Columbia University Global Capital Markets Course Notes 6

TopicsPrice –Yield RelationshipBootstrapping the Yield CurveManaging Interest Rate Risk with DurationDuration1Yields: Day CountConventionsInternational differences on yield conventions.Coupon Characteristics of Major Bond Markets (straight bonds)Usual Frequency of CouponDay Count (month/year)US semiannual30/360 or actual/365Canada semiannual actual/365Japan semiannual actual/365Australia semiannual actual/365i l t l/365UK semiannual actual/365Switzerland semiannual30/360Germany annual30/360Netherlands annual30/360France annual actual/365Eurobonds annual30/3602 FRN’s quarter or semiannual actual/360Yields: ComparingCoupon BondsBond Investment Risk Example: Assuming equal credit quality,which bond should an investor,with a 5year quality, which bond should an investor, with a 5 year horizon, select among the following 4 bonds:B dC M t it tBond Coupon Maturity ytm 3years Reinvestment Risk A 5% 3 years 9.0%B 6%20 years 8.6%Reinvestment Risk Reinvestment Risk Price RiskC 11%15 years 9.2%D 8% 5 years 8.0%Reinvestment Risk Price RiskReinvestment Risk3y Reinvestment RiskYieldTo compare bonds based on prices isimpossible=>One solution is to compute yields (ytm)Yields provide a useful benchmark forcomparing bondsThe yield is the rate of return on a bond only if(1)You hold the bond to maturity(2)You re-invest the coupons at the yield=>Is this realistic?[No reinvestment assumptions are required forspot or ZCB rates]spot or ZCB rates]4YieldsExample: US Treasury note, maturity in 6.5 years, face value = 100years face value=100Price = 101.625pCoupon=6.25%What is the yield?What is the yield?5P i d Yi ld Prices and YieldsIn Excel, there are several ways to Goal Seek Method:IRR formula:y = 2.97%y = 2.97%ytm = 5.94%ytm = 5.94%several ways to compute the YTM: Use Goal Seek byy ytime CF DF CF*DF time CFs 0100-101.6251 3.1250.971133 3.0347921 3.12531250943129471873125Use Goal Seek by setting up the full set of cashflows and discounting or 2 3.1250.9431 2.9471872 3.1253 3.1250.915876 2.8621123 3.1254 3.1250.889438 2.7794924 3.1255 3.1250.863762 2.6992585 3.1256 3.1250.838829 2.6213396 3.125312508146142545673125and discounting, or Use the IRRfunction7 3.1250.814614 2.545677 3.1258 3.1250.791099 2.4721858 3.1259 3.1250.768263 2.4008219 3.12510 3.1250.746086 2.33151810 3.12511 3.1250.724549 2.26421411 3.1253125070363321988543125 12 3.1250.703633 2.19885412 3.125 13103.1250.68332270.4675613103.125Price =101.6257P i d Yi ldPrices and Yields Market convention:ytm = 2 x y y yytm = 2 x 2.97% = 5.94%(ytm = bond-equivalent yield or yield to maturity)Effective Annual Yield (EAY) = (1+BEY*0.5)^2-1 Bonds less than 6 months or 1 year are quoted on a “di t b i ”“discount basis”Note that the price of the bond is above par (it is aNote that the price of the bond is above par (it is a premium bond), so its yield is smaller than thecoupon rate. 8p(Another)Numerical ExampleN i l E lZero coupon bond prices (for $100 face value) for the next15years are given by:the next 1.5 years are given by:6 months97.37501 year94.37501year9437501.5 years91.6875Determine spot rates (zero coupon rates), and price a coupon bond with 5.5% coupons (semi-i b d i h55%(i annual payments) and 18 month maturity. Find the yield of the bond.h i ld f h b d9Bootstrapping the Yield CurveWe assume that there are spot rates/ ZCB yields In practice 3 and 6-month zeroes are available I ti3d6th il bl1-year zeroes used to be availableNo zeroes after 1 year.How do we construct zeros/spot rates beyond six months?12Bootstrapping: Which bondsto include?Newly issued (on-the-run) bonds of regular maturities are used:3months,6months,1,2,3,5,10,and30. are used: 3 months, 6 months, 1, 2, 3, 5, 10, and 30.Can use other bonds in between these maturitiesp pCould this be a problem? How deep are the markets?In practice, each bank uses a different method to bootstrap and will often use some interpolation methods.“Smoothed” or interpolated yieldsProprietary models.P i t d l16P i Yi ld R l ti Price –Yield Relation PYInterest Rate RiskI t t R t Ri kBonds have different characteristics (coupon, maturity, par value). How do we (t it l)H dquantify the interest rate risk for different bonds so that we can compare them?Is there a way to quantify the price/yield Is there a way to quantify the price/yield relationship?=> Duration and/ or PVBP (DV01)19PVBP/ DVO1Present Value of Basis Point of a bond is Present Value of Basis Point of a bond is the change in its price for a 1 basis point (01%)change in its yield(.01%) change in its yield.How does one calculate this?-By changing the yield of the bond by one B h i h i ld f h b d b basis point, recalculating the price, and taking the difference. To convert the units to “per million par”, you multiply by 10000 Duration is a similar concept –related both conceptually and mathematically conceptually and mathematicallyWhat is Duration?Discounted Weighted Average Maturity “Discounted Weighted Average Maturity”–a summary measure of when the bond is going to epais going to repay.y pA measure of sensitivity of the bond price to yield.We know that bonds that are likely to pay We know that bonds that are likely to pay later are more sensitive to yields –How? Polynomial …P l i lD ti M ti tiDuration: MotivationA bond portfolio manager is bullish on intermediate bonds.She would like to add bonds with a maturity of about 5 yearsy y to her portfolio.Unfortunately, there do not seem to be liquid issues in that maturity segment of the market. Can she construct af h k hportfolio of the benchmark 10 year bond and 6 monthTreasury Bills to achieve approximately the same interest rate Treasury Bills to achieve approximately the same interest rate risk?A bank has primarily dollar loans outstanding and funds itselfusing primarily short-term dollar deposits. The maturity of its loans varies between 3 months and 5 years. Can the bank quantify its interest rate exposure?quantify its interest rate exposure?22Interest Rate RiskI t t R t Ri kWhat is the percentage change in bondi D h th i ld h b1%? prices, D, when the yield changes by 1%?We know it is negative: as interest ratesincrease, bond prices fallExample:If the yield increases by1% Example: If the yield increases by 1%, the bond price falls by 2%=>Duration measures the price risk ofD ti th i i k f the bond to a change in interest rates23Interest Rate RiskI t t R t Ri kSimplest Possible Example (ZCB’s)Two zero-coupon bonds with maturities of5and10years Two zero-coupon bonds with maturities of 5 and 10 years ZCB rate rises from 10% to 11% (we assume the term structure is flat)structure is flat)10%11%ΔδΔδ/δΝΝΝ5 years61.3958.54-2.85-4.6% 10 years37.6934.27-3.42-9.1% 10years37693427-342-91%24Interest Rate RiskWhich bond has more interest rate risk?Long-term BondWhat about the relative size of the risk?Proportional to Maturity25Instantaneous Percentage Price Changes for the6 Instantaneous Percentage Price Changes for the 6 Hypothetical Bonds(initially priced at 9%)Percentage Price ChangeCoupon/Maturity in YearsNew Change inBasis9%/59%/256%/56%/250%/50%/25Yield (%)Points7.00-2008.3223.468.7525.4610.0961.738.00-100 4.0610.74 4.2611.60 4.9127.108.50-50 2.00 5.15 2.11 5.55 2.4212.728.90-100.40 1.000.42 1.070.48 2.4289910040100040110050248.99-10.040.100.040.110.050.249.011-0.04-0.10-0.04-0.11-0.05-0.249.1010-0.39-0.98-0.41-1.05-0.48-2.369.5050-1.95-4.75-2.05-5.09-2.36-11.2610.00100-3.86-9.13-4.06-9.76-4.66-21.2311.00200-7.54-16.93-7.91-18.03-9.08-37.8928Numerical Example:ObservationsObservationsThe percentage price changes are symmetric for smallThe percentage price changes are symmetric for small changes in yield, but asymmetric for large changes in yield.g p y,gFor a given coupon rate and initial yield, the longer the term to maturity, the greater the price volatilityPercentage price changes are larger for low coupon bonds and hence low coupon bonds behave as if they have longer and hence low coupon bonds behave as if they have longermaturities.=>Let’s now develop a formal measure that takes all these different affects into account!29Duration: IntuitionDuration quantifies the sensitivity of a bond to an underlying interest rate, typically the bond’s yield.A bond with a high duration (in years) has a lot of risk because ittakes a long time to get your original investment back.A three year coupon bond with a duration of 2 years behaves as a 2year zero coupon bond (in terms of sensitivity to interest ratechanges)MMacaulay DurationMacaulay DurationC1C2C3C4C5C631Numerical Example DurationDuration measures for the 6 bonds:Bond MacaulayDuration Modified Duration4133969%/5-year 4.13 3.969%/25-year10.339.886%/366%/5-year 4.35 4.166%/25-year11.1010.620%/5-year 5.00 4.780%/25-year25.0023.9236。