12. Math Break: The impact of taxation

What is the benefit of letting your savings grow tax free?

Of course, if you never need to pay taxes on your savings, the benefit is obvious. For example, with a 401K program, you invest money pre-tax (i.e., put the money before any taxes are taken from it), and if you withdraw it in retirement (after the age of 59.5), the withdrawals are tax free as well. Even without the matching contributions most employers offer, this is a no-brainer good plan. The first order of the day for anyone who can save money is therefore to max out their 401K.

With a Traditional IRA, things are trickier. You can contribute pre-tax money (if you're below certain income thresholds), but you need to pay taxes when you withdraw the money. And Roth IRA is kind of the opposite: you contribute post-tax money, but the withdrawals are tax-free.

<rant>I find it crazy to have all these different retirement programs available in the tax law, each with its own peculiar rules, and to top it all, complex rules for conversion between plans. And this is before we add other more exotic plans such as Roth 401(K), 529 and others I know little about. Wouldn't it be nice if we had just one plan whose benefits combine all of the above? One pool of savings that can be used in retirement, or for college, or for buying a house?</rant>

Financial advisers will often tell you that you gain more from letting your savings grow tax free, even if you pay taxes at the end. As a mathematician, I wasn't sure there's a real benefit. After all, if you start with, say, $10,000 and expect a gain of 50% over 10 years and taxes of 35%, why should the order matter? After all:

$10,000 * 1.5 * 0.65 (50% gain first, then 35% tax) 

is the same as

$10,000 * 0.65 * 1.50 (pay taxes first, then add the gain).

In math terms, multiplication is commutative - the order of the terms will not change the result. 

And in fact, in some situations this is exactly the result - both kinds of IRAs will give you absolutely no benefit if you invest under these "vanilla" criteria. 

However, for most of us, IRA plans do have a real benefit, as a result of the following factors:

• Your tax bracket when you withdraw the money is typically lower than when you save the money.
• If you invest in mutual funds in a taxable account, you'll have an annual tax bill even if you don't sell them since they have annual distributions
• If you buy bonds directly, when they mature you'll have a tax bill
• And of course, if you change your investment - sell some shares and buy others - you'll have a tax bill.

In other words, the only way that my math above would deem IRA plans worthless is if you use them to buy stocks (not mutual funds) and keep them until retirement without ever selling them, and then sell them when you're still employed with the same tax bracket (or higher.) This is unlikely for most investors.

Which leads us to the next question: suppose you're a typical investor investing in mutual funds (and I hope that means Index funds!) What is the gain from delaying paying taxes or from letting the money grow tax-free?

Let's take for example a traditional IRA and compare it with a taxable account. In this example the IRA is fully invested in bonds, which are taxed as income. Let's also assume that your tax bracket remains constant (if it's lower when you withdraw the money, we already know the benefit). Let's fix your income tax at 35%, and assume a net annual growth of 5% (by net I mean net of inflation - it just makes the calculations easier). Suppose that you start with $10,000 and invest them for 30 years.

Case 1: Tax-free Growth: After 30 years your taxable account is worth 10,000 * 1.05³⁰ = $43,219. You need to pay 35% tax on the gain ($33,219) which leaves you with $31,592.

Case 2: Taxable account: Every year, starting with A, your account grows to A*1.05, but then you need to pay taxes on the gain, A*0.05. This leaves you with A*1.05 - A*0.05*0.35 = 1.0325*A. After 30 years, your account will be worth 10000 * 1.0325³⁰ or $26,104. This is about 18% less.

The difference won't be as dramatic when the taxes are lower - as is the case with long-term capital gain. But if you're buying and selling stocks all the time (a bad idea in any case), your tax bill will quickly eat your gains.

What is my bottom line?

1. Tax-free growth is good. Maximize your 401(K), and if you can open an IRA or a Roth IRA, maximize your contributions to these channels as well.
2. For asset allocation, try to put tax-inefficient investments (such as bonds) into your 401(K). A second choice is long-term aggressive growth funds. 
3. For your taxable accounts, focus on low-turnover index funds. They will minimize your annual tax bill, and when you finally sell your investments at retirement, you'll enjoy all the years of nearly tax-free growth, and, most likely, a lower tax bracket.