Generalized Savings Rate

We’ve covered quite a lot of ground already! We derived a couple of ways to compute your retirement date, but one aspect is still missing.

How to Misuse the Savings Rate

Imagine throughout your entire career you maintain a very respectable savings rate of p=0.5, and as you approach your 17th year, your portfolio is looking to be about \mathbb{X}=25\cdot E_0 or 25 times your annual expenses E_0. Just then, your employer realizes she needs you more than you need her. She offers you a 100\% raise to distract you from leaving. You happily accept the raise (for now at least) and grow into the new lifestyle, wisely keeping your savings rate at p=0.5.

Can you still retire whenever you want, maintaining the new 2x lifestyle?

No. Your portfolio is currently \mathbb{X}=25\cdot E_0 but E_1=2\cdot E_0 and so \mathbb{X}=12.5\cdot E_1. You’re underfunded by 50\% despite maintaining a constant savings rate throughout. You can totally retire at the old expense level of E_0, but you have some time to go before retiring at the new lifestyle E_1. In fact, your effective savings rate with the bigger income I_1 is now the following.

Now with p'=\frac{3}{4}, you can retire in N'=-25\ln(\frac{3}{4}) = 7.2 more years keeping the higher spending level E_1. You’re exactly in the “Half FI” scenario from the last post.

Your boss has you right where she wants you!

All of our current savings rates and savings ratios involve the current level of spending. How can we plan to spend differently in retirement?

Redefining Savings Rate

All of our derivations stem from the savings ratio.

We need a new savings ratio, apparently. Something fancier. Something more general. Suppose we know what we’re currently saving per year (income minus expenses, not counting taxes), and we know what we’ll be spending per year in retirement. Then our generalized savings ratio \mathbf{R} is the following.

This is just a number, a multiple of your target expenses just as before. If you’re currently saving $2000/mo and intend to spend $2000/mo in retirement, then your (fancy new) generalized savings ratio is 1. If you’re currently saving $40k/year but intend to spend double that, then it’s \frac{1}{2}.

Regardless of the time period you’re using or the value of \mathbf{R} itself, every month (or year) you work at savings ratio \mathbf{R} earns you a month (or year) of freedom. When you collect 25 years (or 300 months) of retirement expenses, then you’re retirement eligible (at least for the spending level you specified in the denominator).

Now we just need to find some generalized p that produces the generalized savings ratio above. Your generalized savings rate \mathbf{p} is found by solving for p in R=\frac{p}{1-p}.

Generalized Retirement Schedule

If you’re starting from scratch, then your time to retirement is \mathbf{N}=-25\ln(\frac{\mathbf{R}}{\mathbf{R}+1}). If you’re not starting from scratch, then make sure to remove \mathbb{X}/25 (income from your “invested portfolio”) and whatever other sources of retirement income you already have from your retirement budget, since that portion of the budget is already covered.

The easy way to remember this is numerator = income – expenses, but the denominator = expenses – income. They’re flipped because context demands it.

Some of your retirement income might appear in both the numerator and denominator, and that’s correct. It’s what you’re currently saving divided by your future as-of-yet unaccounted for expenses. Any unspent income is growing your portfolio, and any unaccounted for future spending still needs to be saved for. So your existing portfolio definitely counts in both places.

Comparison

Notice that when you maintain the same level of spending into retirement, the “current expenses” in the numerator equals the “target retirement budget” in the denominator. Also, when starting from zero, “retirement income” is zero. In this special case, the equation captures the same information as the original form in introducing the savings ratio and all the standard discussions of the savings rate.

Progress

A few posts from now, in Beyond Saving, we’ll exclude portfolio income from this calculation, only to include it differently. As the portfolio grows, we’ll update the calculation using a more convenient and insightful method. Thus, when using the update method of Beyond Saving, the numerator and denominator of the generalized savings ratio include everything discussed here except assets from which we withdraw the standard 4%.

Mathematics to Make Your Heart Sing

Why is any of this even relevant?

• People spend less in retirement
• Your house will be paid off
• Darn kids are out of college
• The cost of commuting to work is surprisingly large
• For all but the fatter FIRE programs, federal income tax is essentially zero

For any payments you currently have but intend to have paid off before pulling the retirement trigger, just set the money aside (remove the balances from your “invested portfolio”) and exclude the payments from your current monthly expenses. The calculation above will then also exclude them.

For any retirement income sources that will kick in soon (e.g. you’ll start receiving social security in 36 months), simply remove the sum of those missing payments (e.g. 36 x payment amount) from your “invested portfolio”, set it aside, and go ahead and include it in your retirement income sources (you’re bridging the gap by mentally setting the money aside).

Finally, we see why taxes are always excluded from your savings rate. It’s an expense that generally won’t persist into retirement, so the denominator in the general equation above excludes it.

You might be far closer than you realize.