# Chapter 3 Returns Spreads Yields

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Outline of Chapter 3 § Yield-to-Maturity Definition and Examples Yield and Horizon Return Yield and Relative Value: the Coupon Effect § Spreads Motivation and Definition Spread, Realized Forwards, and Horizon Return Bruce Tuckman - Debt Instruments and Markets 2
Yield-to-Maturity
Definition The yield-to-maturity of a bond is the single discount rate such that the present value of the bond's future cash flows equals the bond's market price. 7.625s of 11/15/2022 § Price (5/14/2021): 111.3969 § Find y such that 3.8125 1 + ࠵? 2 ! + 3.8125 1 + ࠵? 2 " + 103.8125 1 + ࠵? 2 # = 111.3969 y = 0.0252% Bruce Tuckman - Debt Instruments and Markets Definition and Example 4 Yields do not price bonds. Yields quote bond prices.
Easy Formula § P: bond price per 100 face amount § c: annual coupon payment § T: years to maturity § y: yield-to-maturity § 7.625s of 11/15/2022 as of mid-May 2021: 111.3969 = 7.625 0.0252% 1 − 1 1 + 0.0252% 2 !×#.% + 100 1 + 0.0252% 2 !×#.% Bruce Tuckman - Debt Instruments and Markets 5 ࠵? = ࠵? ࠵? 1 − 1 1 + ࠵? 2 !& + 100 1 + ࠵? 2 !&
Easy Formula: Implications § c = 100y P = 100, i.e., bond trades at par § c > 100y P > 100, i.e., bond trades at a premium § c < 100y P < 100, i.e., bond trades at a discount § Breaks out value of coupon payments and value of principal payment Bruce Tuckman - Debt Instruments and Markets 6 ࠵? = ࠵? ࠵? 1 − 1 1 + ࠵? 2 !& + 100 1 + ࠵? 2 !&
Bond Prices at a Yield of 1.5% Bruce Tuckman - Debt Instruments and Markets 7 60 80 100 120 140 160 180 200 220 0 5 10 15 20 25 30 Price Years to Maturity 0.00% 0.75% 1.50% 3.00% 6.00%
Yield and Horizon Return § 7.625s of 11/15/2022 111.3939 1 + ࠵? 2 ! = 3.8125 1 + ࠵? 2 + 3.8125 + 103.8125 1 + ࠵? 2 111.3939 1 + ࠵? 2 " = 3.8125 1 + ࠵? 2 ! + 3.8125 1 + ࠵? 2 + 103.8125 § 2.375s of 5/15/2051; P = 100.6875; y = 2.343% Bruce Tuckman - Debt Instruments and Markets 8 A bond's ex-post return equals its initial yield if i. All of the coupons are reinvested at the initial yield; ii. The yield at an investment horizon before maturity equals the initial yield. Coupon Reinvestment Rate 0% 2.343% 5% Return to Maturity 1.778% 2.343% 3.207%
Yield and Relative Value: the Coupon Effect Coupon Price Yield 0% 82.6446 10.0000% 5% 91.7769 9.7203% 9.5023% 100.0000 9.5023% § 82.6446 = , -.,% ' + -,, -.-,% ( § 82.6446 = , -.-,% ' + -,, -.-,% ( § 91.7769 = 0 -.,% ' + -,0 -.-,% ( § 91.7769 = 0 -.1.23,4% ' + -,0 -.1.23,4% ( § 100.000 = 1.0,34 -.,% ' + -,1.0,34 -.-,% ( § 100.000 = 1.0,34 -.1.0,34% ' + -,1.0,34 -.1.0,34% ( Bruce Tuckman - Debt Instruments and Markets 9 ̂࠵? 1 = 0% ; ̂࠵? 2 = 10% Fairly-priced bonds of the same maturity have different yields!
The Coupon Effect in U.S. Treasuries (5/14/2021) Bruce Tuckman - Debt Instruments and Markets 10 -0.5% 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% May-21 May-26 May-31 May-36 May-41 May-46 May-51 Yield Maturity Date -0.50% -0.25% 0.00% 0.25% 0.50% 0.75% 1.00% 1.25% 1.50% 1.75% May-21 May-23 May-25 May-27 May-29 May-31 Yield Maturity Date Coupon <= 5% Coupon >= 5.25% Maturity Coupon (%) Yield (%) Coupon (%) Yield (%) 5/15/2040 1.125 2.237 4.375 2.107 8/15/2040 1.125 2.245 3.875 2.138 11/15/2040 1.375 2.245 4.25 2.140 2/15/2040 1,875 2.236 4.75 2.133