I was looking across the internet and could not find a simple calculation that will help me understand if I will run out of money over a period of years given a starting amount, fixed yield, inflation rate, and withdrawal rate. Seems simple enough, but in reality you have to tie together two simple concepts: simple interest and future value of annuity with negative payments. The net of the two will provide you the answer: Will I run out of money? If the result is positive, then success. If negative, then you ran out of money.
The variables are the following with the inputs I used:
- Starting Value (pv) = $1,000,000
- Fixed Yield (y)= 5.25%
- Inflation Rate (i)= 2%
- Withdrawal Rate (w) = 4%
- Years Needed (n)= 50
Since I am young and want to retire early, that is why you see 50 for the years needed. Although I know inflation has been high lately, I do expect in the long run that inflation will normalize around 2%, otherwise what would be the purpose of the Federal Reserve. It’s their job to maintain a low inflation rate by influencing market interest rates up or down. The 4% withdrawal rate is what many believe is a safe withdrawal rate, especially those part of FIRE (Financial Independence Retire Early). We can debate whether or not that is appropriate and some advocate a lower rate, but this formula will ultimately allow you to play with the inputs to determine for yourself what is appropriate.
The main reason for building this formula was not because I was concerned about running out of money. I am personally nowhere near one million dollars in assets. However, I do make investments in preferred equity and debt with a fixed rate. The main concern is at what fixed yield would be appropriate if I was retired. The answer for me was at least 5.25%, which is why it was used in this example. Although I target fixed rate investments above this rate, it helps me understand that if I did retire today, I want to lock in a rate above 5.25%, otherwise I would not earn enough to cover ever increasing withdrawals due to inflation.
The first part of the formula is very simple. It’s a simple interest calculation. You take the starting value plus all future fixed interest payments (see numbered bullet points above for symbols):
pv + pv * y * n = $1,000,000 + $1,000,000 * 5.25% * 50 = $3,625,000
Now we need the calculate the withdrawals over the years needed. This is a little tricky because over a long period of time, those withdrawals will have to grow in order to keep your standard of living. Therefore, inflation has to be included in the calculation. The calculation resembles an annuity with payments, but instead with negative payments to represent the withdrawals.
Here is the calculation of withdrawals (see numbered bullet points above for symbols):
- ( pv * w ) * ( ( 1 + i ) ^ n - 1 ) / i = - ( $1,000,000 * 4% ) * ( ( 1 + 2% ) ^ 50 - 1 ) = -$3,383,176
The withdrawal calculation results in a negative value. If you don’t get a negative value, then you did not include the negative sign in the formula (at the beginning of the formula).
Now, if you net the two, the result is a positive value of $241,824. This means within the 50 years, it did not run out money. If it was negative, then it ran out of money some time before the end of the last year. It is important to note that my example could go for another 5 years before running out of money.
There are a few short comings of this calculation that you need to keep in mind:
- It is missing the compound effect of excess cash reinvested. In the first several years, the returns are in excess of the withdrawals and the calculation does not include a return on excess cash.
- The fixed payments are overstated in later years. When withdrawals exceed returns, the balance decreases. If the balance decreases, then subsequent returns would decrease.
Although, there are two short comings with these calculations, they do offset each other to some degree over time.
I went ahead and proved this out by calculating every year of the 50 year time frame. This allowed me to correct the two short comings by investing the excess cash and reducing returns as the cash balance decreased. Through those longer more precise calculations, I ended up with $332,265 at the end of the last year.
I like the simplicity of my formulas. It is not precise, but as you can see, it was fairly close over a 50 year period. It also understated the ending balance, which made it a more conservative estimate.
I am glad I went through this experiment. If you had done it the long way by calculating each year, then you would have discovered that if you invest at a return that is equal to the inflation rate plus the withdrawal rate (6%), then you would not run out of money indefinitely. However, this assumes that you can continually invest at returns that exceed inflation plus the withdrawal rate. It also assumes that your withdrawal rate will only increase by inflation. There may be an unexpected life event that may require a higher withdrawal rate than expected.
If you didn’t understand how inflation affects fixed income investments, then you should now after reading this post. Essentially, if inflation is higher, then fixed income investors will need higher yields to cover their expenses. This is why you may hear people say that investors are “Reaching for yield!”
If you pick an fixed investment that yields less then inflation and your withdrawal rate, then you will run out of money faster in retirement. It is a mathematical certainty. I understand that many choose to allocate a portion of their portfolio to fixed U.S. treasuries (not TIPS: Treasury Inflation-Protected Securities) as a hedge against down markets, but what risk are you trying to prevent? Are you trying to prevent volatility in the value of your portfolio or running out of money? I have found the logic of decreasing volatility in a portfolio as unnecessary especially for someone that is in their early years of investing.
If all investments, including equities, were like fixed investments where their cash flows were certain and you avoided any that may default on those payments, wouldn’t the best investment cost the least for future cash flows? Think of yield on cost. This is why the only thing certain with fixed U.S. treasuries at around a 2% yield is that you are going to run out of money faster. In reality, fixed U.S. treasuries are for governments and banks to hedge short-term risks and meet regulatory guidelines. Rich people use them to preserve short-term wealth while they find a suitable cash flow investment. Hedge funds use them to speculate and hedge changes in the interest rate. If you are none of those, then investing in U.S. treasuries is useless.
This is why I like to look for undervalued fixed investments at a reasonable yield on cost. Since I am far from retiring with a million or more, I am not allocating a lot to fixed investments and prefer undervalued equites that I expect to earn a 12% return or more over time. And if I do find any fixed investments, they are usually preferred stock.
Hopefully this formula was helpful to you in better gauging future expectations. If anyone has any suggestions or an improved formula, then please share in the comments below.
