#### Transcript Fixed Income Basics - Texas Christian University

Introduction to Fixed Income – part 2 Forward Interest rates yield curves spot par forward Introduction to Term Structure Finance 30233 - Fall 2003 Advanced Investments Associate Professor Steven C. Mann The Neeley School of Business at TCU Term structure yield 7.0 6.5 6.0 Typical interest rate term structure 5.5 5.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Maturity (years) “Term structure” may refer to various yields: “spot zero curve”: yield-to-maturity for zero-coupon bonds source: current market bond prices (spot prices) “forward curve”: forward short-term interest rates: “short rates” source: zero curve, current market forward rates “par bond curve”: yield to maturity for bonds selling at par source: current market bond prices Forward rates Introductory example (annual compounding) : one-year zero yield : 0y1 =5.85% ; B(0,1) = 1/(1.0585) = 0.944733 two-year zero yield: 0y2 =6.03% ; B(0,2) = 1/(1.0603)2 = 0.889493 $1 investment in two-year bond produces $1(1+0.0603)2 = $1.1242 at year 2. $1 invested in one-year zero produces $1(1+0.0585) = $1.0585 at year 1. What “breakeven” rate at year 1 equates two outcomes? (1 + 0.0603)2 = (1 + 0.0585) [ 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) = 1.0621 - 1 and $1.0585 (1.0621) = $1.1242. = 1.062103 = 6.21% Forward and spot rate relationships : annualized rates (1 0 y 2 ) f (1 , 2 ) (1 0 y 1 ) f ( n , n 1) f (1 , 2 ) B ( 0 ,1 ) 1 f ( n , n 1) 1 (1 0 y n 1 ) n 1 1 1 (1 y ) n 0 n ; 1 (1 0 y n 1 ) B ( 0 ,1 ) B ( 0 ,1 ) (1 0 y n ) n1 n 1 B (0, n 1) B (0, n ) (1 0 y n ) n 1 1 f (1 , 2 ) B (0, n ) n1 1 B ( 0 , n ) 1 B (0, n 1) B ( 0 ,2 ) f ( n , n 1) (1 0 y n 1 ) 2 B (0, n 1) 1 1 f ( n , n 1) 1 ; 1 Example: Using forward rates to find spot rates Given forward rates, find zero-coupon bond prices, and zero curve n forward rate (year) 0 1 2 3 f f f f f (n,n+1) (0,1) = 8.0% (1,2) = 10.0% (2,3) = 11.0% (3,4) = 11.0% bill price B(0,n+1) B(0,1) = 0.92593 B(0,2) = 0.84175 B(0,3) = 0.75833 B(0,4) = 0.68318 spot rate 0yn+1 8.000% 8.995% 9.660% 9.993% Forward rates Spot rates 0 2 12.0% 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 1 3 Bond paying $1,000: maturity Price yield-to-maturity (1/1) year 1 $1,000/(1.08) = $925.93 0y1=[1.08] -1 year 2 $1,000/[(1.08)(1.10)] = $841.75 0y 2 = year 3 $1,000/[(1.08)(1.10)(1.11)] = $758.33 0y3 =[(1.08)(1.10)(1.11)] year 4 $1,000/[(1.08)(1.10)(1.11)(1.11)] = $683.18 =8% [(1.08)(1.10)](1/2)- 1 =8.995% (1/3) 0y4 =[(1.08)(1.10)(1.11)(1.11)] = 9.660% (1/4) = 9.993% Yield curves rate Forward rate zero-coupon yield coupon bond yield Typical upward sloping yield curve maturity rate Typical downward sloping yield curve Coupon bond yield zero-coupon yield forward rate maturity Coupon bond yield is “average” of zero-coupon yields T Bond Value T B ( 0 , t ) C t B ( 0 , T ) Face t 1 t 1 1 (1 0 y t ) t Ct 1 (1 0 y T ) T Face Coupon bond yield-to maturity, y, is solution to: T Bond Value t 1 1 (1 y ) t Ct T 1 (1 y ) T Face t 1 1 (1 0 y t ) t Ct 1 (1 0 y T ) T bond: 3-year $100 face; annual coupon = 10% B(0,t)Ct T B(0,T) B(0,3)$100 0y T 1 0.92593 8.00% 9.26 2 0.84175 9.00% 8.42 3 0.75833 9.66% 7.58 75.83 Bond Value Bond yield = 9.56% 25.26 75.83 101.09 Face Bonds with same maturity but different coupons will have different yields. bond: 3-year $100 face; annual coupon = 15% B(0,t)Ct T B(0,T) B(0,3)$100 0y T 1 0.92593 8.00% 13.89 2 0.84175 9.00% 12.63 3 0.75833 9.66% 11.37 75.83 Bond Value Bond yield = 9.52% 37.89 75.83 113.72 bond: 3-year $100 face; annual coupon = 5% B(0,t)Ct T B(0,T) B(0,3)$100 0y T 1 0.92593 8.00% 4.63 2 0.84175 9.00% 4.21 3 0.75833 9.66% 3.79 75.83 Bond Value Bond yield = 9.61% 12.63 75.83 88.46 Determination of the zero curve B(0,t) is discount factor: price of $1 received at t; B(0,t) = (1+ 0yt)-t . Example: find 2-year zero yield use 1-year zero-coupon bond price and 2-year coupon bond price: bond price per $100: 1-year zero-coupon bond 94.7867 2-year 6% annual coupon bond 100.0000 yield 5.500% 6.000% B(0,1) = 0.9479. Solve for B(0,2): 6% coupon bond value = B(0,1)($6) + B(0,2)($106) $100 = 0.9479($6) + B(0,2)($106) 100 = 5.6872 + B(0,2)($106) 94.3128 = B(0,2)(106) B(0,2) = 94.3128/106 = 0.8897 so that 0y2 = (1/B(0,2))(1/2) -1 = (1/0.8897)(1/2) -1 = 6.0151% “Bootstrapping” the zero curve from Treasury prices Example: six-month T-bill price 12-month T-bill price B(0,6) = 0.9748 B(0,12) = 0.9493 18-month T-note with 8% coupon paid semi-annually price = 103.77 find “implied” B(0,18): 103.77 = = = 96.0736 = B(0,18) = 4 B(0,6) + 4 B(0,12) + (104)B(0,18) 4 (0.9748+0.9493) + 104 B(0,18) 7.6964 + 104 B(0,18) 104 B(0,18) 96.0736/104 = 0.9238 24-month T-note with 7% semi-annual coupon: Price = 101.25 101.25 = 3.5B(0,6) + 3.5B(0,12) + 3.5B(0,18) + 103.5B(0,24) = 3.5(0.9748+0.9493+0.9238) + 103.5B(0,24) B(0,24) = (101.25 - 9.9677)/103.5 = 0.9016 Holding period returns under certainty (forward rates are future short rates) n forward rate (year) 0 1 2 3 f f f f f (n,n+1) (0,1) = 8.0% (1,2) = 10.0% (2,3) = 11.0% (3,4) = 11.0% bill price B(0,n+1) B(0,1) = 0.92593 B(0,2) = 0.84175 B(0,3) = 0.75833 B(0,4) = 0.68318 spot rate 0yn+1 8.000% 8.995% 9.660% 9.993% One year later: f (0,1) = 0y1 = 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 $100/92.92593 = $108.00 Face value. At end of 1 year, value = $108.00 ; return = (108/100)-1 = 8.0% two-year zero: $100 investment buys $100/84.175 = $118.80 Face value. at end of 1 year, Value = $118.80/1.10 = $108.00 ; return = (108/100) -1 = 8.0% three-year zero: $100 investment buys $100/75.833 = $131.87 face value at end of 1 year, value = $131.87/[(1.10)(1.11)] = $108.00 ; 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 n forward rate (year) 0 1 2 3 f f f f f (n,n+1) (0,1) = 8.0% (1,2) = 10.0% (2,3) = 11.0% (3,4) = 11.0% bill price B(0,n+1) B(0,1) = 0.92593 B(0,2) = 0.84175 B(0,3) = 0.75833 B(0,4) = 0.68318 spot rate 0yn+1 8.000% 8.995% 9.660% 9.993% possible short rate (0y1) evolution: now one year later 11.00% 8.00% 9.00% One year holding period returns of $100 investment in zero-coupons: one-year zero: $100 investment buys $100/92.92593 = $108.00 Face value. 1 year later, value = $108.00 ; return = (108/100)-1 = 8.0% (no risk) two-year zero: $100 investment buys $118.80 face value. 1 year later: short rate = 11%, value = 118.80/1.11 = 107.03 short rate = 9%, value = 118.80/1.09 = 108.99 7.03% return 8.99% 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 rate Forward rate = expected short rate + constant Par Bond yield curve is upward sloping Expected short rate is constant Yield Curve: constant expected short rates constant risk premium maturity Yield curves with liquidity preference rate Forward rate Par bond yield curve Liquidity premium increasing with maturity Expected short rate is declining maturity rate Forward rate Humped par bond yield curve Constant Liquidity premium Expected short rate is declining maturity