2. Fixed Income Basics - part 1 Finance 70520, Spring 2002 The Neeley School of Business at TCU ©Steven C. Mann, 2002 Spot Interest rates The zero-coupon yield curve Bond yield-to-maturity Default-free bond pricing

3. 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

4. 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 94.7867 5.500% 2-year 6% annual coupon bond 100.0000 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 0 y 2 = (1/B(0,2)) (1/2) -1 = (1/0.8897) (1/2) -1 = 6.0151%

5. “Bootstrapping” the zero curve from Treasury prices Example: six-month T-bill price B(0,6) = 0.9748 12-month T-bill priceB(0,12)= 0.9493 18-month T-note with 8% coupon paid semi-annually price = 103.77 find “implied” B(0,18): 103.77 = 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) 96.0736 = 104 B(0,18) 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

6. 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

7. 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 =104.52 per 100. by definition, yield-to-maturity y is solution to: 104.52 = 7/(1+y) + 7/(1+y) 2 + 100/(1+y) 2 initial guess :y = 0.05 price = 103.72(guess too high) second guess:y = 0.045price = 104.68(guess too low) eventually: wheny = 0.04584price = 104.52y = 4.584% If annual yield = annual coupon, then price=face (par bond)

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

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

10. Semi-annual Yield-to-Maturity Define semi-annual yield-to-maturity, y s, as: Price   C t + (Face) t=1 T 1 1 (1+y s /2) t (1+y s /2) T Example: 2-year 7% semi-annual coupon bond, price =103.79 per 100. by definition, semi-annual yield-to-maturity y s is solution to: 103.79 =  3.50/(1+y s /2) t + 100/(1+y s /2) 4 eventually: wheny s /2 = 0.0249 = 2.49% effective annual yield-to-maturity is y A = (1 + 0.0249)2 - 1 = 5.04% Note effective annual yield-to-maturity is y A  (1+y s /2) 2 - 1 If semi-annual yield = semi-annual coupon, then price=face (par bond)

11. Reinvestment assumptions and yield-to-maturity Yield-to-maturity (ytm) is holding period rate of return only if coupons can be reinvested at the same rate as yield-to-maturity Example: 6% semi-annual coupon Par bond (price=100.00) yield-to-maturity, y s, is defined as: So that y s = 0.06 6-month coupon re-invested at ytm becomes 3(1+y s /2) = 3(1.03) in one year. End-of-year value: 103 + 3(1.03) = 106.09. Holding period return: (106.09-100)/100 = 6.09% Effective annual yield: 6% semi-annual yield = (1+0.06/2) 2 -1 = 6.09% When re-investment is compounded semi-annually: re-investmentholding-period rate proceeds at one year return 5.0%103 + 3.075 = 106.0756.075% 7.0%103 + 3.105 = 106.1056.105%

12. Treasury bond quotes and prices Accrued interest = Coupon x [(days since last coupon)/(days in coupon period)] Quotes are “clean prices” (no accrued interest) Actual price is “dirty price” Coupon period coupon

13. Floating rate notes Debt contract: face value, maturity, coupon payment dates Interest payments (coupons) reset at each coupon date. Example: one-year floater, semi-annual payments, Face=$100.00 payment based on six-month simple rate at beginning of coupon period spot six-month ratecoupon paid: end of period date zero (today) 5.25%c = 5.25/2 = 2.625 six months later5.60%c = 5.60/2 = 2.80 Six months from now, value of note is: 102.80/[1+ 0.056 x (1/2)] = 102.80/1.028 = $100 In six months bond will be valued at par. So value of note at time zero is: (100 + 2.625)/[1 + 0.0525 x (1/2)] = 102.625/1.02625 = $100 Note value is at par each reset date.