Tài chính doanh nghiệp - Topic 11A: Default - Adjusted bond return

Financial ModelingTopic #11A: Default-Adjusted Bond ReturnL. Gattis1Learning ObjectivesUse transition matrices to compute multi-period default probabilitiesUnderstand the difference between a bond’s YTM and default-adjusted expected returnCompute default-adjusted expected bond return given a transition matrix and recovery rate23YTM vs. Expected ReturnThe Yield-to-maturity is sometimes referred to as the “promised yield” because it is the discount that equates the market price to the present value of promised coupon and principal payments.The bonds expected return is the discount rate that equates market price to the present value of expected cash flows.To compute expected cash flows we need to know the probability of default and the payment in the event of default (Recovery).To complicate things further, default can happen immediately or through a gradual degradation of the issuer’s creditworthiness4Bond Rating TerminologyBond credit ratings are attempts to assess the probability that a company will default.Credit Eventoccurrence that suggests a default is likelyIncludes: rating change (also called a rating transition), failure to make bond payment, and declaration of bankruptcyRecovery ratefraction of par recovered in default56Historical Credit Rating Transition MatrixRating at beginning of periodRating at End of period The main diagonal shows probability of no transition Sum of rows is 100% (accounts for all possible changes)7Calculating 2-yr Default Probability of “B” bond using 1-yr Transition Matrix82- Period Transition MatrixMMult(tmatrix,tmatrix)Cumulative Prob. of default at t=292- Period Transition Matrix(With Prior Default Rating: E)MMult(tmatrix,tmatrix)Prob. of default at t=2(Conditional on survival in t=1)Prob. of default prior to t=210Expected Bond Returnswith coupons and recoveryCompute the expected cash flow at each coupon payment (t)Expected bond returns = IRR of expected cash flows Expected Cashflow at Time t = +(Pdef,t)(Par x Recovery) +(1-Pdef,t)(Promised Par and/or Coupon Pmt) + 0 -------------------------------------------------------------- Def. Adj. Exp CFDefaults@tSolvent@tPrior Default or maturity3 Possible States @tCumulative probability of survivalTo Compute Expected Return:11Expected Bond Returns1-.0388-.039=.922103.38=.0388*50+.922*110$11.56=3.9%x$50+ (1-3.9%)*$1012Matrix Computation of Payoffs=Mmult(rating, tmatrix)13Function to multiply a matrix n-timesOption Base 1Function matrixpower(matrix, n) If n = 1 Then matrixpower = matrix Else: matrixpower = Application.MMult(matrixpower(matrix, n - 1), matrix) End IfEnd Function1410-Yr Transition Matrix / Copy All=Matrixpower(tmatrix,10)15YTMIRR() and Bondval()Function ytmirr(cr, par, t, freq, mktprice)Dim cf() As Doublen = t * freq ' number of CFsReDim cf(n + 1, 1) 'vector of initial price + n-Cashflows'market pricecf(1, 1) = -mktprice'coupon paymentsFor i = 2 To n cf(i, 1) = cr * par / freqNext I'Final Par Payment and Couponcf(n + 1, 1) = par + cr * par / freqytmirr = Application.IRR(cf()) * freqEnd FunctionFunction bondval(cr, par, t, freq, r)Dim temp, i temp = 0 For i = 1 To (t * freq) temp = temp + (cr * par / freq) / (1 + r / freq) ^ i Next i temp = temp + par / (1 + r / freq) ^ (t * freq)bondval = tempEnd FunctionGeneral Model – Copy All=IF(C27>mat,0,IF(C27=mat,MMULT(rating_row,MMULT(matrixpower(Tmat,C27),payoff_2)),MMULT(rating_row,MMULT(matrixpower(Tmat,C27),Payoff_1))))Copy=>17Adjusting Returns for Semi-annual Coupon PaymentsIn the prior examples, the formulas calculated the returns for annual payment bondsTo adjust for semi-annual payments, divide the coupon rate by 2 and multiply the term by 2.You must also adjust the transition matrix to six-month periodsThe resulting default adjusted return will be a 6-month return. Multiply this number by 2 to report the bond equivalent yield (BEY)See Benninga TextbookRecovery by Industry, Seniority18Payment in Default: Par x Recovery Rate19More Recent Recovery RatesSprint 8¾ 2032, Ba, 22 year (Yahoo Finance as of 3/15/2010)2021Default-Adj Expected Bond ReturnLearning ObjectivesUse transition matrices to compute multi-period default probabilitiesUnderstand the difference between a bond’s YTM and default-adjusted expected returnCompute default-adjusted expected bond return given a transition matrix and recovery rate22