**DAY ONE**

**YIELD CURVES: TYPES OF CURVES AND THEIR CONSTRUCTION**

- 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.

** SPOT RATES AND SPOT RATE CURVES**

- 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

**FORWARD RATES: DETERMINATION, INTERPRETATION AND APPLICATION**

- 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)

**OPTION-ADJUSTED SPREAD ANALYSIS**

- 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

**DAY TWO**

**OPTION-ADJUSTED SPREAD ANALYSIS (cont’d)**

- 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

**BINOMIAL INTEREST RATE TREES**

- 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 (r
_{1d}) - Obtain up-state interest rate r
_{u} = r_{d} x e^{2σ} - 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 AND CONVEXITY OF CALLABLE AND PREPAYABLE SECURITIES**

- 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

**YIELD CURVES AS A TOOL FOR FIXED INCOME MARKET ANALYSIS**

- 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