Inflation adjustment of debt, again

Happy Thanksgiving!

I was looking for something... interest on government debt us... and came upon the nice, professional-looking site of Daniel Amerman, CFA. Pretty graphs, each on black background. One that really stands out is his interest rate graph, yellow on black. This is beautiful stuff.

Some interesting graphs, too -- interesting for what they show rather than how they show it. His first graph, titled "30 Year Earnings on $10,000 Investment", shows earnings for 1% interest versus 6% interest. The difference is staggering.

The graph that got me writing was Amerman's picture of the "US Federal Debt, 1947-1970 (Fiscal Years)". This graph:

Graph #1. Source: Daniel Amerman

It shows the Federal debt (top edge of the red area) and the Federal debt adjusted for inflation (bottom edge of the red area). As it says on the graph, the red area shows the "reduction in real debt" due to inflation.

So in 1970, for example, to pay back about $148 borrowed dollars it would take only about $85.

Well no, not really. It would still take $148 dollars to pay back $148 borrowed dollars in 1970. But $148 would only be worth about $85. I think that's Daniel Amerman's point. It's another way of saying "inflation erodes debt".

In 1947 it took $100 to pay off $100 borrowed ... in 1952 about $84 to pay off $100 borrowed ... in 1966 it took about $88 to pay off $125 borrowed. Yeah, yes and no. Interest aside, it always takes a dollar to pay back a dollar. But if there's inflation, the dollar you're paying back is worth less than the dollar you borrowed in the first place.

So, in 1952, the $100 you paid back was worth about $84. In 1965 the $125 you paid back was worth about $88. And in 1970 the $148 you paid back was worth about $85. That's what the graph shows.

But it is not really right. It assumes you did all your borrowing in 1947, or that you only borrowed dollars that were worth what a dollar was worth in 1947. And of course that is not the case.

Daniel Amerman's graph shows the Federal debt (as a percent of the Federal debt in 1947) and shows the value of that debt in inflation-adjusted dollars. But the inflation adjustment is based on starting in 1947. The graph assumes that all the dollars the Federal government borrowed were 1947 dollars.

Here, I checked my evaluation of the graph:

I downloaded annual data from FRED: GDP Deflator in the second column, and gross Federal debt in the fourth column. In the fifth column (Column E) I calculated "loss of value since 1947" numbers. Not loss of value. Retained value. If you borrowed $100 in 1947 and paid it back in 1952, the dollar was worth a little less, so you only paid back about $86.07 of the purchasing power that was in the $100 you borrowed in 1947. $86.07 is the retained value, what $100 from 1947 was worth in 1952.

See row 18, and column E. I selected Cell E18 and clicked a key to edit the cell, so you can see the calculation I put into that cell:

=100*B13/B18

In that formula, the "B13" is blue -- and cell B13 has a blue box around it to make it easy to see what number is being used in the calculation. (Similarly, the text "B18" is red, and cell B18 has a red box around it.) The spreadsheet program puts those colors there automatically to make things easier to see when you're editing a cell.

I didn't actually want to edit the cell. I just wanted to show you the calculation I used. Further down in Column E (on row 31) there is the number 86 point something (for the year 1965) ... and (on row 36) there is the number 83 point something (for the year 1970). There was a number like that in the cell where we see the calculation, cell E18, but we can't see the number because that cell is in "edit" mode.

Before I activated edit mode, I copied the value from that cell up to the top of column E. On row 1, cells D and E you can read my note: Cell E18 = 86 point something.

The calculation in cell E31 (for 1965) is

=125*B13/B31

The calculation in cell E36 (for 1970) is

=148*B13/B36

Oh, and the calculation in cell E13 (for 1947) is

=100*B13/B13

In each case I took the amount paid back, multiplied by the price index for 1947 (the assumed year-of-borrowing) and divided by the price index for the year-of-payback. Each calculation figures the value lost retained since 1947.

To summarize, the numbers I calculated were $86.07 (for 1952), $86.14 (for 1965), and $83.70 (for 1970). The numbers I got from Daniel Amerman's graph were $84 (for 1952), $88 (for 1965), and $85 for 1970. In each case my calculated number is off a little, compared to my estimate from the graph, but in all cases it is close. (The graph at FRED is also a little off from Amerman's. It's also off from what I expected: I indexed the FRED graph series to make it 100 in 1947, like Amerman's, but both lines came out below 100 for some reason. Close enough for government work, I guess.)

Close enough. I'm satisfied that what I said above is accurate: The assumption underlying Daniel Amerman's graph is that all the dollars borrowed by the Federal government were 1947 dollars.

But in fact not every dollar of Federal debt was borrowed in 1947.

Therefore, there is a error in the graph.


It's not only Dan Amerman's graph. It's almost every graph that shows the inflation adjustment of debt. You can't take a lump sum number like debt -- a number accumulated over many years in an inflationary economy where the value of the dollar changes over time -- and calculate the "real" value the same way you would figure "real" GDP. The calculation doesn't work for debt, because debt is accumulated over many years, and the value of the dollar changes over time.

To calculate the "real" or "inflation adjusted" value of debt, you have to adjust each year's addition to debt separately, using the price index for each year in turn.


Daniel Amerman's graph does provide some useful information. It shows that if we paid off Federal debt today, we'd be paying dollars that are worth less than the dollars we borrowed. But of course we're not going to pay off the Federal debt today. So I don't know how useful his graph is. Except it's useful to me, because it helps me show that the method commonly used to figure "real debt" is incorrect.