1. Fixed Income Basics - part 2 Finance 70520, Spring 2002 The Neeley School of Business at TCU ©Steven C. Mann, 2002 Forward interest rates spot, forward, and par bond yield curves Intro to Term Structure

2. Term structure 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 7.0 6.5 6.0 5.5 5.0 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 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

3. Forward rates Introductory example (annual compounding) : one-year zero yield : 0 y 1 =5.85% ;B(0,1) = 1/(1.0585) = 0.944733 two-year zero yield: 0 y 2 =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) = 1.062103 f (1,2) = 1.0621 - 1 = 6.21% and $1.0585 (1.0621) = $1.1242.

4. Forward and spot rate relationships : annualized rates

5. 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) = $925.93 0 y 1 =[1.08] (1/1) -1=8% year 2$1,000/[(1.08)(1.10)] = $841.75 0 y 2 = [(1.08)(1.10)] (1/2) - 1 =8.995% year 3$1,000/[(1.08)(1.10)(1.11)] = $758.33 0 y 3 =[(1.08)(1.10)(1.11)] (1/3) = 9.660% year 4 $1,000/[(1.08)(1.10)(1.11)(1.11)] = $683.18 0 y 4 =[(1.08)(1.10)(1.11)(1.11)] (1/4) = 9.993%

6. 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

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

8. Par bond yield is yield for bond priced at par: coupon = ytm

9. Example: Assume corporate yield is determined as: Treasury + 300 b.p. Discount bond ( 8.000%) : Treasury ytm = 9.811%  12.811% Par bond ( 9.567% ): Treasury ytm = 9.567%  12.567% Premium bond (12.00 %): Treasury ytm = 9.546%  12.546% $100 million 3-year bond issue: Borrower: use of 8% instead of par: (12.581-12.567) x $100mm = $14,000 annual cost Lender: use of 12% instead of par: (12.567-12.546) x $100mm = $23,000 annual cost

10. 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 $100/92.92593 = $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 investement 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

11. 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 $100/92.92593 = $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 7.03% return short rate = 9%, value = 118.80/1.09 = 108.99 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

12. 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

13. 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