tl;dr⌗

There’s a common method for calculating how much you need to retire called the 4% rule; but blindly trusting some rule can make it hard to really understand what assumptions were made in generating that rule.

I’ve recently written up an analytical mathematical model that accounts for both interest rates and inflation in the context of calculating retirement annuities. I’m going to talk a bit about how some of these basic rules actually derive their numbers, how it applies to this extension, and what we can learn from using these simple analytical tools.

This article has a bit of math in it, but nothing beyond algebra complexity.

Context⌗

Over the past decade, there’s been quite a bit of interest in the Financial Independence, Retire Early movement, where one saves money aggressively in order to have enough saved wealth to retire early.

This post was motivated by a comment in a Reddit thread on what the appropriate FIRE number is:

You’re over complicating this. Just use that range [25x-35x yearly expenses]. Go for that mark. Over time, build the life you want to retire to. And over time, re-total your yearly expenses and adjust your mark as needed.

NOTE: The “25x-35x yearly expenses” informally refers to the 4% rule: If you have some amount of wealth saved, $P$, then by withdrawing 4% of $P$ annually, adjusted for inflation, under basic assumptions of growth in the investment, you should be able to survive on that amount for 30 years.

I agree and disagree with this comment. To some extent – yes, overcomplicating retirement, especially for those who aren’t very familiar with math or finance, is likely going to be about as inaccurate as the 4% rule. The best tool for reducing inaccuracy is to simply wait and recalculate using real data.

On the other hand, blindly trusting such metrics and not considering their limitations/assumptions underestimates the complexity of financial planning and how hard it is to forecast a retirement plan over the course of 30+ years.

Nonetheless, building a complete model that captures all possible variables (as the above post was slowly developing into) is going to be both infeasible and unrealistic. Instead, we can evaluate the merits of the 4% method through the lens of some simple mathematical models.

This isn’t any type of financial advice or even expert financial analysis; I’m writing this more from the interest in some of the math rather than coming in as an expert on this subject. I wouldn’t reuse much of anything I write here in your own planning unless you know what you’re doing, in which case you’ll be smart enough to know not to use this!

The payout annuity model⌗

One strategy for understanding the implications of some rule of thumb is to analyze it through the lens of a simple, explanatory model. The advantage of this analysis is that analytical models often expose simple relationships that give us general and widely applicable insights, as well as allowing us to adapt such rules to new circumstances.

We’re going to start with a very basic model of calculating retirement income, payout annuities. Our model is simplistic: in plain English, we’re going to assume that we start retirement with some amount of money spread across a number of investments, and a series of fixed payments (annuities) where we will gradually withdraw money from that investment account. In math terms:

  1. We are going to start out with some initial amount of money in our investment account, $P$.
  2. Every time period, we are going to withdraw $W$ from the account for $T$ time periods;
  3. Our original investment is going to grow by some interest rate $r$ every time period as well.

The choice of granularity for our time period is arbitrary, but a general rule of thumb is that if you want to work on a month scale, then you would take your annual interest rate $r$ and divide it by 12.

Interestingly under these constraints, payout annuities are identical in formulation to calculation of a fixed payment for a loan in reverse: You’re starting out with a beginning balance that grows at a certain rate, and you’d like to calculate a fixed withdrawal rate such that, in $t$ months, your final balance is 0$.

As such, you can actually use the same formulas to estimate your needed investment balance. Deriving a number is fairly simple: Let $P_t$ be the total amount of money in your account at time $t$. We know that:

  1. $P_0 = P$, i.e., your starting money is the amount of money you’ve got in your investment accounts;
  2. $P_T = 0$, i.e., at the end of your retirement, you’re going to have a 0$ balance, and;
  3. $P_t = (1+r) P_{t-1} - W, 0\le t\leq T$; in other words, during each pay period, your investment grows by $1+r$ and loses a constant $W$.

What we’d like to do for is solve for $P$; to do so we’re going to expand $P_T$ until all of the recursive $P_t$ terms have disappeared.

$$ \begin{aligned} P_T &= 0 = (1+r) P_{T-1} - W \\ &= (1+r) ((1 + r)P_{T-2} - W) - W \\ &= (1+r)^2 P_{T-2} - (1+r) W - W \\ &= … \\ &= (1+r)^T P - \sum_{t=0}^{T-1} (1+r)^t W \end{aligned} $$

You may recognize that $\sum_{t=1}^T (1+r)^t W$ is a finite geometric series; fortunately when $r\neq 0$ we know that this sum is equivalent to:

$$\sum_{t=0}^{T-1} (1+r)^t W = W\frac{1-(1+r)^T}{-r}$$

By substituting in these terms, we can eliminate the summations and get:

$$P_T = 0 = (1-r)^T P - W \frac{1-(1+r)^T}{1 - (1+r)}$$

Since we’re interested in calculating $P$ directly, we can do some basic algebra to arrive at:

$$P = - W \frac{1-(1+r)^T}{(1 - (1+r))(1 + r)^T}$$

For those using Excel, this is actually identical to the Present Value function:

$$P = \text{PV}(r, T, W)$$

Are payout annuities realistic?⌗

To understand what payout annuities calculate, let’s try out a toy example: I want to calculate the amount I would need for retirement for $T=30$, $W=72000$, and $r=10\%$, the approximate average return rate of the S&P 500. In other words, I’m withdrawing \$72,000 per year which translates to actually spending \$57,600 per year and allocating $14,400 in capital gains taxes.

If I wanted to start out withdrawing \$72,000 from my investment account starting at retirement, the amount of money I’d need according to the 4% rule is \$1.8M. Plugging the same numbers into our PV formula:

$$ P = \text{PV}(10\%, 30, 72000) = 678,737.84 $$

In other words, our payout annuities and the 4% rule estimate a difference of over \$1 million dollars! Why is there such a difference?

The major issue is our choice of interest rate. 10% may be the average growth of the S&P 500, but any real retirement portfolio is likely going to be much more diversified to curb against risk. Investments that track the market like this are likely to be highly variable – if you started retirement in 2022 for example, your portfolio could have decreased in value by as much as 20%. The impact of pulling \$72,000 then would take off years from your retirement plan. Thus, if you’re using a simpler model that assumes a constant return, you’ll likely need to choose a rate of return much lower than your actual returns to account for that risk.

The second major issue in payout annuities is the inflation factor. Suppose, for example, that the dollar decreases in value by an inflation rate of about 2% per year. The \$72,000 that you’re withdrawing in 2054 would be worth about \$39,749 in 2024 dollars, meaning that your quality of life is decreasing significantly year-by-year.

The 4% rule’s number seems reasonable, though – so let’s instead fix $P=1.8\text{M}$ and instead try to solve for $r$ instead; that is, what interest rate does payout annuity need to behave similarly to the 4% rule? In this case it turns out to be close to 1.25%! That’s significantly lower than even the most conservative returns on any portfolio, even accounting for inflation. How does this number come about?

Inflation-aware payout annuity⌗

To get a more nuanced view of the 4% rule, let’s modify our initial assumptions. We know that we can account for risk partially by reducing our interest rate number $r$; but to account for inflation, we probably can’t apply the same principle.

Why? Risk directly affects our annual rate of return $r$; thus, even if there’s variability in $r$, we know that there’s going to be some value of $r$ that fits our “investment growth + risk” curve semi-well.

On the other hand, inflation doesn’t affect the growth of our portfolio at all. If inflation is 2.5% one year and our portfolio grows 4.5%, our portfolio is still worth 104.5% its value from last year. Inflation instead affects only our withdrawal rate: If we withdrew $W_{t-1}$ last year and inflation was 2.5%, then our new annual expenses will be around 102.5% of $W_{t-1}$.

Let’s model this by introducing a new inflation rate, $i$, into our model:

$$P_t = (1+r) P_{t-1} - (1 + i)^t W$$

We can similarly solve for $P$ using $P_T$:

$$ \begin{aligned} P_T &= 0 = (1+r) P_{T-1} + (1+i)^T W \\ &… \\ &= P (1+r)^T - \sum_{t=0}^{T-1} (1+r)^t (1+i)^{T-t} W \\ \iff P &= W \frac{\sum_{t=0}^{T-1} (1+r)^t (1+i)^{T-t}}{(1+r)^T} \end{aligned} $$

The series $\sum_{t=0}^{T-1} (1+r)^t(1+i)^{T-t}$ is a bit more complicated than our previous term. I actually looked around online to see if there was a simpler representation of the sum; unfortunately I couldn’t really find anything. I’d love to learn if there is some simpler representation, as an analytical term could give us a bit more insight into how this function behaves.

Nonetheless, this derivation gives us a means of calculating an inflation-aware payout number. Let’s use $r=4.25\%$ and $i=2\%$, a semi-reasonable number for the growth of our portfolio assuming that we have a good amount invested in more risk-averse options and that the inflation rate stops going crazy and holds at around 2% fvor the next 30 years.

Assuming the same \$72,000 initial withdrawal rate per year, the inflation-aware principal we would need to survive for 30 years is around \$1.57M. Note that this is still lower than the 4% rule number of \$1.8M; but we can now analyze what the 4% rule probably assumed when building the intial model.

For example, let’s keep our inflation rate at 2%. For the inflation-aware payout annuity to match the 4% rule, it generates a similar starting amount to the 4% rule at $r=3.24\%$, which is significantly lower than our original conservative estimate.

Model insights⌗

One of the advantages of analytical modeling is that we can often use it to learn about relationships between variables.

For example, one major question I had (which arguably prompted this analysis) was if the 4% rule worked regardless of the yearly withdrawal rate $W$. In other words, if I need \$1.8M to spend an inflation-adjusted \$72,000 anually, do I need \$18M to spend \$720,000 annually? Sometimes these simple rules are based on the unstated assumption of some “regular” spending range, and may not necessarily have a linear relationship.

Fortunately, if we look at the relationship between $P$ and $W$ in both of our payout annuity models, we see that there is a linear relationship and thus the 4% relationship actually does hold no matter what your withdrawal rate is.

One factor that does affect our 4% number, though, is the number of years we plan to be retired. Someone at age 30 may need to plan to retire in 70 years for example – how much would they need to save?

Interestingly, if we fix $r$ and $i$ on our model to those that matched the 4% rule, we can actually get basic multiplier that can be used in your own Excel spreadsheets:

# of Years Initial Withdrawal %
5 20.74%
10 10.68%
15 7.34%
20 5.66%
25 4.66%
30 4.00%
35 3.53%
40 3.17%
45 2.90%
50 2.68%
55 2.51%
60 2.36%
65 2.24%
70 2.13%
75 2.04%
80 1.96%

Of course, these numbers are about as nonsensical as the original “4%” rule of thumb. The point is that depending on how long you plan to be retired, there’s going to be a significant amount of variability in your planned final retirement number.

For example: an individual who plans to retire at age 30 could very well live to 80; in that case, our numbers suggest that they would need \$2.54M to retire at the same time as someone at 50 with the same exact spending plans; a far cry from the initial \$1.8M that the 4% rule suggested they needed.

The final insight – which I haven’t really researched to closely yet – is that it’s actually possible to model my inflation-aware payout annuity model using:

$$\text{PV}(1-\frac{1+i}{1+r}, T, W)$$

Ironically this was actually the number I used to “approximate” the effect of inflation before I had properly derived a model with inflation explicitly included.

I’ve tested this in a number of scenarios by changing $r$, $i$, $T$ and $W$ and have found that the difference between my inflation-aware payout annuity and this adjusted $\text{PV}$ tends to be nearly identical, and linear in $T$ (i.e., the difference isn’t massive but grows linearly as $T$ grows). I found this especially interesting, as it suggests that perhaps there isn’t any loss in representability by using the above term rather than my “proper” inflation-adjusted solution for $P$; it’s so good, in fact, that I’d recommend anyone interested in experimenting with this should use the above $\text{PV}$ term instead simply because it’s easier to write in Excel.

Limitations⌗

Nearly every “rule of thumb” or simplified model approach exists primarily to give us insights, provide ballparks, and understand more deeply the assumptions we make when we start to forecast estimates.

Trusting these models implicitly can be dangerous. Finance is highly complicated, incredibly variable and thus hard to predict in perpetuity; I’d highly recommend readers here play around with some of these models to see just how sensitive they are to the input numbers (particularly, inflation and interest).

If you do have any feedback on this article or related links, feel free to DM me on Reddit (/u/cacrawford). I’d love to hear about any similar types of models or better derivations.