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Chapter 2 Pricing BondsPrice of the bond is the sum of the present value of the cash flows received from holding the bondP = C/(1 + r) + C/(1 + r)2 + C/(1 + r)3 + … + (C + M)/(1 + r)nP = Σnt=1 C/(1 + r)t + M/(1 + r)nNote the subscript on rUse one r or different r’s for each payment?Proper to use multiple r’sPayments 10 years out have different risks than six months out, so different discount ratesCommon to use a single “r” as pricing convention Given a set of cash flows for a bond (C, M) The one interest rate is called the YTMTwo ways to write this:Sum of DCFs FORMULAP = C/(1 + r) + C/(1 + r)2 + C/(1 + r)3 + … + (C + M)/(1 + r)nP = Σnt=1 C/(1 + r)t + M/(1 + r)nPVA + Discounted Maturity FORMULAP = C{[1 – 1/(1 + r)n]/r} + M/(1 + r)nDerive PVA as difference in two perpetuatesDerive FVA as FV of PVAQuoting rules$1,000, 10 year, 6% coupon bond, 5% required return (a.k.a. YTM or discount rate)10 years NPER = 206% Coupon PMT = 0.06/2 x $1,000 = $30r = 5% RATE = 0.05/2 = 0.025Par = $1,000 FV = 1,000EXCEL PV Function 1,077.94What is we didn’t use the S-A convention:10 years NPER = 106% Coupon PMT = 0.06 x $1,000 = $60r = 5% RATE = 0.05Par = $1,000 FV = 1,000EXCEL PV Function 1,077.22 (WRONG)Definition of YTM It is the one discount rate that equates the price to the CFs:r = YTM/2P = C/(1 + r) + C/(1 + r)2 + C/(1 + r)3 + … + (C + M)/(1 + r)nSo YTM is a “pricing convention”It is kind-of-sort-of what you might earn from holding the bond.So sometimes called the “Promised Yield”Later we will calculate total return from holding a bondSee how it differs from YTM and promised yieldYTM is useful for communicating and calculating priceAnd useful for ranking returns for similar bondsPricing a Zero Coupon BondZero Coupon means C = 0P = 0/(1 + r) + 0/(1 + r)2 + 0/(1 + r)3 + … + (0 + M)/(1 + r)nP = M/(1 + r)nStill use semi-annual quoting conventionExample: 10 yr Zero with 8% YTMr = 0.08/2 = 0.04n = 10 x 2 = 20P = $1,000/(1 + 0.04)20 = $456.39Wrong way:r = 0.08n = 10P = $1,000/(1 + 0.08)10 = $463.19 (Wrong)Reprice using EAR:EAR for 8% S-A = (1 + 0.08/2)2 – 1 = 8.16%P = $1,000/(1 + 0.0816)10 = $456.38 (Algebraically the same as above using S-A 0.04 and 20Price-Yield RelationshipYTM = r x 2 is in the denominatorGreater the YTM, the lower the priceSo downward slopingGo to Excel Price-Yield Graphs(Variable %)?(Fixed %)?(Fixed $)?(Variable $)YTM>Coupon RatePar>PriceYTM=Coupon RatePar=PriceYTM<Coupon RatePar<PricePrice over timePrice approaches par from above or belowGo to Excel Price Time GraphsWhy might the YTM change?Think of the YTM as base rate associate with Pure TVM for that bond and a premium for risk.The YTM will change if either The pure TVM changes – all rates change – systematicThis happens all the timeThe risk for that bond changes – idiosyncraticThis happens infrequentlyFloating Rate BondsCoupon is Reference Rate + Spread LIBOR + 5%But the YTM is also the Reference Rate + SpreadSo also LIBOR + 5%Issuer determines the market’s required spread (5%) Sets the coupon spread equal to spread to the reference rateAnd the floating rate bond is issued at parPricing a floating rate bond:Both the coupon and the YTM are the Base Rate + SpreadCoup Rate = LIBOR + Spread YTM = LIBOR + SpreadSince the base rate is in both the numerator and the denominatorboth change together so the coupon rate and YTM stay equalSo floating rate bond usually stay at or near parComplication: The spread for the coupon is in the bond contract – it can’t change!The spread in the denominator can changeCollateral Bond vs Floater vs Inverse FloaterGo to SpreadsheetAccrued InterestBuyer of a bond must pay seller for accrued interest All coupon interest that has been earned, but not paid, since last couponNot so for stocks. Why?Clean price – quoted price – no accrued interestDirty Price – Invoice price – includes accrued interestAccepted method for calculating accrued interest is different for different bonds 30/360 vs Actual/ActualWe will revisitFirst: How Dates work in ExcelPrice, YTM and Accrued Interest for REAL bonds Today is January 16, 2019Bond Matures on March 22, 2025Pays 5.125% couponPriced at 102.56% of parCalc Price, YTM and Accrued Interest using =PRICE() =YIELD()=ACCRINT()These function assume $100 face valueInputs:SettlementMaturityRateYld or PrRedemptionFrequencyBasisUnderstand difference between:=PRICE() vs =PV()=YIELD() vs =RATE() ................
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