The retirement calculations we’ve done so far all apply to someone starting from zero assets invested. What about someone who’s already made some progress? Mathematically, we have a very simple operation that answers lots of questions.
Recall
As discussed in a previous post, your savings rate p is based solely on your income I and expenses E according to the following.
p=\frac{I-E}{I}=\frac{\textrm{what you keep}}{\textrm{what you earn}}Basically, you’re saving p proportion of your take-home. Given that, we then derived that your retirement date is based on p alone (when starting from zero invested assets) and looks like N=-25\ln(p), 0<p\leq1 years.
Effective Saving Rate
Let \mathbb{X} be your invested assets so far. When invested appropriately, \mathbb{X} will supply you with \mathbb{X}/25 = (0.04)\mathbb{X} of retirement income per year (according to the 4% safe withdrawal rate). Simply subtract that retirement income from your expenses E to see your remaining expenses (maybe call it E') not currently covered by your retirement portfolio.
E' = E - \mathbb{X}/25Since we’re using pre-retirement and post-retirement growth rates that match (r=4\% everywhere), this retirement income \mathbb{X}/25 will remain constant over time (inflation-adjusting and probably overly conservative). So you have \mathbb{X}/25 of income that permanently cancels out \mathbb{X}/25 of expenses. Once those are removed from your calculation, you’re starting over from scratch but this time with lower expenses E'. Your effective savings rate based on income I and the lower expenses E' is the following.
p' = \frac{I - E'}{I}Updating Your Retirement Calculation
This is the easy part! You can retire in N=-25\ln(p'), 0<p'\leq1 years, starting now.
Well, that was super easy. What happens when your effective savings rate grows above one, when p'>1? Well, as soon as p'=1, you are retirement eligible; your invested assets completely cover your expenses (by definition). Your personal pension is fully funded, and you are officially financially independent. Mr. Market will probably (temporarily) revoke it the next trading day, but it’s a beautiful thing to see nonetheless.
As p' grows above 1, a couple of interpretations are available.
- You’re fattening up your F.I.R.E.
- You’re lowering your withdrawal rate (e.g. 3.5% is more conservative than 4%)
- You’re stuck in one-more-year syndrome
- You enjoy your work but love that it’s now optional!
Tracking Over Time
Our main use case of this paradigm is to simply give an updated time estimate. Market returns tend to outpace our conservative 4% pre-retirement growth rate, so N=-25\ln(p') will update your estimate whenever reality puts you ahead of schedule.
Here, we notice another use case. Plotting over time, the effective savings rate makes a very nice graph! The more often your p' floats above 100% due to market gyrations, the more often you are financially independent. The graph above plots “actual” (the usual type) and effective savings rates. Notice the effective savings rate poked above 100% a couple of times before moving decisively above, even while the actual savings rate is trending downward. Also, the spread between the two widened over the years as the portfolio \mathbb{X} increased.
Different measurements of progress
A recent ChooseFI episode mentioned an article by Katie titled Why Hitting “Half FI” is More Like 75%. Now that we have the tools, let’s check her math. By “Half FI” she means accumulating half the \mathbb{X} you need to retire, and by 75%, she means having only 25% of the initial time estimate remaining. Calculating the time N_2 to “Half FI” involves a slight modification to an earlier equation (basically accumulating \frac{E}{2\cdot r} instead of the full \frac{E}{r}).
\frac{1}{2\cdot r} = R\frac{e^{r\cdot N_2}-1}{r}~~\textrm{when}~~N_2=\frac{1}{r}\ln\left(\frac{1+p}{2p}\right)Recall (or learn for the first time) that the original was the following.
\frac{1}{r} = R\frac{e^{r\cdot N}-1}{r}~~\textrm{when}~~N=\frac{1}{r}\ln\left(\frac{1}{p}\right)If we take the ratio, do we get 75%?
\frac{N_2}{N} = \frac{\ln\left(\frac{1+p}{2p}\right)}{\ln\left(\frac{1}{p}\right)} \stackrel{?}{=} 75\%No, not always. For p=0.5 savings rate, “Half FI” is really 58%. But for p\approx 0.10 we’re pretty darn close to 75% of the way there. We don’t mean to nitpick a nice article; this is only a silly application. Our methodology assumes a more conservative market return, so of course, Katie’s numbers are further along than ours.
Commuting Diagrams
While the above application is silly indeed, it is very mathematically pleasing. The example shows the process can be “updated” at any time. Half FI plus the other half equals the initial estimate. To see this, first realize the updated savings rate is given by the following (covered more generally here).
\begin{darray}{rcl}p' &=& 1 - \frac{1-p}{2} = 1 - \frac{\textrm{original spending rate}}{2}\\\\&=&\frac{1+p}{2}\end{darray}So the first half N_2 plus the second half N' = -25\ln(p') equals the original estimate N=-25\ln(p).
\begin{darray}{rcl}\underbrace{\ln\left(\frac{1}{p}\right)}_\textrm{original estimate}&=& \underbrace{\ln\left(\frac{1+p}{2p}\right)}_\textrm{first half} + \underbrace{\ln\left(\frac{1}{p'}\right)}_\textrm{second half}\\\\&=&\ln\left(\frac{1+p}{2p}\right) + \ln\left(\frac{2}{1+p}\right)\end{darray}Those are equal exactly because the following are.
\frac{1}{p} = \frac{1+p}{2p}\cdot\frac{2}{1+p}Generalizing to Other Income Streams
The notion of effective savings rate can be generalized to include other sorts of income. Above, we defined it to be your expenses minus whatever your portfolio covers. You might also subtract all other life long income streams from your expenses. What should remain are your expenses E' that still need to be covered by more portfolio income.
Your expenses minus
- Withdrawal rate of your current portfolio
- Rental income
- Passive business income
- Social security (if it’s already started)
- Pension (if it’s already started)
- Annuities (if they’re extremely likely to last your lifetime)
- Royalties, etc
is what you want. Then p'=\frac{I-E'}{I} is your generalized effective savings rate, and N=-25\ln(p') is how long it’ll take to save up the remaining amount.
Summary
There it is, one simple subtraction in the right spot answers so many questions.
- How much longer?
- How does rental income play a role?
- How do I count my home equity? (not at all unless you intend to downsize)
- How much does losing my pension make a difference? (pensions do fail)