## Rule of 72 $ t \approx \frac{72}{r} $ where, $t$ is the number of years to double and $r$ is the rate of return. For example, it takes approximately 7.2 years to double at a 10% compounded annual rate of return. We can show this empirically using the compounding equation: $ (1.10)^{7.2} = 1.98 $ ## Approximating the Annual Rate from MOIC The Drexel University student investment fund started with $250,000 in 2007 and as of 2024 the fund has grown to $5,000,000. That represents a 20x return, but what is the compound annual rate of return? Using the compounding equation, we can solve for the rate $r$ as such: $(1+r)^{17} = 20$ Rearranging we have: $20^{1/17}-1 = r$ Or, $ r=19.26$ You can also arrive at an approximation of this answer using mental math: 1. Realize a 20x return is just over 4 doublings in 17 years. We know that $2^{4}=16$ and $2^{5}=32$ so $4/16=0.25$ which means approximately 4.25 doublings. We can check this with the actual math: 1. To determine how many doublings 20x is we have: $2^{x}=20$ 2. To solve for $x$ we take the log base 2 of both sides: $log_2{2^{x}} = log_2{20}$ 3. Reduces to: $x \cdot log_2{2}=log_2{20}$ 4. Since $log_2{2}=1$ we have $x=log_2{20}$ 5. Therefore $x=\frac{ln{20}}{ln{2}}\approx 4.3219$ 2. Since $\frac{17}{4.25}=4$ we have a doubling every 4 years. 3. Then utilizing the rule of 72 we can just approximate the rate of return by dividing 72 by 4 to arrive at 18% which is pretty close to the computed 19.26%. For a different method to compute this in your head see [Moontower's Growth Rate = 70% times doublings per year post](https://moontower.substack.com/p/growth-rate-70-doublingsyears). It is always good to practice several methods of arriving at an answer to instill mental flexibility. Explore Further: [[The Volatility Drain]] Tags: #math Your support for Cedars Hill Group is greatly appreciated <form action="https://www.paypal.com/donate" method="post" target="_top"> <input type="hidden" name="hosted_button_id" value="74PGN8ZXHQVHS" /> <input type="image" src="https://www.paypalobjects.com/en_US/i/btn/btn_donate_LG.gif" border="0" name="submit" title="PayPal - The safer, easier way to pay online!" alt="Donate with PayPal button" /> <img alt="" border="0" src="https://www.paypal.com/en_US/i/scr/pixel.gif" width="1" height="1" /> </form>