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# P1.T4.901. Exploiting arbitrage opportunities with a replicating bond portfolio (also: clean versus dirty, and day-count conventions) (Tuckman Ch.1)

#### Nicole Seaman

##### Director of FRM Operations
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Learning objectives: Construct a replicating portfolio using multiple fixed income securities to match the cash flows of a given fixed income security. Identify arbitrage opportunities for fixed income securities with certain cash flows. Differentiate between “clean” and “dirty” bond pricing and explain the implications of accrued interest with respect to bond pricing. Describe the common day-count conventions used in bond pricing

Questions:

901.1. Below is displayed the discount function implied by a set of U.S. Treasury Notes (this is Tuckman's Tables 1.2 and 1.3 with refreshed dates): (Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)):

As of May 28th, 2019, your test of the Law of One Price reveals a price discrepancy: the U.S. Treasury Note that pays a 0.750% coupon and matures in 1.5 years on 11/30/2020 (aka, 3/4s of November 30, 2020) trades at a market price of only $99.00. However, its theoretical price per the above discount factors should be$100.255 = $0.3750*0.99925 +$0.3750*0.99648 + $100.3750*0.99135. Consequently, an arbitrage trade is rendered possible and this trade is shown below (this is based on Tuckman's Table 1.5): (Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)): The replicating portfolio is shown in column 5 (highlighted in orange), while the 3/4s of November 30, 2020 bond's cash flows are shown in column 6 (highlighted in blue). If the arbitrage trade scales up to$5.0 million in face value, which is nearest to the profit (aka, net proceeds) implied by this trade?

a. $1.26 b.$31,375
c. $62,750 d.$6.3 million

901.2. Consider a bond with a face value of $100.00, a maturity of three years, and a yield of 7.0% per annum that pays a semi-annual coupon. Similar to Tuckman's Figure 1.4, we can plot this bond's price over time (that is, over the next three years until it matures) if we assume its yield does not change. This scenario is called "unchanged yield(s)". Today, if the bond's coupon rate is 5.0%, then the bond price is$94.67 per the calculator's TVM keystrokes: 6 N, 3.5 I/Y, 2.5 PMT, 100 FV and CPT PV = -94.6714; notice that the coupon rate is always given in per annum terms (to avoid confusion) even as the coupon pays twice per year. When the coupon rate is less than the bond yield, the bond's price is less than par. Similarly, if the bond's coupon rate is 7.0%, then the bond's price is $105.33; because the coupon rate is now above the yield, we expect the price to be above par. (Tuckman's Figure 1.4 Source: Bruce Tuckman, Fixed Income Securities, 3rd Edition (Hoboken, NJ: John Wiley & Sons, 2011)): Given the assumption of unchanged yields, each of the following is necessarily true about the bond's price line plot EXCEPT which statement is NOT necessarily true? a. If the bond's semi-annual coupon rate is 5.0%, the flat price pulls to par such that the flat price line plot is strictly increasing b. If the bond's semi-annual coupon rate is 5.0%, the full price line plot is always below$100.00; i.e., the full price never jumps above $100.00 c. if the bond's semi-annual coupon rate is 9.0%, the flat price pulls to par such that the flat price line plot is strictly decreasing d. If the bond's semi-annual coupon rate is 9.0%, the full price line plot is always above$100.00; i.e., the full price never drops below $100.00 901.3. Barbara is retrieving the flat (aka, clean, quoted) price of bond by subtracting the accrued interest from the full (aka, dirty, cash, invoice) price on its settlement date of March 5th, 2019 (this date can be represented as "2019-03-05" in ISO 8601 format a, see https://en.wikipedia.org/wiki/ISO_8601). Each bond has a face value of$1,000 and pays a semi-annual coupon with a coupon rate of 9.0% per annum. The coupon payment dates are December 31st and June 30th, such that the last coupon (before the settlement date) was paid on 2018-12-31. The bond's yield is 5.0% per annum with semi-annual compound frequency.

The bond position is for a face value of $10,000,000 and Barbara initially computed the accrued interest assuming a 30/360 day count convention, as if these were U.S. municipal bonds. But actually, these are U.S. Treasury bonds and therefore employ an actual/actual day count convention. Which is nearest to her error with respect to accrued interest on the$10.0 million position?

a. $15.92 b.$162.25
c. $2,320.00 d.$3,384.00