Comparing taxable, traditional, and Roth accounts

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

Now that I have 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 are worth discussing. First, 403(b) now allows a Roth account. Second, inflation has been high. These factors beg 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, the 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 down investment performance by the rate \((r+i)\tau\), which is the product of nominal return and tax rate. This observation has a significant implication. If we assume conservatively 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 3% annual difference is enormous in the investment world.

Of course, taxable and retirement accounts have their own advantages and disadvantages, 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 required minimum distributions (RMDs), whereas 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 a 30% tax rate, to invest $10,000, a traditional IRA requires $10,000 of pre-tax income, whereas a 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.