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).
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 firstname.lastname@example.org.
"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.
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.
“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. 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?
"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.
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…
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.
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 email@example.com or 805-319-4145.
"Despite its role in...finance, the expectations hypothesis (EH) of the term structure of interest rates has received virtually no empirical support." - Predictions of Short-Term Rates and the Expectations Hypothesis, Federal Reserve Bank of St. Louis
As we've written on these pages before, forecasting is a necessary evil in finance. It's uncertain by nature and of course the longer the horizon, the more difficult the job. The theory that forward rates are good predictors of future realized rates is called the expectations hypothesis and as one MIT professor put it, "If the attractiveness of an economic hypothesis is measured by the number of papers which statistically reject it, the expectations theory of the term structure is a knockout."
For fun (and to dust off my fast fading coding skills) I went back and looked at how US Treasury implied forward 10Y rates have done in forecasting realized 10Y UST yields from July, 1959 to the present. We used first of month data for 3, 6 and 12 month Tbills as zero rates (making the appropriate daycount adjustments of course) and then 2, 3, 5, 7, 10, 20, and 30-year UST coupon instruments for our implied 10Y forward calculations. And this is what we get...
The red line is the actual 10Y yield over the period and the "hair" is the implied 10Y par yield 1, 2, 3, and 5 years forward. The way to read this then is to look at how often the hair tracks with the actual realization of the 10Y yields as shown by the red line. In general, during this single big rate cycle we've seen over the last 50 years, forward rates have badly underpredicted when rates were going up (note the implied decreasing 10Y forwards during the 70s) and then overpredicted over the last 30 or so years as rates have fallen. How badly do forwards do? Well over this 50 year span, and this holds over most subperiods as well, you'd be better off as a forecaster just assuming today's yield curve stays constant i.e. a perfectly random walk.
Let's look at the tax-exempt market. Analyzing today's current tax-exempt yield curve (non-call) we see an implied increase in the curve over 10 years, though we think not in a particularly realistic way. The bottom line in the chart below is the current non-call tax-exempt curve from 1 month out to 30 years (labeled in green).
Each successive curve above it is the implied forward yield curve in 1 year forward increments from 1 year through 10. Over the 10 year horizon, you can see the 1 month tax-exempt rate smartly moving up over 500 basis points, equivalent to a 7% slam on the monetary brakes by the Fed. However this is accompanied by only a 1.45% move in the long end of the curve from 3.73% up to 5.18%. Realistic? Perhaps, but we'd expect to probably see a higher 30Y rate if the Fed were really that active over the next 10 years.
Don't get me wrong - if you're in a financial services environment as a trader or you're looking to perform a fair price analysis of an interest rate derivative using an interest rate model, you better use forward rates. If you've got complete and relatively efficient markets, you'll get your head removed if you don't. However, if you're an issuer or working with an issuer looking at some sort of scenario analysis on their debt portfolio, forward rates may be a "good to know" but probably not the end of the forecasting road.
If you've taken a break from the news lately, you may have missed the hot water Bloomberg's found themselves in over Bloomberg reporters accessing certain information about Bloomberg users. Finance types scouring for the proverbial free lunch in the markets are understandbly private and the prospect of some Bloomberg journalists looking over their shoulders from those comfy midtown offices is well, unsettling. Of course this is likely overblown by the non-Bloomberg media but we thought the message we got today (below) after logging in to our own Bberg terminal (below) was particularly entertaining and candid...
Doesn't anyone watch Mad Men over there?
"Horse sense is the thing a horse has which keeps it from betting on people" - W.C. Fields
A bookie matches bets. If a bookie sees lopsided interest in bettors taking the Giants by 3 in Sunday’s big game against the Patriots, she’ll ultimately need to adjust the odds she is offering to get a matched book. The instant before the game starts, you could calculate an implied probability for each team winning based upon the bets in the book. Does the bookie care what these specific probabilities are? Absolutely not. Does the bookie even care which team wins? Not if she’s done her job right. The only thing the bookie cares about is that the sum total of the implied probabilities for the teams adds up to more than 100%. Because the amount over 100% is her vigorish – and that’s how she buys dinner. To borrow a term from asset pricing theory, her position is “risk-neutral”.
Contrast the bookie now with bettors, who are in a different position entirely. Bettors might conduct fundamental research into the severity of center Ryan Wendell’s sprained ankle, or how wide receiver Victor Cruz’s hamstring is doing. Bettors are obviously interested in the odds and payout from the bookie, but they are concerned as much or more with the real world likelihood that the Pats will win the day. On this latter analysis of the real world rests the core betting decision. It would certainly be wrong to conclude that the bookie held all relevant analysis of the probabilities based on the bets in her book.
Of course, this describes our financial markets as well. Does a vanilla equity options trader care whether the Jan14 40 MSFT calls expire in the money? Not one bit. His money is made regardless of outcome: he runs a matched book functionally identical to the bookie’s. But how about investors who hold a position in those same options? Absolutely they care. And in the case of options there is an important extension to consider. The somewhat radical and at the time very unintuitive conclusion by Fisher Black, Myron Scholes, and Robert Merton, is that for purposes of pricing an equity option, the trader must assume the growth rate of MSFT to be the risk free interest rate. The same “risk-neutral” term again applies: the options are fully hedgable and as such the underlying growth must be assumed to be the risk free rate or arbitrageurs will enter the market and make it so. The investor of course in order to evaluate a buy or sell decision, must examine the real world expected growth rate of MSFT; it is the real world growth that matters. Is there a difference? Over the last 5 years MSFTs annual ROE has averaged about 40.1%. Over the same time 6M Tbills have yielded 0.24% - big difference.
At Intuitive Analytics we welcome the increased application of asset pricing and valuation theory to the municipal market. We believe it is a very positive and constructive development which ultimately can lead to better decisions under uncertainty. At the same time we believe, particularly in light of recent financial history, that the right models be used the right way, for the right reasons and with well-vetted inputs. There is important context to consider in the use of any financial model. Though it might be nice if we all could live in the bookie's world to make our vig, states, municipalities and the dedicated debt managers who serve them live in the real
one. They have to take real, unhedged positions; their job is harder.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." - John von Neumann
I’ve landed on Earth from outer space, in these here United States (somehow feel right at home in Williamsburg…) and am trying to learn from you Earthlings how to calculate present value savings in public finance. During my journey here I tried to educate myself by reading a variety of materials including:
In the end, I’ve seen 2 methods. The first and more commonly used to date I'll call the Single Rate Method (SRM) which usually uses the arbitrage yield for discounting and the second I call the Zero Method (ZM) where different zero rates from the term curve are used to discount each cash flow. In order to understand these two methods better, I ran some refunding numbers and to my surprise what I discovered is that both methods are WRONG. I’m confused. Here’s what I did…
I took a hypothetical 20-year, 5% municipal bond callable at 102 in 3 years and ran some refundings, 20 to be exact. I used a single maturity, non-call par bond for each refunding from year 1 to year 20. Coupons/yields for the 20 individual refundings are shown in the 2nd column in the the table below.
The maximum hypothetical escrow yield is 2% so the first 4 scenarios limit the escrow yield to the arbitrage yield of the respective refunding bond. To the size of the escrow (remember, 3 year call at 102) we add 1% for costs of issuance to arrive at the new bond size in the 'New Bond Size' column. In order to calculate PV savings we first take the present value of the existing bonds to maturity then subtract the present value of the new bond which is of course just the 'New Bond Size' under both methods.
Now in order to compare the two methods, we need to generate zero rates from the par curve which I discovered is not a common procedure in public finance circles. Nevertheless I do this easily using the handy Bootstrap function I found in these public finance utilities. Results and a proof of equality between discounting at 4% (the arb yield) and zero rates for the 20 year refunding bond scenario are shown in a table at the end of this post.
So how do these methods look? The first SRM method (green above) creates a massively different change in present value of the current debt, ignoring the valuation of the call feature itself for the moment. But the implication that the value of the existing bond to the borrower is as much as $17.2mm or as little as $11.4mm depending on the tenor of the refunding at best, lacks intuitive appeal. But it is exactly this calculated change in value of current debt that leads to the huge $5.7mm amount of PV savings for the 1% 1-year bond versus the paltry $199k savings in the 4%, 20 year bond scenario. This method must be wrong.
So let's look at the Zero Method (blue above). Using the zero rate discount factors (table below) the present value of existing debt is constant, which seems like an improvement. Even incorporating the call would be a constant offest against the noncall debt value for each scenario. But the present value of the new debt isn't changing either, except when the yield restriction kicks in. Therefore PV savings is identical in refunding scenarios from year 5 through year 20 but starts going DOWN as we refund with earlier maturities from year 4 and in. The larger present value of the new bond deal results from the arbitrage yield restriction on the ("100% efficient"!) escrow, hence decreasing savings. Therefore the Zero Method shows that refunding a 20-year 5% bond with a 1% 1-year bond actually creates negative savings. Though it may be in some sense more theoretically consistent, it's performance in aiding refunding criteria selection just doesn't seem spot on.
I'm new to your planet but we're trying to (re)finance a lot of infrastructure back home. What am I missing? Where have I gone astray? Can someone please help? Obi-Wan was busy...
You can download the Excel spreadsheet, 'PV Savings for Refundings-Methods Compared' used for this article here.
PROOF OF ZERO RATE EQUIVALENCE TO 4% PAR RATE, 20Yr BOND
|| Zero Rate
|| PV @
|| Par Rate
|| Zero Rate
|| New DS
|| Zero DFs
|| TOTALS =
Calculated using Bootstrap in these Utilities.
My colleague David de la Nuez (PhD Operations Research) and I build on the linear algebra from our prior video to show how to set up sources and uses and cash flows for a hypothetical three bond deal. Dr. David then solves the linear program using a few simple lines of MATLAB code; doesn't get much juicier than this, don't miss it!