Bond Prices and Yields
Part 2
Yield Curve
• A curve can be developed that shows various annualized
yields-to-maturity, y, against times-to-maturity, T, for bonds
from the same issuing entity or from issuing entities with the
same risk
y
0
T
2
Current US Treasury Yield Curve
www.treasury.gov
3
Bond Credit Rating
From Investopedia
interest coverage ratio =
EBIT
interest expense
4
Yield Curve
• A corporate bond, say a AA rated bond, might
have the following yield curve
AA Corporate Bond Yield Curve
risk premium or credit spread
y
U.S. Treasury Debt Yield Curve
0
T
5
Zero Coupon Yield Curve
• An especially useful version of the yield curve plots
zero coupon bond yields against times-to-maturity
• A zero coupon yield curve depicts pure interest
rates with no ambiguity due to coupon
reinvestment risk
• These curves are not observable since no bonds
with time to maturity greater than one year are
issued, but can be constructed from coupon bond
yield curves
6
Liquidity Risk
• Liquidity risk results from a bond issue having few buyers
and sellers
o The issue is ‘illiquid’ or not ‘liquid’
• Older U.S. Treasuries (referred to as ‘off the run’) can trade
with a liquidity rate premium compared with newer issued
U.S. Treasuries (referred to as ‘on the run’)
y = f(risk free time value of money, credit risk, liquidity risk)
7
Price – Yield Curve
F=$1000
c=7% semiannual
T=4.5 yrs
Each (p,y) point
results from a DCF
m×N
P=å
i=1
CFi
æ yö
ç1+ ÷
è mø
i
Illustrates how price changes as yield-to-maturity changes for a
particular bond ( c, m, N, and F are constant)
8
Homework 13
• Plot the price – yield curve for the following bond
o
o
o
o
o
Par value: $100
c = 3%
m=2
N = 5, T = 5.0
Plot P for yields, y, varying from .5% to 6.0%
• Chose a y increment that results in a smooth curve
• Submit a knitr pdf with echoed code, plot, and
markdown descriptions
9
Determine the Fair Price of a Bond
• In this case c, N, m and the zero coupon yield
curve, zi, are known
m×N
• Compute the fair value, P
CFi
P=å
ti
(1+z
)
i
i=1
CFi
Cash flow diagram
ti for bond cash flows
P
Zero coupon bond yield curve
zi
0
Ti for zero coupon bonds
10
Determine the Fair Price of a Bond
F=$1000
c=7% semiannual
T=4.5 yrs
With the following zero coupon yield curve
11
Homework 14
• Calculate the fair price of the bond from the
previous slide
• Print a table with cash flows, discount factors, and
discounted cash flows
• Submit a knitr pdf with echoed code, discount
factor plot, and markdown descriptions
12
Amortizing Bond or Loan
•
•
Bond principal is repaid periodically, not all at maturity
Examples are home and automobile loans - amortizing loan
•
Given the nominal annual interest rate, r, m, P, and N, what is the
monthly payment, C?
mN
C
C : monthly payment
P=
i
æ
ö
r
o Includes principal repayment and interest
i=1
1+
ç
÷
N : number of years
è mø
m : number of compounding periods per year
(12 for home and auto loans)
r : nominal fixed interest rate for the loan
P : loan principal (the mortgage amount)
Solve for C using Excel Goal Seek, R uniroot, …
o Find the value of C that equates the left and right hand sides
•
•
•
•
•
•
å
13
Amortizing Bond
F
C
F
14
Homework 15
• You wish to borrow $300,000 at 6.5% annual fixed
with monthly payments for 30 years
• What is your monthly payment?
• What is your total payout over 30 years?
• How much total interest will you pay?
• Submit a knitr pdf with echoed code and
detailed markdown descriptions
15
Formulas
• Annuity
o An annuity is a finite sequence of fixed payments, C. If the
nominal yield is y, then the present value, P, is
æ
ö
ç
÷
1
1
÷
P=C × ç
mN
ç æ y ö æ y öæ y ö ÷
ç çè m ÷ø ç ÷ç1+ ÷ ÷
è m øè m ø ø
è
o The formula can be rearranged to compute the fixed
payment, C, if the present value, P, is known
æ æ y öæ y öm×N ö
ç P × ç ÷ç1+ ÷ ÷
è m øè m ø ÷
C= ç
ç æ y öm×N
÷
ç ç1+ ÷ -1 ÷
è è mø
ø
16
Formulas
• Annuity Example
o Home mortgage example
• $300,000 loan at 6.5% fixed rate compounded
monthly for 30 years
æ æ y öæ y öm×N ö
ç P × ç ÷ç1+ ÷ ÷
è m øè m ø ÷
C= ç
ç æ y öm×N
÷
1+
-1
÷
ç ç
÷
è è mø
ø
æ $300,000 × 0.542% × (1+0.542%)360 ö
C= ç
÷ =$1896.20
(1+0.542%)360 -1
è
ø
17
Formulas
• Bonds
o Annuity for coupon payment plus the discounted par
value
æ
ö
ç
÷
1
1
F
÷+
P=C × ç
ç æ y ö æ y öæ y öm×N ÷ æ y öm×N
ç çè m ÷ø ç ÷ç1+ ÷ ÷ ç1+ ÷
è m øè m ø ø è m ø
è
o Example: F=$1000, c=7% semi-annual, T=4.5 yrs,
y (annual nominal yield) = 8%
æ
ö
ç
÷
1
1
÷ + $1000 =$962.82
P=$35× ç
ç æ 8% ö æ 8% öæ 8% ö9 ÷ æ 8% ö9
÷ ÷ ç1+
÷
ç çè 2 ÷ø ç ÷ç1+
è
ø
è
ø
è
ø
2
2
2
è
ø
18
Formulas
• Bonds
o Bond with fractional initial period
previous coupon
next coupon
F+C
C
d-e=139 e=227
6/30/15
F = $1000
c = 5%
C= $50
y = 6%
e = 227 days
d = 366 days
N, M = 6
m =1
6/30/16
6/30/17
6/30/18
6/30/19
6/30/20
6/30/21
Now
11/16/201
5
ö
é æ
ùæ
ö
÷
ê ç
úç
÷
÷
1
1
F
1
úç
÷+
P= êC × ç1+
e ÷
ê ç æ y ö æ y ö æ y öM-1 ÷ æ y öM-1 úç
d
æ
ö
ê ç çè ÷ø ç ÷ × ç1+ ÷ ÷ ç1+ ÷ úçç ç1+ y ÷ ÷÷
m èmø è mø ø è mø û
ë è
èè m ø ø
M = N × m (number of periods)
19
Formulas
• Bond with fractional initial period
previous coupon
next coupon
F+C
C
d-e=139 e=227
6/30/15
6/30/16
Now
11/16/201
5
F = $1000
c = 5%
C= $50
y = 6%
e = 227 days
d = 366 days
N, M = 6
m =1
6/30/17
6/30/18
6/30/19
6/30/20
6/30/21
ö
é æ
ùæ
ö
1
1
F
1
÷
úç
÷+
P= êC × çç1+ N-1 ÷
N-1 ç
e ÷
êë è y y × (1+y) ø (1+y) úûç 1+y d ÷
) ø
è(
é æ
ö 1000 ùæ 1 ö
1
1
÷
úç
÷+
P= ê50 × çç1+ 5÷
5 ç
227 ÷
êë è .05 .05× (1+.05) ø (1+.05) úûç 1+y 366 ÷
) ø
è(
P= [ 260.62+747.26] × (.9645)
P=972.10
20
Formulas
• Bond with fractional initial period
previous coupon
next coupon
F+C
C
d-e=139 e=227
6/30/15
6/30/16
6/30/17
6/30/18
6/30/19
6/30/20
6/30/21
Now
11/16/201
5
F = $1000
c = 5%
C= $50
y = 6%
e = 227 days
d = 366 days
N, M = 6
m =1
21