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Trinity Uses RAY to assess 4s vs 5s - 2nd Gen Refunding Matters!


"Prediction is very difficult, especially if it's about the future."  - Niels Bohr

The other day one of our clients, Melio & Co, helped Trinity Health determine final deal structure on their $1 billion+ issue using Refunding Adjusted Yield (RAY). There were a variety of different coupon and optional redemption structures in play and Trinity needed to know where the market was offering attractive value. Melio & Co used RAY's real-world modeling to deliver exactly that insight to Trinity. And RAY does this by faithfully modeling the sometimes messy economics of refunding.   

To test what the market offered, Trinity decided to use a 7% net present value savings criterion in the RAY calculation. As Mark Melio said, “One of the attractive features of RAY is its ability to incorporate an issuer’s actual refunding criteria.”

Refunding probability

A powerful feature of the market model embedded in the RAY calculation is its ability to create both non-call and callable yield curves. This is in contrast to no-arbitrage yield curve and bond option models that not only fail to capture realistic changes in yield curve shape, but almost invariably force the use of non-call curves - notoriously difficult to estimate in the municipal market.

With callable borrower yield curves in hand, RAY quantified not only first generation refunding savings but also the refunding of the refunding bonds i.e. the 2nd generation refunding. Of course, expected tax law is respected; advance refunding of callable advance refunding bonds can only be done taxably, or precluded altogether as selected by the modeler.  

The table below shows some key results of the RAY analysis for 2 callable bonds. They allow us to draw some interesting conclusions.1  The yield to maturity on the 5s is higher than the 4s as our intuition would tell us. But when looking at RAY1 (the “1” indicating only 1st generation refundings), the 4s looked about 5bps more attractive to Trinity than the 5s, 3.752% vs 3.802%.


Yield to Maturity



4% Coupon Bond, 10Y call




5% Coupon Bond, 10Y call





However we also know that relative to the 4s, the 5s are likely to be refunded earlier and more often. Given our modeled refunding bonds were themselves callable after 10 years, this means the 5s are likely to be refunded with callable bonds more frequently, which in turn will lead to greater 2nd generation savings. And the full RAY calculation, capturing both 1st and 2nd generation refundings, quantifies this greater benefit directly. 

Adding the 2nd generation refundings reduces RAY on the 4s by about 3bps while the 5s fall by about 13bps. This changes the answer entirely and actually leaves the 5s looking about 4bps cheaper and ultimately more attractive to Trinity than the 4s.

How would RAY change with a 5.50 coupon? Or an 8 year call? What about different refunding criteria? Excellent questions all. Stay tuned or better, we'd love to hear from you., 646.202.9446x101

1Results have been modified slightly to preserve confidentiality.

Why Option Pricing Models are Wrong for Tax-exempt Issuers (Part 2)


In Part 1 (a suggested read if you’re getting to this article first) we proposed 3 questions that determine whether bond option pricing models, in contrast to real-world option models, are appropriate for tax-exempt issuers analyzing their callable bonds:

1) Is the market for all components of the option complete (defined in context below)?

2) Are markets for the items in 1) free of arbitrage profit due to trading activity sufficient to drive such profits to zero?

3) Is the purpose for using the model one of pricing and/or hedging?

Reviewing the bidding for our beloved tax-exempt/muni  market, we have a “Maybe” on 1) and a solid “Absolutely Not” on 2).1 This brings us to question 3), one of the least discussed but critically important questions for understanding suitability. Is the purpose for using the model by issuers one of option pricing and hedging? Or is it more aptly described as risk and/or performance management? Are munis simply figuring out the right hedging strategy to implement so they can make an arbitrage-free option price? That doesn’t sound like a muni or tax-exempt borrower to me; that sounds like an options dealer.2

Aren’t munis taking real risk in managing these call features? Of course they are. TheySee the light aren’t running some sort of matched options book (which in munis, doesn’t exist anywhere anyway). Issuers own a bunch of options through the sale of callable bonds, and in trader parlance, they are naked long. Refund too early and the opportunity for greater budgetary benefit may be precluded; refund too late and an attractive interest rate market may never return. It is decidedly not a simple engineering exercise of coming up with an option price based on highly liquid underlying instruments, predicated on conditions that, in muniland, just do not exist. So the answer to 3) is also an unmistakable, “Not remotely.”3 

But here let me circle back to the original LinkedIn question and provide a nice piece of research germane to the topic of fixed-income option model suitability. It is by two of the world’s foremost authorities on yield curve models, Riccardo Rebonato and Sanjay Nawalkha. The article is both free and meaningfully titled, “The Right Interest Rate Models to Use: Buy Side vs Sell Side” published in the Journal of Investment Management, 2011.   

The points made in the paper apply more broadly than to just tax-exempt entities. But as I explained in the first article on this topic, this is where we need to go to learn about lawnmower engines and how they might work in a go-kart. Though the paper focuses on one particular type of option pricing model, the popular LMM-SABR model, the points made about option pricing models being “useless at best and dangerous at worst for the buy-side institutions” are completely general and unnequivocally apply to tax-exempt borrowers.

The Cliff’s Notes: tax-exempt borrowers clearly fit into the paper’s “Buy-side.” Unlike dealers, buy-siders “are not in the business of making money by ’trading’ interest rate derivatives while maintaining zero interest rate exposures. This is the major difference between sell-side dealer banks and the buy-side borrowers and investors.” The authors further go on to describe “less sophisticated buyers such as, cities, counties, foundations, charities,…etc.” So tax-exempt issuers are clearly in the “buy-side.” Easy enough.

Cutting to the chase, the authors then say “…buyside practitioners and sellside banks need different types of interest rate models.” The buy-side institution “is interested in knowing whether the derivative is priced too cheaply or too expensively by a realistic model that can simulate the risk and return trade off under the physical measure.” Translation: issuers need a real-world (synonymous with “physical measure”) model  to provide an estimate of value to compare to what the market might provide. As an aside, we think the best practice version of this uses the issuer’s actual refunding criteria, something that is accomplished naturally with a suitable real-world model.   

To be clear, these models that buyers (tax-exempt issuers) should use are not arbitrage-free pricing models such as Black-Derman-Toy, Black-Karacinski, or Hull-White. Buyers need a real-world (physical measure) interest rate options model like Bernadell, Coche, Nyholm (2005), Rebonato (2002, ppt file), or the more powerful one we use to calculate Refunding Adjusted Yield (RAY), Deguillaume-Rebonato-Pogudin, 2013. 

But most municipal market practitioners know this, even if they don’t have the technical background to explain exactly why. If decades-old option pricing models really applied so neatly to tax-exempts than bankers, financial advisors, salespeople, traders and issuers would’ve been using them successfully for years. But the vast majority of them have not, and rightly so. Now with truly appropriate and far more powerful alternatives available, why should they?


1In the words of MMA's Matt Fabian at today's Municipal Analyst Group of NY lunch, "There's only one way to short munis; and that's don't own munis."

2But of course there can be no real option dealers in the muni market because the answers to questions 1) and 2) are a joint “No” to begin with. 

3The fact is, even if the answer to 1) and 2) had been a resounding “Yes”, if the answer to 3) is a “No”, then option pricing models would still not be appropriate. The answer to all 3 must be “Yes” for option pricing models to be appropriate.

Why Option Pricing Models are Wrong for Tax-exempt Issuers (Part 1)


"It's not how we make mistakes, but how we correct them, that defines us."  - Anonymous

The other day I was asked in the Tax-Exempt Debt Structuring (TEDS) group on LinkedIn to provide some "scholarly" references (outside of our own) that support my comment that “option pricing models are inappropriate for issuer’s use in looking at option value or refundings.” To put a finer point on it, there are two main types of option models: pricing models or more accurately, relative pricing models (some regrettably call these “standard” models as if anything else is non-standard) and real-world models. The question at hand is which type of model is the right one for tax-exempt issuers, or any bond issuer for that matter, to use when analyzing their optional redemption features.   We’ve been dancing around this topic for a while now and think it’s time the matter was set definitively straight.1

The original LinkedIn question to me was odd as it stipulated the reference(s) should be from the narrowWhoops realm of municipal finance alone, a notoriously sparsely researched area (no offense, MFJ). Let me analogize briefly to explain why this is silly. Let’s say we live in a world with only engineless go-karts. Then someone comes along and says, “Hey we have these great things called ‘lawnmower engines’ and they’ll really make these go-karts go!” The LinkedIn question is akin to saying, “Show me some literature from the engine-less go-kart community that says these lawnmower engines are inappropriate for us.”  Doesn’t make much sense, right? Why would anyone from the engineless go-kart community publish such a thing in the first place? In order to properly answer the question we’d need to comb the annals of the Journal of Lawnmower Engines to study up on details of the engines, the conditions under which they work properly, their weight, the torque they deliver, etc. Only with that information in hand, coupled with our knowledge of engineless go-karts, could we ascertain whether small engines and unpowered go-karts would indeed be a winning combination.

All that’s to say we better learn something about bond option pricing models generally if we’re going to determine whether they make any sense for municipal and tax-exempt issuers. And so we have. The fact that option pricing models are the wrong ones for issuers is obvious if you look at three simple questions. The applicability and suitability of option pricing models to any situation rests on a "Yes" answer to all of these basic questions:

1) Is the market for all components of the option complete (defined below)?

2) Are markets for the items in 1) free of arbitrage profit due to trading activity sufficient to drive such profits to zero?

3) Is the purpose for using the model one of pricing and/or hedging?

In the remainder of this article we’ll address 1) and 2) as these speak to the foundational assumptions for any option pricing model to be valid. Question 3) we’ll save for tomorrow’s second and final installment.

  1) Is the market for all components of the option complete?

For standard option pricing models to apply the market must be complete. This means in essence that there are non-call bonds of the issuer in question trading across all relevant parts of the non-call yield curve and without “friction.” What are the relevant parts? The parts that cover the term of the option being modeled. Now some could argue that we can write a non-call yield scale for an issuer, which is true. But writing some best-guess, non-call tax-exempt scale is a far cry from having the full complement of non-call bonds across the curve to trade, both to buy and sell short. Andrew Ang et al discuss the difficulty of selling munis short in their paper, Taxes on Tax-exempt Bonds (more below). And we won’t even mention the “frictions” involved in trading munis. So the answer to 1) is probably a “Not really” but let’s just be charitable and call it a “Maybe...sort of”.  

  2) Are markets for the items in 1) all free of arbitrage profit due to trading activity sufficient to drive such profits to zero?

So with the answer to 1) wobbly at best, let’s look at Question 2). Are muni markets free of arbitrage profits? Pose this question to any muni salesperson, trader, investment banker or professional investor and wait for the belly laugh. This is a non-starter. You can’t short munis due to the tax-treatment. And the one flavor of research you could call plentiful in the muni market is the type that concludes with some variety of, “It’s inefficient”, “It’s fragmented”, “It’s unhedgeable”, “It’s expensive” or “It’s broken” You could imagine someone trying to mistakenly argue that it’s arbitrage free, by virtue of the simple fact that you can’t implement an arbitrage trading strategy that captures a bond’s mispricing to begin with. But that’s not arbitrage free, that’s just prohibitively expensive and insufficiently tradable. To quote Ang et al, in Taxes on Tax-exempt Bonds,

"Unlike Treasury bonds, shorting municipal bonds is very hard because only tax-exempt authorities and institutions can pay tax-exempt interest. An investor lending a municipal bond to a dealer would receive a taxable dividend because that dividend is paid by the dealer, not a tax-exempt institution. Even if an active repo municipal market existed, it may be hard to locate a suitable municipal bond as a hedge because of the sheer number of municipal securities. Shorting related interest rate securities, like Treasuries and corporate bonds, opens up potentially large basis risk. Another reason arbitrage may be limited is because the trading costs are much higher than Treasury markets."

Well, there it is. Let’s review the bidding. The answer to 1) is an iffy “Maybe”, the answer to 2) is an unqualified “No.” The foundational assumptions on which bond option pricing models rest simply do not get there in the municipal market. And it's not just a foundational crack, this foundation's built on sand. Can we just sweep these facts under the rug, hold our nose, and blithely hit the Calc button? To what end? So we can generate some winged-horse numbers using sexy models that work nicely in other markets? Where's the benefit?

We could stop here because all 3 questions need a Yes for standard option pricing models to apply. But wait, there’s more! Question 3) is probably the most important of all.  And we’ll cover it fully in our next article. Spoiler alert – muni issuers aren’t options dealers. 

1We tried to set the matter straight with The Right and Wrong Models for Callable Municipal Bonds published back in 2013. But we understand. It took the Royal Navy fifty years after James Lind definitively showed citrus fights scurvy in 1753 to keep fresh oranges on British ships. These things take time.  

The Right Analysis for Refundings – MSRB, Take Note


We see a lot of confusion in public finance as to how to analyze refundings. Unfortunately I think much of it stems from people outside of public finance coming in without a complete understanding of the environment in which a tax-exempt issuer operates i.e. the muni market. These interlopers get excited when they see option specifications in an official statement, then cry out, “We’ve seen these before. We have fantastic models used everywhere else, they must apply here too!” Unfortunately the foundational assumptions underpinning these models do not exist in the muni market leading this statement to be bunk (technical term my father used to use…). In fact those elegant bond options models do not apply in the muni flea market.

Let’s start with a simple assumption. While we're here it is very important to understand the assumptions of any financial model, particularly so with options. The complexity andbiz peep at whiteboard elegance of the math involved in options can temporarily blind the most clear-thinking practitioner to the fact that important assumptions are just not satisfied in the muni market.   

The assumption I make is simply stated but has far-reaching and critical implications for the right financial economic analysis for refundings:

An Issuer can ONLY achieve value from an optional redemption by performing an (advance or current) refunding

That is, the only way to lock in economics is to actually exercise the option. Some may argue that this is an extreme assumption. Ang et al in their irretrievably flawed paper on advance refundings take issue with this very topic, lamenting that issuers should more efficiently manage the undeniable interest rate risk in their optional redemptions by using derivatives. In the ivory tower they enjoy the luxury of ignoring patently unbalanced exposés like this one on Chicago Public Schools use of synthetic fixed-rate debt. Very real political considerations aside, suffice it to say that the number of states, munis, or tax-exempt entities comfortable with swap contracting and its attendant liquidity and credit risks is unfortunately very small. For the vast majority, swap is currently a four letter word. 

So let’s assume the above statement for the moment; what are the ramifications? In short, it means the beautifully elegant models increasingly brought to bear on municipal optional redemption features do not apply. Option Adjusted Yield, lognormal short-rate models, no-arbitrate conditions – they don't work for munis. In an article on stock options discussing option owners who cannot sell, Paul Wilmott (himself a veritable LeBron James of financial engineering) puts it this way:

“In many situations, the only way of locking in the profit may be to exercise the option early. The ‘theory’ says don’t exercise, but if the stock does fall then you lose the profit. At this stage it is important to remember that the theory is not relevant to you [emphasis added].

On Exercising American Options: The Risk of Making Too Much Money  Ahn Hyungsok and Paul Wilmott, 2003

Issuers are in exactly the same spot. They can’t just sell their options. Nattering about the right volatility or mean reversion input to use in your standard bond option model (BK or BDT or the like) is a lot of misspent energy. As research has detailed recently, for many reasons above and beyond the simple one stated here, these are simply the wrong models for issuers to use to solve the refunding timing problem, which is a risk management problem if ever there was one.

So if we can’t use all that admittedly Nobel-worthy theory in analyzing refundings, what are we to do? In short, get real. Use a real-world market model of both issuer and SLGS yield curves, capturing the complicated way those two markets move. And with that real-world model in hand search for good, robust signals that help issuers decide when to refund. And do this by testing actual issuer refunding criteria: PV savings, escrow efficiency, the NYS/MTA table, the opportunity index used by the state of Wisconsin and any and all combinations thereof.

Two pieces of research do just that, by testing roughly 40 different refunding policies (read “signals”) both on a 50 year historical basis and prospectively on a simulated basis. One result that may surprise is that refunding efficiency (using one of those standard bond option models) doesn’t fare so well under the bright light of both historical and simulated performance. In fact, 100% refunding efficiency was ranked dead last among all policies. And lower percentage refunding efficiencies behave in practice a lot like the far simpler signal of 5-6% present value savings. More interesting, an Alternative Policy was identified, currently not in use to my knowledge, that trumps all others – and not by a little. More on this in future articles.

As the MSRB drafts its curriculum for educating municipal financial advisors, I hope the realities of the municipal market are kept front of mind when they cover options. They have the choice between propagating the mistakes currently happening far too often in the municipal market, or helping advisors and ultimately issuers understand the right types of analysis and models that apply and why. I’m crossing my fingers for the latter. 

Refunding Adjusted Yield (RAY) Shines Light on Issuer Financing Cost


It’s 2015. Watson vanquished humans in Jeopardy 4 years ago and is now rapidly moving towards replacing as many oncologists as possible. Google is just one company running driverless cars and trucks around everywhere. Facebook is trying to monetize every eye twitch you make looking at a web page. Let’s check in on innovation in public finance:

  • Rarely if ever calculate and manage relevant risk metrics. Check
  • Analyses of all stripes performed in spreadsheets or 20+ year old bond software - despite massive limitations. Check
  • Unexamined rules of thumb for when to refund bonds. Check
  • Applying 25+ year old, black-box option models that are simply inappropriate for munis and few understand. Check

The last one might be considered an “innovation” given its somewhat more recent rise but with innovation like that…well, it seems like we could do better.

How about we start with improving the good ol’ yield calculation for issuers?!? We’ve used recent fixed income research, spawned from the obvious shortcomings of models duringRAY the financial crisis, to create a better yield mousetrap. We can now incorporate an issuer’s actual refunding criteria in cash flow calculations to create a lifetime cost of financing including 1st and 2nd generation refundings. For that reason, this is called the Refunding Adjusted Yield (RAY). 

How’d we do it?  Well, it’s just 4 simple steps: 

  1. Implement a real world market model that realistically generates the issuer’s tax-exempt, taxable, and SLGS (escrow) curves in a single consistent model. Ideally make it fully transparent and testable. Examples are here and here. (We chose the latter as it has the benefit of capturing how yields actually change across levels.)  You can test it yourself by playing Curve Quiz (free!) on iphone/ipad here or on your Android device here
  2. Using issuer’s actual refunding criteria, determine when hypothetical refundings occur (make nice refunding probability graph).
  3. Given the timing of refundings in 2) and the future yield curves in 1) adjust the original debt service cash flows based on the new refundings. Do the same for 2nd generation refundings as applicable. Don't forget to enforce the rules: no tax-exempt refunding of advance-refunding bonds, just like the real world.
  4. Average the net cash flows from 3) and calculate a yield back to the purchase price of the bond or issue. This is the Refunding Adjusted Yield (a RAY of shining light on true lifetime project cost and muni bond structuring!)

What does RAY look like for an actual deal structure? For a 20 year level debt issue (amort and pricing at bottom of article) we have these statistics: 

Issue Par                                               $100,000,000.00

Issue Price                                        $113,496,745.45

Arbitrage Yield                                               2.910%

True Interest Cost (TIC)                                       3.382%

RAY with 5% PV Savings Criterion                      3.207%

For those not fully initiated into the mysteries of public finance yield calculations, the arbitrage yield is generally a yield to worst (from the investor’s perspective) calculation and as such is often to the call date for premium callables. The TIC on the other hand is to maturity. Armed with this information, it is intuitive that the RAY would sit somewhere between the arbitrage yield and TIC.  In this case, obviously, the closer RAY is to the arbitrage yield the more likely the bonds are to be refunded and called. In this case the RAY calculation incorporated a refunding if the callable bonds hit a 5% PV savings target. Different refunding criteria lead to different RAYs.  More on that to come...   

This framework has a ton of really nice side benefits too, which we’ll look at over the next few weeks:

  • How RAY Changes with Refunding Criteria/Policy
  • The effect on RAY of 2nd Generation Refundings
  • New stats like Avg Time to Refunding, Refunding Adjusted Avg Life, and % of Escrow Supported Cash flows

The information content in these numbers is staggeringly greater than simple arb yield or TIC. Ding dong...the TIC is dead 

If you’d like to know more or would like to calculate a RAY for a new pricing or to compare structures, drop us a line - or call 646.202.9446.

Year Type Coupon Yield
      1 Serial 1.50% 0.15%
      2 Serial 2.00% 0.36%
      3 Serial 2.00% 0.65%
      4 Serial 3.00% 0.96%
      5 Serial 3.00% 1.27%
      6 Serial 4.00% 1.59%
      7 Serial 4.00% 1.88%
      8 Serial 5.00% 2.14%
      9 Serial 5.00% 2.33%
    10 Serial 5.00% 2.46%
    15 Term 5.00% 3.10%
    20 Term 5.00% 3.50%


Hull-White add Weight to New Interest Rate Modeling Research


Last summer I wrote an article describing a missing link in rate modeling that had been discovered in exciting new research by Nick Deguillaume, Ricardo Rebonato, and Andry Pogudin entitled The nature of the dependence of the magnitude of rate moves on rates levels: a universal relationship. This mouthful offered two simple takeaways. First, accurately capturing how rates are expected to change, particularly over long time horizons, is central to every rate risk management decision we face. And second, that so-called “standard models” that don’t provide for the observed fact that rates tend to change differently depending on their level aren’t so realistic nor as a result, very good at informing interest rate decisions like refunding opportunities.

The other day Mr. Deguillaume emailed me to let me know their research apparently caught the attention of a couple other fellows who’ve been known to dabble in the dark arts of yield curve modeling, John Hull and Alan White.  This duo, a veritable LeBron James and Kevin Durant of financial economics, published a paper entitled A Generalized Procedure for Building Trees for the Short Rate just last month affirming the findings of Deguillame et al though in a completely different setting, no-arbitrage derivative pricing models. With the weight of this new research, what I’ve called this newly discovered missing link just got a heck of a lot bigger.

The Hull-White paper describes a more powerful, generalized Hull-White short-rate model that allows for interest rate changes to behave almost identically as described in the Deguillame et al research above i.e. rate changes are dependent on rate level. Specifically that rates below about 1.50% and above about 6% move in lognormal terms (as % of the rate level itself) and that rates in between this range move normally (in basis point terms). 

 HW Rate Graph

The fascinating result is that when they calibrate their modified model to market traded cap prices, the new “regime-shifting” rate volatility is more than twice as good at fitting all traded cap prices than standard normal (1st generation Hull-White) or lognormal (Black-Karasinski) interest rate models. 

Now their results, unlike Deguillame and company, were based on a grand total of a single day’s market data (2Dec13), hardly a robust data set. But I expect they or others pick up the torch very soon to test these results across more data.  

All of this points towards a sort of “Unified Theory” of interest rate modeling where a regime shifting feature is essential in every setting. Whether for the purpose of real-world option valuation for issuers or investors or no-arbitrage pricing models for interest rate derivatives dealers, a type of level dependent volatility is now known to be a best practice feature. Lest we get too inebriated by our own quantitative Kool-Aid, we must always keep in mind that everything here is a “model” and by definition only fractionally as complex as the real world. That said, it seems we’ve moved one step closer to realism when considering how to tackle the tough rate decisions that (public and private) corporations must face.

For more information on how to use these models in your organizations, give us a call (646.202.9446) or email

Refundings Show Finance Profs How to Colossally Screw-Up Finance


"To confuse the model with the world is to embrace a future disaster driven by the belief that humans obey mathematical rules."  - The Financial Modeler's Manifesto, Emanual Derman and Paul Wilmott

Recently I discussed new research by Andrew Ang (Columbia), Richard Green (Carnegie Mellon), and Yuhang Xing (Rice) that maintained that in the presence of transaction costs, advance refundings always destroy value to the issuer. Based on this conclusion the authors proceed to question the tax policy wisdom of allowing municipal borrowers to advance refund at all, a conclusion I believe is not universally popular in the public finance community for a number of sound reasons.

My first reading of the paper was cursory but on more detailed examination I find they make a mistake, big time. It’s a mistake that strikes to the core of their hypothesis, and it’s not really even a mistake in their public finance either. It’s a fundamental error in plain old financial modeling itself, in how they calculate “losses,” and ultimately in their conclusion. It’s a mistake I frankly am very surprised that finance professors would make. 

Mistake right here

The error shows itself unobtrusively in one seemingly non-controversial but foundational sentence on page 10 which goes both unnoticed and, for the paper's sake, tragically unexamined,

“We can represent the value of any security as the discounted expectation of its payoffs under the risk-neutral measure….”

In a word, WRONG. Dead wrong. Grade F (My dad the navy frogman was tough during homework review...) Where are the grad students and peer reviewers on this stuff? This seemingly innocuous statement naturally avoids detection by public finance practitioners because their jobs simply don’t require an intimate understanding of arbitrage-free (risk-neutral) pricing, for reasons we’ll see more clearly below. And it fails to trip the warning detectors of many financial theorists and academics because that lot is so steeped in the dark arts of asset pricing theory that such a statement is utterly unremarkable and unquestionably obvious. The theory is so established and ubiquitous in academic circles that it fails to even arouse the suspicion of its cultural adherents. And that culture, both academic and practical, chooses at its peril to forget the very tenuous and often unrealistic foundation on which it rests.*

But before unpacking the specific problem with Ang et al’s statement above I would offer that those keepers of the asset pricing mysteries (particularly those teaching our youth!) would do us all a tremendous favor by reading the Modeler’s Manifesto and abiding by its Modeler’s Hippocratic Oath,

  • I will remember that I didn't make the world, and it doesn't satisfy my equations.
  • Though I will use models boldly to estimate value, I will not be overly impressed by mathematics.
  • I will never sacrifice reality for elegance without explaining why I have done so.
  • Nor will I give the people who use my model false comfort about its accuracy. Instead, I will make explicit its assumptions and oversights.
  • I understand that my work may have enormous effects on society and the economy, many of them beyond my comprehension.

Had the authors committed to and followed the Oath, their paper would be either very different or unwritten.

But back to their statement and it’s fundamental, fatal flaw. To correct their statement one would need to add a very simple but crucial qualifier, which for this paper ultimately renders the statement itself simply incorrect,

One can represent the value of any security as the discounted expectation of its payoffs under the risk-neutral measure IFF the risk-neutral measure EXISTS! 

Without going into a lot of unnecessary minutia, the simplest way to assess whether risk-neutral valuation technology might apply to a given problem is to first assess whether or not the security in question is in fact hedgeable.** Hedgeability equals price-ability. Hedgeability is a prerequisite for constructing a risk-free and self-financing hedging strategy and such a strategy is an absolutely necessary prerequisite for any risk-neutral pricing theory to apply.

So how can we determine whether or not a callable municipal bond or bond option is hedgeable? We could ask a dealer. But that work is unnecessary because conveniently we can directly quote none other than the lead author of the paper himself, Andrew Ang, from some excellent research he got published all the way back in 2010. Though the research relates to municipal bonds impacted by the market discount rule, you’ll note that the section below applies to municipal bonds quite generically,

“Municipal issuers are unable to arbitrage the mispricing of…bonds. IRC §148 specifically prohibits arbitrage across municipal bonds and other types of bonds (for example, Treasury and corporate bonds) by tax-exempt institutions…. Unlike Treasury bonds, shorting municipal bonds is very hard because only tax-exempt authorities and institutions can pay tax-exempt interest. An investor lending a municipal bond to a dealer would receive a taxable dividend because that dividend is paid by the dealer, not a tax-exempt institution. Even if an active repo municipal market existed, it may be hard to locate a suitable municipal bond as a hedge because of the sheer number of municipal securities. Shorting related interest rate securities, like Treasuries and corporate bonds, opens up potentially large basis risk. Another reason arbitrage may be limited is because the trading costs are much higher than Treasury markets.”

                        - Ang, Bhansali, and Xing, Taxes on Tax-exempt Bonds Journal of Finance (2010)

Couldn’t have said it better myself. This paragraph provides exactly the reasons why municipal bonds are effectively unhedgeable. But the undeniable and incontrovertible consequence is that risk-neutral pricing theory therefore CANNOT apply. Ang 2010, where were you when Ang et al, 2013 did this research!?!?! The ultimate ramification for the paper is that the “loss” calculations the authors create, which they base entirely on (25+ year old) risk-neutral models using risk-neutral inputs, are figments of the authors’ highly educated imaginations. Unicorns are pretty too but they don’t exist, and existence is (I believe still) a really important feature if you’re using loss estimates to try to influence national fiscal policy.   

I know that to some, the above may largely be high concept gobbledygook. The bottom line is this: "losses" must be relative to some value that is actually attainable, or else it’s very tough to realistically call them “losses.” So where’s the strategy whereby a municipality actually captures the value the authors characterize as loss? The research only mentions the use of devices such as interest rate swaps to hedge the rate risk an advance refunding otherwise eliminates. This is a reasonable answer for those tax-exempt entities who maintain the risk appetite, liquidity reserves, management expertise, legal authority, and political will to execute swaps. And could those borrowers who satisfy all of these criteria please raise your hand? Now how much of the over $1.5 trillion in outstanding, unrefunded, callable, fixed-rate muni bonds is on the books of those swap-happy borrowers with hands up? 10%? 5%? Now there’s a research question worth answering. And that question can be rolled into an investigation of the number of misleading press articles written and careers cauterized or destroyed due to the use of municipal derivatives. Here’s a case study contribution to kick off that new paper – my exchange with the editors of the NYT on their coverage of muni swaps.

Though currently the danger of thoughtful new reform of any sort being considered in Washington is remote (maybe get the SLGS window open first?), I maintain that should it ever occur, this paper demonstrates more an example of professors inebriated by the exuberance of their own quantitative verbosity than it does some new discovery that would inform the prudent revision of the Tax Code.


*See 'The Formulat that Killed Wall Street' for a unique and fascinating bit of research on the culture of financial modeling and its contribution to the financial crisis via the CDO market.  

**From a variety of sources, "The lack of arbitrage is crucial for existence of a risk-neutral measure."

***For a truly excellent article describing the types of rate models that different players should use in their respective situations (including munis explicitly), see What Interest Rate Model to Use - Buy Side versus Sell Side.   

New Research: Advance Refunding Always Destroys Value - Use a Swap


“Real knowledge is to know the extent of one’s ignorance” - Confucius

According to Bloomberg, there are just over $1.5 trillion fixed-rate, callable, unrefunded, fixed rate municipal bonds outstanding today. Helping issuers analyze, evaluate and ultimately execute current and advance refundings on these types of bonds is an integral part of public finance. Brand spanking new research  by Andrew Ang (Columbia), Richard Green (Carnegie Mellon), and Yuhang Xing (Rice) finds that in the presence of transaction costs, advance refundings always destroy value to the issuer. They state,

Advance refunding provides short-term budget relief, but it destroys value for the issuer. By pre-committing to call, the issuer surrenders the option not to call should interest rates rise before the call date. The value lost to the issuer, and transferred to bondholders, is the value of a put option on the bonds. In addition, since the assets in the [escrow] trust are Treasury securities, the transaction provides free credit enhancement for the bondholders, also at the expense of the issuer. Finally, the intermediaries who create the trust and issue the new bonds collect fees to do so.”  

They next empirically attempt to estimate the value lost from advance refundings using MSRB data from January 1995 to December 2009 capturing ~149,000 pre-refunded securities with aggregate par of $454.4 billion.Refunding question Interestingly they find the loss is rather small in both dollar and percentage terms. Hey issuers - nice timing!

One interesting question central to this topic is whether the ability to execute an advance refunding itself actually has value for the issuer. This research posits that the payoff from advance refunding is always zero at best i.e. there is no positive value assigned to the ability to advance refund. In fact, they argue that an interest rate swap is a better tool for the job,

“Even if the goal is to accelerate or borrow against the uncertain future interest savings associated with the call provision, a swap contract could achieve this more efficiently.”

But the question stands, some commercial analytics calculate an “Advance Refunding Option” which explicitly quantifies the ability to advance refund as an issuer benefit. This research clearly takes a contrary view. Who’s right and why? 

2013 Discovery - The Missing Link in Interest Rate Modeling!


"There is no logical way to the discovery of these elemental laws. There is only the way of intuition, which is helped by a feeling for the order lying behind the appearance."  - Albert Einstein

Though many may assume we’ve been studying this finance thing so long we’ve got it all figured out, the fact is that exciting new discoveries in basic things like interest rate modeling still happen. In fact, the period in which we now find ourselves - a relatively long stretch of extremely low rates in a post-Bretton Woods era has led to a recent discovery in understanding interest rate movements. A paper in the March 2013 issue of Quantitative Finance entitled The nature of the dependence of the magnitude of rate moves on the rates levels: a universal relationship by Nick Deguillaume, Riccardo Rebonato, and Andrey Pogudin (sets you back $37 for the download) describes how interest rates move in a way that is incredibly robust across markets, currencies and decades, even centuries, of interest rate history. It’s a missing link discovered!   

The authors investigate the behavior of interest rate changes and whether or not they varied based upon the level of interest rates themselves. This question is of critical importance to hedgers, speculators and frankly anyone who wants to use interest rate models. Most pricing models suffer from a very limited number of factors (often just one!) and as such don’t exhibit particularly realistic yield curve changes over time.  And then the question is, do changes in rates occur as a percentage of rates themselves (log-normal) or do changes happen in basis points terms (normal model).  In the words of the authors,

“Despite the importance of the issue, a direct analysis covering a wide range of rate levels has only recently become possible thanks to the period of very low rates many currencies have entered in the last few years.”

As described below, they essentially discover three answers or “regimes” dependent on the level of rates themselves. The degree to which the model fits, with little or no change to the parameters, decades and even centuries old history of different markets and currencies is really impressive and fascinating. The results are roughly as follows:

Rates below ~1.5% - In easy money environments rates tend to follow a proportional (lognormal) process reflected by a percentage of the rate

Rates between ~1.5% and ~6% - With rates in a more normal range rate changes appear to be independent of the rate level, moving in basis points i.e. normal volatility such as in Vasicek’s original 1977 mean reverting model. 

Rates over ~6% - Rates return to moving in a proportional (lognormal or even hypernormal) manner, scaled by the level of the rate itself

See Figure 11 below from their paper

Movements to level img

What’s this got to do with public finance, you ask? To the extent that the dark art of public finance is in part about managing interest rate risk through new money and refinancing, a whole lot. Stay tuned…

Muni Call Options: Information vs Price


Today on (Un)Calculated Risk we welcome Shaun Rai, a Managing Director at Montague DeRose and Associates, as our guest contributor (and another outstanding IA client!). 

A bond salesman friend recently had his home on the market.  It had been listed for a month or two when I asked him if there had been many showings, or any offers to buy the house.  He indicated that there had been some interest but no actual offers as yet.  I asked him what the house was worth. He responded, “Not sure, but I know what it’s not worth.”

In light of the recent increased focus on using municipal bond call option valuations to assess refunding opportunities and, potentially, award competitive bids via TIC+, this anecdote serves as a useful reminder that it is important to distinguish between “information” and “price.”

A price is the amount a willing buyer will actually pay a willing seller for a good or service.  In this sense, prices do not exist for municipal bond call options, for there is no traded market for these options.  An issuer cannot sell the call option embedded in its bonds.  An investor cannot buy a call option on bonds it owns to cover the call option it has sold to the issuer.  Callable and non-callable bonds of the same maturity with the same credit are very rarely offered to the same investors on the same day.Price vs Value

Thus, market participants can only estimate the “value” of municipal call options using option pricing models.  And in doing so, they must input key pricing parameters which cannot be precisely extracted from actual, traded market prices. For example, there are no actively traded non-callable yield curves, nor is there a forward municipal bond market.  Given these limitations, using theoretical option values to assess a refunding opportunity is, at its core, a convenient “short form” way to do probability-weighted scenario analysis in which the results are a function of the assumptions used.

This leads to the conclusion that the only “price” that can be established for a muni call option on an outstanding bond is the present value savings that an issuer is willing to accept to execute a refunding of that bond.  If the issuer executes a refunding for present value savings of $5 million, that is the “price” of the call option on that day.  If an option pricing model indicates that the theoretical value of the option is $6 million, that is “information,” but it is not a “price.” 

Does this mean that using estimates of muni call option values is not useful?  As a dyed-in-the-wool derivatives guy, my opinion is definitely not.  Estimates of muni call option values can be very helpful in thinking about whether to pull the trigger on a refunding or if it makes sense to use lower coupon bonds to achieve a lower yield-to-maturity.  However, it is important to emphasize that the call option valuation is “information” --  it is not a “price” -- and should be viewed and used in the same way an issuer would use more traditional scenario analysis.

Confusing “information” generated by models with “price” confirmed by the market can lead to poor decision making.  My friend the bond salesman knows the difference – until he sells the house, he doesn’t know what it’s worth.  

Shaun Rai is a Managing Director at Montague DeRose and Associates, a leading municipal financial advisory firm based in California, whose clients include many of the largest issuers of municipal bonds on the West Coast, including the State of California and the State of Washington.  Shaun can be contacted at or 805-319-4145. 

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