Welcome. My name is Rodolfo Riverol and I have over twenty years experience in public finance including 3 and 1/2 years as a director in Citgroup's public finance department. This site along with the companion YouTube channel streetmath is an introduction to FIXED INCOME MATH in PUBLIC FINANCE and MUNICIPAL BONDS. It's a hands on EXCEL SPREADSHEET based TRAINING SITE. The final objective is for you to be able to build a bond sizing spreadsheet that includes 1) a new bond schedules, including amortization, 2) a sources and uses page, and comparisons to either revenue cashflows or refinanced cashflows.
-------
Remember the first rule about fight club in the movie fight club is 'you don't talk about fight club'? So, let's talk math instead. But first just a few war stories that came a little close for comfort and demonstrate why knowing your bond math is imperative. For example, I once saved a bond closing by suggesting we view a DERIVATIVE that was collateralized by treasury bonds as ONE SINGLE SECURITY UNDER THE LAW instead of a collection of individual treasuries. Yep. By the way, collateralized by treasury bonds means the derivative provider (seller) gives treasury securities to the buyer with the agreement that the provider can come back anytime and restructure the portfolio of treasuries by plucking out treasuries and putting in new ones. Well, the derivative provider inverted the interest rates on the treasuries he gave us, and this violated a federal escrow account earnings restriction. However, it worked when viewed as a single super large security with lots of little treasury legs, because in this case the whole agreement (with all of its treasury securities) would go to only one of two escrows, instead of being divided out between two accounts. Sounds boring, but the alternative, making the newspapers for a delayed closing is a humiliating career ender. Here's another story, and this is really great. In1998 when premium bonds past the call date became commonplace, everyone in the muni industry faced the realization that arbitrage yields were about to become exponentially harder to calculate. By law the arbitrage yield for bond issues with premium bonds past the issue's call date is calculated by finding the call date for each of these problematic premium bonds that produces the lowest possible arbitrage yield - and of course the number of permutations is mind boggling - on the order of 20 factorial x 356^20 or about 4.3x10^69 for a 30 year issue, and clearly not the way to go. But inspiration struck, and I proposed a better way by of using our standard bond pricing formula. We'd use the pricing formula to find the dates for each bond that produced the overall lowest yield in a finite and quick number of steps. Specifically, we plug in our arb yield guess (3% or whatever) instead of the market yield into the bond pricing formula for each maturity to get dates that gives us the lowest price (which is always one of the call dates or maturity, and usually the call date if the coupon is substantially higher than the market yield), then use these dates to recompute a new yield, say 3.1415etc, and repeat and keep adjusting our dates and our recomputed arbitrage yields until we hit the PV target. Great example of how valuable knowing your bond math is. Really - all kidding aside, I even impressed a couple software companies. Another is my independent rediscovery of 'Abrew' zero coupon prices for the Black Derman Toy model after reading Gemmill's book Derivatives: An International Approach. And three more good ideas on the NYC account, including the construction of a variable to imitate the minimum of a group of values in a linear optimization model. Because as it turns out (and you can easily prove) the minimum function is not linear, but, by NYS local Finance Law, min is needed for NYC transactions. The technique is easy enough to explain too, so.... If x(n) is a linear expression what we're focusing on, and d(n) is related to x(n), then at time n + 1 we set the two constraints x(n + 1) <= d(n + 1) and x(n + 1) <= x(n). From this construction we see that x(n + 1) <= minimum {d(1), d(2), ..., d(n + 1)}. In the case of NYC the quantity d(n) was the debt service in year n for thier105 test, and d(n) was the principal for their 50% rule test.
And never panic, think of it as a game. Take the anxiety out of it and think of it as a game. No consequences - just a game. You're better than you think you are, you can make it easy, think of it as a game.
One caveat, this site is for teaching and training and though I love my spreadsheets - using them yourselves for actual deals is at your own risk.
However, you can hire me to run numbers for you. I freelance, and I can train people too.
Please don't forget to visit my YouTube channel streetmath, it's integral to the training. My contact info is Rodolfo Riverol / rodolforiverol@gmail.com / 917 498 4150. I have a BA in mathematics from New York University, I was a PhD graduate student at Rutgers University in math, and I'm a graduate of Bronx Science.
And if you like my site please consider supporting me on Patreon. Thank you.
Oh, and one last thing about the movie fight club again, remember the last scene where all the buildings came down? The whole house of cards burst into flames. Some people like to think of this as an illusion or diversion, but trust me stick with the math - it's a solid foundation/alibi.
YOUTUBE CHANNEL streetmath (Videos)
From Youtube Channel Part 1 First Steps in building a Refunding Spreadsheet (Video)
Part 2 building a Refunding Spreadsheet (Video)
Part 3 building a Refunding Spreadsheet (Video)
Part 4 building a Refunding Spreadsheet (Video)
IMPORTANT SPREADSHEETS ON THIS WEBSITE
Arbitrage Yield Calculation Spreadsheet
How to use the spreadsheets. Save them to your computer and make sure you have Excel. Open them up, enable macros, then use. I've minimized VBA coding and everything is open code.
A Fixed Rate Bond is a loan from one party, YOU, called "the Buyer" of the bond, and also known as the "Investor" (the lender or creditor), to another, "the Issuer", like the State of California or the Commonwealth of Massachusetts etc., and issuer is a fancy word for debtor or borrower. In these cases the Issuer (CA, MA, etc.) promises to make fixed payments on fixed dates over time to YOU, the lender (aka the Buyer). So, issuing bonds means borrowing money (from YOU), and issuer mean borrower or debtor.
Municipal Bond Issuances are large borrowings by states, municipalities, and authorities, from insurance companies, investment & bond companies, and others, that are marketed by investment banks that act as brokers for the bonds and don't usually want to buy them for themselves, but will buy (underwrite) and inventory them and sell them over time if necessary.
Large Investment banks like Citigroup and Bear Stearns have large 'Sales and Trading Desk' that are full of brokers who sell municipal bonds to insurance companies (like State Farm, AllState and GiECO, Prudential, MetLife and TIAA) , as well as investment companies and bond funds (like Nuveen, Pimpco and Blackrock etc) and to retail (ie, to the general public).
A comprehensive list of municipal bond issuers (States, Municipalities and Authorities who borrow money) and their issuances (borrowings) can be found on the emma, website from msrb (municipal securities rulemaking board). To use emma just fill in the issuer name (NYS, CA etc.) and click. Click here to see an OS example .
A Bond's Official Statement (description), aka OS, prospectus provides specs (click here to view an OS) (I chose this deal because it was my last Citigroup deal). The specs in an OS always include:
The Purpose or Reason for the borrowing/issuance (WHY you're borrowing money - WHAT IT'S FOR...roads, bridges, buildings, schools, ...., pocket money, Ferrari, legal settlements). Page 7 in our (OS example)
Credit, Security, Revenues used to pay the bond debt (ie, What you use to pay the bonds, eg, taxes, revenues from say tolls, gimmicks), Page 1 in our (OS example) .
Additional Bonds Test (aka ABT) and Coverage Constraints are test that allow for issuance of more bond so as not to panic current bondholders. Because there's nothing worse than your creditors making you look bad by dumping your loans and debt on the market making it so you can never borrow again.
Credit Ratings from an independent rating agency like Moodys, S&P or Fitch (answering HOW SAFE IS THIS). And, whatever you may think of them, almost everybody gets one.
Reserve Funds used to avert defaults if revenues tank.
A flow of funds that follows every dollar from the bond issue into other funds like the Construction Fund, Refunding Escrow Fund, Capitalized Interest Fund, Debt Service Reserve Fund, Debt Service Fund, Underwriter's Discount (bankers), Upfront Cost, and last but not least, contingency or petty cash as well as a pecking order for Senior and Sub Lien Bonds.
Bond Payments Dates, Interest Rates, and Amortization Schedule (which answers WHEN and HOW MUCH).
Sources and Uses page tracing the money going to the Construction Fund, Escrow Fund, Capitalized Interest Fund, Debt Service Fund, various cost of Issuance like bond counsel and rating agencies, Underwriters, etc from bond proceeds, equity contributions, etc. Kitchen table balancing.
Call Option payment dates or Call Dates and prepayment penalties known as Call Premiums are found in OS's. Unlike mortgages, municipal bond principal cannot be prepaid at any time. Munis have a lockout periods, and principal can only be prepaid after the end of the lockout period which is marked by the "call date", and which is typically ten years after the delivery date for fixed rate bonds. Page 7 in our (OS example) One more thing, though you can't prepay principal before the call date, you can borrow money years before the call date to set up dedicated interest-bearing escrow accounts that will pay off the principal on the call date. These financings are called advance refundings, and are done when borrowing rates are low and issuers don't want to gamble and possibly lose savings if they wait to refund until the call date.
More details about Municipal Bond Debt issuances are described in a State's Constitution or an authority's Bond Indenture.
Bond Indentures are agreements between state/municipal borrowers (debtors) and their investors (creditors / lenders).
Information summarized in an Official Statement (OS), include Revenue Sources (ie Credit for bond borrowing), Flow of Funds, ABTs or Additional Bonds Tests, and Reserve Funds, with even more detail found in the bond indenture
Other useful bond documents include:
Comprehensive Annual Financial Report (CAFR) detailing projects, revenue allocations and future bond borrowings within the next year. For more on CAFRs
Verification Reports are independent accountants' reports confirming a bank's refunding calculations, especially the arbitrage and escrow yields on advance refundings.
General Obligation Bonds (GO) are generally backed (paid for) by property taxes.
Revenue Bonds are backed by revenues from projects like toll roads and bridges.
Appropriation Bonds are backed by State appropriations (funds earmarked by the municipality or state with stronger credit), for example the New York State Local Government Assistance Corporation LGAC.
Asset Backed Bonds borrow against a stream of payments or revenues, like mortgage payments, student loan payments, and Tobacco securitization payments, like California Tobacco for example. For example, the money from an asset back borrowing may be used to originate mortgages, and in turn, the mortgage payments used to pay the principal and interest on the asset back issue.
Conduit Bonds are backed by payments from loans to small towns and municipalities that might not normally or be able to issue bonds due to credit. For example, State Revolving Funds SRFs like Mass Trust make loans to dozens of small towns.
Private Activity Bonds borrow to finance private projects like sports stadiums.
Construction funds
Are interest bearing accounts that pay construction draws for projects like roads and bridges over time.
Interest earnings on construction funds are limited by the arbitrage yield and are "subject to rebate" if the monies are not used within 24 months of closing.
Capitalized Interest Fund (Revenue Bonds).
Also called Cap I funds, are interest bearing accounts drawn on to pay interest on bonds until construction is complete and project revenues (like tolls) kick in.
Interest earnings are subject to rebate, which means that interest earnings above the arbitrage yield must be paid back to the federal government.
Debt Service Reserve Funds (Revenue Bonds).
A DSRF is a self insurance policy where one year's worth of principal and interest (debt service) is set aside to stop a default should revenues lag.
The amount of the DSRF that can be financed by tax-exempt bond proceeds is the minimum of the three below, and anything above it has to be funded by taxable bonds or equity.
Maximum annual debt service.
10% of Par.
125% Average annual debt service.
The Issuer gets the DSRF deposit back at the end of the bond issue and usually uses it to pay back the last year's Debt Service.
The DSRF amount can earn up to the arbitrage yield, any earnings over the arb yield have to be rebated back (paid back) to the federal government.
If the issuers don't want or can't fund a DSRF, then a non refundable Surety Payment to an insurer equal to 2% to 3% of the DSRF can be used instead. But issuers never recover the cost, compared to the DSRF where it does get the DSRF back plus interest.
Debt Service Fund (Revenue Bonds).
Is an account where deposits equal to1/6 of the next upcoming Bond Interest payment and 1/12 of next Bond Principal payment are made every month from Revenues to pay the Debt Service on time.
General Obligation Bonds, aka GO Bonds, usually have level annual payments and are repaid using taxes like property taxes, sales taxes, etc., and.
Last principal payment cannot exceed 120% of the life of the underlying project.
Revenue Bonds (which are funded by project revenues like tolls, special taxes, etc.) have payments that follow / parallel revenues.
If revenues bottleneck then to get through tough times when ordinary interest on the debt might exceed revenues, issuers include CABs (zero coupon bonds that defer all interest until maturity) and convertible CABs (a hybrid of CABs and current interest bond), optimized to minimize their added cost.
Some asset backed bonds like housing, student loan, and tobacco bonds have "super sinker" and "turbo" principal maturities that expand and contract like a toy slinky ie, which can be prepaid or sometimes postponed depending on whether the cashflows that back them come in or not.
Refundings are a large part of the business. They are bond refinancings, akin to mortgage refi's. Money from new 'refunding' bond issues (ie, from new loans akin to new refi mortgages) is put into an interest-bearing account or escrow which is then used to pay off the Refunded Bonds' principal, interest and call premium (penalty) by the bond's call date. So instead of putting the money into a construction account, like a hew money bonds, the money is put into an escrow account to eventually pay off the old bonds. But unlike mortgage refinancings, where the old principal is paid off immediately, refunding escrows can last for years paying principal and interest on the old bonds (analogous to paying construction draws) until it can finally pay off the outstanding principal on the call date.
New Refunding principal and interest payments are structured to produce either level savings, or upfront savings compared to the old refunded principal and interest payments.
Types of Refundings:
A Current Refunding has an escrow that is 90 days or less, ie, the refunded bond's call date is within 90 days of the closing date.
Advance Refundings (refundings with escrows longer than 90 days) are no longer allowed, at least for not for now, but when they were...
Unless exempt, the earnings on the money or proceeds associated with any tax-exempt bond issue, regardless of whether it came directly from the bond issue, like construction fund accounts, DSRFs, cap I accounts or refunding escrow accounts, OR indirectly, like DSRF and escrow accounts that have 'transferred' over from another bond issue, WAS restricted by the bond issue's arbitrage yield (see below for definition and calculation). All interest earnings above the arbitrage yield has to be paid back to the federal government.
Starting on 1/1/2018 you cannot use tax-exempt bonds to Advance Refund any bonds, but you can use taxable bonds to advance refund tax-exempt bonds.
Refunding Analysis.
Savings and PV Savings.
Annual savings are usually set up to be either level or upfront.
Aggregate PV savings criteria vary by issuer, but most require at least 3% total PV savings as a percentage of refunded principal.
Issuers usually use some form of bond by bond savings threshold, and all issuers require at least a little savings per maturity.
Interest.
Compound interest is when interest is paid on interest as well as on principal or said differently, when interest keeps getting added back into the principal.
If an interest rate of i is compounded semi-annually then the interest after n semiannual six month periods = ((1+i/2)ⁿ - 1)x(the original loan amount).
The total amount of money the lender gets back in the end is = ((1+i/2)ⁿ) x (the original loan amount).
Simple interest is when interest is paid only on the original loan amount.
If an interest rate of i is paid semi-annually but non compounded then the interest after n six month periods = n x i/2 x (the original loan amount).
The total amount of money the lender gets in the end is = (1+n x i/2) x (the original loan amount).
Compounding effect increases interests: proof: when n>1, (1+i/2)ⁿ > 1+ n*i/2 + n(n - 1)*i2/4, and the ratio between payments above = (1+i/2)ⁿ/(1+n x i/2) increase as n increases.
Common definitions, formulas and jargon.
Given an interest rate of i, an amount of money M and a time period n, the Future Value (FV) of M is = M*(1 + i)n, and is just the original principal M plus interest. If n is the number of years then i is an annual rate, if n is the number of months then i is a monthly rate, etc.
Given an interest rate i, an amount of money M and a time period n, the Present Value or PV of M is the amount of money you need to deposit into an interest account with rate i to get M dollars at time n and = M/(1 + i)n. The PV
From this point on we will take all interest rates to be annual rates compounded semiannually, and that's because interest rates on municipal and treasury bonds are quoted that way.
The Semiannual PV or Semiannual PV Factor = 1/(1+i/2)(number of days between today and maturity/180)
More on Present Value.
If C1,C2,...,Cn are any n future debt obligations/draws/payments that are due d1, d2,...,dn days from now, then their Present Value = C1/(1+i/2)(d1/180) + C2/(1+i/2)(d2/180)+...+ Cn/(1+i/2)(dn/180)
If the C1,C2,...,Cn are construction draws, then the above formula tells us how much we have to deposit in the construction account.
If the C1,C2,...,Cn are the capitalized interest draws, then the above formula tells us how much we have to deposit in the Cap I account.
If the C1,C2,...,Cn are semiannual bond coupons and par, then the above formula gives us the total price of a bond including any accrued interest.
Sometimes you're being asked to solve for an interest rate i when the PV given or known, and by the way that known PV number is called the PV Target.
This is very important, because you will be doing this to compute the borrowing rate for the bonds (arbitrage yield), and the investment rates on accounts it funds (like escrow, construction fund, Cap I fund) to insure the investment rates do not exceed the borrowing rate, ie that the issuer is making money at the federal government's expense.
Observe that as i increases each PV component Cn/(1+i/2)(dn/180) strictly decreases (think of y = 1/x from high school). This means the Present Value sum C1/(1+i/2)(d1/180) + C2/(1+i/2)(d2/180)+...+ Cn/(1+i/2)(dn/180) also decreases as i increases.
This relationship - that PV decreases as interest increases, means that if all the cash flows C1,C2,...,Cn are positive, and all the d1, d2,...,dn are positive and we want to solve for i in C1/(1+i/2)(d1/180) + C2/(1+i/2)(d2/180)+...+ Cn/(1+i/2)(dn/180) = PV Target, then i is UNIQUE - there is only one single i for a given PV when all the cashflows are positive.
By daycount we mean how we compute the difference in days between any two dates.
It's important to keep earnings rates and borrowing rates on the same basis in order to compare them. In municipal fiancé we've selected '30/360 daycount' to compute borrowing rates on bond issues, as well as the earnings rates on accounts they fund like construction funds, refunding escrows, DSRFs, etc.
Municipal bond calculations are on a '30/360 daycount' - which we like to say means a year is comprised of twelve 30 day long months, but which precisely means
For municipal bonds the number of 30/360 days or '30/360 daycount' between two dates m1/d1/y1 and m2/d2/y2 is = (y2-y1)*360 +(m2-m1)*30 +d2-d1, where
1) d1 is the lesser of 30 and d1, and
2) if d2 = 31 and d1 = 30 then change d2 = 30.
Most issues do not have a 30 or 31st delivery date and for them the first formula (y2-y1)*360 +(m2-m1)*30 +d2-d1 will do.
For example, if the dates are 8/15/2027 and 8/19/2025 then the 30/360 daycount is (27 - 25)*360 + (8 - 8)*30 + (15 - 19) = 716, and if the dates are (rarely) 8/30/2027 and 8/31/2025 then the daycount is 720 = (27 - 25)*360 + (8 - 8)*30 + 30 - (30 instead of 31).
State and Local Government Securities (SLGS) and Treasury Bonds (used in refunding escrows, construction funds, cap I funds, etc.), have actual/actual daycounts for interest computation, which still peculiarly differ from each other. However, for the escrow or construction fund or capI fund, etc, yield computation you must still use the 30/360 method above.
To compute the accrued period for Treasury bonds we count the actual number of days between the two coupon payment dates that straddle the delivery date.
Treasury Bond example. A 5% 8/15/2027 treasury bond with delivery on 8/19/2025 has coupon payment dates on 8/15/2025 and 2/15/2026. The accrued period = (8/19/2025 - 8/15/2025)/(2/15/2026 - 8/15/2025) = 4/184.
For treasury bonds the accrual period is generally used only in the pricing formula (see below), the coupon payments are almost always full payments and never prorated. The only time coupons on treasury bonds are prorated is when the treasury is a newly issued brand new treasury. An example of pricing (1 + 2%/2)^(4/184)*{ 2.5/(1 + 2%/2)1 + 2.5/(1 + 2%/2)2 + 2.5/(1 + 2%/2)3+ 102.5/(1 + 2%/2)4 } - 2.5*(4/184)
SLGS example: SLGS are different.
Unlike treasury bonds where you choose from whatever maturities exist, YOU set the maturity dates for your SLGS.
Unlike treasury bonds, SLGS are purchased at face value or par value, and the first interest is always prorated.
The first interest payment is prorated according to if the SLGS has a maturity greater than 1 year and 1 month, and the payment occurs within 1 month of delivery or later than 1 month from delivery.
For example,
If you have a requirement 10/1/2045 you can buy a SLG maturing on 10/1/2045. Your first interest will be on 10/1/2025 and your proration factor for that payment will be (10/1/2025 - 8/19/2025)/(10/1/2025 - 4/1/2025) = 43/183 and applied to interest on 10/1/2025.
If you have a requirement of 9/1/2045 you cab buy a SLG maturing on 9/1/2045. Your first interest payment will be on 3/1/2026 and your proration factor will be 1 + (9/1/2025 - 8/19/2025)/(9/1/2025 - 3/1/2025) = 1 + 13/184 and applied to the interest on 3/1/2026.
If you have a requirement of 10/1/2025 (less than six months from delivery), then you will get all your interest at maturity and the factor in this case will be (10/1/2025 - 8/19/2025)/(8/19/2026 - 8/19/2025) = 43/365.
Delivery Date - Is the date where money changes hands, ie, where investors (ie, lenders) give money to issuers (ie, debtors) who in turn give bonds (ie, IOUs) to investors.
Maturity or Redemption Date - Payment date for a given bond.
Coupons - Are unchanging fixed semiannual payments that are in multiples of .125% (1/8th of 1%) or of .10% ( 1/10 of 1%). Although coupons are sometimes referred to as interest rates, coupons are NOT interest rates - a point of common confusion, yields are the interest rates.
Par Amount (sometimes called Principal Amount) - Is the dollar amount the bond holder (bond buyer, investor, lender) gets back at maturity, and uses to compute semiannual interest payments by multiplying it by the coupon over 2.
Accrued Interest - Interest accrued (computed) from the last possible payment date right before Delivery to the Delivery date itself.
Call Dates - Are dates prior to maturity where the bond can be paid off at a 'Call Price' + Accrued Interest. The first call date is usually ten years after the issuance date, and that's when the issuer can cease making the old bond principal and interest payments and replace them by lower principal and interest payments from the new issue. Any changes to the debt payments before the call date, for example from an advanced refunding, is basically a reshuffling or restructuring of the debt payments. All real savings comes after the call date, the rest is hocus-pocus
Call Price - Is the amount paid to bond holders from securities in the refunding escrow on the call date so the bonds are finally and forever paid off and gone and finito. No more expensive old bonds, just cheaper new ones.
Why did I choose the bond pricing formula? Because it encompasses all the calculations we use in public finance calculations in a nice neat little nutshell. As above, the bond pricing formula is just the sum of a bunch of present values. Like any other interest-bearing account (a construction account or refunding escrow account, etc.), except it's much neater because almost all the payments are equal.
The Price = lowest Present Value of the bonds cashflows over all possible Redemption Dates including Maturity.
If c is the coupon, p is the par or redemption or call price, and y is the yield or interest rate then the Bond Price to a given Redemption Date is =
= Sum over all Coupon Payment Dates {(c/2)/(1+y/2)(#days between [Delivery Date to Coupon Payment Date]/180)} + p/(1+y/2)(#days between [Delivery Date to Redemption Date]/180) - Accrued Interest , where c is the coupon, p is the par or call price, and y is the yield or interest rate.
= c/2 * {∑ 1/(1+y/2 )(#days between [Delivery Date to Coupon Payment Date]/180)} + p/(1+y/2)(#days between [Delivery Date to Redemption Date]/180)- Accrued Interest
= (1 + y/2)(Accrued Period) * [ c/2 * { 1/(1 + y/2)1 + 1/(1 + y/2)2 + … + 1/(1 + y/2)n } + p/(1 + y/2)n ] - Accrued Period * c/2
For example, if we have a 5% bond with a yield of 2% maturing on 8/1/2027 and delivery on 8/19/2025 then the PRICE = 2.5/(1 + 2%/2)(#daysdifference(2/1/2026 - 8/19/2025)/180) + 2.5/(1 + 2%/2)(#days(8/1/2026 - 8/19/2025)/180) + 2.5/(1 + 2%/2)(#days(2/1/2027 - 8/19/2025)/180)+ 102.5/(1 + 2%/2)(#days(8/1/2027 - 8/19/2025)/180) - 2.5*#days(8/19/2025 - 8/1/2025)/180 = 105.708, where the difference in days is computed on a 30/360 basis.
= (p - c/y)/(1 + y/2)(#days between [Delivery Date to Redemption Date]/180) + (c/y) * (1 + y/2)(Accrued Period) - Accrued Period * c/2. (and this last formula comes from using the sum of a geometric series).
And using this last little formula we get (100 - 5/.02)/(1 + .02/2)(#[days[8/1/2027 - 8/19/2025]/180) + (5/.02) * (1 + .02/2)(#days(8/19/202 - 8/1/2025)/180) - (#days(8/19/2025 - 8/1/2025)/180) * c/2 = 105.708 (same price as above, as expected, TG).
Although everyone, including Excel, has pricing formulas, the advantage of this last formula is how compact it is.
Some Comments.
It turns out that you need to calculate the Price only to only about 4 dates: 1) the maturity date 2) the first call date and 3) any other call dates where the call price changes, like for example the par call date. And it's usually no more than 4 dates including maturity, hence only about 4 calculations.
You can prove this to yourself by observing that (1 + c*x)/(1 + y)^x is either strictly increasing or decreasing as a function of x when coupon c and yield y are constant. Easiest way, elementary calculus: differentiate to get (c - log(1 + y))/(1 + y)^x, which is always positive (increasing) or always negative (decreasing) because c - log(1 +y) is independent of x and is either positive or negative, or you can look at this memo for a no calculus approach.
The Price only depends on the Delivery Date, and never on the Dated Date.
The price formula does NOT depend on whether the actual first Coupon is short or long. By convention, all bonds (even bonds from new issues with prorated interest) are priced as if they were already in existence, regardless of when the actual first Coupon payment is made or any proration. The first possible Coupon payment in the formula is always taken to be the first possible payment date after Delivery, and the coupon is always assumed to be a full semiannual coupon payment.
Bond Prices are truncated to three decimals.
If the Coupon = Yield then Price is forced to = 100%, and the bond is a Par Bond.
If all the Redemption Prices are at Par then:
If Coupon > Yield then the Market Price = Price to the First Call Date, and Price > 100 making it a Premium Bond.
If Coupon < Yield then the Market Price = Price to Maturity, and Price <=100 making it either a Par or Discount Bond. .
By definition, the Average Life of a bond is equal to Time Weighted Cash flows divided by the Total Cash Flows.
Time Weighted Cash Flows = Sum over all Payment Dates {#days between [Delivery Date to Payment Date])/360} *Coupon/2 + Par * #days between [Delivery Date to Maturity Date])/360
Total Cash Flows = #Coupon Payments * Coupon/2 + Par
A 2 1/2 year 5% bond has cashflows 2.5, 2.5, 2.5, 2.5, 102.5 at times .5, 1, 1.5, 2, 2.5 in years making its average life = (2.5*.5 + 2.5*1 + 2.5*1.5 + 2.5*2 + 102.5*2.5)/(2.5 + 2.5 + 2.5 + 2.5 + 102.5) = 268.25/112.5 = 2.39 years.
A 2 1/2 year 4% bond has cashflows 2, 2, 2, 2, 102 at times .5, 1, 1.5, 2, 2.5 in years making its average life = (2*.5 + 2*1 + 2*1.5 + 2*2 + 102*2.5)/(2 + 2 + 2 + 2 + 102) = 265/110 = 2.41 years.
Bonds with higher coupons have shorter average lives than those with lower coupons, said differently the higher the coupon the more quickly an investor is getting back their money, hence the shorter the loan.
Zero coupon bonds like municipal Capital Appreciation Bonds (CABs)and Treasury STRIPS, have average life = maturity.
A 2 1/2 year zero coupon bond has cashflows 0, 0, 0, 0, 100 at times .5, 1, 1.5, 2, 2.5 in years making its average life = (0*.5 + 0*1 + 0*1.5 + 0*2 + 100*2.5)/(0 + 0 + 0 + 0 + 100) = 250/100 = 2.5 years.
The shorter the average life the faster the debt is repaid.
The shorter the average life, the shorter the loan, hence the lower its interest rate should be in an ascending yield curve environment.
Bonds with high coupons should have lower interest rates and those with lower coupons should have higher interest rates.
Capital Appreciation Bonds (CABs), also called zero coupon bonds, are bonds that pay interest only at maturity.
For example, a $5,000, 3% CAB (zero coupon bond) maturing in 3 years cost $4,572.7 (principal amount or price) and pays $427.3 in interest in three years + principal for a total of $5,000 with no other interest payments in between.
By the definition of average life, the average life of a CAB is its maturity date - all other coupon bearing bonds have shorter average lives (look at the examples above).
CABs have higher rates than coupon bearing bonds, as they should, because they have higher average lives, ie, are longer loans. However, CAB rates are generally higher than the theoretical rates they should have.
Because their rates are so much higher than ordinary bonds, CABs aren't used unless absolutely necessary.
CABs are used when issuers have steep revenue curves or revenue bottlenecks where ordinary current coupon bonds fail when coupon interest fills the bottleneck too soon.
For example toll road authorities like TCA and OOCEA.
When CABs have to be used their maturities are optimized to minimize interest cost, ie to maximize bond proceeds.
The Yield to Maturity (YTM) is the yield YOU solve for in the Bond Price Formula above that makes your Bond Price to Maturity = Given Market Price.
The Yield to Call (YTC) is the yield YOU solve for in the Bond Price Formula above that makes your Bond Price to the Call Date = Given Market Price.
Some comments.
Yield to Maturity (YTM) is greater than the Market Yield and Yield to Call (YTC) is greater than the Market Yield.
YTM is equal to the Market Yield only when the bond is a Par or Discount Bond.
Yields are usually calculated using iterative numerical methods like Newton's Method or the secant method.
Bond issuers buy insurance when they believe it will result in lower interest cost even after taking insurance cost into account.
To calculate cost, insurers generally apply a fixed Fee to every dollar they insure (principal and interest computed to the stated maturity).
The Insurance Cost for any specific bond maturity = (Bond Principal + Interest to maturity) x (Insurance Fee).
In many cases you can cherry pick which bond maturities to insure, you don't necessarily have to insure the whole bond issue.
Savings from Insured vs. Uninsured.
When the coupons fpr the insured and uninsured bonds are equal, the Insurance Benefit or Savings is simply = difference between the insured price and uninsured bond prices less the insurance cost, and if the difference is positive, then the insurance is worth it.
I put this section here in this spot (before the section on bond structuring) to clarify and better explain the 'theoretical rates' or ideal zero coupon interest rates in zero coupon bond rates section above.
Arbitrage is the difference in price when the same set of securities is priced in two different ways. For example consider the price of ONE single two year bond that pays out $5 in year one and $105 in year two, compared with the price of TWO zero coupon bonds that pay exactly the same. In an arbitrage free market these two bond portfolios have the same price.
An 'efficient' market is one where there is NO arbitrage. The treasury bond market is a relatively efficient market, municipals are not.
In an efficient market, the proceeds (or value) of any cashflows generated by coupon bonds is always equal to the proceeds of the same cashflows when generated by only zero coupon bonds.
In an efficient market the zero coupon yields/rates are derived from noncallable bond yields and also known as "Spot Rates".
Zero coupon prices can be used to price any noncallable bond.
To go from zero coupon yields to current coupon yields is easy, just multiply the current interest bonds cashflows (coupons over time & par at maturity) by the corresponding zero coupon prices then add them up and compute the rate of return.
We have to use a bit of algebra to go from coupon bonds to zero coupon prices and yields. The technique is called bootstrapping (see spreadsheet) and we back into zero coupon prices from current interest prices by using all zero coupon prices before maturity to PV their coupons then subtract it from the price to determine a (zero coupon) price for the maturity + interest.
Wikipedia.com reference on bootstrapping and zero coupon yields
Forwards and zero coupon interest rates (above section) are very closely related. Short term forward rates that pay only at maturity can be seen future zero coupon rates and can be used to derive current zero coupon bond rates, and vice versa the ratio of zero coupon prices gives us forward rates.
Forward rates are interest rates agreed on today that start paying interest at some time in the future, but not today, and which make investors indifferent between investing in two things:
A long term interest rate starting today, versus
A short term interest rate starting today and ending on the forward start date plus the Forward Rate.
Forward Rates are breakeven rates, and are arbitrage free, in other words, they are set up or structured so you can't profit from different investment strategies in the same market.
Interest rates can be decomposed into forward rates the same way matter gets broken up into atoms, or whole numbers into prime numbers.
For example, a 5 year rate can also be broken down into a (1 year rate) + (1 year forward starting in 1 year) + (1 Year forward starting in 2 years) + (1 year forward starting in 3 years) + (1 year forward starting in 4 years).
Here is a very simplified, very elementary example (with no PVs) to conceptually illustrate what forwards are.
Suppose a 1 year rate starting today is 1% and a 2 year rate is 2%, then a 1 year forward rate staring 1 year from today and ending 2 year from today is 3%. That's because the 2 year 2% rate pays a total of 4% at the end of 2 years, and a 1 year 1% rate + a 1 year forward 3% rate starting in 1 year and ending in 2 years, pays a total of 1% + 3% = 4% at the end of 2 years.
Furthermore, if the 3 year rate is 3% then using the same logic the 1 year forward starting 2 years from today = 3*3% - 2*2% = 5%.
Notice that 1 year + 1year forward in 1 year + 1 year forward in 2 years = 1% + 3% + 5% = 9% = 3*3% foots perfectly.
Adjusting for PV effects the forwards would actually be 3.03% and 5.13% in this example.
Though the municipal bond market is NOT efficient, there are municipal bond forward rates that can be locked in today to perform current refundings in the future on a bond's call dates.
Although forwards refundings are not advance refundings, we can reenvision them as advanced refundings at current muni rates with a phantom advance refunding escrows earning at the short muni rates that pay off the interest on the advance refunding bond issue (called a cross over refunding where the interest on the new issue is capitalized).
The difference between municipal forward rates and the theoretical forward rates (where the market is actually efficient) is called the 'illiquidity fee' (which is just a fancy word for there's a difference, and there's nothing we can do about it because the market is what it is and we just can't price good forwards in muniland)
Credit, Gordon Gemmill's 1993 book 'Option Pricing: An International Perspective'.
Credit, Richard Flavell's book 'Swaps and other Derivatives'
A statistical model used to value Call Options that assigns probabilities to forward rates.
The Forward Rates are used to compute refunding savings which in turn are weighed by the forward rate probabilities to compute the Option's Value.
Credit, Gordon Gemmill's 1993 book 'Option Pricing: An International Perspective'.
Credit, Richard Flavell's book 'Swaps and other Derivatives'
BDT is a very popular Call Option Valuation Model developed by Fisher Black, Emmanuel Derman and William Toy in 1990.
The Model's computed bond prices are easily fitted with the actual observed market bond prices (ie, fits the yield curve with relative ease).
Breaks a yield curve up into a series of "future" six month interest rates (called short rates) along with probabilities in a binomial tree structure.
The short rates have a binomial probability distribution with equal 50/50 probability of rising and falling.
When the volatility is zero (in other words the bell curve collapses to a line), the short rates are nothing more than the six month breakeven rates computed every six months that makes the original yield curve arbitrage free, which are also the six month forward rates.
Forward Rate contracts do exist in the derivatives market, and option pricing models like BDT etc. can be used to price municipal SIFMA rate swaptions (options on LIBOR swaps) in that market, but it's hardly ever done. In part because the savings are PVed back at higher taxable LIBOR rates instead of low muni rates.
BDT is a popular model because of its relative ease of use. It's a binomial lattice model with one interest rate scale and one volatility scale.
Credit, Gordon Gemmill's 1993 book 'Option Pricing: An International Perspective'.
BDT Example Spreadsheet (download then righclick on properties and unblock the file to use)
Since refunding savings are generated after the call date some folks are reluctant to advance refund bonds preferring a wait and see strategy, believing the market will hold out until the call date.
Instead of PV savings per maturity being greater than some fixed percent, some issuers started using the "Refunding Efficiency Ratio = PV Savings / Option Value".
While an option value does exist in the SIFMA/LIBOR swap derivatives market it is almost never under consideration. Option value was/is used only to reject refundings - not as an alternative to one.
The option value computed is a theoretical number computed using the municipal interest rate curve along with a volatility set to get desired per maturity PV savings results.
Longer bonds had to generate more savings than shorter ones to get picked.
Bond with longer call dates had to have more savings than bonds with shorter calls to get picked.
A problem with these model is they could reject a high PV savings bond because its call date was out too long, but accept the same bond with lower PV savings when its call date got closer.
There's also confusion that these models pretend to predict interest rates, which of course is a huge NO.
Option value far from being a stagnant, varies as much as refunding savings with changes in rates.
Derivatives pricing models can be used to augment PV saving policy by narrowing in on specific savings thresholds per maturity. However, it must be stressed that THERE ARE NO GUARANTEES, and that NOT EXECUTING A REFUNDING BASED ON ANY MODEL'S RESULTS MAY RESULT IN LOWER SAVINGS IN THE FUTURE.
More on option value and refunding efficiency
Changing a bond's coupon changes its option value, which also changes its option adjusted yield - a yield which is computed to Maturity by setting the PV target = Market Price + Option Value.
In the spirit of balance and fairness we'd like to keep the option adjusted yield constant as we look at different proposed coupons, and that means different coupons will carry different market yields.
The Arbitrage Yield is a number set up by the federal government whose sole purpose is to limit earnings on money from tax-exempt bonds. Of course this means that the federal government wants a calculation making the Arbitrage Yield as LOW as possible in exchange for the tax exemption it is giving to bond buyers.
Multipurpose Allocations.
Is a way of dividing the bonds by purpose for tax reasons, like selecting the best savings bonds to refund.
Although the way they're set out in the OS cover is the easiest way to allocate bonds between purposes, there are cases where a different allocation, like a prorata allocation for better refunding candidates, is preferable. For example, say you had a bond issue that was comprised of a short refunding and a long new money, and a few years latter you wanted to refund the 'new money' portion using tax exempt bonds, then a prorata allocation is a better choice, and there may be even better ways than prorata.
Transferred Proceeds.
When bonds get refunded we have to deal with the earnings on any existing accounts they might have laying around, like any escrows, or DSRFs or even long construction funds they may have funded. Because of the tax-exemption given to bond holders, the federal government doesn't want those old accounts earning at the old interest rate if the bonds that funded those accounts were refunded at lower interest rates and are now gone. The fed wants the earnings somehow reduced. To reduce earnings we say the old accounts 'transfer' to the new refunding issue, and the earnings hit they take depends on how they 'transfer'. The federal government's nightmare scenario is for an issuer to issue a new 5% interest refunding issue, invest the borrowed money in a long term 5% interest bearing escrow; and then a few years later to refund the 5% issue with a new 3% issue that it uses to buy a short escrow that peters out before the first 5% escrow does. Net net, that leaves a 5% escrow and an issue with 3% interest cost for what appears to be a 2% gain. That's why these outstanding escrows and DSRF accounts have to have their earnings reduced.
Money and cashflows from the old issue 'transfer' to the new issue when the new refunding escrow pays old issue bond principal - this is what triggers a 'transfer'. This is the start of the story on how the issuer is going to lose earnings, because all earnings on anything that's 'transferred' gets reduced or restricted and the bigger the transfer the worse it is for the issuer.
The amount transferred on any given date = ({old principal being paid off on that date by the escrow}/{total outstanding old principal as of that date}) x refunded issue's untransferred escrow cashflows as of that date, securities, money, etc.
This ratio above, the ({retired old principal being paid off}/{total outstanding old principal}) is called the transfer factor.
The difference of the transferred cashflows PVed at the new arbitrage yield minus the same cashflows PVed at the old escrow yield is the penalty computed for every transfer on evey transfer date. The sum of those penalties PVed at the new arbitrage yield is the transfer proceeds penalty.
In some cases, like DSRFs or current refundings (where there is no escrow or other account), the issuer cuts a check to the fed. In other cases, like advance refundings, the issuer reduces their max earnings rate to lower than the arbitrage yield by making the PV target for the max allowable earnings yield to = the transfer proceeds penalty + the PV of the escrow requirements at the arbitrage yield.
Universal Cap.
A bond issue's Universal Cap limits how much money and securities transfers into an issue. That's why it exist and is a benefit to the issuer, because an issuers doesn't want its money restricted.
The Universal Cap on any given date = {Value of Outstanding bonds as of that given date} - {Value of all accounts purchased with the bonds, which includes escrows, DSRFs etc., as of that given date}. For example, if all proceeds of a bond issue have been spent, the Universal Cap = value (or accreted value ) of the bonds as of that date (the sum of the individual PVs of the bonds at their individual market yields), and if it's a par issue then the Universal Cap is the outstanding par in this case.
True Interest Cost (TIC).
The TIC is the PV Rate / internal rate of return "measuring" an issue's average interest rate accounting for fixed cost.
The PV of the debt service at the TIC rate is supposed to be equal to the TIC Target. And the TIC target (by definition) = the Sale proceeds (also called Bond Production) - Insurance Cost - Underwriter's Cost - Other Issuance Cost.
Arbitrage Yield (and calculation spreadsheet).
Established by the 1986 Tax Code to prevent issuers from borrowing in the low interest rate tax -exempt market and investing in higher taxable interest securities thus making risk free money (arbitrage) off the federal government.
In spirit the Arbitrage Yield is the average interest rate adjusted for any and all fixed costs (akin to points on a mortgage) that affects interest rates, like insurance (which - in theory - should make investors more comfortable they won't lose their money, and thus lower rates), but not accounting for takedowns (brokerage fees) or issuance cost which don't really make the bonds any more secure or attractive.
The Arb Yield Target is = (by definition) the Sale proceeds (also called Bond Production), less Insurance Cost, less Hedge Termination, and is what the PV of the adjusted debt service when discounted back at the Arbitrage Yield is supposed to equal.
The debt service being PVed for the arb yield must be adjusted for each out of safe harbor callable bond by choosing the call date with lowest PV is as Redemption Date (Maturity Date) for that out of safe harbor bond.
A callable bond is out of safe harbor if its price is greater than 100 + 25bps times {the number of years from delivery to call date rounded down to the nearest integer}. For example, if a bond maturing on 9/1/2038 with a delivery of 10/1/2025 has a call date of 9/1/2035 and a price of 112 then it is an out of safe harbor bond because 112 > 100 + .25x9.
Choosing the call date that produces the lowest PV for each out of safe harbor bond at each yield iteration will lead to the lowest (and most restrictive) yield.
This can be equivalently achieved by (1) ADDING the bond price at the assumed arb yield (instead of market yield) LESS (2) the price to maturity at the assumed arb yield TO (3) the target PV for each out of safe hour bond while leaving the initial debt service out to maturity alone.
The call date producing the lowest PV is either (1) the par call date or (2) the maturity date, and is found by computing the run of the mill, regular bond price (the same formula found in Excel or DBC eg) at maturity and at the par call date using the Arbitrage yield instead of the market yield. More on arb yield Arb Yield Memo , Explicit Arb Yld Calculation Spreadsheet
Arbitrage Yield Calculation Spreadsheet (please take a look, there's no substitute for experience)
Another useful yield is the TIC modified for negative arbitrage where the new Target = TIC target - negative arbitrage.
The best way to learn is by doing - which means trying to structure a few simple bond issues after watching the videos below and looking through the spreadsheets. As an exercise, try to structure a new money $100 million bond issue with 20 year level annual debt service with 5% interest rate par coupon bonds and $1 underwriter's fee.
Remember the objective is to
(1) setting up a payment schedule for the new bonds (which should be keyed off a stream of either revenues or refunded bond payments), and
(2) putting the money we borrow into interest bearing funds, whatever they are, construction funds or refunding escrow funds, DSRF, etc.
The schedules for construction funds, capitalized interest funds, refunding escrows, are all pretty much the same. It's always the same cookie cutter formula: a column of dates, followed by a column of requirements or draws (be they construction draws or refunded bond principal and interest requirements, or...etc.), followed by a PV column. Just remember, don't panic - think of it as a game, and this as being a glorified mortgage broker, LOL.
YOUTUBE CHANNEL streetmath (Videos)
Part 1 (Video) SETTING UP CASHFLOWS (REFUNDED CASHFLOWS AND/OR REVENUE STREAMS) First Steps in building a Refunding Spreadsheet in Excel
Part 3 (Video) Continuation of 2
Part 4 (Video) Continuation (INCLUDING YIELD CALCULATIONS AND SLGS SECURITIES ESCROW)
Below are links to two files:
A new money sizing spreadsheet that can be used for most simple new issues, and
A refunding spreadsheet. It's a good idea to look through them, learn to manipulate them, and ultimately be able to reconstruct them from scratch.
New Money Features include:
The ability to enter construction fund and cap I fund draws.
The ability to solve for level debt service or revenue coverage.
Note, the only real difference between the two spreadsheets is that one is borrowing for a 'construction fund', the other a borrowing for a 'refunding escrow fund' - but both are funds. The bond principal and interest are structured off cashflow streams, in one cash the refunded bonds, in the another 0 each year, or a stream of revenues that pay for the bond principal and interest. Always think general to specific, high to low.
There are several ways to select bonds to refund.
1) There are bond by bond, aka, maturity by maturity PV savings criteria that select bonds base on a static percentage savings number, for example, all bonds with more than 2% PV savings.
2) Option Value Pricing models.
Option value models like BDT above, are used to set up a framework / theory (or story) for making sense of and evaluating call options, just like the bond pricing formula is used to make sense of bond prices. Option models require a steep learning curve, and can sometimes mislead some folks into erroneously believing they forecast interest rates.
So, you have to make sure that whomever you're presenting this to understands that these models in no way ever try to predict interest rates.
What these models actually do is jiggle around a bunch of assumed interest rates under a bell curve (draw a curve).
Most of the time the interest rates being jiggled and assumed are "forward rates". That is to say, they are nothing more than break even rates based on today's interest rate scales (see forward rates section above). And because they're break even rates they're called non-arbitrage models (ways to profit without risk...see below for more).
They are bond by bond PV savings models that use current interest rates and a 'volatility' (similar to a standard deviation) to create a bell curve of probabilities with associated interest rates. Probabilities and rates that are used to compute the option value and threshold PV savings percentage for each maturity. It's still arbitrary but it can be argued to be more dynamic because it depends on today's current interest rates and not some invariant criteria that probably should have some wiggle room for different markets.
'Eligible Investment Securities.'
The definition of eligible investment securities as well as defeasance/escrow securities is found in the bond indenture.
Eligible investment securities are Treasuries, SLGS or other direct obligations of the federal government or high grade Agencies.
Escrow Securities (Defeasance Securities).
Escrow securities are Treasuries, SLGS or other direct obligations of the federal government.
Agency securities are generally not allowed.
SLGS(State and Local Government Securities).
A special government program proving securities tailored to an issuer's maturity requirements.
SLGS rates come from the prior days treasury bond rates, which can sometimes lead to opportunities.
SLGS certificates mature within the first 13 months and pay interest at maturity.
SLGS notes mature after the first 13 months up to forty years and pay interest semiannually.
Interest is computed on an actual - actual basis.
Treasury Securities.
Agency Securities.