Fascinating math in trading and investing

 Rule of 72

The rule of 72 is a useful rule to calculate how long it takes for a particular investment to double given a certain return. The rule  is very simple: time to double = 72  divided by the annual return

• A portfolio with a 12% annual return will roughly take ~ 6 years to double (72 divided by 12)
• 8% return ~9 years
• 6% return ~12 years

The mathematics of investment losses

No one wants to sell at a loss, but sometimes it is better to take a small loss, than take a devastating decline that is impossible to recover from. To get back from a 50% loss, you will require a 100% gain and from there the numbers simply get more depressing...

Compounding returns

Below table shows the yearly returns for a number of stock indices and a worldwide index ETF.

To calculate the compounded return from 2013 to 2025 - you need to multiply the  the individual period returns (Use the Product() function in Excel), as long as the returns in the "1+R" format. The result is itself in "1+R" format, so we need to subtract 1.

Compounding is a hard concept to grasp  especially over longer time horizons. Let's take an initial investment of 1.000 EUR which is invested in 3 assets with 4%, 6% and 8% return rates and calculate the final values after 40 years.

The curves stay relatively close for the first 10 years, but then they begin to separate and by year 40 the gap becomes enormous:

• 4% return - 4.801 EUR 
• 6% return - 10.286 EUR
• 8% rerurn - 21.725 EUR

6% ends up more than double of 4%, 8% ends over 4x higher than 4% all from the same 1000 EUR starting point.  You can also turn this around and calculate the impact of cost compounding. Two funds for which the gross performance is the same but the yearly total expense rate (TER) is quite different will have significant different end results over a longer time period.