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 2Part 2

2. Yield CurveYield 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

3. Current US Treasury Yield CurveCurrent US Treasury Yield Curve
3www.treasury.gov

4. Bond Credit RatingBond Credit Rating
From Investopedia
4
interest coverage ratio =
EBIT
interest expense

5. Yield CurveYield Curve
• A corporate bond, say a AA rated bond, might
have the following yield curve
y
0 T
AA Corporate Bond Yield Curve
U.S. Treasury Debt Yield Curve
risk premium or credit spread
5

6. Zero Coupon Yield CurveZero 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

7. Liquidity RiskLiquidity 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

8. Price – Yield CurvePrice – Yield Curve
F=$1000
c=7% semiannual
T=4.5 yrs
Illustrates how price changes as yield-to-maturity changes for a
particular bond ( c, m, N, and F are constant)
Each (p,y) point
results from a DCF
8
P=
CFi
1+
y
m





÷
i
i=1
m×N
∑

9. Homework 13Homework 13
• Plot the price – yield curve for the following bond
o Par value: $100
o c = 3%
o m = 2
o N = 5, T = 5.0
o 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

10. Determine the Fair Price of a BondDetermine the Fair Price of a Bond
• In this case c, N, m and the zero coupon yield
curve, zi, are known
• Compute the fair value, P
P=
CFi
(1+zi )ti
i=1
m×N
∑
zi
0 Ti for zero coupon bonds
ti for bond cash flows
CFi
Cash flow diagram
Zero coupon bond yield curve
10
P

11. Determine the Fair Price of a BondDetermine the Fair Price of a Bond
F=$1000
c=7% semiannual
T=4.5 yrs
With the following zero coupon yield curve
11

12. Homework 14Homework 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

13. Amortizing Bond or LoanAmortizing Bond or Loan
• Amortizing bond principal is repaid periodically, not all at maturity
• Examples are home and automobile loans - amortizing loans
• Given the nominal annual interest rate, r, m, P, and N, what is the
monthly payment, C?
• C : monthly payment
o Includes principal repayment and interest
• N : number of years
• 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
P=
C
1+
r
m





÷
i
i=1
mN
∑
13

14. Amortizing BondAmortizing Bond
14
F
F
C

15. Homework 15Homework 15
• You 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

16. FormulasFormulas
• Annuity
o An annuity is a finite sequence of fixed payments, C. If the
nominal yield is y, then the present value, P, is
o The formula can be rearranged to compute the fixed
payment, C, if the present value, P, is known
P=C×
1
y
m





÷
-
1
y
m





÷ 1+
y
m





÷
mN








÷
÷
÷
÷
C=
P×
y
m





÷ 1+
y
m





÷
m×N
1+
y
m





÷
m×N
-1








÷
÷
÷
÷
16

17. FormulasFormulas
• Annuity Example
o Home mortgage example
• $300,000 loan at 6.5% fixed rate compounded
monthly for 30 years
C=
P×
y
m





÷ 1+
y
m





÷
m×N
1+
y
m





÷
m×N
-1








÷
÷
÷
÷
C=
$300,000×0.542% ×(1+0.542%)360
(1+0.542%)360
-1





÷=$1896.20
17

18. FormulasFormulas
• Bonds
o Annuity for coupon payment plus the discounted par
value
o Example: F=$1000, c=7% semi-annual, T=4.5 yrs,
y (annual nominal yield) = 8%
P=C×
1
y
m





÷
-
1
y
m





÷ 1+
y
m





÷
m×N








÷
÷
÷
÷
+
F
1+
y
m





÷
m×N
18
P=$35×
1
8%
2





÷
-
1
8%
2





÷ 1+
8%
2





÷
9








÷
÷
÷
÷
+
$1000
1+
8%
2





÷
9
=$962.82

19. FormulasFormulas
• Bonds
o Bond with fractional initial period
P= C×1+
1
y
m





÷
-
1
y
m





÷×1+
y
m





÷
M-1








÷
÷
÷
÷
+
F
1+
y
m





÷
M-1












1
1+
y
m





÷
e
d









÷
÷
÷
÷
÷
M = N×m (number of periods)
19
d-e=139 e=227
previous coupon next coupon
6/30/15 6/30/16 6/30/17 6/30/18 6/30/19 6/30/20 6/30/21
F = $1000
c = 5%
C= $50
y = 6%
e = 227 days
d = 366 days
N, M = 6
m =1
Now
11/16/201
5
C
F+C

20. FormulasFormulas
• Bond with fractional initial period
P= C×1+
1
y
-
1
y×1+y( )
N-1






÷
÷+
F
1+y( )
N-1








1
1+y( )
e
d






÷
÷÷
P= 50×1+
1
.05
-
1
.05×1+.05( )
5






÷
÷+
1000
1+.05( )
5








1
1+y( )
227
366






÷
÷÷
P= 260.62+747.26[ ] ×.9645( )
P=972.10 20
d-e=139 e=227
previous coupon next coupon
6/30/15 6/30/16 6/30/17 6/30/18 6/30/19 6/30/20 6/30/21
F = $1000
c = 5%
C= $50
y = 6%
e = 227 days
d = 366 days
N, M = 6
m =1
Now
11/16/201
5
C
F+C

21. FormulasFormulas
• Bond with fractional initial period
21
d-e=139 e=227
previous coupon next coupon
6/30/15 6/30/16 6/30/17 6/30/18 6/30/19 6/30/20 6/30/21
F = $1000
c = 5%
C= $50
y = 6%
e = 227 days
d = 366 days
N, M = 6
m =1
Now
11/16/201
5
C
F+C