## 2013 Municipal Finance Conference

The Bond Buyer and Brandeis International Business School will host the 2013 Municipal Finance Conference on August 1-2, 2013 at the Boston Park Plaza Hotel.  The conference will bring together academics, practitioners, regulators and issuers to discuss the functioning of the municipal bond market.  It will be a unique opportunity for practitioners to stay current with the latest research and for academics to ensure that their research is meaningful.

Some very interesting and timely papers have been submitted for the conference.  This is all independent scholarship.

I serve on the conference organizing committee.  I hope to see you there!

## The Reinhart-Rogoff Film Festival

You can observe a lot by watching.”  - Yogi Berra

The biggest story in economics this year revolves around a study by Harvard economists Carmen Reinhart and Kenneth Rogoff (“RR”).  Their work has been credited (or blamed) for efforts around the world to reduce deficits and slow the growth in sovereign debt.  A recent critique of their work found a now-infamous Excel error and challenged RR’s methodology, prompting a furious debate between those advocating restraint in deficit spending and those calling for more spending to stimulate economic growth.  The spending doves have latched onto the real and perceived flaws in RR’s work to call for an end to “austerity” measures, while the deficit hawks argue that not enough has been done to curtail spending.

It seemed to me that this noise might be obscuring an opportunity to learn from the data behind the study.  What if we just let the data speak for itself, instead of using it as a weapon?  I have attempted to do that by animating the data. I think the results are interesting, if unlikely to change any minds.

Background

RR’s 2010 paper, “Growth in the Time of Debt”, had been widely cited by policymakers for concluding that countries with a Debt/GDP ratio above the “threshold” of 90% experience dramatic declines in their growth rates.  There were several sections to the paper, but only one part has attracted controversy.  It concerns the experience, since World War II, of twenty advanced economies:

 Australia (AU) Italy (IT) Austria (AT) Japan (JP) Belgium (BE) Netherlands (NE) Canada (CA) New Zealand (NZ) Denmark (DE) Norway (NO) Finland (FI) Portugal (PT) France (FR) Spain (ES) Germany (DE) Sweden (SE) Greece (GR) UK Ireland (IE) US

For each of these countries and for each year from 1946 to 2009 (except for some gaps), RR compiled the ratio of sovereign debt to GDP, and the growth rate in real GDP.  There were nearly 1,200 such observations.

RR grouped the observations by Debt/GDP ratio:

• 0-30%
• 30-60%
• 60-90%
• Above 90%.

We will refer to the last category as “high debt”, and we will use “debt” as shorthand for the ratio of sovereign debt to GDP.

For each debt bucket, they found the mean and median rates of growth in real GDP.  Here are the mean calculations from RR’s Appendix Table 1:

 Debt/GDP Ratio: 0-30% 30-60% 60-90% 90%+ Average Real Growth Rate: 4.1% 2.8% 2.8% -0.1%

RR concluded that average growth falls considerably when the debt ratio exceeds a “threshold” of 90%.

This conclusion was recently challenged by graduate student Thomas Herndon,  and professors Michael Ash and Robert Pollin (“HAP”), all from the University of Massachusetts.  They disputed RR’s selection of data and RR’s method of weighting the data for average calculations.  Pretty dry stuff.  But HAP jolted economists, pundits, and spreadsheet users everywhere with their discovery that RR had made an embarrassing spreadsheet error.  RR had not included all of the countries in their average calculations.  This kind of error is common but easily prevented, as I wrote here.

The impact of the spreadsheet error depends on your perspective.  If you accept RR’s approach to selecting and averaging data, then the average growth for debt above 90% changes from -0.1% to +0.2%.  Not very troubling.  But HAP argue that when RR’s other mistakes (in HAP’s view) are corrected the average growth rate for high debt should have been 2.2% instead of -0.1%.  Now critics of austerity use the HAP paper to argue that high levels of debt should not be feared since they do not seem to dramatically slow growth rates.

In response, RR assert that the Excel error was minor and that their general conclusions remain valid.

Limited Vision

It seemed to me that this uproar has kicked up a cloud of dust, obscuring the reality behind the original study.  I wondered what could be learned by taking a close look at the actual data.

RR didn’t help matters by boiling down a thousand data points into a few buckets.  If the data were a distant galaxy, RR’s summary would be like a grainy speck seen through a weak telescope.

HAP deliver much higher resolution with a nice scatter chart in Figure 3 of their paper.  Here is my version of their chart (two points are missing because I limited the range of the y axis):

HAP call the observations “Country-Years.”  Each marker represents the debt ratio and growth rate for some observation.  But for any point, we can’t tell which country and which year.  Is it Ireland in 1950 or Finland in 2005?

I suspected that the country and the year actually matter.  Each country has its own culture, politics, resources, demographics, institutions and history.  On the other hand, each country is subject to the same laws of economics.  For example, hyperinflation has never ended well.

Different years may not be fully comparable.  The recovery from World War II might have been a fundamentally different era from our own.

Certainly, statistical analysis could help scrub away the noise of special circumstances if we had a sufficiently large data set.  To some critics, the data set in this case is inadequate.  There may be more than a thousand data points, but these correspond to only twenty countries and to even fewer business cycles.

Taking averages, as RR and HAP have done, is a way to wash out distortions caused by individual circumstances.  But perhaps it is useful to actually look at individual cases.  An anthropologist who visits real families might learn more than one who just studies an abstract family of 2.3 children and 0.6 dogs.

The scatter chart is like an overhead x-ray of a complex archaeological site.  Scattered shards of pottery from many different eras are jumbled together.  The x-ray is nice, but reassembling and categorizing the objects is better.

I decided to dig deeper into the data to compare the performance of the different countries.  I began by considering the data’s four dimensions:

• Country
• Year
• Debt/GDP
• Real GDP Growth

It is possible to work all four dimensions into a single chart, but it is not very readable:

Animations

Animating the data shows the four dimensions more clearly.  The following video shows the experience of all twenty countries from 1946 to 2009.

Like all great movies, there is too much to see in one viewing.  Here are some suggestions for watching the video:

• Follow one or two countries at a time.
• Watch for historic events that interest you.
• Look for countries moving together.
• Look for when changes in growth and debt seem to be related or unrelated.
• Look for evidence that high debt causes slow growth or vice versa.
• Look for evidence supporting or contradicting a threshold at 90% Debt/GDP.

To me, there is no obvious change in behavior when countries cross the 90% Debt/GDP line.  The “threshold” seems to be an artifact of how RR set up their debt categories.  On the other hand, much of the highest growth was experienced at low debt levels.

It is also clear that growth rates are often volatile and are affected by more factors than the debt level.

The RR Rogue’s Gallery

Now let’s look at the countries at the heart of the RR controversy.  RR identified eight countries that experienced more than 90% debt for at least one year in the 1946-2009 range (although Belgium was inadvertently excluded from their overall average due to the Excel error):

For three of these countries, the U.S., the U.K., and New Zealand, the only experience with high debt levels came after World War II.  All three pulled their debt down to more normal levels (although very slowly in Britain’s case).  I wonder if these cases of debt reduction can inform us about recent debt expansion around the world.  The old data are relevant only if a reversible process is involved.  Does leveraging look like deleveraging in reverse?  We will return to that question.

HAP pointed out that RR included only one year of New Zealand’s post war experience in their 90%+ calculation, distorting that country’s contribution to average growth.  HAP also objected to RR’s exclusion of Australia and Canada from the high debt group.  Apparently, the data for these two countries had not been available to RR when they wrote their paper.

After the War

The next animation highlights the countries that reduced their high debt levels after World War II.  This time we include Australia and Canada, for which fast growth seemed to helped them reduce their debt.  Only Belgium ever crossed back over the 90% level  (Belgium crossed repeatedly).  ”Tails” show the previous five years for each country.

A Greek Tragedy

HAP objected to the way RR averaged the data.  RR averaged the growth rates for each country in each debt bucket, and then took the simple average of these country averages.  For example, the average growth rate for Greece’s nineteen years of high debt was 2.9%.  To RR, this was as important as the -2.0% growth rate for the U.S., which experienced only four years of high debt.

On the other hand, HAP contend that each Country-Year observation should carry equal weight.  In their view, Greece’s high debt growth rate is 4.75 times as important as the American rate.

HAP’s averaging method implicitly assumes that each Country-Year delivers the same amount of information.  To test this assumption, consider the next animation. It includes Greece and other countries whose debt levels were not the worst in the whole data set.

For the seven years from 2000 to 2006, Greece’s economy was remarkably consistent.  The Greek data points stayed within ranges that were small compared to the data set as a whole.

Were these all truly independent observations that should be given equal weight in the analysis, like separate test tube experiments?  Is it just chance that these data points are clustered so closely together?  Or were they joined together in a single episode of economic history?    If so, treating each data point as an equally valuable source of information would be misguided.

It is ironic that HAP’s greater emphasis on the Greek experience would have made high levels of debt look more benign.  While it is true that Greece enjoyed some healthy growth along with high debt, no one would want to emulate Greece’s recent experience.

Japan in the English Channel

Finally, I want to share an observation that surprised me.  I noticed that as the U.K. recovered from its high debt load after World War II, its growth rate was almost always between 0% and 5%.  I also noticed that Japan’s growth was generally in the same range as its debt grew in the 2000s.  These are only two cases, but perhaps the processes of getting into debt and getting out of debt can be mirror images after all.

Former trader Bruce Krasting  argues that “Japan is all that the economists need to look for the evidence of the failure of More Debt is Better.”  I agree.

Summary of Observations

After studying these animations, my impressions are that:

• There is no hard “threshold” of debt where economic performance suddenly changes.
• The debt level isn’t always the strongest influence on the growth rate.
• No method of weighting the data is without flaws.
• Countries follow their own paths and great care should be taken in drawing conclusions from the experience of a few countries.
• The shortcomings of Reinhart and Rogoff’s paper are not proof that high debt is benign.

Acknowledgments

I would like to thank Kirk Monteverde of QuantifyRisk.com and Michael Ash, one of the HAP coauthors, for making the RR data available to me in a form that I could use.

## A Tool That Would Have Helped Reinhart and Rogoff

Economists Carmen Reinhart and Ken Rogoff (“R&R”) have been widely cited for a study that concluded that growth rates for countries with public debt over 90% of GDP are lower than for countries with less debt.  A recent critique by Thomas Herndon and two other economists examined R&R’s analysis and challenged their conclusions.  Their most sensational finding was that R&R’s calculations were skewed in part by a spreadsheet error.  R&R acknowledge the error but defend their conclusions.

The spreadsheet error is described here in detail.  Instead of averaging the results for all twenty countries in the analysis, R&R included only the first fifteen in the list (which is in reverse alphabetical order for some reason).  This has been widely described as “coding” error, as if someone had typed

=Average(F30:F44)

when they meant to type

=Average(F30:F49).

I don’t think this is what happened.

It is more likely that whoever built the spreadsheet began with a smaller list of countries, and created an average formula that included all of those countries.  Later on, more countries were added in new rows, and the analyst assumed that the calculation would adjust to include the additional countries.  Sometimes Excel updates formulas like this automatically.  But not always.  This time, Excel did not adjust the formula, making it appear that highly indebted countries had negative average growth when the average was actually positive.

Similar problems have led to spreadsheet disasters in the past.  In Good Fences Make Good Spreadsheet, Part 1, I explained how to use “fences”, or bracket rows, to prevent this sort of error.

Another tool of preventative medicine is the Excel table, as I described in Tables vs. Fences.  Tables are a good way to organize data and calculations.  Once you set up a block of data or calculations, use CTRL+T to make it into a table.  Make sure it has header rows and total rows.  The total row allows not only for regular totals, but also for averages and other ways to summarize data:

Just click on each cell in the bottom row to select the appropriate summary calculation.

A nice feature of tables is that the summary calculations remain stable even if the data evolves and you need to insert or delete rows.

I find it good practice in Excel to use tables as much as possible.  If Reinhart and Rogoff had done so they could have avoided their error.

Note: my spreadsheet is for illustrative purposes only, and it may not exactly replicate R&R’s work because of rounding.

## Talk at the Chicago Fed

I was pleased to speak on a panel last week at the Sixth Annual Risk Conference sponsored by the Federal Reserve Bank of Chicago and the Driehaus College of Business, DePaul University.

I appeared on a panel on Business Model Risk, and the title of my presentation was “The Treasury and the Fed: A Symbiotic Relationship, and What it Means for Interest Rates.”

The theme of the conference was “Navigating the New Playing Field”:

In order to navigate today’s increasingly complex global markets, financial institutions and supervisors alike must consider challenges posed by current economic conditions, an evolving regulatory environment, new market participants and increasingly demanding technology capabilities and controls.

I enjoyed the conference.  It was informative, the participants were engaging, and the hosts were gracious.

My slides, and those of some of the other presentations, can be found here (click on “Agenda” after you get to the conference home page).  I discussed how the playing field of interest rates is affected by the actions of the Treasury and the Fed.  One slide showed a “spectrum” of the Treasury’s marketable securities:

All 337 securities are represented by a succession of thin bands in order of maturity.  The thickness of each band corresponds to its outstanding principal amount, and the bands are color-coded by type of security (blue for T-Bills, green for T-Notes, etc.).  You can see how front-loaded the debt is.  Half of the debt matures within three years, an observation also made in The National Debt is Closer than it May Appear.

Other slides showed the Fed’s share of each of these securities, weighted by par amount or by duration.  The format for these charts was inspired by a visualization created by the research firm Stone & McCarthy.

## Puzzle #9: Island of Games

Life is fun on the Island of Games.  The thousand inhabitants enjoy competing at chess, checkers, and contests to solve the Rubik’s Cube puzzle as fast as possible.  The islanders are rated for their skill at each of the three games.  The ratings fall between 0 and 1.

The ratings for each category seem to follow a uniform distribution. For example, here is a histogram for the Chess ratings:

There isn’t much correlation between the skills for the three games:

Correlations between Skills

 Chess Checkers Rubik’s Cube Chess 1.0000 Checkers 0.0530 1.0000 Rubik’s Cube 0.0452 -0.0049 1.0000

The next three charts confirm this lack of correlation.  The first chart compares the ratings for Chess and Checkers, and includes a linear regression:

Here is Chess vs Rubik’s Cube:

And Checkers vs Rubik’s Cube:

The raw data and some statistical analysis can be found here.

All of this looks like pure noise.  But there is a hidden structure.  Can you find it?

Posted in Fun, Math | | 3 Comments

## Puzzle #7: Treasury Island, Part 1

In Puzzle #7: The Mysterious Islands, I told the story of two islands where each individual’s remaining lifespan can be perfectly predicted.

The lifespans on Uniformia are evenly distributed, and their average remaining life expectancy will gradually decline over time.

Ignoramia is different.  Their average life expectancy is actually going to increase for a few years, but only Franz knows it.

He uses this knowledge to gain power over his fellow islanders.  They will credit Franz for the improved life expectancy, but he did not cause the change.  He was like an ancient astronomer predicting an eclipse to everyone else’s amazement.

The Ignoramians are so pleased by the extension of their average life expectancy that they ignore its alarming cause: the die off rapidly.  At the beginning of the story, there are one hundred Ignoramians, with an average life expectancy of 1.6 years.  The lifespans of the Ignormanians were distributed as follows:

 Lifespan (years) Number of Ignoramians 1 70 2 20 3 5 4 2 9 3

There are 30 Ignoramians with lifespans of two years or more.  The average of their initial lifespans is 3 years.  If you combine this with the one-year lifespan of the other 70 islanders, you get an overall average life expectancy of 1.6 years.  The unlucky 70 pull down the average.

After a year, the first 70 have perished.  But for the survivors, this is great news.  True, the average life expectancy of the 30 declines from three years to two, but the other 70 are gone and no longer suppress the average.  So the average life expectancy improves from 1.6 years to 2 years.  This is cause for celebration on the island.

The Ignoramians are happy because they only think about a single statistic.  As long as this number improves, they will ignore their own rapid extinction.

Solutions

I also posted the puzzle at Wilmott.com, where they tore through it faster than piranha on a drunken squid.  On this blog, the first solution came from Jamie Wolfe, an old friend.  ΣΗΦ used linear algebra, which got me thinking about a way to use matrices to explain the relationship between average life expectancy and the mortality distribution.   Aman  gave a very clear explanation of how to solve the problem.

There is more than one solution to the puzzle.  For example, instead of three deaths in the ninth year, we could have one each in years eight, nine and ten.

In Part 2 of this post, I will discuss what the puzzle is really about.

## Fast Formulas #5: Quasi-Geometric Series (The Other Trick)

In Fast Formulas #3: Pool Average Life with CPR Prepayments,  I described Formula 3.1., a compact formula for the average life of a prepaying loan pool, and  I wrote that the formula had been found by using  “a couple of tricks.”    I explained one of the tricks in Fast Formulas #4: Geometric Series (One of the Tricks).  This post describes the other trick, which concerns a variation on the geometric series.

A geometric series is a sum such as

$Q=q+q^{2}+q^{3}+...+q^{n-1}+q^{n}$

If n is large, it would be tedious to calculate the terms of $Q$.  Fortunately, there is a shortcut, as discussed in Fast Formulas #4:

$Q=\dfrac {q-q^{n+1}}{1-q}$                                                                                                            (Formula 4.4)

If you multiply each term in the geometric series by its exponent, you get this variation:

$S=q+2q^{2}+3q^{3}+...+(n-1)q^{n-1}+nq^{n}$

The series $S$ doesn’t have its own name, but it is an example of a quasi-geometric series.  It has applications to the duration of a bond or annuity.

We can find a shortcut formula for $S$ with the same strategy we used to find Formula 4.4. We construct a new series that shares most of its terms with the original series, and then cancel out the matching terms on our way to a compact formula.

The first step is to multiply $S$ by $q$:

$qS=q^{2}+2q^{3}+3q^{4}+...+(n-1)q^{n}+nq^{n+1}$

Most of these terms nearly match the corresponding terms of $S$.  For example, $S$ has the term $3q^{3}$ while $qS$ has the term $2q^{3}$.  We need one more $q^{3}$ in the new series.  We also need a $q$, one more $q^{2}$, one more $q^{4}$, and so on.  In short, we need to add $Q$ to the new series.  This gets us a perfect match for all of the terms of $S$, with an additional term:

$qS+Q=q+2q^{2}+3q^{3}+4q^{4}+...+nq^{n}+nq^{n+1}$

Subtracting $S$ from this, almost everything cancels out:

$qS+Q-S=nq^{n+1}$,

$S=\dfrac {nq^{n+1}-Q} {q-1}$.

Applying Formula 4.4 and a little more algebra, we find the shortcut formula for $S$:

$S=\dfrac {q-(n+1)q^{n+1}+nq^{n+2}} {(q-1)^{2} }$                                                                                 (Formula 5.1)

We could also find Formula 5.1 by exploiting some basic calculus.  You might have noticed that $S$ is very similar to the derivative of the geometric series $Q$:

$Q'=1+2q+3q^{2}+...+{n}q^{n-1}$

Multiplying by $q$ we see that  $S=qQ'$.  So all we need is a short formula for $Q'$.  By Formula 4.4, we can find $Q'$ by taking the derivative of $\dfrac {q-q^{n+1}}{1-q}$, using a rule for the derivative of the quotient.  If you work it out, you will end up with Formula 5.1.

Formula 5.1 is the other trick I used to find Formula 3.1.

Posted in Fast Formulas, Math, Uncategorized | | 1 Comment