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The True Rate of Return

Inflation is often called a "hidden tax," and rightly so. But inflation's impact on savings and investment is much worse than commonly known.

It is not correct to define inflation as the general rate of change of prices. Prices rise and fall for a great many reasons. Little can be understood by considering all the factors contributing to price changes in an aggregate called "inflation." Instead, the concept of inflation should be more narrowly focused to allow the study of the influence of the quantity of money on prices.

Economic study requires a clear mental separation between "real" wealth such as consumer's goods, factories, etc., and "nominal" wealth which is real wealth's expression in monetary terms. Increasing the quantity of money does not create real wealth; it only creates little pieces of paper. If you gave everyone a million dollars they wouldn't all be able to rush out and buy a recreational boat — there physically aren't that many boats. They would have to be produced first. This could not happen quickly, because there aren't enough factories to build that many boats, nor are there enough "spare" materials to construct them with, nor enough skilled people to put the boats together. In order to create more boats, resources would have to be diverted from other areas of the economy, and this would mean fewer houses, cars, and everything else. Real wealth cannot be increased by increasing nominal wealth. The concept of inflation defined as the change in the quantity of money permits the study of the economy in response to purely nominal (as opposed to real) influences.

In the United States, Congress is responsible for taxation and the Federal Reserve is responsible for inflation. Investors need to be aware of the effects of both taxation and inflation. Let's do a little math. I promise it's simple, and I'll use small steps.

First, a few definitions:

  • r: the nominal rate of return on an investment
  • t: the rate of taxation on investment earnings
  • i: the rate of inflation
  • m0: the original amount of money invested
  • m1: money after one investment period (year)

Inflation does not affect all prices uniformly. Some are impacted earlier, some later, and different prices may be affected to a greater or lesser extent. Inflation is frequently assumed to be uniform simply because it makes the math easier. I'm using that assumption here.

Let's create a formula for m1. The investment earnings are (m0×r). The taxes on those earnings are (m0×r×t). After one investment period, you'll have your original principal, plus earnings, less taxes:

  • m1 = m0 + m0×r - m0×r×t
  • m1 = m0 × (1 + r - r×t)
  • m1 = m0 × (1 + r × (1 - t))
  • m1 = m0(1+r(1-t))

For example, assume you're investing $1,250 at 8% and taxes are 24%. Earnings would be $1,250×0.08=$100. Taxes would be $100×0.24=$24. $1,250+$100-$24=$1,326. Verifying the formula above, m1=$1,250(1+0.08(1-0.24))=$1,326.

Inflation will reduce the purchasing power of your principal. In order to keep up with inflation, your money needs to grow at the same rate. Expressed symbolically:

  • e: equivalent purchasing power after one period
  • e = m0 + m0×i
  • e = m0(1+i)

Using the example of $1,250 principal from above, and assuming a 4% inflation rate, after one year you need to have $1,250(1+0.04)=$1,300 to maintain your purchasing power.

It's important to realize that inflation affects the prices of investments as well as the prices of goods. The unfortunate consequence of this is that you will owe taxes on the portion of investment gains attributable to inflation, despite those gains having no economic relevance at all. In terms of the examples above, you were taxed on the whole $100 of earnings, not just on the $50 over and above what you needed in order to keep pace with inflation. Not only is inflation a "hidden tax," but you pay it on money that you haven't really (pun intended) made!

Using the examples above, m1=$1,326 and e=$1,300. You only made $26 after considering both taxes and inflation! You're only 2% ahead ($26/$1,300) of where you started on an after-tax, equivalent-purchasing-power basis. This is a much smaller rate than the 8% nominal rate assumed for the examples.

The "real rate of return" is simply the nominal rate of return minus the inflation rate, (r-i). But if you use this figure for projecting the performance of your investments, you'll underestimate the impact of taxes, because you're taxed on your nominal (not real) returns. But even without taxes, the "real rate of return" still overstates your returns: if there were no taxes and your $1,250 grew to $1,350 over one period, you're only $50/$1,300 = 3.85% ahead of break-even, not 4% ahead!

Your "true rate of return" — a definition I'm stipulating, not a recognized term of art — is (m1-ee, e.g. ($1,326-$1,300)÷$1,300=2%.

Let's express the "true rate of return" in terms of the original variables:

  • (m1-ee
  • [m0(1+r(1-t))-m0(1+i)] ÷ m0(1+i)
  • m0[(1+r(1-t))-(1+i)] ÷ m0(1+i)
  • [(1+r(1-t))-(1+i)] ÷ (1+i)
  • (1+r(1-t))÷(1+i) - 1

Verifying the formula, (1+0.08(1-0.24))÷(1+0.04) - 1 = (1+0.08(0.76)÷1.04 - 1 = 0.02, or 2%.

Because this is a function of three variables, it's difficult to visualize. But here are my attempts to do so. Inflation primarily changes the height of the line, with a very small effect on slope. Taxation changes only the slope. The lines show the regions where the real rate of return is between 0% and 10%. I do this because economic growth tends to be in this range in the long run, and nominal interest rates tend to move in tandem with inflation.

Effect of inflation on the True Rate of Return Effect of taxation on the True Rate of Return Effect of taxation and inflation on the True Rate of Return

As you can see, inflation by itself doesn't have a very significant impact on the true rate of return. And taxation by itself doesn't have a very significant impact on the true rate of return. But the combination of the two are very potent — it becomes possible to lose money even though the real rate of return is positive!

This graph shows the ranges most relevant to me, personally:

Effect of likely taxation and inflation on the True Rate of Return

The 24% tax rate results from Federal long term capital gains or dividend taxes of 15% plus the Oregon state tax of 9%. The 37% tax rate results from a Federal wage, interest, and short term capital gains rate of 28% plus the Oregon state tax of 9%. I expect inflation in the 3-5% range and real rates of return in the 2-8% range. This tells me I should expect a true rate of return (as I've defined it) probably between 1% and 4%, roughly half the real rate of return.

The effect of the combination of taxation and inflation is a powerful argument in favor of using tax-advantaged vehicles such as Roth IRAs and 401(k)s. It's also a powerful argument in favor of cutting taxes and returning to the gold standard.

Tiny Island