Quarter the Cross: An Elegant Construction

Last month, I presented an infinite pattern of nested crosses as a response to the  #QuarterTheCross challenge on Twitter.  The idea is to find interesting ways to shade one-quarter of the area of the cross built from five squares.nested_crossNow David Butler, a math lecturer at the University of Adelaide, has found an elegant way to construct this pattern with the classic tools of compass and straightedge (I used some simple MATLAB code).

As the crosses get smaller, each new one must shrink by \sqrt{3}.

I asked David to explain his construction.

The segment marked “1” is \dfrac{1}{3} of the “radius” from a corner of the outer cross to the center, so the construction generates an inner cross with the required radius, \dfrac{\sqrt{3}}{3}=\dfrac{1}{\sqrt{3}} as long as the original radius.

A nice feature of this construction is that the compass is only used near the beginning.  After that, the construction always provides a scaffold for the next inner cross.


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Quarter the Cross

Yesterday, I noticed that some of the math teachers I follow on Twitter were challenging their students, and themselves, with the #quarterthecross problem.  The problem is simply to find interesting regions of a five-square cross that take up exactly one-quarter of its area.  This looks like a great way to get kids to explore fractions and geometry.


A solution that occurred to me would be to use an infinity of nested crosses.  If the area of the cross shrinks at each step by one-third, then the combined area of the colored regions converges to one-quarter of the large cross.  This works because the alternating geometric series

\dfrac{1}{ 3} - \dfrac{1}{ 3^{2}}+\dfrac{1}{ 3^{3}}- \dfrac{1}{ 3^{4}}+\dfrac{1}{ 3^{5}}...

converges to \dfrac{1}{4}.  Note that if the area shrinks by one-third, the width of the cross must scale down by the square root of 3.  So every two steps the width shrinks by one-third, placing a little cross precisely inside the middle square of a larger cross.

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My FT Alphaville Commentary on Managing U.S. Treasury Debt

My commentary Go Long, Mr Mnuchin has just been posted at FT Alphaville of the Financial Times.

I address whether the U.S. Treasury debt should be lengthened and whether it should sell 50-year or 100-year bonds.  I think that “ultra-long” bonds are a good idea but that there won’t be enough demand for them to significantly reduce the Treasury’s interest rate risk.  This should be accomplished by reshaping the whole distribution of maturities, which is currently very front-loaded.  I also argue that the Treasury should update its measures of interest rate risk, and keep a close eye on how the Fed manages its Treasury portfolio.

Please check out the post and let me know what you think!

I am grateful to my wife Corrina for her essential suggestions and thoughtful editing through countless revisions. 

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I am excited to go to New Orleans this week to speak at the annual conference of the National Association of Municipal Advisors (NAMA). I will be on a panel about bond structuring with Lori Raineri of Governmental Financial Strategies Inc., Dave Abel of William Blair, and Shelley Aronson of First River Advisory. We will discuss some of the important structuring questions that face municipal advisors, such as how to analyze  the tradeoffs involved with callable premium bonds.

Click here to see our slides.  They include a conceptual bond glossary that I prepared.

I look forward to a great conference (with maybe a little jazz on the side).



Updated 10/3/2016 with link to presentation slides.

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Taming Premium Bonds

Premiums 15-Year
The Bond Buyer published my commentary Taming  Premium Bonds earlier today.  Although callable premium bonds are very popular in the municipal market, I argue that they hurt issuers and the market.  Market rules dictate that the proceeds to the issuer compensate only for high coupons to the call date.  The compensation for high coupons past the call date is buried inside the call option, making a refunding almost inevitable.  These bonds also make the market more opaque.  As a solution, I propose that the call premiums be set to breakeven levels so that the price-to-call matches the price-to-maturity.  Please see the article for the details.

Update: This not a new idea.  I applied it as a financial advisor many years ago, and it is also the basis for one of the puzzles on this blog.

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Fifth Annual Municipal Finance Conference

On July 12, this year’s Municipal Finance Conference will be held for the first time at the Brookings Institution in Washington, D.C.   The other sponsors are the Brandeis International Business School and the Olin Business School at Washington University in St. Louis.

The purpose of the conference is to bring municipal finance professionals together with academics to encourage useful research and better practice.  This year’s program will include several papers on financial distress and other challenges for municipalities.  The keynote speaker will be Governor Alejandro Garcia Padilla of Puerto Rico.

As a discussant, I will respond to a paper  that attempts to account for the potential benefits of future refundings within the interest cost of a bond or bond issue.

I understand that there may still be openings for the conference.  Click here for more information, and to register.  I hope to see you there.

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Generating Sitemap Links with Excel

We have become good friends with the Ysais family of Kyle, Texas.  They visited us in Colorado last summer and we all had a great time.  Corrina stays in touch with Marie almost every day, our daughter Skypes and plays Minecraft with the two boys, and Michael taunts me with mouthwatering photos of his smoked brisket.

Marie runs Ysais SEO, which helps businesses optimize their search engine results.  She asked me to help with an Excel workbook to create Google sitemap links.  I took the file she sent and worked it into a tool that can create several links for each specified website.

I thought I would share the workbook because it makes a good example of how to use Excel’s table feature and its HYPERLINK function.  With Marie’s consent, here is the workbook. Click on the image to open the file.

sitemap generator

There are four sheets, each with its own table (note that these are not “data tables”).  The first three tables provide the building blocks for the links: the Google sitemap text (the “prefix”), the domain names, and part specifiers (the “suffix”).  On “SiteMaps”, the last tab, a table generates hyperlinks for all possible combinations of the three text blocks.

To use the workbook, make sure that the prefix text meets your needs.  As with all inputs in this workbook, the prefix text is colored blue.  You can insert rows above if you need more than one prefix. Next, enter the domain names.  There is room for 500 domain names.  Blanks are fine, and you can always insert more rows if you need more.  The final input step is to review the suffix text.  You can insert more suffix items if you need more.

Now go to the sitemap generator on the last sheet.  Click on the “Good?” filter and select “TRUE” only.  This will hide the blank links.  Be sure that the bottom of the “Last?” column shows a “TRUE” value.  This insures that the last possible combination of your text blocks has been reflected in the sitemap list.  If not, you will need to insert more rows into this table.

Marie reports that the workbook also made it easy for her to create Bing sitemaps.

Note: a good way to insert rows into any of these tables is to click on the cell labelled “Count” at the bottom of the first column.  Insert the number of needed rows and copy down any formulas from above as needed.

Please let me know if you have any questions.

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