# Comparing taxable, traditional, and Roth accounts

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Three years ago, I wrote a blog post in which I mentioned that I stopped (strictly speaking, significantly reduced) contributing in 403(b) and 457(b) retirement savings plans because I could no longer afford it.

Now that I got a new job (with a higher salary), I have the luxury of investing again. There are several factors that have changed over the past few years, which is worth discussing. First, 403(b) now allows a Roth acount. Second, inflation has been high. These factors begs a fresh analysis. For simplicity, let $$\tau$$ be the tax rate, $$r$$ the real rate of return on investment (continuously compounded), $$i$$ the inflation rate, and $$t$$ the investment horizon. Suppose I have a pre-tax income of $$Y$$ available for investment.

If the money is invested in a retirement account, because capital gains are not taxed, investment grows at a nominal rate $$r+i$$. Therefore, regardless of whether it is invested in a traditional account (you pay taxes when withdrawing) or in a Roth account (you pay taxes before investment), the resulting amount in real terms is $$(1-\tau)Ye^{(r+i)t}/e^{it}=(1-\tau)Ye^{rt}$$. Thus, inflation has no impact on investment.

In contrast, if the money is invested in a taxable account, assuming that capital gains are realized continuously, investment grows at a nominal rate $$(1-\tau)(r+i)$$. Therefore the resulting amount in real terms is $$(1-\tau)Ye^{(1-\tau)(r+i)t}/e^{it}=(1-\tau)Ye^{(r-(r+i)\tau)t}$$. Thus, taxes and inflation drag the investment performance by rate $$(r+i)\tau$$, which is the product of nominal return and tax rate. This observation has a significant implication. If we assume conservativelly that the nominal return is 10% (say, real return 7% plus inflation 3%) and the marginal tax rate is 30%, then the investment performance decreases by 3% (because $$0.1\times 0.3=0.03$$) annually. A difference of annual 3% is enourmous in the investment world.

Of course, taxable and retirement accounts have their own advantages and disadtanvages, so taxes should not be the only consideration. With a taxable account, you have liquidity. With a retirement account, you give up liquidity. Another consideration is the difference between traditional and Roth accounts. Traditional accounts have a required minimum distribution, wheras Roth accounts do not. Furthermore, both traditional and Roth accounts have the same annual contribution limits (currently $23,000 for 403(b)), so Roth allows a larger investment in pre-tax dollars. For instance, at 30% tax rate, to invest$10,000, traditional requires $10,000 of pre-tax income, whereas Roth requires$14,286. With my current employer, 9% of my salary is matched and contributed to a traditional account, and 457(b) is also traditional. Therefore it seems wise to max out the remaining investments in Roth only.

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