Constructing Spot and Forward Yield Curves: Essential Concepts and Market Conventions

1. We Open DoorsSM Building Spot and Forward Curves(prepared for BMGT 798, University of Maryland) Alyce Campbell, Ph.D. February 14, 2001

2. Class Objectives • Review class objectives • Review definitions of yield • Review some market conventions • Review basic bootstrap method for spot yield curve construction • Review theories of credit risk term structure Freddie Mac

3. Class Objectives • Objective is to understand some basic terminology from the fixed income markets and understand different approaches to constructing continuous curves from discontinuous prices Freddie Mac

4. Spot and Forward Curve Modeling • Spot and forward yield curves are used to compute discount functions, so it is extremely important to understand how to construct and interpret constructed curves • Standard methodology is to take a series of prices of liquid non-callable fixed rate instruments (Treasury bonds, corporate bonds, interest rate swaps) of different maturities and use these to “bootstrap” the rest of the curve • Typically only a very few data points are available, so spot curve construction requires selecting interpolation and/or smoothing methodologies. • When applied, the curve used must match the risk characteristics of the cashflows Freddie Mac

5. Spot and Forward Curve Modeling http://www.bondbasics.com/slidshow/parcylds.gif http://www.bloomberg.com/wealth/tools/C13.html Freddie Mac

6. Spot and Forward Curve Modeling http://www.bondsonline.com/docs/treas-yieldcurveset.html http://www.bes.co.id/idrcurve.htm http://www.bondbasics.com/slidshow/parcylds.gif http://www.bloomberg.com/wealth/tools/C13.html http://www.ots.treas.gov/ Freddie Mac

7. Concepts of Yield • Spot Yield to Maturity • constant rate at which cashflows of a long term interest bearing security must be discounted to obtain the instrument’s present value. • equivalently, rate of return an investor will receive if a long term interest bearing security is held to maturity and the interest payments are reinvested at this rate • Spot Yields on Zero Coupon Bonds • yield on an instrument that pays no coupon until maturity • implies the discount curve • Credit Yields • rates for loans on products with credit risk • approximately: credit yield = riskfree rate + credit spread • tied to default rates and credit ratings Freddie Mac

8. Concepts of Yield • Forward Rates • rates at which loans can be made at some future date, as implied by the term structure • calculated through mechanical calculation using relationship: (1+R(t+k))(t+k) = (1+Rt)t (1+tFk))k • R = spot rate; F = forward rate • Expected Future Spot Rates • unknown but expected rates (in probabilistic sense) at which loans will be made at some future date Freddie Mac

9. Some Market Conventions • Pricing Conventions • Day-count bases for calculating accrued interest • Settlement rules • Non-business day rules • Compounding conventions • References: • Stignum and Robinson (Money Market and Bond Calculations) • ISDA Documentation (www.isda.org) • BMA Documentation (www.bondmarkets.com) Freddie Mac

10. Some Market Conventions--Pricing Example: bond prices, mortgage bonds Remember that bond prices move inversely to interest rates. As bond prices increase, interest rates and points decrease. As bond prices decrease, interest rates and points rise. When the market is "rising", bond prices are rising and interest rates and points are falling (a rising market is desirable). When the market is falling", bond prices are falling, and interest rates and points are rising (a falling market is bad). Freddie Mac

11. Some Market Conventions-Pricing Bonds trade in "ticks" or increments of 1/32 of a point. This conversion table may help: Freddie Mac

12. Some Market Conventions-Pricing Freddie Mac

13. Some Market Conventions- Pricing Freddie Mac

14. Some Market Conventions- Pricing Freddie Mac

15. Some Market Conventions- Pricing Freddie Mac

16. Some Market Conventions • Daycount Bases and Fractions • daycount fraction = number of interest accrual daysnumber of days in coupon period • conventions based on actual days or 365 days • conventions based on 30 day months and 360 day years 360(Y2-Y1)+30(M2-M1)+(D2-D1) • can result in accrual amount not equal to coupon payment amount • used in calculation of “dirty price” (vs “clean” or “flat” price) Freddie Mac

17. Some Market Conventions • ACT/ACT • actual days held/actual days; denominator will vary from period to period (“6 months” varies between 181 and 184) • ACT/365 • denominator is always 365 even if year is a leap year • ACT/365(J) • leap year is ignored in both numerator and denominator • ACT/365(ISDA) • adjusts for leap year by using average • 30E/360 • if D1 and/or D2 =31, set D1 and/or D2 = 30 Freddie Mac

18. Some Market Conventions • 30/360 (ISDA) • if D1=31, set D1=30, else use actual and • if D2=31 and D2=30, set D2 = 30, else use actual • 30/30(PSA) • if D1=31 or D1=last day of Feb, set D1=30 else use actual • if D2 =31 and D1 = 30 set D2 = 30, else use actual • 30/30 (SIA) • if D1 = 31 or D1=last of Feb, and bond pays on last of Feb, set D1=30; else use actual • if D2=31 and D1=30, set D1=30; else use actual Freddie Mac

19. Some Market Conventions • Settlement rules • rules that define how many until payment and delivery of security • varies by instrument • T+3 • trade settles three days from trade date • http://www.jpmorgan.com/MarketDataInd/GovernBondIndex/GovernBondIndex.html • shows conventions for bonds in different markets Freddie Mac

20. Some Market Conventions • Non-business day rules • rules that define which days are good business days given holidays and weekends • No date adjustment • cycle dates are not adjusted for weekends or holidays and are forced to land within a cycle month • Next good business day • dates adjusted for weekends and holidays to the next good business day • Previous good business day • dates adjusted for weekends and holidays to the previous good business day Freddie Mac

21. Some Market Conventions • Modified following business day • dates adjusted to the next good business day unless that day falls in the next calendar month in which case the date is adjusted to the previous good business day • End of month - no adjustment • dates adjusted to land on last day of the month • End of month - next good business day • dates adjusted to the last day of the month but if that day is a weekend or holiday, then it is adjusted forward to the next good business day Freddie Mac

22. Some Market Conventions • End of month - previous good business day • dates adjusted to the last day of the month but if that day is a weekend or holiday, then it is adjusted to the previous good business day • Two business days prior to third Wednesday of month • dates are two business days prior to the third Wednesday of the month (used for Eurodollar futures) • Deposit rollover method • each date set so it occurs on the same day of the month as the previous date. Each date is set to the next good business day but no dates may be adjusted past the last good business day of the month Freddie Mac

23. Some Market Conventions • Compounding • annual; semi-annual; daily; continuous • Converting from one basis and compound frequency to another • need to know quote convention and basis • find future value and then solve for missing value Freddie Mac

24. Some Market Conventions • Calculating accrued interest • dirty and clean prices • short, long, normal coupon periods • dated dates • accrued interest vs coupon payments • calculating AI on US Treasuries • semi-annual; ACT/ACT • use correct settlement date rules Freddie Mac

25. Term Structure: Curve Construction • Standard methodology is to take a set of points and “bootstrap” up from knowns to unknowns • All methods use the bond equation in one form or another. • Bond Equation: P(0) + AI(0) = [(C/f)/(1+Z(t))^ft] + [(1+Z(T))^T] • whereP(0) = price at time 0, AI(0) = accrued interest at time 0,C = quoted coupon rate, f = compounding frequency, Z(t) = zero rate function,R = redemption value of the bond (normally par, but not always; for example in an amortizing note, R may depend on time of redemption); T = terminal date (maturity date for bonds, swaps) • Note that we can also express this in a form in which the Z(t) function is replaced by the discount function: • P(0) + AI(0) = [(C/f)(t)]+ [R(T)* (T)] • Goal is to find the “best-fit” function for either Z(t) or (t). Freddie Mac

26. Term Structure: Curve Construction • Interpolation Methodologies • Explicit assumptions about functional form: (eg. linear interpolation on rates or cubic splining) • Explicit assumptions about properties (eg. constant forwards) • Statistical fitting (eg.cubic or quadratic regression splines) • Multifunction methods (differently shapes parts of curve • Can have significant impact on the shape and implied NPVs • Some “Best-fit” criteria • How well bonds not in sample are priced • Smoothness of the forward or spot curves • Absence of negative forwards/fits “no arbitrage” condition • Flexibility of functional form (can it model a variety of shapes) Freddie Mac

27. Term Structure: Curve Construction • Methodology • We cannot observe discount functions directly—in fact we typically use the prices of coupon-bearing instruments and infer these. We do the inference by making assumptions about the relationship between the pricing points and use these assumptions to fill in the blanks (curve-fitting techniques, interpolation techniques, extrapolation techniques Key issues in data selection are: 1) choose price points for liquid instruments (meaning the instrument trades frequently with very narrow bid-ask spread 2) choose instruments of all the same risk class and same structure; for example, constructing a Z(t) curve that mixed callable and non-callable debt would lead to erroneous results, because the callable pricing points have embedded in them the implied value of the option. 2) choose points that reflect the type of instruments that are going to be priced using the inferred functions Freddie Mac

28. Term Structure: Curve Construction • Methodology: Linear in the Par Rates • For coupon-bearing instruments the most basic method of curve construction is to take the rates on par instruments with no accrued interest and fit through the yields. That is we take the bond equation for several bonds which are identical in all respects except they have different maturities, and solve for missing unknowns. In this methodology we assume 1) the Z(t) function is the same for all bonds 2) the functional form between two known points in linear in the par rates • This methodology means that we do not have a “smooth” function; that is there are points where the first derivative is undefined. Also this does not impose any constraints on the implied forward rates, so these may be negative (which is a “no-no”). This methodology also does not fully capture the additional assumption of a smoothly increasing risk premium (if this is the model of markets we believe to be most accurate). • Example: Suppose 1 yr par swap yield = 5%, 3yr = 5% and 10 yr = 8% • Under this method we find the piece-wise functional form for par rates to be: • Y(t) = ½(t) + 9/2 for 1 <t<3Y(t) = 2/7(t) + 36/7 for 3<t<10 • This forces the 2 yr point to be 5.5. • Once we have the interpolated points for swaps we don’t observe we can calculate the yield for swaps on a regular grid of say, semi-annual points, and then use the bond equations to solve for discount functions Freddie Mac

29. Term Structure: Curve Construction • Methodology: Bootstrapping Assuming Constant Forwards • The set of yields also implies a set of forward rates. The assumption is based on a no-arbitrage model in that we assume that the long rate is the geometric average of a series of short rates. That is: • [1+Z(2)]^2 = [1+Z(0)][1+F(1)] • {1+Z(3)]^3 ={1+Z(0)]{1+F(1)]{1+F(2)] = [1+Z(2)]^2[1+F(2)] • and so on, where F(k) = the forward rate for a loan starting at time k-1 and ending at time k • NOTE: Forward rates are an estimate of the unknown future spot rate. In general, these are biased estimates, if there is a liquidity premium built into rates. • From this definition, • F(k) = [(k-1)-(k)]/(k) • In bootstrapping, we can impose the constraint that the implied forwards are constant between the two known points. • In the example above, we find the unknown 2 yr rate by solving the following equation: • [(1.06)^3]/[(1+R(2)]^2 =[1+R(2)]^2/(1.03) • This will result in a slightly different yield than in the case of linear through the par rates. This will also translate into a slightly different discount function. Freddie Mac

30. Term Structure: Curve Construction • Methodology: Cubic Splining • The problems with piece-wise linear functional forms are that they are not very flexible so cannot accommodate very well the complex shapes we observe in markets. Also negative forwards are not excluded. • This leads to fitting higher-order functional forms. The simplest functional form that has smoothness is the cubic function and in cubic splining we fit cubic functions between points, plus constrain the derivatives at the “joins” to be continuous. General set of equations is as follows: • R(j+k) = A R(j) + B R(j+1) + CR(j)’’ + D R(j +1)’’ • where • R (j) and R (j+1) are the rates at time point t(j) and t(j+1), and the notation (‘’) indicates second order derivatives. • That is we assume that the equation for the unknown point is a weighted average of the two known points and the second order derivatives at those known points. Freddie Mac

31. Term Structure: Curve Construction • Methodology: Cubic Splining • The values for A, B, C, D are: • A = [t(j+1) -t(j+k)]/[t(j+1) -t(j)] • B = [t(j +k) -t(j)]/[t(j+1) -t(j)] • C = 1/6[A^3-A](R(j+1)-R(j)]^2 • D = 1/6[B^3-B](R(j+1)-R(j)]^2 • Since the derivatives themselves are unknown, we solve for these over the interval [t(j+1), t(j)] and [t(j-1), t(j)] setting them equal at t(j). • For N points we have N-2 equations in N unknowns, so we add boundary conditions. • For example, we can set R(1)’’ and R(N)’’ = 0 (so-called natural spline) • or set specific values for R(1)’’ and R(N)’’. • Reference: Numerical Recipes in C Freddie Mac

32. Term Structure: Curve Construction • Methodology: Regression Analysis • In this approach, a regression is run against a set of actual yields or the discount points to find the best-fit, then this is used for out-of-sample prediction • Some functional forms that have been proposed are: • (1) polynomial functional forms: Z(t) = a+bt+ct2 + dt3 +... • (2) exponential polynomials: (t) = exp[-(a+bt+ bt+ct2 + dt3 …)] • (3) exponential on the forwards: F(t) = ajexp(-kjt) • The value of this approach is that the data shapes the curve. Freddie Mac

33. Term Structure: Curve Construction • Testing to Achieve A Goal • Need to define criteria • Flexibility--can it handle a variety of forms • Smoothness • What happens to outliers (predicted, overused?) • Predicts actual prices • Some measures • Descriptive statistics for comparing predicted to actual yields include Mean Absolute Error, Root Mean Squared Error Freddie Mac

34. Term Structure • Plot of yields against time, par swap curves; variety of shapes observed Freddie Mac

35. Discount Functions DF = 1/(1+Zt)t, where Zt = zero rate to t Freddie Mac

36. Implied Forwards Calculated from (1+Zt )t/ (1+Z(t-n) )(t-n) Freddie Mac

37. Term Structure • Plot of yields against time, par swap curves Freddie Mac

38. Implied Forwards Calculated from (1+Zt )t/ (1+Z(t-n) )(t-n) Freddie Mac

39. Term Structure of Credit Spreads • Credit Spread is the difference in yield between risky bonds and riskless bonds of the same maturity • Term structure of risky yields is constructed using the same methodology as described previously • Typically bonds of the same risk class as defined by S&P or Moody’s are used • Classes are AAA down to D; based on measures which indicate probability of default • Problems in estimation are (1) bonds are often not very liquid (2) most corporates have embedded options (3) risk classification is imperfect Freddie Mac

40. Term Structure of Credit Spreads • Credit spread is a function of the probability of default (which is also related to probability of change in grade), so credit spread increases as grade goes down • Two basic theories • Risk premium, like liquidity premium, is a function of the increasing uncertainty about the future (in this case, probability of default) so credit spreads will increase with increasing maturity; applies to low risk bonds • Risk premium tied to size of cashflow, so is greatest near the time that principal needs to be rolled over, so decreases all else equal with time to maturity (“crisis at maturity” model) • Currently modeling of credit spreads and default probabilities under very active investigation, because credit derivatives and risky ABS are increasing in volume Freddie Mac

41. Option-Adjusted Spreads • Option-adjusted spreads are calculated for bonds, particularly ABS with a lot of optionality • No theory • Generated by solving for the OAS using a Monte Carlo approach • OAS values used for pricing spread products • Currently modeling of OAS values and prepayment risk are extremely active areas of applied research Freddie Mac

42. Equilibrium Modeling Steps • Define characteristics of return distributions • Define behaviors of investors • Define trading environment (particularly transactions costs) • Determine what happens after investors have completed all rounds of trading and will not trade even if markets are open; in this event all information has been disclosed and demands are satisfied Freddie Mac

43. Theories of Term Structure • Expectations Hypothesis • Investors form expectations of future unknown interest rates and embed them in interest rates through arbitrage activities; assumes accurate forecasting, no transactions costs • Underlies most forecasting/pricing models • Model assumes that expected future spot rates are equal to the forward rates so that the expected yields are determined by the one-period rates • (1+Rt)t = (1+R1)1 (1+1F1)1 (1+2F1)1(1+3F1)1….(1+t-1F1)1 • expected one-period return is the same regardless of bond maturity • expected n-period return is the same for all investment alternatives • yield curve can have any functional form Freddie Mac

44. Theories of Term Structure • Liquidity Premium Hypothesis • Same trading environment and no arbitrage conditions as in the pure expectations model, but assumes that investors require a liquidity risk premium for lending long • Model assumes that forward rates equal expected spot rates plus a liquidity premium • (1+Rt)t = (1+R1)1 (1+1F1)+1L1)1 (1+2F1+2L1)1(1+3F1 2L1)1….(1+t-1F1 +t-1L1 )1 • 1L1 < 2L1 < 3L1 …< t-1L1 • expected one-period return differs according to the bond maturity • expected n-period return is not the same for investment alternatives because of the “extra” return built into the model • predicts upward sloping yield curves, which are, in fact observed roughly 2/3 of the time Freddie Mac

45. Theories of Term Structure • Market Segmentation Hypothesis • Based on supply and demand modeling, with buyers/sellers having preferred maturities • Example is that some issuers prefer lending long; some buyers such as insurance companies prefer holding long • Related assumption is that arbitrageurs have limited ability to arbitrage between different segments • Implication is that there is no relationship between short-and long-term rates or between forward rates and expected future spot rates Freddie Mac

46. Theories of Term Structure The liquidity premium, if it exists, biases the slope upwards, and can even make a downward-sloping expectations curve upward-sloping Freddie Mac

47. Term Structure Implications • Empirical testing • Methodology • Define the spot yield curve (take points, bootstrap, and interpolate) • Formulate estimates of forwards • Determine whether or not the forwards are unbiased estimates of future spot rates • Results • The results almost universally reject the pure expectations model; that is it appears that there are risk premia in spot rates • The amount and term structure of liquidity premia is not known, because there is not a good model for predicting these and they are likely time-varying • Data noise may be an issue--there is measurement error for some points on the curves Freddie Mac

48. Term Structure As A Forecasting Tool • Spot rates are a function of the real rate of return and expectations of inflation (not including any risk or liquidity premia) • Generally the most variable part of this is the inflation expectation • If rates rise, generally interpreted as an indicator of rising inflation, which is a feature of boom economies • If rates fall, generally interpreted as an indicator of declining inflation, which is a feature of economies in the mature phase of the business cycle • Generally credit risk premia decline in boom economies and decline in worsening economies, because probability of default generally increases in recessions • A lot of uncertainty because of the time-varying properties of liquidity and credit risk premia Freddie Mac