### Calculating your Portfolio Target with the 4% Rule

You are saving money and learning about the difference between assets and liabilities. As you invest your savings into assets, the real magic begins. Your investments start to grow, whether it be from appreciation, cash flow, dividends, or all of the above. Then the growth starts to growth. Then the growth on the growth on the growth starts to grow. It’s a runaway chain reaction, but it does take time. This is the magic of compounding returns and leads to exponential growth of your wealth, the namesake of this blog.

However, the opposite effect is just as powerful. Anyone who has ever been in credit card debt with interest rates over 20%, or even worse situations with payday or title loans, understands how hard you have to swim against the current just to stay still. The interest keeps racking up and many people never escape the debt. If this describes your situation, check out Dave Ramsey’s Total Money Makeover for steps to dig yourself out of debt before you start building your wealth.

So how can we use exponential growth to accelerate our goal of Financial Independence? Some insight comes from the “Trinity Study,” a study in which three professors from Trinity University looked at portfolio withdrawal rates over periods up to 30 years with different mixtures of stocks and bonds, represented by the S&P 500 index and a long-term, high-grade corporate bond index respectively. Their results showed the various success rates for different withdrawal amounts over different time periods, but the often quoted one is the 4% withdrawal rate, which gives at least 98% success in the historical scenarios examined.

This has led to the 4% withdrawal rate rule of thumb, which states that if you withdraw 4% of an initial investment portfolio and continue to take that amount adjusting for inflation, it will last for at least 30 years with near certainty.

The inverse of this is the 25x rule of thumb and gives us a Target Financial Independence Number; just multiply your annual expenses by 25. Once you reach this Target FI Number, you have a very high likelihood of never running out of money.

But why does this work? First of all, saving 25x your annual expenses is the equivalent of saying that you’ve saved up 25 years’ worth of expenses, so that’s a pretty good start! Unfortunately, the picture is not as rosy as it first seems, as inflation is working against us. How can we calculate what the has been?

For a single year, take the return and divide it by the inflation. For example, let’s say the market had a decent year and the S&P 500 was 16.4%. Inflation in the same year was a modest 1.6%. Finding the real return then is simply:
$16.4\%\ nominal\ return = 116.4\% = 1.164$ $1.6\%\ inflation = 101.6\% = 1.016$ ${1.164\over 1.016} = 1.146 = 114.6\% = 14.6\%\ real\ return$ So, the real gain in this year was around 14.6%. Let’s say the following year was pretty mediocre with a 3.4% total gain while inflation was a little above average at 2.9%. Repeating the above calculation: ${1.034 \over 1.029} = 0.005 = 0.49\%\ real\ return$ Not so great, but how did the real returns fare over the combined period? Remembering that both returns and inflation compound, we have to multiply the results to find out: ${1.164 \over 1.016} * {1.034 \over 1.029} = {1.204 \over 1.045} = 1.151 =15.1\%\ real\ return$ However, this 15.1% real return is the total over two years. Most often, we are interested in the annualized number, referred to as the Compound Annual Growth Rate or CAGR. Annualizing the growth rate makes it easier to compare over different lengths of time as the total growth rate for a few years versus a few dozen years will be vastly different due to compounding, but their CAGR’s can be compared directly.

To find the CAGR, we use exponents just like when compounding, but this time in the inverse. Since the period in this example is 2 years, we will . $1.151^{\left( 1 \over 2 \right)} = 1.073 = 7.3\%\ CAGR$ For example, from the start of 1926 to this writing near the end of 2020, the has returned a total of about 988,925%. To find the CAGR: $988,925\%\ total\ growth = 9,889.25^{\left(1\over2020-1926\right)} = 9,889.25^{\left(1\over 94\right)} = 1.103 = 10.3\%\ CAGR$ So, the S&P 500 has returned 10.3% annually, but this is the nominal return, dollar for dollar. Over the same 94 year time period, inflation was about 1,416%, degrading the spending power of that money even while it was growing. Using the same calculation: $1,416\%\ total\ inflation = 14.16^{\left( 1 \over 94 \right)} = 2.9\%\ annual\ inflation$ To calculate the real return, we must divide out the inflation: ${10.3\%\ nominal\ growth \over 2.9\%\ inflation} = {1.103 \over 1.0286} = 1.072 = 7.2\%$ This means that while nominal return has been 10.3% the real return has been about 7.2%. The important thing to notice is that the real return is greater than our 4% rule of thumb! Taking 4% out each year while adding 7.2% in growth puts you on a track to end with more money than you started with.

Since the market returns are anything but smooth and constant, however, that gap between the expected real growth of the market and the withdrawal rate is important. In unfavorable sequences of market returns, it will serve as a buffer to get your through the tough times before the exponential growth comes to your portfolio’s rescue. While that buffer is needed for the small percentage of worst-case scenarios, most of the time the gap will actually serve to grow your wealth faster than you are spending it.