Fixed Income Basics Finance 30233, Fall 2010 The Neeley School of Business at TCU ©Steven C. Mann, 2010 Spot Interest rates The zero-coupon yield curve Bond yield-to-maturity Default-free bond pricing Forward Rates Term Structure Theory

Term structure yield Maturity (years) Typical interest rate term structure “Term structure” may refer to various yields: “spot zero curve”: yield-to-maturity for zero-coupon bonds ( 0 y t ) source: current market bond prices (spot prices) “forward curve”: forward short-term interest rates: “forward rates: f(t,T)” source: zero curve, current market forward rates “par bond curve”: yield to maturity for bonds selling at par source: current market bond prices

Determination of the zero curve B(0,t) is discount factor: price of $1 received at t; B(0,t)  (1+ 0 y t ) -t. Example: find 2-year zero yield use1-year zero-coupon bond price and 2-year coupon bond price: bondprice per $100:yield 1-year zero-coupon bond % 2-year 6% annual coupon bond % B(0,1) = Solve for B(0,2): 6% coupon bond value= B(0,1)($6) + B(0,2)($106) $100= ($6) + B(0,2)($106) 100= B(0,2)($106) = B(0,2)(106) B(0,2) = /106 = so that 0 y 2 = (1/B(0,2)) (1/2) -1 = (1/0.8897) (1/2) -1 = %

“Bootstrapping” the zero curve from Treasury prices Example: six-month T-bill price B(0,6) = month T-bill priceB(0,12)= month T-note with 8% coupon paid semi-annually price = find “implied” B(0,18): = 4 B(0,6) + 4 B(0,12) + (104)B(0,18) = 4 ( ) B(0,18) = B(0,18) = 104 B(0,18) B(0,18)= /104 = month T-note with 7% semi-annual coupon: Price = = 3.5B(0,6) + 3.5B(0,12) + 3.5B(0,18) B(0,24) = 3.5( ) B(0,24) B(0,24) = ( )/103.5 =

Coupon Bonds Price =  C t B(0,t) + (Face) B(0,T) where B(0,t) is price of 1 dollar to be received at time t or Price=  C t + (Face) where r t is discretely compounded rate associated with a default-free cash flow (zero-coupon bond) at time t. Define par bond as bond where Price=Face Value = (par value) t=1 T T 1 1 (1+r t ) t (1+r t ) T

Yield to Maturity Define yield-to-maturity, y, as: Price   C t + (Face) t=1 T 1 1 (1+y) t (1+y) T Solution by trial and error [calculator/computer algorithm] Example: 2-year 7% annual coupon bond, price = per 100. by definition, yield-to-maturity y is solution to: = 7/(1+y) + 7/(1+y) /(1+y) 2 initial guess :y = 0.05 price = (guess too high) second guess:y = 0.045price = (guess too low) eventually: wheny = price = y = 4.584% If annual yield = annual coupon, then price=face (par bond)

Coupon bond yield is “average” of zero-coupon yields Coupon bond yield-to maturity, y, is solution to:

Bonds with same maturity but different coupons will have different yields.

Forward rates Introductory example (annual compounding) : one-year zero yield : 0 y 1 =5.85% ;B(0,1) = 1/(1.0585) = two-year zero yield: 0 y 2 =6.03% ;B(0,2) = 1/(1.0603) 2 = $1 investment in two-year bond produces $1( ) 2 = $ at year 2. $1 invested in one-year zero produces $1( ) = $ at year 1. What “breakeven” rate at year 1 equates two outcomes? ( ) 2 = ( ) [ 1 + f (1,2) ] breakeven rate = forward interest rate from year 1 to year 2 = f (1,2) (one year forward, one-year rate) 1 + f (1,2) = (1.0603) 2 /(1.0585) = f (1,2) = = 6.21% and $ (1.0621) = $

Forward and spot rate relationships : annualized rates

Example: Using forward rates to find spot rates Given forward rates, find zero-coupon bond prices, and zero curve Bond paying $1,000: maturityPriceyield-to-maturity year 1 $1,000/(1.08) = $ y 1 =[1.08] (1/1) -1=8% year 2$1,000/[(1.08)(1.10)] = $ y 2 = [(1.08)(1.10)] (1/2) - 1 =8.995% year 3$1,000/[(1.08)(1.10)(1.11)] = $ y 3 =[(1.08)(1.10)(1.11)] (1/3) = 9.660% year 4 $1,000/[(1.08)(1.10)(1.11)(1.11)] = $ y 4 =[(1.08)(1.10)(1.11)(1.11)] (1/4) = 9.993%

Yield curves maturity rate Forward rate zero-coupon yield coupon bond yield Coupon bond yield zero-coupon yield forward rate Typical upward sloping yield curve Typical downward sloping yield curve

Holding period returns under certainty (forward rates are future short rates) One year later: f (0,1) = 0 y 1 = 10% f (1,2) = 11% f (2,3) = 11% One-year holding period returns of zero-coupons: invest $100: one-year zero: $100 investment buys $100/ = $ Face value. At end of 1 year, value = $ ; return = (108/100)-1 = 8.0% two-year zero: $100 investment buys $100/ = $ Face value. at end of 1 year, Value = $118.80/1.10 = $ ; return = (108/100) -1 = 8.0% three-year zero: $100 investement buys $100/ = $ face value at end of 1 year, value = $131.87/[(1.10)(1.11)] = $ ; return = (108/100) -1 = 8.0% If future short rates are certain, all bonds have same holding period return

Holding period returns when future short rates are uncertain One year holding period returns of $100 investment in zero-coupons: one-year zero: $100 investment buys $100/ = $ Face value. 1 year later, value = $ ; return = (108/100)-1 = 8.0% (no risk) two-year zero: $100 investment buys $ face value. 1 year later: short rate = 11%, value = /1.11 = % return short rate = 9%, value = /1.09 = % return Risk-averse investor with one-year horizon holds two-year zero only if expected holding period return is greater than 8%: only if forward rate is higher than expected future short rate. Liquidity preference: investor demands risk premium for longer maturity

Term Structure Theories 1) Expectations: forward rates = expected future short rates 2) Market segmentation: supply and demand at different maturities 3) Liquidity preference: short-term investors demand risk premium maturity rate Expected short rate is constant Forward rate = expected short rate + constant Yield curve is upward sloping Yield Curve: constant expected short rates constant risk premium

Possible yield curves with liquidity preference rate Expected short rate is declining Forward rate Yield curve Liquidity premium increasing with maturity maturity rate Expected short rate is declining Forward rate Humped yield curve Constant Liquidity premium