In Sunday’s Getting Paid To Flip Million Dollar Coins, I mentioned that exponential functions such as investment compounding are best displayed on a semi-log chart. Let’s do another example of that step-by-step for anyone that wants to learn or anyone who has struggled to teach it to someone else.
Suppose your wealth grows according to this compounding formula:
Wealth = a(1+r)ᵗ
where:
a = starting wealth
r= compounding rate (ie 10%)
t = time in years
For our examples we just use a = 1, so our charts are “growth of a dollar”.
For rule of 72 fans, you know that at a 10% growth rate wealth doubles every 7 years.
Wealth = (1.10)⁷ = 1.95
If you started with $10,000 after 30 years you’d have about $175k.
This chart is not necessarily hard on the eyes, but the fact that time is the exponential variable is a clue that over long stretches an exponential chart is going to become low resolution.
Here’s a 90 cumulative return history for the SP500
2 observations:
- The later years where you are compounding on a larger base of wealth stretch the chart so the earlier years’ changes are invisible.
- The resolution of the chart and the ‘larger base effect’ obscure what you probably care about — how the rate of return is changing.
Here’s the log chart:
The log chart now shows the resolution of zigs and zags in the early years by making the Y-axis distance between wealth levels of 10 and100 the same as 100 to 1,000 or 1,000 to 10,000.
To create our own log chart, we transform the wealth function:
Wealth = a(1+r)ᵗ
Log(Wealth) = Log(a) + Log(1+r)ᵗ
Log(Wealth) = Log(a) + t * Log(1+r)
This fits the form of a line:
Y = b + mX
Set “starting wealth” to a = $1.
That reduces the equation to:
Log(Wealth) = t * Log(1+r)
t, time, is our independent variable and Log(1+r) is a constant slope that depends on the rate of return.
Rule of 72 enjoyyyers know compounding at 10% for 7 years doubles wealth:
Wealth = (1.10)⁷ = 2
We take the log of both sides:
Log(Wealth) = Log(2) = 7 * Log(1.10)
You can just use a calculator to see that log(2) rounds to .29 and slope of the log chart will be Log(1.10) = .041
To interpret the log chart we observe, if:
- Log (Wealth) = .29 that represents a doubling of wealth
- Log (Wealth) = 1 that represents a 10x increase in wealth aka an order of magnitude increase
Let’s now chart the wealth function as Log(Wealth):
Note: each of the 10 series corresponds to a rate of return of 10%, 9%, 8% and so on. The middle series (purple) is 5% per year and the flattest line corresponds to 1% per year.
- If log (wealth) = .3, wealth has approximately doubled
- If log (wealth) = .48, wealth has tripled
- If log (wealth) = .7, wealth has 5x
- If log (wealth) = 1, wealth has 10x
Also note that at 10% growth per year we computed the slope of the log chart earlier to be Log(1.10) = .041.
And voila, it takes about 25 years (1/.04) to 10x your wealth, aka Log(Wealth) = 1, a whole order of magnitude.
Moving your eyes to the right along the line where Y=.3, to the light blue numbers. Those numbers represent a rate of return of 4%. You can see that it takes 4 extra years to get to the same level of wealth if you compound at 4% instead of 5%.
What you can generally observe is that earning 2% instead of 1%, is vastly more important than going from 9% ror to 10% ror. This idea is captured in the fact that 2% is double the rate of return of 1% and 10% is only 11% bigger than 9% but in practical terms it is a reminder that:
- a 1% difference in performance is a big deal
- taxes are a big deal
- fees are a big deal (“oh it’s just 1%”)
- inflation rates (and real returns) are a big deal
- but all these “big deals” matter more when the difference is a compounded rate of 2% vs 3% as opposed to 9% vs 10%
Here’s the chart zoomed in to holding periods of at least 10 years:
At 5% per year, you’ll double wealth in 14 years. At 3% it will take almost a decade longer.