Right and Wrong Model for Callable Muni Bonds

7/23/2019 Right and Wrong Model for Callable Muni Bonds

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The right and wrong models for evaluating callable municipal bonds

Peter Orr, David de la Nuez

ABSTRACT

Fixed rate municipal bonds are often sold with an optional redemption feature giving

issuers the right to call the bonds prior to maturity. The application of no-arbitrage bond

option models to help assess the value of these optional redemption features though not

common has been increasing. Despite the availability of these models, widespread public

finance industry adoption has not occurred. This paper outlines theoretical and practical

problems with no-arbitrage models employed for the purpose of analyzing embedded

options in municipal bonds. We also highlight recent research in yield curve modeling

and show an example of a real-world approach to analyzing municipal bond options

which introduces the concept of expected present value (EPV) savings.


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1. Introduction

Of the $3.7 trillion in municipal and tax-exempt bonds outstanding1, approximately $1.54

trillion2, or 41.6%, are long-term, fixed-rate, unrefunded and subject to redemption prior to maturity at the

option of the issuer. These optional redemption features are call options purchased by the issuer at the

inception of a bond transaction. Issuers usually fund an optional redemption through a refinancing issue

called a refunding. Per current tax code3there are two types of refundings, current and advance. The latter

occur more than 90 days in advance of the call date and involves using proceeds from the refunding issue

to purchase a portfolio of eligible securities (State and Local Government Securities (SLGS), US

Treasuries or agency securities) designed to match the cash flows of the refunded bonds to their

respective first optional redemption dates.4,5

A stubbornly intractable problem faced by practitioners yet addressed infrequently by researchers

relates to determining the optimal timing of exercise (e.g. see (Gurwitz, Knez, & Wadhwani, 1992),

(Kalotay, Yang, & Fabozzi, 2007), and recently (Ang, Green, & Xing, 2013)). For American and

Bermudan options, the definition of an optimal exercise strategy is synonymous with valuation; without a

clearly defined optimal exercise, there can be no correct option valuation. (Glasserman, 2003) states,

The value of an American option is the value achieved by exercising optimally. Findingthis value entails finding the optimal exercise ruleby solving an optimal stoppingproblemand computing the expected discounted payoff of the option under this rule.

So how do issuers in practice solve this optimal stopping problem? How do they select when to

refund bonds? Many issuers use long-standing heuristics to determine whether or not to refund a bond

1Source:Report on the Municipal Securities Market, U.S. Securities and Exchange Commission2Source: Bloomberg, August, 20133

Section 148 of the Internal Revenue Code covers rules governing tax-exempt bond issuance including refundings.4For a detailed description of refunding mechanics see Ang , , (2013) and Kalotay (1998)5There is disagreement as to whether the ability to advance refund a bond enhances or detracts value from the issuer.Ang et al (2013) maintains, Advance Refunding provides short-term budget relief, but it destroys value to theissuer. By pre-committing to call, the issuer surrenders the option not to call should interest rates rise before the calldate. Kalotay (1998) on the other hand characterizes this differently saying, The advance refunding option (ARO)

is intimately tied to the call option. In the absence of a call option, the ARO has no economic value. An advancerefunding locks in the call exercisean issue that is advance refunded will be called at the first call date (at thepreviously set call price). The value of the ARO should therefore be measured incrementally above that of the calloption.


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such as present value interest cost savings (PV savings) as a percentage of the par amount of the refunded

bonds. Once this percentage exceeds a specified threshold, usually between 2 and 6%, the bonds are

deemed to be candidates to include in a refunding bond issue.

But American and Bermudan bond call options are one of the most common forms of exotic

interest rate derivative (see (Rebonato, 2003)).6And a variety of term structure models are readily

available commercially that are suitable for pricing these options. Why has there not been widespread

application of these technologies to the municipal market?

Though some may point to a dearth of Street quants engaged in addressing problems in public

finance, another plausible reason is that term structure and bond option models designed for relative

pricing in arbitrage-free environments do not apply. Perhaps they are round pegs that do not fill neatly

into the square holes of embedded municipal bond options. In the remainder of this paper we address this

possibility. We first describe two broad classes of users of term structure models, hedgers and speculators,

with an example highlighting terminology and more importantly modeling approaches applicable to each.

We next provide a brief taxonomy of term structure models with a discussion of their purposes and

features, followed by a discussion of municipal bond options, their characteristics, and the best term

structure models to use in their analysis and valuation. Last we apply a regime-switching yield curve

model derived from recent research which sheds important new light on how to evaluate these features

include the concept of expected present value (EPV) savings.

2. Hedgers, Speculators and the Terminology of Models

For the purpose of analyzing term structure models the world is split into two types of market

participants: hedgers and speculators. Sometimes these players go by other names: sell-side and buy-side

or dealers and end-users. For our purposes the characteristics are the same: speculators are those that take

and manage naked long or short positions, while hedgers are those that are also exposed to risks but

6Some may treat American options as a vanilla product but as a practical matter, determining the optimal exercisestrategy for American or Bermudan bond options involves solving the same free-boundary problem. For this reasonwe refer to both of them as exotic here.


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attempt to apply offsetting positions that lead to (ideally) a net zero risk exposure. The canonical example

of a hedger is a sell-side dealer.

Though the term speculatortends to carry a loaded and pejorative meaning in public finance

circles (state and local laws abound prohibiting what is defined as speculation), for purposes of evaluating

term structure models both issuers and investors fall into exactly this camp: they each hold naked,

unhedged risk positions, short and long bonds, respectively. In the fixed income context speculators or the

buy-side are fixed income mutual funds or hedge funds, pension funds, insurance companies,

commercial banks, corporate borrowers, municipal borrowers, and others. (Rebonato & Nawalkha,

2011)

Functionally, hedgers are concerned with the rightpricefor instruments. Dealers attempt to make

prices that are consistent with other traded instruments in the market. If inconsistency exists, then

arbitrage profits may follow. However, the theory that takes this idea and extends it to an ability to price

assets generally7relies fundamentally on two market characteristics: completeness and efficiency. This

approach can be shown equivalent to one where the price of any asset under these conditions is the

discounted expectation of the assets payoff under what is referred to as the risk-neutral measure.The

counterintuitive discount rate to be used in allsuch cases, irrespective of the riskiness of the underling

instruments, is the risk-free rate (see example below). This approach, relying heavily on measure theory,

has led to a new lexicon in asset pricing. In the literature one now finds reference to risk-neutral

pricing, the risk-neutral density, risk-neutral valuation, the risk-neutral world, and risk-neutral

inputs. All are different ways in which to describe the requisite arbitrage-free environment.

Speculators, on the other hand, are concerned with price but are also very interested in value.

Valuation may be driven by a host of real world considerations that may involve economic or financial

forecasts, holding periods, and other unique needs or circumstances that drive the assessment of costs or

7The most recent and general incarnation of this theory can be found in (Delbaen & Schachermayer, 1994).


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benefits relative to a specific risk. The modeling world for this type of arbitrage-rich valuation setting has

its own nomenclature in contrast to the risk-neutral terminology above; the speculatorsdomain is

variously called the real, physical or objective world or measure. Note that if arbitrage exists, the

only recourse is to analyze assets in this real or physicalworld environment.

3. Risk-Neutral versus Real Worlds An Example

To illustrate the distinction between risk-neutraland realor physicalworlds, we take a

simple numerical example from the equity markets. Start with technology stock ABC currently trading at

price SABC= 100. You are asked to make a price for purchase of non-dividend stock ABC on a forward

basis in one year i.e. to offer a forward contract on ABC at price FABC. The consensus annual return on

equity (ROE) forecast for ABC is 35% and the one year risk-free rate of interest is Rrisk-free= 5%. At what

price do you set your forward contract on ABC? Given the consensus forecast, you might think the

appropriate forward price on ABC is

FABC= (1+ROE) SABC = (1+.35) 100 = 135.

However, other market participants could then borrow 100 at 5% to purchase the stock today.

Next year they would take the 135 forward price you pay them, pay back the loan plus interest (105 total),

deliver the stock they purchased today, and pocket 30 per share. Ignoring delivery risks, this transaction

generates riskless profits.

Clearly 135 is not the correct price if arbitrage profits are to be precluded. In order to set the

arbitrage-free price you must determine the risk-free trading strategy that replicates the terms of the ABC

forward contract. You are obligated to purchase ABC stock in one year though in the end you do not want

to own it. If you sell a share of ABC stock short today for $100, invest the proceeds at the risk-free rate,

in one year you will have 105 plus a short sale of ABC stock that needs to be covered. This self-financing

trading strategy generates both cash and the requisite short position in ABC stock to immunize the overall

position. Therefore the arbitrage-free price of the forward contract is in fact 105. Our implicit assumption

is that ABC grows at the risk-free rate of interest,


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FABC= (1+Rrisk-free) SABC = (1+.05) 100 = 105.

In this example we presumed ABC was a technology stock with a consensus ROE of 35%. What

if it were a utility stock with an expected ROE of 12%? Or a bank stock with an expected return of 17%?

Where did the real world assumption for the growth in ABC enter our calculation of the forward price? Of

course it did not. Given our trading strategy ignores the real world returns entirely, we say this is a risk-

neutraltrading strategy; the underlying riskiness of the assets in question is completely irrelevant. A

trader at a dealer bank making prices on these types of forwards is concerned solely about the cost of

financing the position. We can assume, irrespective of the real-world growth of the underlying asset, that

it grows at the risk-free rate.

On the other hand, for an investor who wants to buy or sell this forward contract in an unhedged

environment, the real-worldreturn on the stock very much matters. It is the real-world return that

ultimately drives the payoff and that is the environment in which she performs her analysis.

4. Types of Term Structure Models and their Roles

Though a variety of excellent surveys and taxonomies of term structure models already exist (e.g.

see (Brigo & Mercurio, 2001), (Rebonato, 2003), (Rebonato & Nyholm, 2008), (Narwalkha, Beliaeva, &

Soto, 2007)), we offer one more in the hopes that it be at once more concise and intuitive for public

finance practitioners then those that have come before, while still managing to serve the technical

purposes of this paper. To this end we divide term structure models into three categories: deterministic;

pricing; and real-world. Those either well-versed in term structure modeling or less interested in such fare

may proceed to Section 5 without loss of overall meaning.

Deterministic

It could easily be argued that deterministic models do not belong in the category of term structure

models at all. However, we think it important to point out that deterministic models are the most common

yield curve models used in public finance today. Deterministic models are a special case of a more


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complete term structure model i.e. they simply display zero volatility. As an example, in order to create a

forecasted schedule of principal and interest for variable rate demand obligations (VRDOs), a public

finance analyst must employ an interest rate model, usually of the deterministic variety. The rate selected

is often constant and is frequently a simple historic average of short-term yields over some period.

Pricing Models

Pricing models, or as they are frequently called relative pricing models, are designed to do just

that: price interest rate derivatives using an over-arching assumption that markets are arbitrage-free. In

order to price an instrument such as a bond option, these models rely on the ability of the user to eliminate

the risk involved in being short or long such an instrument, just as described in our simple forward stock

example above. In order for a pricing model to become successful or perhaps even a market standard,it

must balance certain competing objectives,

Some of the prerequisites for industry acceptance are ease, quickness and stability of

calibration, speed of pricing for reallive quotes and for hedging, ability to give anintuitive interpretation to the model parameters, market consensus about the modellingapproach, etc. Not surprisingly, it is rare for a theoretically justifiable model to displayall, most, or even a few of these features. (Rebonato & Nawalkha, 2011)

And further,

unless the model allows fast and stable calibration it cannot be used for industry purposes, no

matter how attractive. (Rebonato & Nawalkha, 2011)

Pricing models are founded on the dual principles of market completeness and efficiency i.e. all

derivative payoffs are replicable with other usually more basic instruments and there are no free lunches.8

(Rebonato, 2003) further subdivides relative pricing models into two camps: fundamental modelsthat

explicitly estimate the functional form of the market price of risk along with the true dynamics of yield

curve and reduced formmodels that evolve yields directly in a risk-neutral setting. Early examples of

fundamental models include (Vaek, 1977), (Brennan & Schwartz, 1982), (Cox, Ingersoll, & Ross,

1985), and (Longstaff & Schwartz, 1992).

8(Rebonato & Nawalkha, 2011) go so far as to say Complexderivatives models are technological tools mainly

used to price and hedge structured products.


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In reduced form models, forward rates and implied volatilities from the current prices of vanilla

swaps, caps, and swaptions are used to calibrate the model directly and ultimately evolve the yield curve

only in the risk-neutral world. Examples of reduced form relative pricing models include (Ho & Lee,

1986), (Hull & White, 1990), (Black, Derman, & Toy, 1990), (Black & Karasinski, 1992), and (Brace &

Musiela, 1997) and (Jamshidian, 1997). In 1990s the reigning paradigm for derivatives pricing became

the LIBOR market model ( (Brace & Musiela, 1997) & (Jamshidian, 1997)). The most salient feature was

its ability to recover the log-normal (Black) prices of caplets and swaptions (de Guillaume, Rebonato, &

Pogudin, 2013). Currently, a variation on the LIBOR Market Model, the LMM-SABR model is

considered by many to be the gold standard in relative pricing models (Nawalkha, 2009).

Real World Models

Real world yield curve models have shown perhaps the most development over the course of the

last decade. These models are designed for the purpose of creating yield curve evolutions in the real world

measure, in contrast to risk-neutral ones described above in pricing models. Emphasis for these models is

not on calculation speed or calibration ease but rather on capturing the driving econometric and statistical

features observed in actual yield curves. Sample applications for this class of term structure model

include:

asset and liability management for corporations, banks and pension funds;

strategic asset allocation decisions;

counterparty credit risk assessment;

testing the effectiveness of derivatives pricing models. (Rebonato & Nyholm, 2008)

One of the earliest types of real world rate models developed involves Principal Component

Analysis (PCA) (see (Martellini & Priaulet, 2001) for a review). The PCA method involves transforming

historic yield curve changes into principal components, a small number of which often explain the vast

majority of historic yield curve movements. The first three principal components also offer an intuitive

interpretationrespectively level, slope and curvature. Unfortunately, simulating yield curves using PCA

can quickly lead to unrealistic yield curve shapes, even if one retains all of the principal components in

the data (see (Rebonato R. , Mahal, Joshi, Bucholz, & Nyholm, 2005)). Other examples of real world


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yield curve models include VAR modeling (Kugler, 1990) and (Lanne, 2000), VAR and regime switching

(Engsted & Nyholm, 2000), (Rebonato R. , Mahal, Joshi, Bucholz, & Nyholm, 2005), (Bernadell, 2005),

(Diebold, Rudebusch, & Aruoba, 2006) and (Diebold & Li, 2006).

One of the more exciting recent discoveries in real-world yield curve modeling is reflected in (de

Guillaume, Rebonato, & Pogudin, 2013); they find an interest rate modeling missing link which

successfully explains how yield changes relate to yield levels. The relationship holds across currencies

and decades, even centuries, of observed yield curves within a regime switching model that lends itself to

generating history-consistent yield curves. We use this research as a basis for modeled results in Section

8.

5. Pricing ModelsFundamental Problems

Given the term structure model taxonomy above, it is clear that deterministic models are ill-suited

for evaluating the call options in municipal bonds: the uncertain degree of opportunity lost from

exercising a bonds redemption feature must be evaluated and this requires modeling and capturing the

stochastic nature of yield curves. Pricing models are the next candidate though it should come as little

surprise that they are not the answer either. We first explain why, ignoring the quirks of the municipal

market while focusing on the purpose and nature of the models themselves.

As discussed above, relative pricing models are designed to determine the price of assets in the

risk-neutral measure. In practice this measure is embodied in a tree, lattice or other simulated

environment. However, in a fixed income setting do yield curves within this risk-neutral world resemble

real, history-consistent yield curves? To the surprise of many, the answer is no. The drift term within

pricing models leads to yield curve behavior that simply has never occurred:

These drift terms, as is perfectly appropriate for the relative -pricing application forwhich they have been devised, drive a wedge between the real-world and the risk-adjusted evolution of the yield curve. As mentioned above, over long horizons, theevolutions produced by these approaches become totally dominated by the no-arbitragedrift term and bear virtually no resemblance to the real-world evolution. (Rebonato R. ,Mahal, Joshi, Bucholz, & Nyholm, 2005)


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Further, in discussing the applicability of pricing models to topics such as the ones mentionedabove,

There is a more fundamental problem with these approaches, that makes themunsuitable for our purposessince these models have been designed to price derivativessecurities, the modelling choices of the real-world dynamics of the yield curve tend to be

dictated by computational considerations: one-factor models, which imply approximatelyparallel moves of the yield curve, for instance, are common and the diffusive assumptionsfor the driver(s) is almost universally made. For these reasons, fundamental pricingmodels are not appropriate for the [valuation] applications highlighted in Section 1.(Rebonato & Nyholm, 2008)

And though the section below is in relation to the currently popular Libor Market Model, the

critique applies equally to the analysis of municipal call options with any relative pricing model,

The virtues of the LMMSABR model which allow it to perfectly fit the observed prices

of bondsunder the riskneutral measure, are of not much use to most sophisticated

institutional users of interest rate derivatives and structured products, if these virtues donot allow a meaningful riskreturn analysis under thephysical measureTheseinstitutions [end-users and municipalities] are not in the business of making money by

trading interest rate derivatives while maintaining zeroexposures. This is the majordifference between sellside dealer banks and the buyside borrowers and investors.

Hence, the buyside users of interest rate derivatives need to perform risk return analysis

under the physical measure to understand the risk and return tradeoffs[emphasisadded]. (Rebonato & Nawalkha, 2011)

It is interesting to note that even within the purview of pricing models, there is currently some

debate as to whether or not market-implied (i.e. arbitrage-free) quantities are entirely appropriate. If there

is reason to believe that structural imbalances may exist in markets, and that arbitrage activity may be

curtailed by residual and material risks, then even in a relative pricing context, market-implied values

may not be the best choice (see (Rebonato, 2003)).

6. Pricing ModelsProblems Specific to the Municipal Market

Though an issuer may exercise an optional redemption for a variety of reasons including retiring

bonds with restrictive covenants, replacing insured bonds, or simply applying excess revenues to shrink

the balance sheet, the most common objective is to achieve interest cost savings through financing with

lower interest rates. This is called a high-to-low refunding. This last motive is the one we focus on here.

Traditional no-arbitrage bond option models rely on the existence of a non-callable underlying

equivalent bond which can be traded in a continuous fashion with unfettered ability to be bought or sold.


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In the municipal bond market this situation simply does not exist. (Ang, Bhansali, & Xing, 2010) describe

in some detail the reasons why it is so difficult to hedge a municipal position in the context of market

discount bonds even for dealers, though the analysis applies equally in the broader municipal bond

context to both issuers and investors,

Municipal issuers are unable to arbitrage the mispricing of market discount bonds. IRC 148specifically prohibits arbitrage across municipal bonds and other types of bonds (for example,Treasury and corporate bonds) by tax-exempt institutions. Dealers, however, should still view thetaxation of market discount bonds by individuals as a profitable trading opportunity. However, amajor impediment is that dealers have little ability to hedge the purchase of market discountbonds on their books. Unlike Treasury bonds, shorting municipal bonds is very hard because onlytax-exempt authorities and institutions can pay tax-exempt interest. An investor lending amunicipal bond to a dealer would receive a taxable dividend because that dividend is paid by thedealer, not a tax-exempt institution. Even if an active repo municipal market existed, it may behard 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 uppotentially large basis risk. Another reason arbitrage may be limited is because the trading costsare much higher than Treasury markets.

We can put a finer head on this if we look at the hedge strategy for a municipal bond option with

an eye towards which ones are traded and observable. For a complete analysis we would need the

following:

(1) Option free tax-exempt yields for the issuer

(2) Implied volatilities for the yields in 1)

(3) Market yield curve for escrow securities (SLGS or other)

(4) Implied volatilities for escrow yields in 3)

(5) Option free taxable yields for the issuer (for callable, non-advance refundable bonds)

(6) Implied volatilities for the taxable yield curve in 5)

(7) Implied correlations between option free tax-exempt yields and escrow yields

(8) Implied correlations between option free taxable yields and escrow yields and,

(9) Implied correlations between tax-exempt and taxable issuer yields

Few of these items are observed directly as traded instruments. SLGS are published at the end of

each trading day on the Treasurys web site. Given that SLGS are designed to be a basis point lower than

comparable US Treasury yields we might imply (4) from the prices of traded Treasury options, though at

present no study exists comparing the two markets or the validity of that assumption. Items (1) and (5) we

might try to estimate using secondary market data and a methodology like one described in (Nelson &


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Siegel, 1987), though this data is choppy at best and frequently entirely unobservable, particularly for (5).

Volatilities in items (2) and (6) depend upon an active market or accurate option pricing which, as

discussed in this paper, presents an interesting chicken or egg problem. How should one imply volatilities

without market-tested, reliable, trading-based models? Correlations for (7)-(9) simply do not exist on an

implied basis in any market.

We agree with (Kalotay & May, 1998) that in order to completely analyze the municipal

refunding decision, both treasury and borrower yield curves must be understood. However, no-arbitrage

term structure models are ill-equipped for accommodating the dynamics of full yield curves from multiple

markets and the resulting calibration issues that ensue. Given the importance of both the replacement

yield of the new bond, which usually matches closely to the maturity of the refunded bond, and the yield

on the escrow, which is to the call date,9any single factor model is inadequate for this purpose.10

The logical conclusion of the above analysis is that models designed for relative pricing in

general provide little benefit and are simply inappropriate.

7. Characteristics of a Good Model for Analyzing Municipal Bond Call Features

In order to properly evaluate municipal call options, we find the following characteristics

desirable in a markets model:

1. Positive nominal rates

2. Full description of term structure dynamics including volatilities and correlations for at least

three markets: tax-exempt, taxable, and escrow investments

3. Control over yield curve shapes while allowing for variation of slopes, humps, and twists

4. Recovery of the distributions of historical yield changes

5. Replication of historic yield correlations both within and across markets

6. Long high end and fat low end distribution tails for the yields themselves

7. Accurate analytic approximations of yield distributions

8. Rapid calculation without excessive memory load

9Note these are not only two different points on the yield curve, but also two entirely different markets.

10It doesnt make sense to use a single factor model to value any security whose cashflow is linked to the shape of

the yield curve. (Kalotay A. , 1995)


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With a market model fitting the above criteria we could then analyze and evaluate different

exercise criteria currently in practice as well as ultimately establish refunding criteria to determine

valuation.

8. Real-World ExamplesIntroduction to Expected Present Value (EPV) Savings

We find a model derived from recent research by (de Guillaume, Rebonato, & Pogudin, 2013) to

be a best-fit which, with some enhancements and adjustments, satisfies all eight criteria above.11We

created 10,000 simulations of three interdependent yield curve markets over 30 years using daily yield

curve data from 1970 to 2013.12Inputs include a starting yield curve as well as infinite horizon yields

towards which yields will revert over the analytic horizon. The latter are usually derived from a forecast

or long-term average for each tenor in the yield curve over an historical period deemed relevant.

The starting yield curves (solid lines) and their respective infinite horizon yields (dotted lines) are

shown inFigure 1andTable 1:

11See forthcoming research Evaluating Municipal Refunding Criteria: A Simulation-Based Approach.12SOURCE: Delphis Hanover Corporation


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Figure 1

Table 1

Tax Exempt SLGS Taxable

Tenor Spot

Infinite

Horizon Spot

Infinite

Horizon Spot

Infinite

Horizon

3 Month 0.175% 2.791% 0.071% 3.158% 0.321% 3.408%

6 Month 0.112% 3.613% 0.362% 3.863%

1 Year 0.320% 3.000% 0.130% 3.994% 0.432% 4.296%

2 Year 0.450% 3.450% 0.240% 4.315% 0.668% 4.743%

3 Year 0.660% 3.833% 0.350% 4.584% 0.785% 5.019%

4 Year 0.870% 4.158% 0.520% 4.811% 0.855% 5.146%

5 Year 1.020% 4.434% 0.750% 5.001% 1.414% 5.665%

7 Year 1.520% 4.669% 1.210% 5.162% 1.983% 5.935%

10 Year 2.180% 4.869% 1.860% 5.297% 2.731% 6.168%

12 Year 2.640% 5.038% 2.130% 5.410% 3.171% 6.451%

15 Year 3.010% 5.183% 2.390% 5.505% 3.461% 6.576%

20 Year 3.490% 5.305% 2.700% 5.586% 3.762% 6.648%

30 Year 4.030% 5.409% 3.090% 5.653% 4.092% 6.655%

40 Year 4.160% 5.498%

0%

1%

2%

3%

4%

5%

6%

7%

TE Infinite Horizon SLGS Infinite Horizon Taxable Infinite Horizon

Tax Exempt Spot SLGS Spot Taxable Spot


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Fifteen samples of tax-exempt borrower and escrow yield curves are shown inFigure 2below.

Note that each simulated environment includes a complete set of correlated taxable, tax-exempt, and

escrow yield curves. The colors of tax-exempt and escrow yield curves are matched to their respective

simulated environments and as expected, borrowing and escrow yield curves are consistently high or low

depending on the simulated environment.

Figure 2


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Though the simulations are performed daily to more faithfully capture real yield curve dynamics,

we calculate the value of refunding certain hypothetical bonds on a quarterly basis using an assumption of

one percent for costs of bond issuance. The net present value (NPV) savings calculation reflects negative

arbitrage13where it exists and ensures the escrow yield is no greater than the bond yield for tax-exempt

refundings.14The objective is to create a standard NPV savings value in each simulated environment, akin

to an option payoff. As an example, the graph below shows the distribution of NPV savings for a 5%

bond maturing July 1, 2031 callable July 1, 2016.

Figure 3

The bond inFigure 3 is a bond callable but not eligible for refunding on a tax-exempt basis until

90 days in advance of the call date. The black line reflects expected PV savings across all simulated paths

at each quarter. The green line extends from one standard deviation above the mean to the 95thpercentile.

The top of the red line begins at one standard deviation below the mean and drops to the 5thpercentile of

the distribution. Note that prior to the call date the taxable advance refunding leads to lower NPV savings

overall relative to when the tax-exempt refunding is possible starting 90 days prior to the call date. We

also see the initial average reduction in NPV savings as rates rise on average within the model. Last it is

13Negative arbitrage occurs when the yield on the escrow is lower than the refunding bond yield. 14

Section 148 of the Internal Revenue Code precludes issuers from earning a higher yield on the escrow than the

arbitrage yield on tax-exempt refunding bonds.


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0

5

10

15

20

25

EPVS

avings/$100

Refunded Bond Coupon

20Y Maturity, Callable in 3Y

Refunding Criteria - 5% PV Savings Threshold

Tax-Exempt Adv Refunding

Taxable Adv Refunding

Current Refunding Only

important to note that the refinancing rate is rolling down the simulated yield curve as the bond

approaches maturity. This last fact explains why there is declining savings variability as we approach

maturity.

Assuming the borrower

refunds the bonds at a 5% NPV

savings threshold (common in

practice), we can calculate

expectedpresent value (EPV)

savings by appropriately

discounting and averaging

across all simulated

environments. InFigure 4,EPV savings is shown for three different bonds types: one allowing for tax-

exempt refunding (blue); one that can be advance refunded only taxably (orange); and one that we specify

may only be refunded on a current basis i.e. no more than 90 days in advance of the first call date (green).

At lower coupon levels, EPV savings for all bond types is similar indicating that refunding is happening

at similar times, in this case close to maturity. In the coupon range over 5.25%, exercise happens very

quickly or even immediately. In this range, the taxable advance refunding fares poorly relative to either

the tax-exempt advance refunding or waiting for the current refunding near the call date. Note that there is

a middle region, however, with a coupon range between 4.75% and 5.25%. Here the refunding criteria of

5% PV savings impacts EPV savings in unintuitive ways. Despite higher coupons, EPV savings actually

falls in certain cases because the refunding decision in the lower coupons is delayed such that NPV

savings at a later time is higher, perhaps due to less negative arbitrage in the escrow.

It is important not to generalize too much from this example. The 5% NPV savings refunding

criterion chosen, though common, is arbitrarily applied to all bonds regardless of coupon or terms. As

such, there is little we can resolve dispositively about these three bonds or their embedded options. For

Figure 4


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instance, it would be incorrect to take these results as proof that either advance refundings always destroy

value, or alternatively that it is always more advantageous to wait until the call date to refund.15Without

optimal exercise criteria, these assessments cannot be accurately made.16

InFigure 5 we provide visualization of an optimal exercise boundary for a municipal bond option

given a certain set of

decision criteria. The

coloring of this chart

shows the percentage of

simulated markets where

certain optimal refunding

criteria have been

satisfied. Refinancing

rates, along the y-axis,

are bucketed such that if

100% of the refinancing

rates in a rate bucket for a given time period correlate to satisfying the refunding criteria, then the

rectangle is green. If in turn the number of rates satisfying the refunding criteria is 0%, the rectangle is

red. A color gradient corresponds to the intermediate values. Note that the greater the time to the call date

remaining, the greater the area of either yellow or red, indicating little or no refunding. This results from

the role of the other important variable not directly observed in this graph, the escrow yield. Negative

arbitrage is significant in those areas where, despite a very low refinancing rate (even below 2%), the

region is still not one where refunding occurs frequently. This region shrinks as we move closer the call

15(Ang, Green, & Xing, Advance Refundings of Municipal Bonds, 2013) believe they have answered this question,

though we find much of their reasoning problematic.16

See forthcoming research Evaluating Municipal Refunding Criteria: A Simulation-Based Approach

5% Bond - 18Y Maturity, 8Y Call

Optimal Exercise Boundary

NO REFUNDING REGION

REFUNDING REGION

Figure 5


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date. After the call there is a clear distinction as the escrow yield is no longer a factor and there is a direct

correspondence between refinancing rate and savings.

9. Conclusion

We have reviewed the problem of valuing embedded optional redemption features in callable,

fixed-rate tax-exempt bonds. From the investor and issuer perspectives as takers of nonzero risk positions,

we show that appropriate analysis of these features should be done using real-world term-structure

models, not relative pricing models. The latter are designed with other essential objectives in mind with

an emphasis on calculation speed and ease of calibration to market-traded instruments. Recent research

offers promising new ways of simulating history-consistent tax-exempt, taxable, and escrow yield curves.

We introduce the concept of expected present value (EPV) savings and describe how it is calculated using

a regime-switching, multiple-market yield curve model. In forthcoming research we will use these

methods to shine new light on the $1.5 trillion in callable, fixed-rate municipal bonds outstanding.


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Right and Wrong Model for Callable Muni Bonds