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 at Citgroup. This site along with the companion YouTube channel streetmath is an introduction to the FIXED INCOME MATH used in PUBLIC FINANCE for MUNICIPAL BONDS. It is a hands on EXCEL SPREADSHEET based TRAINING SITE. The topics covered are things people in the industry should know, and have probably seen... and forgotten.
Now, just a couple of stories to show why knowing your calculations matters. For example, I once saved a bond closing. and I did it by suggesting we view a DERIVATIVE that was backed by treasury bonds as ONE SINGLE SECURITY UNDER THE LAW instead of a bunch of individual treasuries. The derivatives provider gave us an inverted yield curve for the treasuries that blew an escrow test but was fine if viewed as one security. Sounds boring, but there's nothing more humiliating than making the newspapers for a delayed closing. Another story. In1998 when everyone realized that premium bonds past the call date were here to stay, we all faced the daunting fact that arbitrage yields had to be calculated by finding the combination of call dates that produced the lowest arbitrage yield - and the number of permutations was staggering. But inspiration struck, and instead of going the route of countless permutations I proposed using our standard run of the mill bond pricing formula to iteratively find the dates for each bond that produced the overall lowest yield. To do so, 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, then repeat and keep adjusting our cashflow stream and our recomputed arbitrage yields until we hit the PV target. This shows how skillful and powerful you can be at solving thorny problems if you understand how these calculations work.
One thing, this site is for teaching and training, and though I like my spreadsheets - using them yourselves for actual deals is at your own risk. However, you can hire me to run numbers for you - I've freelanced before. Also, please visit my YouTube channel streetmath. My contact info is Rodolfo Riverol / rodolforiverol@gmail.com / 917 498 4150.
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
A Fixed Rate Bond is a loan from one party, "the Buyer" of the bond aka the "Investor" (the lender), to another, "the Issuer" (the debtor, the borrower), where the Issuer promises to make fixed payments on fixed dates over time to the Buyer.
Municipal Bond Issuances are large borrowings by states, municipalities, local governments and authorities, from investment companies and bond companies. insurance companies, and others, and marketed by investment banks that act like brokers for the bonds but will buy (underwrite) the bonds if necessary, ie, if they can't sell them.
Large Investment banks like Citigroup and Bear Stearns have large Sales and Trading Desk filled with brokers that 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 list of all municipal bond issuances can be found on emma, a 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) and which includes:
The Purpose or Reason for the borrowing/issuance (ie, 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 bonds (without panicking current bondholders).
Credit Ratings from an independent rating agency like Moodys, S&P or Fitch (answering HOW SAFE IS THIS).
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, Escrow Fund, Capitalized Interest Fund, Debt Service Fund, 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, Cost of Issuance, Underwriters, etc from bond proceeds, equity contributions, etc.
Call Option payment dates known as Call Dates and prepayment penalties known as Call Premiums. Unlike mortgages for example, municipal bond cannot be prepaid at any time, only after the call date which for fixed rate bonds is typically ten years after issuance. Page 7 in our (OS example)
More details about Municipal Bond Debt issuances are described in a State's Constitution or an authority's Bond Indenture.
Which are agreements between investors (lenders) and state/municipal borrowers (debtors).
Information summarized in an Official Statement (OS), include Revenue Sources (ie Credit for bond borrowing), Flow of Funds, ABT, 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.
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General Obligation Bonds (GO) are 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.
Conduit Bonds are backed by loan repayments from loans made to smaller towns and municipalities that might not normally issue bonds on their own or have strong credit. For example State Revolving Funds SRFs like Mass Trust make loans to dozens of small towns.
Asset Backed Bonds borrow against a stream of payments (revenues), like mortgage payments, student loan payments, and Tobacco securitization payments, for example California Tobacco.
Private Activity Bonds borrow to finance private projects like sports stadiums.
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Principal and interest PAYMENTS
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 and parallel revenues.
If revenues bottleneck, revenue bond issues might even include CABs (zero coupon bonds that defer all interest until maturity) and convertible CABs (a hybrid of CABs and current interest bond).
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.
USES of Money.
Construction funds for building projects.
Are interest bearing accounts that pay construction draws 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).
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 recovers the DSRF deposit at the end of the bond issue and can use it to pay back the last year's Debt Service.
The DSRF amount is invested at up to the arbitrage yield.
A non refundable Surety Payment to an insurer equal to 2% to 3% of the DSRF can be substituted, but you never recover the cost.
Debt Service Fund (Revenue Bonds).
Is an account where deposits equal to1/6 od the next upcoming Bond Interest payment to bond holders + 1/12 next Bond Principal payment to bond holders are made every month from Revenues to pay the Debt Service on time.
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Refundings are a large part of the business. They are bond refinancings, akin to mortgage refi's, where money from new bonds (loans) is put into interest-bearing accounts called escrows that are 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, as is done with hew money bonds, the money is put into an escrow account to eventually pay off the old bonds.
New Refunding principal and interest payments are structured to produce either level or upfront savings versus 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 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.
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Interest.
Compound interest is when interest is paid on interest as well as on principal or alternatively, when interest keeps getting added back into the principal.
If an interest rate of i is compounded semi-annually then the interest after n 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 = M/(1 + i)n. This means the PV is the amount or money you need to deposit into an interest account with rate i to get M dollars at time n.
From this point on all of our interest rates will be annual rates compounded semiannually, and that's because municipal and treasury bonds pay semiannually.
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 obligations/payments/draws 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.
When you are given the PV or known the PV, and you are being asked to solve for the interest rate i then the given Present Value number is called the PV Target or Target PV.
Why is this 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 not creating a money machine 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.
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Daycount
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, and in municipal fiancé we've selected '30/360 daycount' to compute borrowing rates on bond issues and the earnings rates on the accounts they fund like their 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 days or '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 (lenders) give money to bond issuers (debtors) who in turn give bonds (IOUs) to investors.
Maturity Date - Final payment date for the bond.
Coupons - Are unchanging fixed semiannual payments that are in multiples of 1/8 f 1% (.125%) or of 1/10 of 1% (.10%). Although coupons are sometimes referred to as interest rates, coupons are NOT interest rates - a point of common confusion, yields are interest rates.
Par Amount (sometimes called Principal Amount) - Is the value off which the interest is computed using the coupon, and which is paid along with the Coupon on the Maturity date.
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 basically cancel the old bond payments and replace them by lower principal and interest payments from the new issue, any changes before the call date, even from an advanced refunding, is basically a reshuffling of the debt.
Call Price - Is the amount paid to bond holders when the bonds are finally and forever paid off on by the money in the refunding escrow on the Call Date.
The Price = lowest Present Value of the bonds cashflows over all possible Redemption Dates including Maturity.
The Bond Price to a given Redemption Date is, where c is the coupon, P is the par or call price, and y is the yield or interest rate.
= 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.
For example, if we have a 5% bond yielding 2% that matures on 8/1/2027 with delivery 8/19/2025 then the PRICE = 2.5/(1 + 2%/2)(#days(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.
= 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
= (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).
Using this last formula we get (100 - 5/.02)/(1 + .02/2)(8/1/2027 - 8/19/2025]/180) + (5/.02) * (1 + .02/2)((8/19/202 - 8/1/2025)/180) - ((8/19/2025 - 8/1/2025)/180) * c/2 = 105.708
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 1) the maturity date 2) the first call date and 3) any other call dates where the call price changes, and it's usually no more that 4 dates including maturity, thus only about 4 calculations.
The Price only depends on the Delivery Date, and never on the Dated Date.
All bonds, even brand new bonds from new issues, are priced as if they were already in existence in the secondary market. Regardless of when the actual first Coupon payment is made, the first possible Coupon payment in the formula is always taken to be the first possible payment date after Delivery, and always assumed to be a full payment. So the price formula does NOT depend on whether the actual first Coupon is short or long.
Bond Prices are truncated to three decimals.
If 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. .
Investopedia.com reference on bond pricing formula
Every bond has an Average Life defined as the 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.
Bonds with a zero coupon, eg, municipal Capital Appreciation Bonds (CABs), Treasury STRIPS, have maximum average life, and the average life = maturity.
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) are zero coupon bonds, ie, they are bonds that pay interest only at maturity, ie, they are sold at a discount.
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 yields than bonds, and should because they have higher average lives, ie, are longer loans, but their yields are also generally higher than their computed theoretical yields.
Because they're so expensive, CABs aren't used unless absolutely necessary.
CABs are used when issuers have steep revenue curves or revenue bottlenecks where regular current coupon bonds fail.
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.
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 proceed 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.
To go from coupon bonds to zero coupon prices and yields, we use algebra. 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
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.
Youtube video on Newton's method
Wikipedia.com reference on Newton's Method
Bond issuers insure their bonds because they think insuring them will results in lower interest cost, and banks will accommodate by supplying different interest rates for insured versus uninsured bonds analysis.
As to cost, insurers apply a Fee to every dollar they insure, principal and interest both, and interest computed to maturity, never to an earlier call date.
Insurance Cost = (Total Bond Principal + Interest) x (Insurance Fee).
You can cherry pick which bond maturaties to insure, you don't have to insure the whole bond issue.
Savings from Insured vs. Uninsured.
When the insured and uninsured coupons are the same, the Insurance Benefit or Savings = insured bond price - uninsured bond price - insurance cost.
The Arbitrage Yield's sole purpose is to limit your earnings. The Arbitrage Yield is a number created by the federal government which it uses to limit earnings on tax-exempt proceeds (money borrowed using tax-exempt bonds). Of course this means that the federal government wants a calculation that makes the Arbitrage Yield as LOW as possible. That's because the government wants states, municipalities and local governments and authorities to earn as little 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 the bonds between purposes there are cases where a different allocation, like a prorata allocation for even better refunding candidates, might save the issuer money. For example, say you had a bond issue that was comprised of a short refunding and a long new money, and 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.
Transferred Proceeds.
When bonds get refunded we have to deal with the earnings on any existing accounts, like escrows and DSRFs they might have funded. Because of the federal tax-exemption granted to bond holders, the federal government doesn't want those old accounts earning at the old interest rate if the old interest rate is higher that the new refunding bond borrowing rate (arbitrage yield), especially if the old refunded bonds are gone. To reduce earnings we say these accounts 'transfer' to the new issue, and the earnings hit they take depends on how they 'transfer'. The federal government's nightmare scenario is if an issuer issues a new 5% interest refunding issue and invest the money in a long term 5% interest bearing escrow, then a few years later refund this 5% issue with a 3% issue that it uses to buy a short escrow that peters out before the 5% escrow does. Net net, that leaves a 5% escrow and an issue with 3% interest cost for a gain of 2%. So, 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 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 = ({retired old principal}/{total outstanding old principal}) x refunded issue's untransferred escrow cashflows, securities, money, etc.
The ratio ({retired old principal}/{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 can transfer into that issue. That's why it exist, and it is a benefit to the issuer because the issuer wants to limit how much money gets its earnings restricted.
The Universal Cap on any agiven dates = {Value of Outstanding bonds as of the given date} - {Value of all accounts the bonds purchased which includes escrows, DSRFs etc., as of the given date} as of that 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 its a par issue the Universal Cap is the outstanding par in this case.
True Interest Cost (TIC).
The TIC is the PV Rate "measuring" an issue's average interest rate accounting for fixed cost.
The PV of the debt service at the TIC rate is equal to the TIC Target (which 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 low and investing high.
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 x (the number of years from delivery to call date rounded down to the nearest integer).
Choosing the call date with lowest PV at each yield computation iteration automatically leads to the lowest (and most restrictive) yield. This can be equivalently achieved by ADDING the bond price at assumed arb yield (instead of market yield) MINUS the price to maturity at the assumed arb yield TO 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)
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 and for us that means structuring a few a bond issues, so please look through some of the videos immediately below, as well as the two links below the vids.
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)
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 peruse 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.
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1) There are bond by bond, aka, maturity by maturity PV savings criteria that select bonds base on a constant unchanged percentage savings, for example, all bonds with more than 2% PV savings.
2) "Option Value" models to select candidates.
The learning curve for option value models is steep, and you have to make sure that issuers understand that these models in no way predict actual future interest rates. We all know interest rates are unpredictable, especially long term rates. What these models do is jiggle around a bunch of assumed interest rates under a bell curve. Most of the time the interest rates being jiggled are "forward rates", which are nothing more than break even rates based entirely on today's rates, and not having any arbitrage opportunities (ways to risklessly profit today from trading rates...below for more).
Also sometimes called "Refunding Efficiency" models, these models are bond by bond PV savings models which use the current interest rate curve and a volatility (to create a bell curve) to determine the threshold PV savings percentage for each maturity by computing each maturity's option value. It's still arbitrary but it can be argued that it's more dynamic because it depends on today's current interest rates and not some invariant criterion, that probably should have some wiggle that accommodates changing markets.
Forward Rates are interest rates produced by option value models, and which depend on today's market rates and model "volatility" (fatness of the bell curve) assumptions.
In the market, 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 essentially breakeven rates. They are arbitrage free, ie, they are structured or set up to make all investment paths starting today to any future date monetarily indifferent.
Forward rates can be used to breakup or decompose or factor interest rates into shorter lego like building block rates, like 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 rough very elementary conceptual example with no PVs.
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.
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 (were 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)
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.
Wikipedia reference on Fischer Black's option pricing model
A 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 BDT and other models can be used to price municipal SIFMA rate swaptions (options on LIBOR swaps) in that market.
BDT is a popular model because of its relative ease of use.
Wikipedia reference on Black Derman Toy
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 used only to reject refundings - not as an alternative to one.
The option value computed is a theoretical number computed using the municipal yield curve instead of the SIFMA/LIBOR swap curves used in the derivatives market, 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 model and refunding efficiency
Refunding Efficiency Spreadsheet
Changing a bond's coupon changes its option value, which also changes its option adjusted yield.
If bond's option adjusted yield is to remain unchanged after a coupon change then its price must also change, so it's yield must also change.
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 are Treasuries, SLGS or other direct obligations of the federal government.
Agencies are generally not allowed.
A special government program proving securities tailored to an issuer's maturity requirements.
SLGS rates are set off the prior days treasury bond rates.
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.
Agency Securities.
SIFMA/LIBOR Swap Basics.
Swaptions instead of swaps.
Basis Risk.
Tax Risk.
Regression Analysis and LIBOR swap ratios.
Pricing a Swap: Boot strapping a yield curve and cash flow models.