Playing with the missing dimension in finance

2025-10-27

Correctly reporting the units of dimensional quantities is indispensable to precise thinking and writing in technical domains. In physics and chemistry, we understand the importance of dimensional analysis, which is why the common confusion between kilowatts and kilowatt-hours sticks out so badly. A key step when manipulating dimensional quantities is making each factor explicit and not relying on convention. There's no convention that an hour is the appropriate time unit when dealing with energy, so it's necessary to specify it every time.

Finance is not like this. In finance, the conventional unit of time is one year. As long as this is followed consistently there is no real trouble or risk of a mixup. But in the interest of completeness, let's rephrase some finance concepts in the proper dimensions.

To begin with, consider the interest rate on a money market account. With an interest rate of 4%, your principal will double in 17.7 years, or will multiply by a factor of \( e \) in 25.5. By a convention borrowed from physics, let's take this number, the "\( e \)-folding time", to be the characteristic timescale of a given growth process. Imagine shopping around for a better yield: 23.6 years is a slight improvement over 25.5, but is it enough to justify the hassle?

A commonly cited figure in corporate finance is a company's P/E ratio: price over earnings. Price is the market capitalization in dollars, while earnings is dollars per year. Forming the ratio P/E gives a quantity in units of years. The value of a share of a company is derived from the market's estimate of the value of all of its future cash flows, net of debt obligations, for the rest of time. We can interpret the P/E ratio as putting a very rough number on how long "forever" is. For Toyota, according to the market, it's 10 years. For Tesla at the time of writing, it's 290 years---or, one year at current revenue and one subsequent year at 289x current revenue.

In macroeconomics, it's common to discuss a country's debt to GDP ratio. You know where this is going: debt is a plain dollar amount, while GDP is reported in dollars per year. The debt to GDP ratio is therefore equal to the amount of time it would take a country to pay off its debt, were it to direct all of its GDP to the task. Greece's debt to GDP ratio reached 15 months in 2009 after the Great Financial Crisis, and remained elevated above 18 months throughout the 2010s. I was shocked to learn that in 2025, Japan had a debt to GDP ratio of 28 months, the highest in the world. If nothing else, this is a remarkable demonstration of the power of monetary soverignty.

In physics it is very common to speak of the characteristic timescale of some process, without worrying too much about the details of how it is measured. Once a characteristic scale has been identified for some process, it is natural to ask, how is it set?