1. Bond Markets Investments Chapter 7 QFE Section 1.1—1.2 & 20.1

2. Bond Markets2 Payments: Redemption value, M, paid at maturity, n, and Coupons, C t, paid at specified dates, t, until t = n. C t 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).

3. Bond Markets3 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, r f, commensurate in size to preceived risk level of income stream (Security of C t + time-to-maturity related interest rate risk).

4. Bond Markets4 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.

5. Bond Markets5 Pure Discount Bonds Recall, Thus, the price today of money to be received in n periods time should be commensurate with the discount rate: rs (n) is the price of n-period money (annual) Spot Rate for n-period money: rs n = f(P, M, n)

6. Bond Markets6 Example: Zeroes & Spot Rates P = 62,321.30, M = 100,000, n = 6years Spot rate?

7. Bond Markets7 Coupon Paying Bonds? Stream of coupon payments C t, 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.

8. Bond Markets8 Current Yield A.k.a. running/flat/interest yield: Quick summary of simple interest, i.e. annual income relative to expenditure Caveats: –No capital gain –Time to Maturity & Face Value? –Interest on coupons?

9. Bond Markets9 Yield to Maturity (YTM) YTM can be calculated ex ante: YTM = y = f(P,M,C,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.

10. Bond Markets10 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?

11. Bond Markets11 Aside: Pricing an Annuity Recall, the price of a perpetuity: Thus, an annuity in n periods time should cost: The difference should be the price of a T-period annuity.

12. Bond Markets12 Breaking-Up a Coupon Paying Bond 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.

13. Bond Markets13 Example: YTM of Semi-Annual Coupon Paying Bond P = 900C = 10% of M = 1,000, 3 years to maturity semi-annual. y? y = 0.142 = 14.2% p.a.

14. Bond Markets14 Example: Price when YTM given? 20-year, C = 10% of M = 1,000, YTM = 0.11 p.a. semi-annual P? or

15. Bond Markets15 (One Period) Holding Period Return (HPR) Ex post measure of return

16. Bond Markets16 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:

17. Bond Markets17 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:

18. Bond Markets18 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.

19. Bond Markets19 Bond Pricing 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.

20. Bond Markets20 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: r 1 = 0.05 r 2 = 0.06

21. Bond Markets21 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.

22. Bond Markets22 Example: Bootstrapping Spot rates (annual return) given for first six months, r 1 = 8%, and year, r 2 = 8.3%. Calculate the 18-month spot rate given an 18- month coupon paying bond with C = 8.5% of M = 100 semi-annual.

23. Bond Markets23 Coupon Stripping C = 12.5% of FV = 100 = P, semi-annual, 20 yrs YTM? Spots: r 6months = 0.08 and r 12months = 0.083 PV(C 1 ) = 6.25/1.04 = 6.0096 PV(C 2 ) = 6.25/(1.0415) 2 = 5.7618 Profits?

24. Bond Markets24 Equilibrium Price Spot 1-year r 1 = 0.1 Spot 2-year r 2 = 0.11 Consider 2-year coupon bond, C= 9% of M = 1000 & P = 966.4866 Stripping coupons: PV(C 1 )= 90/1.1 = 81.8182 PV(C 1 )= 1090/1.11 2 = 884.668 What is your guess as to the YTM?

25. Bond Markets25 Accrued Interest Cum-dividend: clean + accrued Ex-dividend: clean - rebate

26. Bond Markets26 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

27. Bond Markets27 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

28. Bond Markets28 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.

29. Bond Markets29 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.