Tài chính doanh nghiệp - Topic 11B: Accrued interest, clean and dirty prices, yield curve models, and forward rates

Financial ModelingTopic #11b: Accrued Interest, Clean and Dirty Prices, Yield Curve Models, and Forward RatesL. Gattis1Learning ObjectivesCompute bond accrued interest and invoice priceConstruct yield curves using an empirical curve fitting and theoretical modelCompute forward interest rates given zero-coupon spot interest rates2Clean and Dirty PricesWhen you buy a bond between coupon payment dates, you must pay the seller the quoted price (a.k.a, clean price) plus accrued interestAccrued Interest: Coupon Payment x Accrual Factor Accrual Factor = Days since last pmt / Total days between pmtsDirty “Invoice” Price (Paid to Bond Seller)Quoted price plus accrued interestPresent Value of all future coupon payments and parE.g.: $100 Par, 10% Annual Coupon Bond immediately prior to maturityPV = $110, Quoted Price = $100, Dirty (Invoice) due seller is $110Clean Price = Dirty Price - Accrued InterestClean price is also the Quoted priceExcel’s price function compute the Clean PriceClean (Quoted) prices are less volatile because it not go up and down with coupon payments3Excel’s “Clean” and “Dirty” Prices4Bloomberg IBM Bond DES530/360 Day Count Convention360: Assume 360 Days/Year, 180 Days between coupon pmt dates30: Assume 30 days per month in counting days since last couponExcel’s Days360(date1, date2) function returns the number of days between two dates assuming 30/monthAccrual Method: IBM6IBM Bond YIELD and MDURATION7Term StructureThe Term Structure of Interest Rates shows securities yields (of similar credit risk) across maturities8Polynomial Curve Fitting(Yahoo Finance Data)9Select Maturity and YTM, Insert xy scatter (markers only), right click on series, add trend line – 3rd order PolynomialPolynomial Curve Fitting(Yahoo Finance Data)10Select last 2 columns, Insert xy scatter, right click on series, add trend line – 3rd order PolynomialPolynomial Curve Fitting(Yahoo Finance Data)11Regression Results – Use regression for exam and assignment12Regress YTMs on T, T^2 and T^3 to get a more precise equationUse regression results to get “Model Yield”: YTM = a + b*T+c*T^2+d*T^3Use Model Yield to compute Model Price (PRICE): Using settlement of 8/11/2006 and freq=2WarningThese models do not work well on very short term instrumentsSome yields from Yahoo have negative or no yieldsomit these observationsYTM is not the same as expected return (The expected returns may plot closer to line)Rating agencies are sometimes a bit late to change ratingsThe undervalued bond may soon be downgradedThe differences may be due to differences in recover rates1314Term Structure ModelsRegression (Empirical) ModelsRegress historical term structures on maturityPros: can estimate today’s term structure with high r2Cons: does not describe why the shape may existBond analysts tend to use other yield curve (a.k.a. interest rate) modelEquilibrium ModelsStart with assumption about economic variables (long term average interest rate, rate volatility, mean reversion) and develop a process for short term rates (i.e., rate changes are normally distributed and follow a random walk process) that then determines long term rates and the term structurePros: based in economic assumptionsCons: may not fit today’s term structuresExamples: Vasicek, Cox-Ingersoll-RossNo Arbitrage ModelsCurrent term structure is an input that is used to estimate volatility and other economic factorExamples: Ho-Lee, Black-Karasinki, Hull-White15The Vasicek model can create upward sloping, down ward sloping, and slightly humped yield curvesEquilibrium Model ExampleZero, Spot and Forward Interest RatesSpot rates are interest rates on instruments that start today and mature at a future date (t).Written r(0,t)Forward rates are interest rates on instruments that start at a future date (t1) and mature at a later future date (t2).Written r(t1,t2)Forward interest rates are used to “lock into” future investment or borrowing interest ratesYou can calculate implied (theoretical) forward rates by assuming there is no arbitrage when comparing forward and spot instrumentsZero rates are the yields on Zero-Coupon instruments. The Zero yield is the discount rate the equates the PV of bond payment to the market price16Spot vs. Forward Interest Rates2-Yr Spot Vs. 1-Yr Spot + 1-Yr Forward Alternative 2-Year $1 Investment Strategies: 1. 2-year Zero Payoff = $1x[1+r(0,2)]2 2. 1-year Zero + 1-year forward payoff = $1x[1+r(0,1)]x[1+r(1,2)] Arbitrage must ensure: [1+r(0,2)]2=[1+r(0,1)] [1+r(1,2)] r(0,1) 6.0%Zero Coupon Spot Rate = r(0,t) (return on bond issued today maturing at t) r(0,2) 6.5% r(0,3) 7.0%1230 r(1,2) ????% r(2,3) ????%Forward Rate = r(t1,t2)(return on bond issued at t1 maturing at t2)Implied Forward Interest RateArbitrage must ensure: $1[1+r(0,2)]2=$1[1+r(0,1)] [1+r(1,2)] And the implied forward rate starting in 1 year, maturing at end of Yr 2 r(1,2)=[1+r(0,2)]2/[1+r(0,1)] – 1 r(1,2)=(1.065^2)/(1.06)-1=7.002% r(0,1) 6.0%Zero Rate (return on bond issued today) r(0,2) 6.5% r(0,3) 7.0%1230 r(1,2) ????% r(2,3) ????%Forward Rate(return on bond issued at t)Forward Interest RatesIn general:1 year forwards =r(1,2)The price of n-year Zero = 1/(1+r)^n20Appendix – Just for Fun!Vasicek ModelModel (Formula) used to derive the entire yield curve based on the evolution of short-term interest ratesAssumes rate changes are normally distributed and follow a random walk process with mean reversion21The Vasicek ModelChanges in the short-rate (Δr), r, can be written as Δr = a(b-r)Δt + σ Δt.5ZWhere Δt = change in time r = spot rate at time t (current short-term, spot interest rate)b = the long-term mean short-term, spot ratea = the speed of mean reversionσ = standard deviation of rate changeZ is a random number22The Vasicek ModelUsing Vasicek’s model of the short rate, he shows that the price of a zero coupon bond (with $1 par) at maturity t is:And the spot (zero) rate is:Where:23The Vasicek model can create upward sloping, downward sloping, and slightly humped yield curves24Vasicek ModelCurrent Rate Long-run RateHighVolLow Vol25Vasicek FunctionFunction vasicek(a, b, r, v, t)Dim pt, bt, atbt = (1 - Exp(-a * t)) / aat = Exp(((bt - t) * (a ^ 2 * b - (v ^ 2) / 2) / (a ^ 2)) - ((v ^ 2 * bt ^ 2) / (4 * a)))pt = at * Exp(-bt * r)vasicek = ptEnd Function