Bond Mathematics & Valuation - Suite LLC
Derivatives EducationFreedom from the Black boxSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comBond Mathematics & ValuationBelow is some legalese on the use of this document. If you’d like to avoid a headache, it basicallyasks you to use some common sense. We have put some effort into this, and we want to keep thecredit, so don’t remove our name. You can use this for your own edification. If you’d like to give this toa friend for his or her individual use, go ahead. What you can not do is sell it, or make any money withit at all (so you can’t “give” it away to a room full of your “Friends” who have paid you to be there) ordistribute it to everybody you know or work with as a matter of course. This includes posting it to theweb (but if you’d like to mention in your personal blog how great it is and link to our website, we’d beflattered). If you’d like to use our materials in some other way, drop us a line.Terms of UseThese Materials are protected by copyright, trademark, patent, trade secret, and other proprietaryrights and laws. For example, Suite LLC (Suite) has a copyright in the selection, organization,arrangement and enhancement of the Materials.Provided you comply with these Terms of Use, Suite LLC (Suite) grants you a nonexclusive, non transferable license to view and print the Materials solely for your own personal non commercial use.You may share a document with another individual for that individual’s personal non commercial use.You may not commercially exploit the Materials, including without limitation, you may not createderivative works of the Materials, or reprint, copy, modify, translate, port, publish, post on the web,sublicense, assign, transfer, sell, or otherwise distribute the Materials without the prior written consentof Suite. You may not alter or remove any copyright notice or proprietary legend contained in or on theMaterials. Nothing contained herein shall be construed as granting you a license under any copyright,trademark, patent or other intellectual property right of Suite, except for the right of use licenseexpressly set forth herein.Bond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 1 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationFreedom from the Black boxSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comPrice Yield Relationship·Yield as a discount rate·Pricing the cash flows of the bond·Discount Factors based on Yield to Maturity·Reinvestment risk·Real World bond prices- Accrual conventions- Using Excel’s bond functions- Adjusting for weekends and holidaysBond Price Calculations·Price and Yield·Dirty Price and Clean PricePrice Sensitivities·Overview on measuring price sensitivity, parallel shift sensitivity, non parallel shift sensitivity, and individual market rate sensitivity·Calculating and using Modified Duration·Calculating and using Convexity·Individualized Market Rate SensitivitiesBond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 2 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comFreedom from the Black boxBond Mathematics & ValuationPrice Yield RelationshipYield as a Discount RateThe price of a bond is the present value of the bond’scash flows. The bond’s cash flows consist of couponspaid periodically and principal repaid at maturity.The present value of each cash flow is calculatedusing the yield to maturity (YTM) of the bond. Yield tomaturity is an internal rate of return (IRR). That is,yield to maturity is an interest rate that, when used tocalculate the present value of each cash flow in thebond, returns the price of the bond as the sum of thepresent values of the bond’s cash flows.We can picture the price yield relationship asfollows:100%7%7%7%7%7%95%PrincipalAll coupon and principal PV’s are calculated using the yield of the bond.CouponCouponCouponCouponCouponPVPVPVPVPVPVAll coupon and principal PV’s are calculated using the yield of the bond.PricePricing the Cash Flows of the BondSuppose the bond above has annual coupons of 7%and a final principal redemption of 100%. The principalis sometimes referred to as the face value of the bond.The market price of the bond—the PV of the fivecoupons and the face value—is 95% (95% of Par, butin practice no one will include the ‘%’ when quoting aprice). This is a given. Market prices are the startingpoint.We can picture the bond’s cash flows as follows:The coupons are cash flows—not interest rates. Theyare stated as 7% of the principal amount. The % onlymeans a cash flow of 7 per 100 of principal. The sameis true of the price, which is stated as a per cent of theprincipal.We do not yet know the yield to maturity of thisbond. Remember that we defined yield to maturity asthe IRR of the bond. We have to calculate the yield tomaturity as if we were calculating the bond’s IRR.IRR stipulates the following relationship betweenprice and yield. The yield to maturity is the interest rateof the bond. There is only one interest rate (I%) whichreturns 95% as the sum of the PV’s of all the cashflows.95 % 7% 7% 7% 7% 7% 100 %(1 I%)1 (1 I% )2 (1 I%)3 (1 I%)4 (1 I%)5 (1 I%)5Notice how we calculate the PV of each coupon one byone. It is as if we are investing cash for longer andlonger periods and earning the yield (the IRR) on eachinvestment.The future value of our investment each period iscalculated by adding the yield to 1 and thencompounding it to the number of periods.For Year 1 our imaginary investment looks like this:Bond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 3 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comFreedom from the Black box95% PV (1 I% ) 1 7% PV of 1st coupon invested at I% for 1 yearThis is the same as saying that we can invest anamount of money today earning a rate of I% for oneyear. When we get back our invested cash and theinterest it has earned for the year, the total will beworth 7%.For Year 2 our imaginary investment looks like this:PV (1 I% ) 2 7%PV of 2nd coupon invested at I% for 2 yearsAgain we assume we can invest an amount of moneytoday earning a rate of I% for two years. When we getback our invested cash and the interest it has earnedafter two years, the total will again be worth 7%.Simple algebra gives us the formula for PV given afuture cash flow and the number of periods:Coupon PVYear 1 (1 I%)1 (1 I%)7%(1 I%)4 (1 8.2609%)(1 I%)227%5(1 I%) 3(1 I%)100%(1 I%)5In this case I% turns out to be 8.2609%. This is theinterest rate which prices all the cash flows back to95%: 7%(1 8.2609%)2 7%(1 8.2609%)3107%(1 8.2609%)5Discount Factors Based on Yield toMaturityDividing 1 by 1 plus the yield raised to the power of thenumber of periods is how we calculated the annualdiscount factors above. These are discount factorsbased on the bond’s yield.DFYear 3 7%4 Calculators cannot solve for IRR directly. They find itby trying values over and over until the calculatedpresent value equals the given price. This method ofcalculating is called iterative. IRR is an iterative result.Using a financial calculator to calculate yield is easy.In this case we use a standard Hewlett Packardbusiness calculator:ValueKeyDisplay5 [N]5.000095 [CHS][PV] 95.00007 [PMT]7.0000100 [FV]100.0000[I%]8.2609%The IRR or yield to maturity of the above bond is8.2609%.7%7%(1 I%)7%DFYear 1 Extending this logic to the rest of the cash flows givesus the price yield formula we saw above.1(1 8.2609%)DFYear 2 Coupon PVYear 2 7%17%and95% 7%DFYear 4 DFYear 5 1(1 0.082609)11(1 0.082609)21(1 0.082609)31(1 0.082609)41(1 0.082609)5 0.923695 0.853212 0.788107 0.727970 0.672422There is no real life explanation for this. It is simplyhow IRR works. There is no promise that we can earna rate of interest in the market for one year or twoyears or three years, etc., equal to the yield. In fact, it isentirely implausible—even impossible—that we couldearn the yield on cash placed in the market.Bond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 4 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comFreedom from the Black boxDespite this problem, we still use IRR to calculatebond yields. The key is to always start with a marketprice and use it to calculate the yield. Never go fromyield to price—unless you are absolutely certain thatyou are using the correct yield for that very bond.Reinvestment RiskIn fact, the IRR problem is even more interesting. Inorder to earn the stated yield on the bond, IRRassumes that the bond owner can reinvest thecoupons through maturity at a rate equal to the yield.This is never possible. As a result, no investor has everactually earned the stated yield on a bond paying himcoupons.The so called reinvestment assumption says thatwe must be able to reinvest all coupons receivedthrough the final maturity of the bond at a rate equal tothe yield:bring with any certainty, this is a mostly fruitlesscalculation.Only one kind of bond carries no reinvestment risk.This is a bond that does not pay any coupons, a so called zero coupon bond.If you hold a zero coupon bond through finalmaturity, you will earn the stated yield without any risk.The only cash flow you will receive from the bond is thefinal repayment of principal on the maturity date.Nothing to reinvest means no reinvestment %8.2043%8.8820%9.6158%141.2804%All coupon s re ce ived a re reinve ste d through maturity at a rateequal to the yield of the bond—8.260 9% in thi s exa mple.The IRR reinvestment a ssumption re quires the inve sto r ha ve141.2804% at maturity if he inve sts 95 % up front—in order toearn the sta te d yield to ma turity.If we can reinvest at the yield, the return for theentire five years is 8.2609%:æ 141.2804% öç 95%èø( 15 )- 1 8.2609%If we cannot reinvest at the yield, the return over theperiod does not equal the stated yield. This is the riskof reinvestment.It is possible to calculate the yield of a bond (its IRR)using a different reinvestment rate—if it makes senseto claim that we know what the actual reinvestmentrate will be. Since we do not know what the future willThe return on this zero coupon bond is 8.2609%:æ 100% öYield ç è 67.2422% ø( 15 )- 1 8.2609%Real World Bond PricesWhen we move into the real world of the market weencounter baggage and distortions to the abovecalculations in the form of accrual conventions,weekends and holidays. Incorporating these real worldissues into the price and yield of a bond is our nexttask.Accrual ConventionsAccrual of interest is the first topic when we talk aboutbonds. In fact, this is a question of how we count timemore than how we accrue interest.Interest accrues over periods of time, and there area lot of different ways to count time in use in financialmarkets. Counting time with government bondsbecame simpler in 1999, as all of Europe’s governmentbonds adopted an approach similar to that already inuse in France and the United States.Bond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 5 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comFreedom from the Black boxThe other major issue is the number of couponspayable each year. In the UK, the U.S. and in Italy,government bonds pay semi annual coupons. In mostother countries, coupons are paid annually. A summaryof the accrual conventions and coupon payments for aselection of government bond markets galSpainSwedenSwitzerlandUnited /AA/A30E/36030E/360A/ACoupon lSemi AnnualAnnualAnnualAnnual or Semi AnnualAnnualAnnualAnnualAnnualSemi AnnualIf this function is not available, run the Setupprogram to install the Analysis ToolPak. After youinstall the Analysis ToolPak, you must select andenable it in the Add In Manager.SyntaxStart dateEnd dateBasisBasis0 pean 30/360If any argument is non numeric,YEARFRAC returns the #VALUE! errorvalue.If start date or end date are not validserial date numbers, YEARFRAC returnsthe #NUM! error value.If basis 0 or if basis 4, YEARFRACreturns the #NUM! error value.Using Excel’s Bond FunctionsDay Count FunctionsYEARFRAC(start date, end date, basis)is a serial date number that represents thestart date.is a serial date number that represents theend date.is the type of day count basis to use.Day count basisUS (NASD) 30/360All arguments are truncated to integers.Excel offers several functions for calculating thenumber of days between any two dates according todifferent day count conventions. YEARFRAC returns afraction of a year. COUPDAYBS returns the number ofdays from the beginning of the current coupon periodto the settlement date. COUPDAYS returns thenumber of days in the current coupon period.COUPDAYSNC returns the number of days betweenthe settlement date and the next coupon date.COUPNCD returns the next coupon date. COUPPCDreturns the previous coupon date before the settlementdate. All of these functions require similar inputs asexplained for the YEARFRAC function immediatelyfollowing.YEARFRAC returns the year fraction representingthe number of whole days between start date andend date. Use YEARFRAC to identify the proportion ofa whole year's benefits or obligations to assign to aspecific VALUE("06/30/2006"),2) 01/2006"),3) 0.49589Adjusting for Weekends and HolidaysCoupons cannot be paid on weekends or holidays.Bonds normally do not adjust the size of the couponpaid to reflect this, and thus the investor simplyreceives the stated coupon one or two—or eventhree—days late. Contrast this to swaps, where theamount of coupon paid is usually adjusted to reflectwaiting days.Bond yield calculations also normally ignoreweekends and holidays, although it is perfectly easy tocalculate the yield considering the exact days eachBond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 6 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comFreedom from the Black boxcoupon will be paid. Such calculations are sometimescalled true yields.Bond Price CalculationsFVPVPrice and YieldWe can check the math of bonds using the followingU.S. Treasury bond:Issuer: U.S. TreasurySettlement:09 Jan 06Coupon:4.5%Issue Date:15 Nov 051st Interest:15 May 06Maturity:15 Nov 15Mkt. Price:101 1/64%Mkt. YTM:4.37133%Accrued Int.:0.6837%“Dirty” Price:101.6993%You can check all of these calculations using a typical;HP business calculator.What is the calculator actually doing? It iscalculating the price of each of the bond’s cash flowsusing the YTM as a discount rate.The market convention uses the yield to maturity asthe discount rate, and discounts each cash flow backover the number of periods as calculated using theaccrued interest day count convention. In the case ofTreasuries, this is the A/A s.a. convention, which treatseach year as composed of 2 equal periods. Days to theend of the current 6 month period are counted in termsof how many days there actually are. This number ofdays is divided by the number of actual days in the full6 month period.The number of days to the first coupon, for example,is 126:09 Jan 06 – 15 May 06:126 days15 Nov 05 – 15 May 06:181 daysExpressing this in periods:126 0.696133181The price of the first coupon (its present value) can becalculated in the following way:N0.696133I%YR4.37133 2 2.1857PMT04.5 2 2.25ï – 2.2164All the other cash flow present values are calculatedin the same manner. Adding them up gives us the priceof the bond:DatesA/A/ DaysPeriodsCash FlowCF PV15 Nov 059 Jan 065515 May 06126101.6993%0.6961332.2500%2.2164%15 Nov 061.6961332.2500%2.1690%15 May 072.6961332.2500%2.1226%15 Nov 073.6961332.2500%2.0772%15 May 084.6961332.2500%2.0328%15 Nov 085.6961332.2500%1.9893%15 May 096.6961332.2500%1.9467%15 Nov 097.6961332.2500%1.9051%15 May 108.6961332.2500%1.8643%15 Nov 109.6961332.2500%1.8245%15 May 1110.6961332.2500%1.7854%15 Nov 1111.6961332.2500%1.7473%15 May 1212.6961332.2500%1.7099%15 Nov 1213.6961332.2500%1.6733%15 May 1314.6961332.2500%1.6375%15 Nov 1315.6961332.2500%1.6025%15 May 1416.6961332.2500%1.5682%15 Nov 1417.6961332.2500%1.5347%15 May 1518.6961332.2500%1.5018%15 Nov 1519.696133102.2500%66.7909%Dirty Price and Clean PriceNotice that the price of the bond is 101.6993%, not101.0156%. The so called “dirty price,” i.e. the price ofthe bond including accrued interest, is the “true” priceof the bond.The dirty price is the sum of the present values ofthe cash flows in the bond.The price quoted in the market, the so called “clean”price, is in fact not the present value of anything. It isonly an accounting convention. The market price is thetrue present value less accrued interest according tothe market convention.The accrued interest from 15 November 2005 to 09January 2006, is the fractional period remainingBond Mathematics & ValuationCopyright 2006 All Rights ReservedSuite, LLCPage 7 of 13United States – 440 9th Avenue, 8th Floor – New York, NY 10001 - Tel: 212-404-4825Email: info@suitellc.com Website: www.suitellc.com
Derivatives EducationSuite LLC Derivatives EducationAnalytics, Trading Tools & Serviceshttp://www.suitellc.comFreedom from the Black boxthrough the next coupon date subtracted from 1 fullperiod, times the coupon amount:(1- 0.696133) 4.5% 2 0.6837%This is the same accrued interest figure we calculatedabove.Subtracting the accrued interest from the truepresent value gives us the price as quoted in themarket:101.6993%- 0.6837%101.0156%This is the market price we saw above.Bond Yields and the Influence of the Coupon SizeImagine the following yield curve made up of bondswith liquid market prices:DateCouponPriceYield19 Sep 0619 Sep 0719 Sep 0819 Sep 0919 Sep 1019 Sep %98.50%6.0150%6.5496%6.8802%7.5985%7.8744%We can strip out the discount factors from this marketusing the bootstrap methodology (outlined in detail inProduct Analysis: Interest Rate Product Structures)and calculate the par coupon yields for this curve.The discount factors are calculated using thefollowing relationship: (PVft ; discount factor, Cpn:coupon payment, P: present price)n -1PVPVfn FVP - Cpnn å PVf1 CpnnPar coupon yields are calculated using the followingrelationship:1 - PVfnnå PVftt 1Discount factors and par coupon yields are as follows:DatePVf19 Sep 0619 Sep 0719 Sep 0819 Sep 0919 Sep otice that the par coupon yields are not equal to theyields on the market bonds. In this yield curve all bondshave prices less than par and all bonds also haveyields
Bond Mathematics & Valuation Below is some legalese on the use of this document. If you’d like to avoid a headache, it basically asks you to use some common sense. We have put some effort into this, and we want to keep the credit, so don’t remove our nam e.
As bond order increases, bond length decreases, and bond energy increases H 2 bond order 1 A bond order of 1 corresponds to a single bond Bond order (number of-bonding e ) - (number of antibonding e ) 2 electrons/bond 38 MO Energy Diagram for He 2 Four electrons, so both and *
The bond order, determined by the Lewis structure, is the number of pairs of electrons in a bond. Bond length depends on bond order. As the bond order increases, the bond gets shorter and stronger. Bond length Bond energy C C C C C C 154 pm 134 pm 120 pm 835 kJ/mol 602 kJ/mol
Set up a new Bond - Connect the BOND to your home WIFI network a. Make sure you have a working wireless network b. Plug the Bond to an outlet c. The Light ring will be white while the BOND is in boot up mode d. BOND light ring will flicker green when the BOND is ready to be used BOND is launching - get notified! First Name Last Name Enter .
Part I: Basic Bond Topics. covers basic issues you are likely to encounter when representing your client in a bond case. Specifically, we will cover how to: 1. Locate your client and assess bond eligibility 2. Enter your representation and request a bond hearing 3. Present your bond case 4. Pay bond . Part II: Advanced Bond Issues.
Chapter Three Interest Rates and Security Valuation. . Bond Valuation The present value of a bond (V b) can be written as: . Relation between Interest Rates and Bond Values Interest Rate Bond Valu
Calculate the value of a bond and relate it to the yield to maturity on the bond. 3. Bond Valuation: Four Key Relationships Describe the four key bond valuation relationships. 4. Types of Bonds: Identify the major types of corporate bonds. 5. Determinants of Interest Rates Explain the effects of inflation on interest rates. 2 FIN3000, Liuren Wu
2 Business Valuation Update February 2018 Business Valuation Resources Market Multiple adjustMents: Get a Grip On Grp Business Valuation Update (ISSN 2472-3657, print; ISSN 2472-3665, online) is published monthly by Business Valuation Resources, LLC, 111 SW Columbia Street, Suite 750, Portland, OR 97201-5814.
Automated Valuation Models (AVMs) are computer-based systems which encompass all data concerning real estate in a particular area and are capable of producing more consistent valuation reports within a short time. Traditional valuation methods employed by valuers somewhat delay the valuation process. For
