Chapter 3 Returns Spreads Yields

Bruce Tuckman Fall 2023 Clinical Professor of Finance ©2023 All rights reserved. Debt Instruments and Markets Spreads, Yields, and Returns

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

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

Motivation § BTPs * and Bunds, May 2021 § Intuitive to quote yield spread of 1.198%, or about 120bps. * BTPs: Buoni del Tesoro Poliennali § Johnson & Johnson, August 2020 § Issued 2.10s of 9/1/2040 § Yield quoted as 75bps above a Treasury benchmark • Tsy 1.25s 5/15/2050 Bruce Tuckman - Debt Instruments and Markets 12 Coupon Maturity Price Yield BTP 0.60% 8/1/2031 95.502 1.066% Bund 0.00% 2/15/2031 101.300 -0.132% But yield spreads difficult to interpret; i. Often no benchmark w/ same maturity; ii. Coupon effect

Bond Spreads § Tsy 7.625s of 11/15/2022. P = 111.3969; PV @ benchmark curve: 111.2797 111.2797 = 3.8125 1 + 0.0154% 2 + 3.8125 1 + 0.0154% 2 1 + 0.1008% 2 + 103.8125 1 + 0.0154% 2 1 + 0.1008% 2 1 + 0.1833% 2 • Bond spread s defined such that 111.3969 = 3.8125 1 + 0.0154% + ࠵? 2 + 3.8125 1 + 0.0154% + ࠵? 2 1 + 0.1008% + ࠵? 2 + 103.8125 1 + 0.0154% + ࠵? 2 1 + 0.1008% + ࠵? 2 1 + 0.1833% + ࠵? 2 • ࠵? = −0.0727% or about -7bps • ࠵? < 0 ⟺ bond trades rich to benchmark curve § Spread can be interpreted as extra return if rates "stay the same" or if interest rate risk is hedged away. Bruce Tuckman - Debt Instruments and Markets 13

Scenario of "Realized Forwards" Term 5/15/2021 11/15/2021 5/15/2022 0.5 0.0154% 0.1008% 0.1833% 1.0 0.1008% 0.1833% 1.5 0.1833% Bruce Tuckman - Debt Instruments and Markets 14 Term Structure of Forward Rates as of Future Dates • In this scenario, forward rates perfectly predict future short-term rates.

Spread, Realized Forwards, and Horizon Return, I § Realized forwards • 6-month forwards, mid-May, 2021: 0.0154%, 0.1008%, 0.1833% • Realized 6-month forwards, mid-November, 2021: 0.1008%, 0.1833% § 7.625s of 11/15/2022 111.3969 1 + 0.0154% + ࠵? 2 1 + 0.1008% + ࠵? 2 1 + 0.1833% + ࠵? 2 = 3.8125 1 + 0.1008% + ࠵? 2 1 + 0.1833% + ࠵? 2 + 3.8125 1 + 0.1833% + ࠵? 2 + 103.8125 Bruce Tuckman - Debt Instruments and Markets 15 A bond's return to maturity is equivalent to rolling over short-term investments at forward rates plus its spread if i. Forward rates are realized; ii. The spread is unchanged over the horizon.

Spread, Realized Forwards, and Horizon Return, II § Realized forwards • 6-month forwards, mid-May, 2021: 0.0154%, 0.1008%, 0.1833% • Realized 6-month forwards, mid-November, 2021: 0.1008%, 0.1833% § 7.625s of 11/15/2022 111.3969 1 + 0.0154% + ࠵? 2 = 3.8125 + 3.8125 1 + 0.1008% + ࠵? 2 + 103.8125 1 + 0.1008% + ࠵? 2 1 + 0.1833% + ࠵? 2 0.0154% + ࠵? 2 = 3.8125 + 3.8125 1 + 0.1008% + ࠵? 2 + 103.8125 1 + 0.1008% + ࠵? 2 1 + 0.1833% + ࠵? 2 − 111.3969 111.3969 Bruce Tuckman - Debt Instruments and Markets 16 A bond's one-period return equals the short-term rate plus its spread if i. Forward rates are realized; ii. The spread is unchanged over the period.

Spread, Realized Forwards, and Horizon Return, III Term Realized Forwards High-Rate View Low-Rate View 0.5 0.0154% 0.0154% 0.0154% 1.0 0.1008% 0.1500% 0.0500% 1.5 0.1833% 0.2500% 0.1000% 7.625s 11/15/22 Realized Forwards High-Rate View Low-Rate View Price as of 5/15/21 111.2797 111.2797 111.2797 Price as of 11/15/21 107.4758 107.4148 107.5462 6m Annualized Return 0.0154% -0.0941% 0.1420% Cash as of 11/15/22 111.4464 111.4499 111.4423 Return to 11/15/22 0.1498% 0.1529% 0.1461% Cash from Rolling 1yr bonds to 11/15/22 111.4464 111.5110 111.3718 Return to Rolling 1yr bonds to 11/15/22 0.1498% 0.2078% 0.0827% Bruce Tuckman - Debt Instruments and Markets 17 0.00% 0.05% 0.10% 0.15% 0.20% 0.25% 0.5 1 1.5 Realized Forwards High-Rate View Low-Rate View For this slide, the spread of the 7.625s is set to 0.

Yield; Relative Value; Horizon Returns 1. Calculate the yield of the 1.625s of 11/15/2022 as of May 15, 2021, at a price of 102.2862. Use a calculator or Excel's solver. 2. Can you conclude from the yields in the table below that the 0.625s are cheap relative to the 6.25s? Why or why not? 3. An investor buys a bond paying annual coupons at a yield of 2% when the term structure of interest rates is flat. From then to the bond's maturity, all realized 1-year rates are greater than 2%. Is the realized bond return greater or less than 2% per year? 4. The first three annual forward rates implied from bond prices are 2%, 3%, and 3.50%. An asset manager with an investment horizon of three years strongly believes that the 1-year rate will be 2.50% in 1 year and 3% in 2 years. Which of the following is this manager's best strategy? a. Buy and roll 1-year bonds; b. Buy a 2-year bond and roll into a 1-year bond; c. Buy a 3-year bond; d. Hold cash. 5. On slide 17, using the rates in table on the left, reproduce the prices, cash flows, and returns in the table on the right. Bruce Tuckman - Debt Instruments and Markets 19 Coupon Maturity Price Yield 0.625 5/15/2030 99.270 0.642% 6.250 5/15/2030 153.042 0.587%