The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Market: The Bond Model The Oxford Guide to Financial Modeling Thomas S. Y. Ho and Sang Bin Lee Copyright © 2004 by Thomas Ho and Sang Bin Lee. All rights reserved.

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Mathematics Principal and coupons –Maturity –Coupons as a % of the principal –Perpetual bonds Accrued Interest –Quoted price and invoice price –Accruing linearly

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Mathematics (2) Yield –Yield to maturity –Compounding yield: annual, quarterly, monthly… continuously

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bonds and Bond Market Money Market –LIBOR rates –Fed Fund rates –Overnight Repo rates LIBOR(month)13612 % Discount rate1.24 Fed Funds1.50 Repo1.68 Banker's acceptance 1.86 Prime rate4.75

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bonds and Bond Market (2) Treasure Securities –Bill, notes, and bonds –STRIPS –TIPS Other Bonds –Corporates, Municipals, Mortgages…

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Swap Market Counter parties in an exchange of payments, over the tenor of the swap Notional amount Vanilla swap: floating rate for the fixed rate The swap rate: the fixed rate for each swap tenor

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 7 Yield Curves Time value of money depending on the time of the payments: risk free rates Discount function: the present value factor Nominal yield curve Spot yield curve Par yield curve

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 8 Spot Yield Curve Figure 3.1 Treasury market spot curve

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Economics of the Yield Curve Real Rate and Nominal Rate –The Fisher Equation Yield Curve Shapes Yield Time to Maturity Yield Time to Maturity

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Economics of the Yield Curve (2) Expectation Hypothesis –The expected interest rate = the forward rate

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Economics of the Yield Curve (3) Liquidity Premium Hypothesis –Upward sloping yield curve as explained by the premium Preferred Habitat Hypothesis –Market structure affects the shape and movement of the yield curve

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Model The bond cash flow is the combination of the coupon payments and the principal The cash flow is viewed as a portfolio of single payments The portfolio is the sum of the present value of each payment

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Model (2) No Arbitrage Opportunity Law of One Price

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Model (3) The cash flow for each year is given by: Term123 Coupon10 Principal 100 Cash flow The price, by the law of one price, is

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Model (4)

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Forward Prices and Forward Rates Futures and Forward Contracts –The marking to market mechanism of the futures market –Forward delivery of a bond –Arbitrage condition and the pricing of a forward contract Forward rate movement –Forward rate under different shapes of the yield curve

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 17 Forward Pricing Model T* is the delivery date of the forward contract T is the maturity of the zero coupon bond to be delivered P(·) is the discount function F is the forward contract price base on $1 principal

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 18 Forward Pricing Model (2) Time0T*T*+T Holding a T*- year bond  P(T*) Holding a T- year forward contract with a delivery date at year T* Cash Flow  P(T*) 0

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 19 Forward Pricing Model (3)

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 20 Forward Pricing Model (4) TimetoMaturity t n u o c s i D e t a R Spot Forward TimetoMaturity t n u o c s i D e t a R Spot Forward TimetoMaturity t n u o c s i D e t a R Spot Forward

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 21 Forward Pricing Model (5) Maturity(T)123 Spot rates(%)=8%=9%=10% forward rates=10.01%=11.01%N/A forward rates=12.03%N/A

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 22 Forward Pricing Model (6) Forward rate movements

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Bond Analysis Cheap/Rich Analysis –The valuation model determines the fair value of a bond. –Cheap/rich = the observed price – the fair price Spot Yield Curve, Par Yield Curve, and Nominal Yield Curve –The spot curve determines the par curve –The par curve determines the spot curve –The discount function determines the spot and par curves –Nominal yield curve derived from the observed prices

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 24 STRIPS US STRIPS MaturityTypeBid Price Aug 02ci99 24/32 Aug 02np99 23/32  Nov 02ci99 14/32 Nov 11ci61 02/32

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 25 A standard statistical curve fitting methodology Cubic spline function

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 26 Modified, Effective, Key Rate Duration Modified duration is related to the weighted average life of a bond Effective duration is the price sensitivity of a bond to the yield curve shifts The 2 risk measures are the same if the yield curve is flat and the bonds have no embedded option (ie the bond is a cash flow.) Key rate duration is the price sensitivity of a bond for each key rate shift

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 27 (Effective) duration

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 28 Modified Duration

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 29 Key Rate Durations

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 30 Key Rate Duration Profile The risk of a bond is measures by a set of key rate duration numbers The sum of key rate durations = effective duration Key rate duration of a zero coupon bond equals the duration at the bond maturity

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 31 Convexity Convexity provides the 2 nd order approximation to the price behavior of a bond to the shift of the yield curve Convexity of a bond can be simulated using a bond valuation model

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 32 Performance Profile Performance profile relates the bond price to a range of parallel shifts of the yield curve The profile depicts the behavior of the bond, which can be complex, as described in later chapters. Parallel Shift

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model Applications of the Bond Analytics Barbell trade to enhance returns when the yield curve shifts in a parallel fashion Replicating a Treasury portfolio –For indexation –For enhance indexation –For asset liability management

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 34 A Barbell Trade Bond position value MaturityDurationConvexity A$ B$ Total$ Short-selling$

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 35 Appendix A: Taylor Expansion Remainder

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 36 Appendix A (2)

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 37 Appendix B: The Derivation of Macaulay Duration and Convexity

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 38 Appendix C: Duration & Convexity in measuring price sensitivity

The Oxford Guide to Financial Modeling by Ho & Lee Chapter 3. Bond Markets: The Bond Model 39