Lecture 12: Fixed Income Securities

1. 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 Securities

2. 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 Pnt is the time t price of a discount bond that makes a single payment of $1 at time t+n, and Ynt is the bond yield to maturity. We have, • Expressed in log form, we have: L12: Fixed Income Securities

3. 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 Snt=Ynt-Y1t, or in log term snt=ynt-y1t, measures the shape of the term structure. • Yield curve plots Ynt or ynt against some particular date t. L12: Fixed Income Securities

4. Return for Discount Bonds (1) • Define Rn,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 Securities

5. 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 Securities

6. Forward Rate • The forward rate is defined to be the return on the time t+n investment of Pn+1,t/Pnt 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 Securities

7. 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 Securities

8. 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+Ycnt) • Modified duration: L12: Fixed Income Securities

9. 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 Securities

10. 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 Securities

11. Loglinear Model for Coupon Bonds • Starting from the loglinear approximate return formula, we have L12: Fixed Income Securities

12. Estimating Zero-coupon Term Structure • If the prices of discount bonds P1…Pn 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 Securities

13. 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 Pn 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 Securities

14. 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 Securities

15. Pure Expectation Hypothesis (PEH) • PEH L12: Fixed Income Securities

16. 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 Securities

17. Term Structure Models -- Motivations • Starting from the general asset pricing condition introduces: • 1=Et[(1+Ri,t+1)Mt+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. • Pnt=Et[Pn-1,t+1Mt+1] • It can be solved forward to express the n-period bond price as • Pnt=Et[Pn-1,t+1Mt+1] L12: Fixed Income Securities

18. Affine-Yield Models • Assume that the distribution of the stochastic discount factor Mt+1 is conditionally lognormal • Take logs of Pnt=Et[Pn-1,t+1Mt+1], we have L12: Fixed Income Securities