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Bond Arithmetic

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Bond Arithmetic Debt Instruments Set 2 Backus/Octob er 29, 1998 Bond Arithmetic 0. Overview Zeros and coup on b onds Sp ot rates and yields Day count conventions Replication and arbitrage Forward rates Yields and returns Debt Instruments 2-2 1. Why Are We Doing This? Explain nitty-gritty of b ond price/yield calculations Remark: \The devil is in the details" Intro duce principles of replication and arbitrage Debt Instruments 2-3 2. Zeros or STRIPS A zero is a claim to $100 in n p erio ds price = p n t + n t j j Pay p Get $100 n A spot rate is a yield on a zero: 100 p = n n 1 + y =2 n US treasury conventions: { price quoted for principal of 100 { time measured in half-years { semi-annual comp ounding Debt Instruments 2-4 2. Zeros continued A discount factor is a price of a claim to one dollar: p 1 n d = = n n 100 1 + y =2 n Examples US treasury STRIPS, May 1995 MaturityYrs Price $ DiscountFactor Sp ot Rate 0.5 97.09 0.9709 5.99 1.0 94.22 0.9422 6.05 1.5 91.39 0.9139 2.0 88.60 0.8860 6.15 Debt Instruments 2-5 3. Comp ounding Conventions A yield convention is an arbitrary set of rules for computing yields like sp ot rates from discount factors US Treasuries use semiannual comp ounding: 1 d = n n 1 + y =2 n with n measured in half-years Other conventions with n measured in years: 8 n > > 1 + y annual comp ounding > n > > > > > kn > > 1 + y =k \k" comp ounding n > > < ny n e continuous comp ounding k !1 d = n > > > > 1 > > 1 + ny \simple interest" > n > > > > > : 1 ny \discount basis" n All of these formulas de ne rules for computing the yield y from the discount factor d , but of course they're all n n di erent and the choice among them is arbitrary. That's one reason discount factors are easier to think ab out. Debt Instruments 2-6 4. Coup on Bonds Coup on b onds are claims to xed future payments c ,say n They're collections of zeros and can b e valued that way: Price = d c + d c + + d c 1 1 2 2 n n c c c n 1 2 + + = + 2 n 1 + y =2 1 + y =2 1 + y =2 1 2 n Example: Two-year \8-1/2s" Four coup ons remaining of 4.25 each Price = 0:9709 4:25 + 0:9422 4:25 +0:9139 4:25 + 0:8860 104:25 = 104:38: Two fundamental principles of asset pricing: { Replication: twoways to generate same cash ows { Arbitrage: equivalent cash ows should have same price Debt Instruments 2-7 5. Sp ot Rates from Coup on Bonds We computed the price of a coup on b ond from prices of zeros Now reverse the pro cess with these coup on b onds: Bond MaturityYrs Coup on Price A 0.5 8.00 100.97 B 1.0 6.00 99.96 Compute discount factors \recursively": 100:97 = d 104 d =0:9709 1 1 99:96 = d 3+d 103 1 2 = 0:9709 3+d 103 d =0:9422 2 2 Sp ot rates follow from discount factors: 1 d = n n 1 + y =2 n Debt Instruments 2-8 5. Sp ot Rates from Coup on Bonds continued Do zeros and coup on b onds imply the same discount fac- tors and sp ot rates? Example: Supp ose Bond B sells for 99.50, implying d = 2 0:9377 B seems cheap { Replication of B's cash ows with zeros: 3 = x 100 x =0:03 1 1 103 = x 100 x =1:03 2 2 { Cost of replication is Cost = 0:03 97:09 + 1:03 94:22 = 99:96 { Arbitrage strategy: buy B and sell its replication Riskfree pro t is 99.96 { 99.50 = 0.46 Prop osition. If and only if there are no arbitrage op- p ortunities, then zeros and coup on b onds imply the same discount factors and sp ot rates. Presumption: markets are approximately \arbitrage-free" Practical considerations: bid/ask spreads, hard to short Debt Instruments 2-9 5. Sp ot Rates from Coup on Bonds continued Replication continued { Replication of coup on b onds with zeros seems obvious { Less obvious but no less useful is replication of zeros with coup on b onds { Consider replication of 2-p erio d zero with x units of A a and x units of B: b 0 = x 104 + x 3 a b 100 = x 0+x 103 a b Remark: we've equated the cash ows of the 2-p erio d zero to those of the p ortfolio x ;x ofAandB a b Solution: x =0:9709 = 100=103: hold slightly less than b one unit of B, since the nal payment 103 is larger than the zero's 100 x = 0:0280: short enough of A to o set the a rst coup on of B We can verify the zero's price: Cost = 0:0280 100:97 + 0:9709 99:96 = 94:22: { Remark: even if zeros didn't exist, we could compute their prices and sp ot rates. Debt Instruments 2-10 6. Yields on Coup on Bonds Sp ot rates apply to sp eci c maturities The yield-to-maturity on a coup on b ond satis es c c c 1 2 n Price = + + + 2 n 1 + y=2 1 + y=2 1 + y=2 Example: Two-year 8-1/2s 4:25 4:25 104:38 = + 2 1 + y=2 1 + y=2 104:25 4:25 + + 3 4 1 + y=2 1 + y=2 The yield is y =6:15. Comments: { Yield dep ends on the coup on { Computation: guess y until price is right Debt Instruments 2-11 7. Par Yields We've found prices and yields for given coup ons Find the coup on that delivers a price of 100 par Price = 100 = d + + d Coup on + d 100 1 n n The annualized coup on rate is Par Yield = 2 Coup on 1 d n = 2 100 d + + d 1 n This obscure calculation underlies the initial pricing of b onds and swaps we'll see it again Debt Instruments 2-12 8. Yield Curves A yield curve is a graph of yield y against maturity n n May 1995, from US Treasury STRIPS: 7.4 7.2 7 Spot Rates Par Yields 6.8 6.6 6.4 Yield (Annual Percentage) 6.2 6 5.8 0 5 10 15 20 25 30 Maturity in Years Debt Instruments 2-13 9. Estimating Bond Yields Standard practice is to estimate sp ot rates by tting a smo oth function of n to sp ot rates or discount factors 7.4 7.2 7 6.8 6.6 6.4 Yield (Annual Percentage) 6.2 6 Circles are raw data, the line is a smooth approximation 5.8 0 5 10 15 20 25 30 Maturity in Years \Noise": bid/ask spread, stale quotes, liquidity on/o -the- runs, coup ons, sp ecial features Reminder that the frictionless world of the prop osition is an approximation Debt Instruments 2-14 10. Day Counts for US Treasuries Overview { Bonds typically have fractional rst p erio ds { You pay the quoted price plus a pro-rated share of the rst coup on accrued interest { Day count conventions govern how prices are quoted and yields are computed Details { Invoice price calculation what you pay: Invoice Price = Quoted Price + Accrued Interest u Accrued Interest = Coup on u + v u = Days Since Last Coup on v = Days Until Next Coup on { Yield calculation \street convention": Coup on Coup on Invoice Price = + w w +1 1 + y=2 1 + y=2 Coup on + 100 + + w +n1 1 + y=2 w = v=u + v n = Numb er of Coup ons Remaining Debt Instruments 2-15 10. Day Counts for US Treasuries continued US Treasuries use \actual/actual" day counts for u and v ie, we actually count up the days Example: 8-1/2s of April 97 as of May 95 Issued April 16, 1990 Settlement May 18, 1995 Matures April 15, 1997 Coup on Freq Semiannual Coup on Dates 15th of Apr and Oct Coup on Rate 8.50 Coup on 4.25 Quoted Price 104.19 Time line: Previous Settlement Next Coup on Coup on Date j j j 4/15/95 5/18/95 10/15/95 u = v = n = Debt Instruments 2-16 10. Day Counts for US Treasuries continued Price calculations: Accrued Interest = Invoice Price = Yield calculations: w = d = 1=1 + y=2 to savetyping w 2 3 w +3 104:95 = d 1 + d + d + d 4:25 + d 100 d =0:97021 y =6:14 The last step is easier with a computer Debt Instruments 2-17 11. Other Day Count Conventions US Corp orate b onds 30/360 day count convention roughly: countdays as if every month had 30 days Example: Citicorp's 7 1/8s Settlement June 16, 1995 Matures March 15, 2004 Coup on Freq Semiannual Quoted Price 101.255 Calculations: n = u = v = w = Accrued Interest = Invoice Price = 0 1 n 1 d w w +n1 @ A Invoice Price = d Coup on + d 100 1 d y =6:929 Remark: the formula works, don't sweat the details! Debt Instruments 2-18 11. Other Day Count Conventions continued Eurob onds 30E/360 day count convention ie, countdays as if every month has 30 days Example: IBRD 9s, dollar-denominated Settlement June 20, 1995 Matures August 12, 1997 Coup on Freq Annual Quoted Price 106.188 Calculations: n = u = v = w = Accrued Interest = Invoice Price = 0 1 n 1 d w w +n1 @ A Invoice Price = d Coup on + d 100 1 d y =5:831 Remark: the formula works for all coup on frequencies with the appropriate mo di cation of d [here d =1=1 + y ] Debt Instruments 2-19 11.
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