !

\(ln (1 + r)^t = ln(2) \)

\(t * ln (1 + r) = ln(2) \)

\(t = ln(2)/ln(1+r), remember this\)

We rearrange this by multiplying and dividing by r, so the equation for all practical purposes, remains the same

\(t = [ln(2)/r] * [r/(ln(1+r)]\)

Let us think about the second part of the equation for a second. Let us plot r/ln(1+r) for values of r ranging between 0.01 and 0.15 (essentially, 1% to 15 % of yield per year, which is the range for most investments in the long run)

For r between 1% and 15%, the average of r/ln(1+r) tends to be 1.04 (from the plot)

So, we can rewrite the equation above as

\(ln(2) / ln(1+r) = [ln(2)/r] * 1.04\)

The natural log, or ln (2) = 0.69 (you can check it out in excel using ln (2))

\(ln(2) / ln(1+r) = (0.69/r) * 1.04\)

\(ln(2) / ln(1+r) = (0.69 * 1.04)/r\)

\(ln(2) / ln(1+r) = 0.72/r\)

And when we express r as a %

\(ln(2) / ln(1+r) = 72/(100*r)\)

Since ln(2)/ln(1+r) = t, the time in years to double your current investment at an average rate of return of r is

\(t = 72/(100*r)\)

For the last ten years, NASDAQ has had an annual average return of 13.8% (NASDAQ). So, if you invested in NASDAQ years ago, you could have doubled your investment in

\(t = 72/(100*0.138) = 5.2 Years\)

~ half a decade

Time is the most important variable in investing, much more than how much you invest or the rate of return … and understanding 72 will help you understand your trajectory in the future

You know … ice-cold tub of corona by the Pacific and tacos at 10 AM on a Monday

I know my readers, some of them are unapologetic type As, and they scorn at, non-productive downtime, watching a tall and handsome dude throw a pigskin around on Sundays for like eight hours, and a non-zero inbox. You know who you is, don’t you ?

Coronas and tacos by the crystal-blue and glistening Pacific at 10 AM on a Monday probably goes against every fiber of their being …

My beloved type As - live a little

… D O Double G approves of it