The math behind retirement planning
A companion post.
To those who might read this post, I salute you. We covered how to plan your retirement investing in the main post.
The first part walks through a few compound growth problems that everyone should know how to do.
The second part discusses the assumptions behind our retirement calculations.
The third part shows how to derive the retirement formula.
If you’re reading in the mobile app, the equations below might look ugly or even incorrect. It looks much better if you copy the post link into your browser. When reading in the mobile browser, you have to rotate your phone to horizontal orientation to view the longer equations.
Compound growth
If a $500 investment earns an average annual return of 7.5% for 30 years, how much is that investment worth at the end of the period? To indicate 7.5% growth in one year, we would multiply the initial amount by 1.075. To indicate an average 7.5% annual return over 30 years, we multiply by 1.075 30 times: 1.075^30 = 8.75496. Now we know that 7.5% average annual growth over 30 years is equivalent to multiplying by 8.75496, so we calculate the final balance as $500 × 8.75496 = $4,377.48.
In one expression:
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The next one is harder.
If a $300 investment grew to $9,083.15 over 40 years, what was the average annual return?
We reverse the process above. First we find the factor by which the investment grew: 9,083.15/300 = 30.2772.
Now we can ask, what average return is needed to multiply an investment by 30.2772 over 40 years? We want the number which solves the equation:
You may know that to isolate r, we can neutralize the exponent of 40 with another exponent of 1/40. So we calculate r = 30.2772^(1/40) = 1.089. A multiplier of 1.089 indicates a return of 8.9%.
In one expression:
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Over four years our investments provide annual returns of 3%, 25.9%, -2%, and -11%. What is the average annual return?
We convert these percentage changes to multipliers: 1.03, 1.259, .98, and .89. The geometric mean is calculated as:
A multiplier of 1.0313 indicates a 3.13% return. The geometric mean of annual returns is often called the compound annual growth rate (CAGR). If it’s unclear why the calculation must be done this way, see this footnote.1
Everything is expressed as a multiplier, so we use geometric means, not arithmetic means.
Assumptions of this approach
We use a simple model of retirement investing in the main post. We contribute money at the beginning of each year. The contributions change at a constant rate determined by inflation and the growth factor, and the investments grow at a constant rate until retirement.
The most questionable assumption by far: investment returns will be constant. Of course that isn’t true! But does it matter? Does it make the calculation misleading?
Let’s run a test using two opposite scenarios that partly use real data. In both cases, our average return is 8%. In Sequence A, we get:
US stock returns from 2000-2009 (terrible) in Years 1-10.
Constant 8.24% returns in Years 11-30, to force an 8% average.
US stock returns from 2012-2021 (fantastic) in Years 31-40.
This is ideal, because we get to buy at low prices for the first ten years, then get a ton of appreciation as we approach retirement.
In Sequence B, we get:
US stock returns from 2012-2021 (fantastic) in Years 1-10.
Constant 8.24% returns in Years 11-30, to force an 8% average.
US stock returns from 2000-2009 (terrible) in Years 31-40.
That’s bad, since we don’t have much invested during the fantastic years, then our investments lose money in the final decade before retirement.
At the end of 40 years with Sequence A, we have a nest egg of $5.52M. That’s 83% larger than the $3.02M nest egg from a constant 8% return. It’s the equivalent of getting a constant return of 10.3%! Sequence B gives us a nest egg of only $1.90M, the equivalent of a constant 6.1% return. The nest egg is 37% smaller than $3.02M.
That was deliberately set up as an extreme contrast. It would be hard to replicate with real data placed in its historical sequence — especially for a globally diversified portfolio — because returns tend to revert to the mean. A below-average decade is often followed by an above-average decade, and vice versa. However, it shows that even if you experience an average return of 8%, your results can be higher or lower than expected. It’s yet another reason to have buffers in your plan.
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Another (more conceptual) assumption underlying the calculation is that inflation will be low and fairly stable. I can’t guarantee that, but any investment plan could be thrown in the trash if the US mirrored Argentina’s inflation. There’s nothing we can do about that.
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Tax is another issue. But if we invest for retirement largely in tax-advantaged accounts, it’s okay to downplay it. In a taxable account, most people would be taxed 15% on the gains of a stock ETF when they finally sell it. That’s easy to anticipate and not the worst haircut, given that we had the advantage of keeping that asset liquid the whole time. Before selling it, a small amount of tax would be owed on dividends each year.
Hardcore math (it’s algebra)
We’re planning for retirement and want to know how much to save and invest each year. We’ll use six variables to determine our savings: R for the real annual return on investment, T for years until retirement, D for desired retirement income in today’s dollars, W for initial withdrawal rate, and A for amount of savings already set aside for retirement. Later we’ll include G for annual growth of contributions beyond inflation.
Our task will be easier if we temporarily convert real return into nominal return B divided by inflation F. So R = B/F.
Let’s start with the flat or time-invariant approach to retirement investing. We have a contribution target for the first year C determined by the variables above, and the contribution targets for later years are increased by anticipated inflation F. This means that the first year’s contribution is C, the second year’s contribution is CF, the third year’s contribution is CF^2, and so on. In the final year before retirement, the contribution is:
The next step is to find the value to which each contribution will grow by retirement age. The first year’s contribution C has T years to appreciate, so its value at retirement is CB^T. The second year’s contribution CF has T-1 years to appreciate so its value is CFB^(T-1). The third year’s contribution CF^2 has T-2 years to appreciate so its value is:
So the total value of our nest egg N at retirement age will be:
We can factor out C, and let’s call the rest of it Z:
Let’s work on simplifying Z for a moment. The first term is already simple: B^T. The second term is FB^(T-1), but it could be reframed as:
This may seem pointless, but notice that we can define the next term as:
The additional terms just increment the exponent on (F/B). So here’s Z in a different outfit:
Let's return to CZ. We can now factor out B^T as well, and we’ll call the rest V:
We know R = B/F, so 1/R = F/B:
We now have a workable definition of our nest egg from future contributions N (in nominal dollars). Until now we’ve neglected the savings already set aside for retirement (A). Similar to the first year’s contribution, the anticipated value of A at retirement will be AB^T. The nest egg we’re aiming for is actually composed of the nest egg we build from future contributions (N) plus the value our current savings will have at retirement (AB^T). Let’s call this total nest egg Q.
The desired income in today’s dollars D divided by the initial withdrawal rate W gives us our total nest egg in today’s dollars. Multiplying D/W by the inflation factor over the full period F^T gives us Q, the total nest egg in inflated dollars.
Now we have two expressions of Q that we can set equal:
We established above that:
So we loop that into the equation with Q:
We can finally solve for C in terms of the variables we care about. Divide by B^T:
Since F/B = 1/R:
Then:
Subtract A and divide by V to isolate C:
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And finally, what if we want to increase or decrease contributions each year after adjusting for inflation? The variable G expresses this growth approach, in contrast to the flat approach.
The first year’s contribution C will not change. The second year’s will be CGF, the third year’s will be C(GF)^2, and so on. The modified version of V will be U:
As F/B was converted to 1/R, GF/B will be converted to G/R:
We can isolate C with the additional variable G by replacing V with U:
What if we want to reverse this equation and convert a contribution pattern into retirement income? We solve for D.
Multiply by U and add A:
Multiply by WR^T:
So we can summarize:
To pursue a flat approach, set G equal to 1. To contribute the same dollar amount every year — regardless of inflation — set G as the reciprocal of inflation F. If F is 1.025, then G would be 1/1.025.
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That’s all for today, thank you for reading! Here’s a link back to the main post.
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For some people it may not be obvious why the more common arithmetic mean is unsuitable. Perhaps we could have found the average return like this: (3+25.9-2-11)/4 = 3.975.
To consider this, we can imagine an unlikely sequence of returns, in which we double our money the first year and halve it the following year. Multiplying an asset by 2 and then .5 returns it to its original value. We can calculate the geometric mean of these multipliers, which we know intuitively should be 1: (2×.5)^(1/2) = 1.
That’s right, but the arithmetic mean gives us an incorrect answer, (2+.5)/2 = 1.25.
Another check is to find that the actual return matches the average return to the power of the number of years. 1.03×1.259×.98×.89 = 1.13, which equals 1.0313^4. The arithmetic mean does not satisfy this test. Because we’re averaging multiplicative factors, the geometric mean must be applied.