Yield Curves and Spreads

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Course Features

  • Course date:06/04/2020
  • Course Duration: 2 Day
  • Level: Beginner-intermediate
  • Prerequisites: None
  • Method: Live & Virtual
  • Venue: 115 Broadway, NY, NY 10006
  • Time: 9:00 am – 5:00 pm
  • Dress Code: Business Casual
  • Category:
  • Certificate: Yes
  • CPE Credits: 14
  • Course Code: 806



  • Types of Yield Curves and Methods of Construction
    • Current coupon (on the run) yield curves – plot observed yields
    • Par bond yield curves
    • Spot rate curves – infer discount rates from prices and contract cash flows
    • Forward rates, forward rate curves and forward yield curves
    • On-the-Run U.S. Treasury Curve
  • Rates on 10 regularly issued maturities (4 bills, 5 notes and only 1 bond)
    • Convert T-bill discount rates to their bond equivalent yields
    • Infer a 20-year rate from T-bonds issued about 10 years earlier
    • Use a hermetic splining process to capture implied curvature
    • Why this curve may not be best to assess relative value of Treasuries
  • Other U.S. Treasury Curves
    • Off-the-run Treasury curve
    • Strip rate curve
    • TIPS curve
  • Other Sector Yield Curves
    • Corporate: investment grade and high yield curves
    • Other sector curves: MBS (agency and non-agency), ABS, municipal, etc.



  • Definition and Interpretation of Spot Rates
    • Yields to maturity on hypothetical zero-coupon bonds
    • Strip/zero coupon bond rates versus spot rates
    • Reasons for indirect method of obtaining spot rates
  • Calculation of a Spot Rate
    • Identify a bond having the appropriate maturity
    • Determine the price and contractual cash flows
    • Discount all cash flows prior to maturity by rates already obtained
    • Solve for the discount rate on the final cash flow so PV = price
  • Construction of Spot Rate Curve
    • Take one or more short term rates from the market as a given
    • Bootstrap curve to desired horizon using the above described process
    • Issues related to bond selection for calculating individual spot rates
    • Filling in gaps between rates computed as described above
  • Using Spot Rate Curve to Estimate a Bond’s Fair Value
  • Other Applications of Spot Rates



  • Definition and Interpretation of Forward Rates
    • Future period interest rates
    • Rates can be locked-in through use of appropriate instrument/strategy
    • Consistency of forward rates with spot rates (enforced by arbitrage)
  • Conflation of Forward Rates with Expectations
    • Common (mis)interpretation of forward rates
    • Source of confusion between forward rates and expected future spot rates
    • Empirical evidence
  • Calculation of Forward Rates and Forward Yield Curves
    • Forward rates derived from comparing spot rates of differing terms
    • Obtaining spot rates given prices of zero coupon bonds
  • Trading and Risk Management Strategies Using Forward Rates
    • Forward pricing – interest rate/bond futures prices
    • Hedging future borrowing rates
    • Riding the yield curve (forward rate as a “breakeven” rate)



  • Evaluation of Spread Between Treasury and Other Sector Spot Rate Curves
    • Reason for using spread between curves rather than security yields
    • Higher rates implies lower cost per dollar of future cash flows
    • OAS: spread to Treasury rates due to greater credit and liquidity risk
    • Remainder of spread – premium for selling option to call/prepay
  • Calculation of Z (Zero Volatility) Spread
    • Constant spread to risk-free spot rates
    • Different methodologies for corporates/municipals versus MBS
    • MBS cash flows projected at vector of prepayment speeds
    • Corp/muni contractual cash flow valued on binomial interest rate tree
    • Constant spread to Treasury spot rates/related binomial tree so price = PV




  • Monte Carlo Simulations for OAS analysis of MBS
    • Project numerous (100-300) interest rate paths
    • Assume some volatility and appropriate rules (e.g. mean reversion)
    • Random generation of monthly risk-free rates
    • Use to generate mortgage rates, in turn used to obtain cash flows
    • OAS based on spread to projected rates the equate average PV to price
  • Binomial Interest Rate Trees for Corporate/Municipal Callable Securities
  • Cost of Option = Z Spread – OAS
  • Analytic Applications of OAS Analysis



  • Base of the Tree: One-Period Bond, Cash Flows Discounted By Spot Rate
  • One-period Rate One Period Forward
    • Identify two-period bond to use in process
    • Make assumption about interest rate volatility
    • Guess a down-state interest rate one period in the future (r1d)
    • Obtain up-state interest rate ru = rd x e
    • Discount T=2 cash flows to get present value in up and down states at T=1
    • Discount those up and down state present values by spot rates
    • Compare to observe price of two-period bond
    • Repeat process until present value calculated equals price


  • Tree Can Be Extended to Limit of Available Bonds
    • Extend tree one period at a time
    • Select bond maturing one period beyond length of tree
    • Only added period rates adjusted, others left as previously determined
  • Applications of Binomial Rate Trees
    • Valuation of bonds
    • OAS analysis of callable corporate or municipal bonds



  • Duration of Callable Bonds: Effective Duration
    • Conceptually same as modified – sensitivity of price to change in yield
    • Mechanically different – arc slope versus an instantaneous rate of change
    • Effective duration calculation for callable corporate/municipal bonds
    • OAS duration of MBS (and other MBS duration methodologies)
  • Convexity of Callable/Prepayable Securities
    • Callable bond = similar non-callable bond – call option
    • Negative convexity at yields below to a bit above coupon rate
    • Positive convexity at yields far enough above coupon rate
    • Connection of option value to convexity of callable/prepayable bonds
  • Case Studies Evaluating Impact of Embedded Optionality
    • Rate volatility impact on investment grade versus high yield bond values
    • Option values of callable bonds trading at a premium to par
    • Factors impacting the degree of negative convexity of agency MBS



  • Connection Between Yield Curve Slope and Sector Spreads with Real Economy
    • Slope of the Treasury curve and economic cycles
    • Fed policy impacts on Treasury yield curve
    • Risk factors reflected in yield curve spreads
    • Sectors spread changes correlations with economic strength/weakness
  • Yield Curve Spread Applications in Evaluating Relative Value and Risk
    • How yield curve spreads reflect market’s pricing of expectations
    • Spreads used in quantifying the relative value of bond market sectors
    • Interpreting spreads to identify sectors trading rich or cheap
    • Spread changes impact on relative performance of market sectors



  • George Bollenbacher spent 20 years as a trader in the fixed income markets at such firms as Paine Webber, Chase Manhattan Bank and Donaldson, Lufkin and Jenrette. Then he spent ten years in the technology industry, at such firms as IBM and Unisys, designing systems for the trading industry. He holds a Patent Achievement Award …
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