I've never particularly liked the way many people describe the tradeoffs between different investment accounts. You'll hear phrases like

- "Pay the tax now, enjoy tax-free distributions later." (Roth contributions)
- "Don't pay today and let a bigger principal compound, but face a huge tax bill." (deductible traditional contributions)
- "Pay tax on contributions; earnings grow tax-deferred." (nondeductible traditional contributions)

The issue I take with these is that they're not very *usable*. It's not clear how to compare one type of tax structure to another. Indeed, it's all too easy to draw incorrect conclusions from loaded, prose approximations of what are actually mathematical statements. Take as examples the following two claims and their alleged proofs:

You should minimize the number of investment accounts you have. Why? To start, consider that the value of investments rises exponentially, and also that the rate of growth of an exponential curve increases as you move along it. Therefore, it's more powerful to have fewer accounts with bigger balances (so they'll accelerate upward faster) than to have more accounts, all growing slowly.

or

Roth contributions are preferable to traditional ones, since they incur a smaller tax bill. Why? Suppose you have $10,000 to invest, and you know that over the lifetime of your investment you'll see a return of 10x. Given the option, would you rather pay 25% tax on your principal today—that'd be $2,500—and enjoy the distributions tax free, or would you rather invest all $10,000 today and pay $25,000 in tax when you take distributions? I'd take a $2,500 tax liability over a $25,000 one any day.

One or both of these may seem valid arguments to you; they did at one time either to me or to people I know. They sound plausible! We'll see, however, that both are invalid.

These claims both attempt to suggest answers to the question "In what kind of account(s) should I put my savings?". In our conversations on that question my father is fond of saying "There are a lot of variables", and he's not wrong. To think clearly about such kinds of questions, it's incredibly helpful to be comfortable with letting variables be, well, *variables!* In the rest of this post, we'll explore using simple math to describe investment growth, with important quantities left as variables (e.g., time: years). You'll find that in so doing, the value of a given type of investment will be clear and readily comparable to other types of investments.

Suppose an investor has $1,000 of cash in her pocket that she'd like to invest. She puts it in a bucket (also known as an account), within which she then trades it for ** securities**: tradable financial assets like shares of a company's stock, or government bonds. Assuming that the securities are priced such that all she can trade every cent of her cash for some securities, she can and does make that trade:

Her investments, in the aggregate, appreciate 7% over the next year. The contents of her bucket are now valued at

The expressions on the right-hand side are all equivalent ways of writing the same thing. Take a moment to reacquaint yourself with some of the notation if you need to!

The investor lets it grow another year. Because she starts the year with $1,070, the math works out like

You may see a pattern forming! If we let stand for her principal ($1,000), stand for the annual rate of return (0.07), and stand for the number of years elapsed since she invested the principal, then we have

where the multiplicative factors of , of which one more appears with each additional year, are collapsed together via the exponent ("1.07 to the power of "). This is where the saying "in investing, time is your best friend" comes from, since the account value increases *exponentially* with time.

You may see this equation called the "compound interest with reinvestment" formula. Though we've framed it as an investment scenario here, consider how compound interest might be modeled mathematically. Though the processes are not the same in a literal sense, the math describing the two is the same!

We now have all the tools we need to precisely describe basic investing scenarios. With them, let's disprove the claims given at the outset.

*Claim #1:* "It's more valuable to have fewer accounts with bigger balances. " Suppose dollars are invested, and the assets they were traded for grow for one year. Then, more dollars are invested. Another year passes. The result:

Does it make a difference whether we invested in the same or a different account as ? Notice that we actually didn't make any assumption about exactly which account was put in! You may have thought we were doing one or the other, but the math of either choice is the same. In general, an investment of dollars is valued at after years. **Time invested matters, colocation with other assets does not.** This can be understood by regarding each of your securities (a single share of stock, a bond, etc.) as being on its own personal journey of appreciation, simultaneous to one another. There's no reason that cohabitation in the same bucket should affect those separate journeys.

*Claim #2:* "Roth contributions are preferable, since they incur a smaller tax bill." Suppose we have pretax dollars. In the traditional-vs.-Roth question, we're choosing when to subject our money to taxation: contribution-time, or distribution-time. Call the tax rates at those times and , respectively. What's the post-tax value of either choice, after years of growth at annual rate of return ?

What do you notice about these results? Recall that in math, multiplication can happen in any order and not change the result. Bearing that in mind, you might see that the results are equal when ! In other words, if the tax rates are the same, taxing before growth or after leaves you with the same final amount in your pocket in either situation. This can be seen in the argument for Claim #2, for example, where you'll see that both contributions yield the same final post-tax value of $75,000 (the claim used a sleight of hand, suggesting that the minimum tax bill was the goal rather than maximum post-tax value). Finally, notice that if —i.e., you're in a higher tax bracket earlier than you will be later—then the traditional contribution comes out ahead; vice versa, the opposite is true.

This result won't be surprising if you've already heard and accepted the common rule of thumb regarding traditional vs. Roth contributions:

Assuming you're investing the same portfolios for the same amount of time,

you care most about the relation (>, <, =) between and .

We've just derived that rule.

Key takeaways:

- The degree to which investments are colocated is inconsequential.
- Under common assumptions, the post-tax values of a Roth and traditional contributions depend only on your marginal tax rates at contribution- and distribution-time, respectively.
- You have a renewed appreciation for the tool that allowed us to convincingly prove these claims: math!

With this foundation, we can further compare actual account types (IRAs, 401(k)s, HSAs, etc.), while also incorporating some of the real-world considerations like contribution limits. All that and more in a future post! Stay tuned.