Used with my Principles of Finance course at University of Nebraska http://www.raik381h.com/index.html

1. Bond Prices and YieldsBond Prices and Yields
Part 1Part 1

2. Learning ObjectivesLearning Objectives
• Types of bonds
• Bond parameters including yield
• Bond price as discounted expected cash flow
o Applications
• Compute bond yield from a known price
• Compute bond price from a known yield
• Plot bond price v. yield
• Compute bond price when yield is not known
• Plot bond price v. time to maturity
• Compute mortgage payments
• Bond formulas
• Bond price quotes
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3. Bond OverviewBond Overview
• Corporations and governments can raise capital by selling
bonds
o Long term liability (accounting)
o Debt capital (finance)
• The bond has
o Principal, par, or face value: F
o Price: P
o Yield: y (also the discount rate )
o Maturity date, time to maturity, term, or tenor: T
• Date at which the bond principal, F, is returned to investors
• This is different than the duration
o In the case of a coupon bond (not a zero coupon bond)
• Coupon rate: c (annual, simple, nominal rate)
• Annual payment frequency: m; or period ∆t
o In the U.S. semiannual coupons is typical: m = 2 or ∆t = .5
3

4. Zero Coupon BondsZero Coupon Bonds
• ZCBs do not pay a coupon
o No intermediate cash flow
• The rate of return or ‘yield’ is due to the bond being bought,
at P, a discount to face value, F
• U.S. Treasury bills (T – bills) are zero coupon bonds
o Time-to-maturity at issue is 4, 13, 26, 52 weeks
o Face value $100 to $5,000,000
• A ZCB yield is the interest rate,
(and the discount rate) denoted z
4
F
P
t=0
t=T

5. Zero Coupon BondZero Coupon Bond
• For T ≤ 1 year:
where z is the annual yield – a simple rate
• For T > 1 year
where z is the annualized yield – an effective rate
If a bond has a term of a year or less, simple interest is used,
otherwise compound annual interest is used
5
P=
F
(1+z×T)
P=
F
(1+z)T
F
P
t=0
t=T

6. Zero Coupon Bond ExampleZero Coupon Bond Example
• The face value is $1000, the market price is $850, and the time
to maturity is 3.5 years. What is the annualized yield ?
• The face value is $1000, the market price is $975, and the
time-to-maturity is 0.5 years. What is the annualized yield?
6
$850=
$1000
(1+y)3.5
y=
$1000
$850





÷
1
3.5
-1=4.753%
P=
F
(1+z×T)
$975=
$1000
(1+0.5×y)
y=2×
$1000
$975
-1





÷=5.128%
P =
F
(1+z)T

7. Zero Coupon BondZero Coupon Bond
• A bond dealer can split a coupon bond into ZCBs
o one for the principal and
o one for each coupon
o This is called ‘stripping’ the bond
• The advantage of a ZCB is that there is no reinvestment risk
• For a ZCB, the yield, y, is the zero coupon rate denoted as z
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8. Coupon BondCoupon Bond
8

9. Coupon Bond: MSFTCoupon Bond: MSFT
9
June 30

10. Coupon Bond: MSFTCoupon Bond: MSFT
10
June 30

11. Coupon Bond: MSFTCoupon Bond: MSFT
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12. Coupon BondCoupon Bond
12
P =
current
price
C = coupon payment
c = coupon rate (simple interest)
F = face
or par
value
t=0.0 t=∆t t=2∙∆t t=m∙N∙∆t=T
i=0 i=1 i=2 i=m∙N
t0=0.0 t1=∆t t2=2∆t tm∙N= m∙N∙∆t =T
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13. Coupon PaymentCoupon Payment
• Bond coupon cash flows, C, are defined by a nominal, simple
coupon rate, c, and a compounding frequency per year, m,
or coupon period measured in years, ∆t
o m is always 1 or 2 for a bond
o ∆t = .5 or 1 year
• The total cash flow at time ti, CFi, is defined as:
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CFi = C for i < m×N
CFm×N = C + F
C = c×F×∆t =
c×F
m
example
c=1.625%
F=$1000
∆t=.5
C = $8.125
Effective coupon rate
c= 1+
c
m





÷
m
-1
c= 1+
1.625%
2





÷
2
-1=1.632%

14. Current interest and yield summaryCurrent interest and yield summary
• Bloomberg
• Yahoo
• Bankrate
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15. Coupon Bond YieldCoupon Bond Yield
• Yield to maturity is the actual yield achieved for a coupon bond if
o The bond is held to maturity, and
o Each coupon payment is reinvested at the rate of return y through time T
• The risk that coupons cannot be reinvested at a rate greater than or equal to y
due to market conditions is called “reinvestment risk”
• The yield to maturity, y, is the investor’s expected return on the
investment of P and is thus the issuer’s rate cost
o It’s the issuer’s ‘cost of debt’, kD, for the bond
• The yield reflects both the time value of money and the credit risk of
the borrower
o The expected variance of the cash flows is reflected in the yield, y
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16. Bond PriceBond Price
• The discount rate y is the yield to maturity or simply the yield
on a coupon bond
• It’s the internal rate of return that sets the discounted cash
flow on the right hand side to the market price of the bond, P,
on the left hand sidex4
16
P=
CFi
(1+y)ti
i=1
M
∑P=
CFi
1+
y
m





÷
i
i=1
M
∑
y is the nominal annual
yield to maturity
y is effective annual
yield to maturity
M is the number of periods or “cash flows”
M = N * m if N is an integer number of years
M = T * m if T is floating point years

17. Homework 10Homework 10
• Compute the price, P, of the following bond and
the nominal yield to maturity, y
o Time to maturity 4.5 years
o Annual, nominal coupon rate is 7%
o Semi annual coupons and compounding
o Par value is $1000
o Yield to maturity 8% (effective annual rate)
• Submit a knitr pdf with explanation and echoed
code
o Print the data frame or table for clarity
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18. Homework 11Homework 11
• Compute the nominal and effective yields to
maturity for this bond
o Time to maturity 4.5 years
o Annual, nominal coupon rate is 7%
o Semi annual coupons and compounding
o Par value is $1000
o Price is $900.00
• Submit a knitr pdf with explanation and echoed
code
o Print the data frame or table for clarity
18

19. • For a fractional initial coupon period: t1 < ∆t
Fractional Initial Time PeriodFractional Initial Time Period
For a bond with semi-annual coupons, assume that the next
coupon payment is in 3 months. The coupon payments occur at
t0=0.0, t1=0.25, t2=0.75, t3=1.25, t4 = 1.75, …
i=0 i=1 i=2 i=M
t0=0.0 t1 t2=t1+∆t tM= T
C = coupon payment
F = face
or par
value
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20. Homework 12Homework 12
• Compute the price of this bond
o Time to maturity 4.25 years
o The next coupon is paid 3 months from present
o Annual, nominal coupon rate is 7%
o Semi annual coupons and compounding
o Par value is $1000
o Yield to maturity 8% (effective annual rate)
• Submit a knitr pdf with explanation and echoed
code
o Print the data frame or table for clarity
20

21. Bond Price to MaturityBond Price to Maturity
$825
$850
$875
$900
$925
$950
$975
$1,000
$1,025
$1,050
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Time
Price
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• For a bond with price $840.34 at time 0, here’s a plot of price as
time progresses from 0 to 4.5 years assuming a constant yield to
maturity of 12%
• The plot is computed using the DCF formula. These prices are
referred to as ‘dirty’ prices. There are also ‘clean’ prices.

22. Homework 13Homework 13
• Plot price v. time for the following bond
o Time to maturity 10 years
o Annual, nominal, simple coupon rate is 5%
o Par value is $1000
o Effective annual yield to maturity is a constant 6%
o (It’s a saw tooth curve)
• Submit a knitr pdf with explanation and echoed
code
o Print the data frame or table for clarity
22