School

New York University Stern School of Business **We aren't endorsed by this school

Course

FINANCE 3103

Subject

Finance

Date

Nov 20, 2023

Type

Other

Pages

19

Uploaded by pakho1999 on coursehero.com

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

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

Spreads

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.

Breakout Sessions

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%