L12: Fixed Income Securities1 Lecture 12: Fixed Income Securities The following topics will be covered: Discount Bonds Coupon Bonds Interpreting the Term Structure of Interest Rates Basic of Term Structure Models Materials from Chapter 10 and 11 (briefly) of CLM

L12: Fixed Income Securities2 Zero-coupon Bonds – basic notations For zero-coupon bonds, the yield to maturity is the discount rate which equates the present value of the bond’s payments to its price. where P nt is the time t price of a discount bond that makes a single payment of $1 at time t+n, and Y nt is the bond yield to maturity. We have, Expressed in log form, we have:

L12: Fixed Income Securities3 Yield Curve of Zero-coupon Bonds Term structure of interest rates is the set of yields to maturity, at a given time, on bonds of different maturities. Yield spread S nt =Y nt -Y 1t, or in log term s nt =y nt -y 1t, measures the shape of the term structure. Yield curve plots Y nt or y nt against some particular date t.

L12: Fixed Income Securities4 Return for Discount Bonds (1) Define R n,t+1 as the 1-period holding-period return on an n- period bond purchased at time t and sold at time t+1 Writing in the log form, we have Holding period return is determined by the beginning-o- period yield (positively) and the change in the yield over the holding period (negatively).

L12: Fixed Income Securities5 Return for Discount Bonds (2) The log bond price today is the log price tomorrow minus the return today. We can solve this difference equation forward and get: We can also get: The log yield to maturity on a zero-coupon bond equals the average log return period if the bond is held to maturity

L12: Fixed Income Securities6 Forward Rate The forward rate is defined to be the return on the time t+n investment of P n+1,t /P nt where, in the forward rate, n refers to the number of periods ahead that the 1-period investment is to be made, and t refers to the date at which the forward rate is set.

L12: Fixed Income Securities7 Coupon Bonds Coupon bonds can be viewed as a package of discount bonds There is no analytical solution for yield to maturity of coupon bonds Unlike the yield to maturity on a discount bond, the yield to maturity on a coupon bond does not necessarily equal the per-period return if the bond is held to maturity. –The yield to maturity equals the per-period return on the coupon bond held to maturity only if coupons are reinvested at a rate equal to the yield to maturity. Two cases –Selling at par –perpetuity

L12: Fixed Income Securities8 Duration Macaulay duration: See the example on page 402 Duration is the negative of the elasticity of a coupon bond’s price with respect to its gross yield (1+Y cnt ) Modified duration:

L12: Fixed Income Securities9 Immunization Implications: firms with long-term zero-coupon liabilities, such as pension obligations, they may wish to match or immunize these liabilities with coupon-bearing Treasury bonds. –Zero-coupon Treasury bonds are available, they may be unattractive because of tax clientele and liquidity effects, so the immunization remain relevant. If there is a parallel shift in the yield curve so that bond yields of all maturities move by the same amount, then a change in the zero-coupon yield is accompanied by an equal change in the coupon bond yield

L12: Fixed Income Securities10 Limitations A parallel shift of the term structure Works for small change in interest rates Cash flows are fixed and don’t change when interest rate changes. –Callable securities

L12: Fixed Income Securities11 Loglinear Model for Coupon Bonds Starting from the loglinear approximate return formula, we have

L12: Fixed Income Securities12 Estimating Zero-coupon Term Structure If the prices of discount bonds P 1 …P n maturing at each coupon date is known, then the price of a coupon bond is: If coupon bond prices are known, then we can get the implied zero-coupon term structure:

L12: Fixed Income Securities13 Spline Estimation When there are more than one price for each maturity, statistical methods should be used. One way is regression: In practice the term structure of coupon bonds is usually incomplete. McCulloch (1971, 1975) suggest to write P n as a function of maturity P(n): Assume P(n) to be a spline function. The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing.

L12: Fixed Income Securities14 Tax Effect US Treasury bond coupons are taxed as ordinary income while price appreciation on a coupon bearing bond purchased at a discount is taxed as capital Thus there is a tax effect Page 411, CLM

L12: Fixed Income Securities15 Pure Expectation Hypothesis (PEH) PEH

L12: Fixed Income Securities16 Alternatives to Pure Expectation Hypothesis Expectation hypothesis –Considering term premia Preferred habitat –Different lenders and borrowers may have different preferred habitats Time varying of term premia

L12: Fixed Income Securities17 Term Structure Models -- Motivations Starting from the general asset pricing condition introduces: 1=E t [(1+R i,t+1 )M t+1 ] Fixed-income securities are particularly easy to price. When a fixed- income security has deterministic cash flows, it covaries with the stochastic discount factor only because there is time-variation in discount factors. P nt =E t [P n-1,t+1 M t+1 ] It can be solved forward to express the n-period bond price as P nt =E t [P n-1,t+1 M t+1 ]

L12: Fixed Income Securities18 Affine-Yield Models Assume that the distribution of the stochastic discount factor M t+1 is conditionally lognormal Take logs of P nt =E t [P n-1,t+1 M t+1 ], we have