# Investment and taxation

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Since I started working in 2013, I haven’t thought much about investment. Because I know that investing in an index fund is the theoretically right thing to do and I do not like paying taxes, in addition to participating in my employer’s mandatory defined benefit retirement plan, I have just been maxing out investment in 403(b), 457(b), and Roth IRA (all invested in low cost index funds).

But now my situation has changed a little bit and I had to think about it. Although my (meager) nominal salary has increased by 34% over the past 8 years, my expenditures have increased by even more. There is a lot of inflation. The kids are entering teenage and require expenses for extra curricular and after school activities. Furthermore, my family has become a group of eager tennis players and nowadays tennis-related expenditures (lessons, tournaments, racquets, strings, shoes, etc.) is our number one expenditure, even surpassing taxes.

So I had to make a decision. My options are

1. find a higher-paying job,
2. move to a cheaper house,
3. spend less,
4. draw down savings from taxable investment accounts, or
5. decrease contributions to retirement accounts.

The first two options involve a big decision so let’s set it aside for now. Spending less is not really an option - we all love tennis and we can’t just reduce consumption. So the choice is between spending money from taxable accounts or saving less in retirement accounts.

Which brings me to the comparison of various investment accounts. For simplicity, let $\tau$ be the tax rate, $r$ be the rate of return on investment (continuously compounded), and $t$ be the investment horizon. Suppose I have a before-tax income normalized to 1.

If I invest my income in a traditional retirement account (traditional IRA, 403(b), 457(b)), then I can invest pre-tax but I need to pay tax when I withdraw, so by time $t$ the investment grows to $1\times e^{rt}\times (1-\tau)=(1-\tau)e^{rt}$ after-tax. If I invest in a Roth IRA, then I invest after-tax but do not owe any tax when I withdraw, so the investment grows to $1\times (1-\tau)\times e^{rt}=(1-\tau)e^{rt}$ after-tax, which is the same as the traditional account. This calculation suggests that a traditional (Roth) retirement account is better if the current marginal tax rate is higher (lower) than in the future. Personally, I expect to be in a higher tax bracket in the future, which is why I invest in Roth.

If I invest my income in a taxable account and continuously realize capital gains (e.g., invest in a short-term bond fund), then the investment of 1 (before-tax) grows to $1\times (1-\tau)\times e^{r(1-\tau)t}=(1-\tau)e^{r(1-\tau)t}$, which is clearly worse than an investment in a retirement account (because investment grows tax-free in a retirement account). If I invest my income in a taxable account but realize capital gains only when I withdraw, then the investment of 1 (before-tax) grows to $1\times (1-\tau)\times(1+(1-\tau)(e^{rt}-1))=(1-\tau)(\tau+(1-\tau)e^{rt})$. The strict convexity of the exponential function implies $e^{r(1-\tau)t}=e^{\tau\times 0+(1-\tau)\times rt}<\tau+(1-\tau)e^{rt}$, so realizing capital gains at the end is always better than continuously realizing capital gains.

In the end, I have decided to stop contributing to 403(b) and 457(b) for a while (though I need to pay attention to keep my adjusted gross income below 150k to qualify for “child tax credit”). With various deductions (I was happily surprised to learn that tennis summer camps qualify for “child and dependent care credit”) I am already in a relatively low tax bracket, so there is not much tax benefit of investing in 403(b) and 457(b). On the other hand, you give up liquidity by investing in these accounts, which is more important to me now.

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