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Bond Markets Investments Chapter 7 QFE Section 1.1—1.2 & 20.1 Bond Markets • Payments: Redemption value, M, paid at maturity, n, and Coupons, Ct, paid at specified dates, t, until t = n. • Ct a form of interest and typically expressed as a percentage of M. • Typically longer term to maturity compared to money markets (1<n<30yrs) • Possiblilty of Capital gain/loss (trade at discount/premium). Bond Markets 2 • Issued by government (state/ county/ municipality) or (large) corporations. • Domestic currency issued bonds assumed risk-free (No exchange risk + right to print money). • Assumed risk-premium, rp, above safe rate, rf, commensurate in size to preceived risk level of income stream (Security of Ct + time-to-maturity related interest rate risk). Bond Markets 3 Prices & Rates of Return? • From the redemtion value, M, the size and number of coupon payments, C, and time- to-maturity, n, markets determine a price, P, given other instruments. • P should provide a rate of return commensurate with the return on similar assets. • Many ways to calculate a return, depending on needs. • Conventionally involves compounding. Bond Markets 4 Pure Discount Bonds • Recall, FVn M DPV P 1 rs n n n (1 rs ) n • Thus, the price today of money to be received in n periods time should be commensurate with the discount rate: 1 1 M P n M n rs rs 1 P P • rs(n) is the price of n-period money (annual) • Spot Rate for n-period money: rsn = f(P, M, n) Bond Markets 5 Example: Zeroes & Spot Rates • P = 62,321.30, M = 100,000, n = 6years • Spot rate? 1 6 100, 000 6 rs 1 0.082 8.2% p.a. 62,321.3 Bond Markets 6 Coupon Paying Bonds? • Stream of coupon payments Ct, which are ‘known’ at issue. • Government bonds’ coupons are generally fixed. May actually be indexed or variable [in corporate bond case]. • Redeemable at €M at some specified time in the future [perpetuities aside]. • Prices quoted clean. Prices paid involve accrued interest, i.e. dirty price. Bond Markets 7 Current Yield • A.k.a. running/flat/interest yield: C Annual PClean • Quick summary of simple interest, i.e. annual income relative to expenditure • Caveats: – No capital gain – Time to Maturity & Face Value? – Interest on coupons? Bond Markets 8 Yield to Maturity (YTM) • YTM can be calculated ex ante: YTM = y = f(P,M,C,n) C C C C M P (n) ... n 1 (1 y) (1 y) 1 2 (1 y) (1 y) (1 y) n n • YTM the discount rate (rate of return) that sets the price of a n-period bond, P(n), equal to the PDV of its income stream. • YTM is the internal rate of return of the bond’s cash-flow. Bond Markets 9 YTM: Assumptions & Caveats • Assumes bond held to maturity. • Assumes reinvest C at rate y. Why? • The rate of return is constant at y. • If several payments per period (year) the rate is grossed-up in simple annual terms. [See examples on semi-annual coupons.] • Inverse relationship among P and y. • P = f(y), and that function is convex in y (non- linear). • If y = C => P = M. Why? Bond Markets 10 Aside: Pricing an Annuity • Recall, the price of a perpetuity: P 0 C Perpetuity y • Thus, an annuity in n periods time should cost: C 1 P n y 1 y Perpetuity n • The difference should be the price of a T-period annuity. Bond Markets 11 Breaking-Up a Coupon Paying Bond C 1 M P 1 y 1 y 1 y n n • For simplification, we can view the coupon paying bond as having a one-off lump-sum payment at redemption, M, and a series of periodic payments, the coupons, which can be priced as an annuity. Bond Markets 12 Example: YTM of Semi-Annual Coupon Paying Bond • P = 900 C = 10% of M = 1,000, 3 years to maturity semi-annual. y? 900 50 50 50 50 50 50 1000 1 2 3 4 5 6 6 y y y y y y y 1 1 1 1 1 1 1 2 2 2 2 2 2 2 • y = 0.142 = 14.2% p.a. Bond Markets 13 Example: Price when YTM given? • 20-year, C = 10% of M = 1,000, YTM = 0.11 p.a. semi-annual • P? 50 50 50 1000 P ... 0.11 0.11 2 0.11 40 0.11 40 1 1 1 1 2 2 2 2 50 1 1000 • or P 1 802.31 0.055 1 0.055 1 0.055 40 40 Bond Markets 14 (One Period) Holding Period Return (HPR) • Ex post measure of return n 1 n n Pt 1 Pt Ct H t 1 n Pt Bond Markets 15 Realised Compound Yield (RCY) • A.k.a. Total return/effective holding period return. • Ex post measure of return. • Assumes the interest earned on each coupon is known, plus resale price. • The RCY of a bond held for n-periods: 1 TV TV n 1 rRCY n rRCY 1 P P Bond Markets 16 Example: RCY • 5 year bond with C = 10% pf M = 1,000, trading at par. • RCY assuing 2 year horizon, interest rate r = 8% and a YTM after 2 years of 9%? • TV of Coupons: 100 100(1.08) 208 3 100 100 100 1000 P2 1025.31 1.09 1.09 1.09 1.09 2 3 3 1233.31 1 rRCY 2 rRCY 0.1105 1000 Bond Markets 17 Pricing a Bond • A coupon paying bond must be priced such that each its payments is discounted by the pertinent spot rate. • Deviations from this policy will result in arbitrage opportunities from coupon stripping. • Hence, if arbitrage opportunities exist traders will exploit these, thus exerting pressure on prices. This behaviour will eliminate the arbitrage opportunities. Bond Markets 18 Bond Pricing C C CM P ... 1 r1 1 r2 2 1 rn n • Each coupon represents a single payoff at a certain time in the future. • Each payment can thus be treated as comparable to a zero of equal maturity. • If provided with spot rates you should be able to find a price for a bond. Bond Markets 19 Bond Pricing: Spots • Bond A: coupon 8¾% of FV = 100 annual, 2 years to maturity • Bond B: coupon 12% of FV = 100 annual, 2 years to maturity • Spots: r1 = 0.05 r2 = 0.06 8.75 108.75 PA 2 105.12 1.05 1.06 12 112 PB 2 111.11 1.05 1.06 Bond Markets 20 Calculating Spot (Bootstrapping) • Riskless deep discount securities only have short maturities. Spot rates of longer maturities have to be imputed. • Take the spot rates you have, say up to a year, then calculate the spot rate for the next period (e.g. six months, year) using comparable (riskless) instruments, such as coupon paying government bonds of that maturity. Bond Markets 21 Example: Bootstrapping • Spot rates (annual return) given for first six months, r1 = 8%, and year, r2 = 8.3%. • Calculate the 18-month spot rate given an 18- month coupon paying bond with C = 8.5% of M = 100 semi-annual. 4.25 4.25 104.25 99.45 1 2 3 r1 r2 r3 1 1 1 2 2 2 104.25 99.45 4.0865 3.9180 3 1 r3 2 r3 0.0893 Bond Markets 22 Coupon Stripping • C = 12.5% of FV = 100 = P, semi-annual, 20 yrs • YTM? 6.25 6.25 106.25 P 2 ... 20 1.0625 1.0625 1.0625 P 5.88 5.54 ... 31.61 100 • Spots: r6months = 0.08 and r12months = 0.083 • PV(C1) = 6.25/1.04 = 6.0096 • PV(C2) = 6.25/(1.0415)2 = 5.7618 • Profits? Bond Markets 23 Equilibrium Price • Spot 1-year r1 = 0.1 • Spot 2-year r2 = 0.11 • Consider 2-year coupon bond, C= 9% of M = 1000 & P = 966.4866 • Stripping coupons: • PV(C1)= 90/1.1 = 81.8182 • PV(C1)= 1090/1.112 = 884.668 • What is your guess as to the YTM? Bond Markets 24 Accrued Interest • Cum-dividend: clean + accrued • Ex-dividend: clean - rebate Bond Markets 25 Example: Accrued Interest (Cum Dividend) • 31.03.1993 a 9% T-Bill 2012 quoted at £106(3/16) for settlement 1.04.1993. • Last coupon on 6.02.1993 • Accrued Interest? • 22 days in February + 31 in March + 1 April. • N = 54 9(54/365) = 1.3315 Bond Markets 26 Example: Accrued Interest (Rebate) • 31.03.1993 a 9% Treasury 2004 quoted at £111(5/32)xd • Next coupon date is 25.04.1993 (i.e. 24 days) • Rebate? 9(24/365) = 0.592 • Dirty Price? £111(5/32) – 0.592 = 110.56 Bond Markets 27 Convertible Bonds • A bond that can be converted to a specified number of shares from a certain date on. • Allows for a lower initial cost of capital, since the option to convert provides the holder with upside potential. Bond Markets 28 Call Provisions • Bonds are described as callable if they can be redeemed from a certain date on at (above) a specified strike price. • The bond will tend not to trade above the strike price. • Implies that if interest rates fall the company can refinance. Bond Markets 29

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