Introduction to Fixed Income - Part 1: Money Market Rate Definitions and Spot Zero-Coupon Yields

1. Introduction to Fixed Income – part 1 Money market rate definitions discount factors spot zero-coupon yields using the zero curve to price coupon bonds Finance 30233 Fall 2004 Advanced Investments Associate Professor Steven C. Mann The Neeley School of Business at TCU

2. Bill prices and interest rate definitions Default free bonds (Treasuries) zero coupon bond price, stated as price per dollar: B(t,T) = price, at time t, for dollar to be received at T Interest rates discount rate (T-bill market) simple interest discrete compounding continuous compounding Rate differences due to: compounding day-count conventions actual/actual; 30/360; actual/360; etc.

3. Discount rate: id (T) T 360 B(0,T) = 1 - id (T) Example: 30-day discount rate id = 3.96% B(0,30) = 1 - (0.0396)(30/360) = 0.9967 Current quotes: www.bloomberg.com 360 T id = 100 (1 - B(0,T)) Example: 90-day bill price B(0,90) = 0.9894 id (90) = 100 (1- 0.9894)(360/90) = 4.24%

4. Simple interest rate: is (T) 1 1 + is (T)(T/365) B(0,T) = Example: 30-day simple rate is = 4.03% B(0,30) = 1/ [1+ (0.0403)(30/365)] = 0.9967 Current quotes: www.bloomberg.com 1 B(0,T) 365 T is= 100 [ - 1] Example: 90-day bill price B(0,90) = 0.9894 is (90) = 100 [(1/0.9894) -1](365/90) = 4.34%

5. Discretely compounded rate: rt(h) compounding for h periods 1 [1 + rt(h)/h] h B(t,t+h) = rt(h) = h [ (1/B)(1/h) - 1 ] Example: 1 year zero-coupon bond price = 0.9560 semiannually compounded rate r1(2) = 2 [ (1/0.9560) (1/2) - 1 ] = 4.551%

6. yield 7.0 6.5 6.0 5.5 5.0 Term structure (yield curve) Typical yield curve 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: At first we will focus on the “spot zero curve”: yield-to-maturity for zero-coupon bonds source: current market bond prices (spot prices) Direct relationship between zero coupon spot yield, 0yt , and the price today of a riskless dollar delivered later, B(0,t): Define 0yt such that B(0,t) = (1 + 0yt)–t Remember: B(0,t) is the present value (time 0) of $1.00 at some later time (t)

7. Example A: use spot zero coupon yields to find B(0,t) vector Use B(0,t) = (1 + 0yt)–t

8. Example B: use B(0,t) vector (discount factors) to find zero-coupon yields Solve B(0,t) = (1 + 0yt)–t for 0yt: 0yt = [B(0,t)] –(1/t) -1

9. Example C: use B(0,t) vector (discount factors) to value a 10% annual coupon bond $10 $10 $10 $110 0 1 2 3 4 time Note that the bond yield (10.96%) is found FROM THE PRICE via the excel function “yield(arguments)” Make sure you know how the excel yield function works You also need to learn the “price” function. Use the excel help function for details.

10. Coupon Bonds T Price = S Ct B(0,t) + (Face) B(0,T) where B(0,t) is price of 1 dollar to be received at time t or Price = S Ct + (Face) where rt 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 1 1 (1+rt)t (1+rt)T t=1

11. Yield to Maturity: Annual interest payment Define yield-to-maturity, y, as: Price = S Ct + (Face) T 1 1 (1+y)t (1+y)T t=1 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 = 0.045 price = 104.68 (guess too low) eventually: when y = 0.04584 price = 104.52 y = 4.584% If annual yield = annual coupon, then price=face (par bond)

16. Semi-annual Yield-to-Maturity Define semi-annual yield-to-maturity, ys, as: Price = S Ct + (Face) T 1 1 (1+ys/2)t (1+ys/2)T t=1 Note effective annual yield-to-maturity is yA = (1+ys/2)2 - 1 Example: 2-year 7% semi-annual coupon bond, price =103.79 per 100. by definition, semi-annual yield-to-maturity ys is solution to: 103.79 = S 3.50/(1+ys/2)t + 100/(1+ys/2)4 eventually: when ys/2 = 0.0249 = 2.49% effective annual yield-to-maturity is yA = (1 + 0.0249)2 - 1 = 5.04% If semi-annual yield = semi-annual coupon, then price=face (par bond)

17. Reinvestment assumptions and yield-to-maturity Yield-to-maturity (ytm) is holding period rate of return only if coupons can bereinvestedat the same rate as yield-to-maturity Example: 6% semi-annual coupon Par bond (price=100.00) yield-to-maturity, ys, is defined as: So that ys = 0.06 6-month coupon re-invested at ytm becomes 3(1+ys/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-investment holding-period rate proceeds at one year return 5.0% 103 + 3.075 = 106.075 6.075% 7.0% 103 + 3.105 = 106.105 6.105%

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

19. 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 rate coupon paid: end of period date zero (today) 5.25% c = 5.25/2 = 2.625 six months later 5.60% 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.