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Bonds Prices and Yields
Bonds
Corporations and government entities can raise capital by selling bonds Long term liability (accounting) Debt capital (finance)
The bond has Principal, par, or face value: F Price: P Yield: y (actually “yield to maturity” and the discount rate) Maturity date, time to maturity, term, or tenor: T
Date at which the bond principal, F, is returned to investors In the case of a coupon bond (as opposed to a zero coupon bond)
Coupon rate: c (annual, simple, nominal rate) Annual payment frequency: m; or period Dt
In the U.S. semiannual coupons is typical: m = 2 or Dt = .5
2
Zero Coupon Bonds
ZCBs do not pay a coupon The return and ‘yield’ (rate) is due to the purchase price at
a discount to face value U.S. Treasury bills (T – bills) are zero coupon bonds
Time-to-maturity at issue is 4, 13, 26, 52 weeks Face value $100 to $5,000,000
A ZCB yield is the interest rate, and the discount rate denoted z
3
F
P
t=0
t=T
Zero Coupon Bond
For T ≤ 1 year:
where z is the annual simple rate or yield
For T > 1 year
where z is the annualized effective rate or yield
If a bond has a term of a year or less, simple interest is used, otherwise compound annual interest is used by convention
T)z(1F
P
Tz)(1F
P
4
F
P
t=0
t=T
Zero 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?
5
T)z(1F
P
Tz)(1F
P
Coupon Bond
P = current
price
C = coupon payment F = face or par value
t=0.0 t=Dt t=2∙Dt t=M∙Dt=T
i=0 i=1 i=2 i=M
t0=0.0 t1=Dt t2=2Dt tM= M∙Dt =T
6
Coupon 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, Dt
The total cash flow at time ti, CFi, is defined as:
7
$8.125 C.5t1000$F
%625.1cexample
tFc C
%y12y
1
%632.112
1.625%1
y rate, coupon Effective
2
2
T=num of years (floating)N=num of years (integer)
m=periods per year
In this course, generally M=Nm
360= 30 12
Coupon Bond Yield
Yield to maturity is the actual yield achieved for a coupon bond if The bond is held to maturity, and Each coupon payment is reinvested at a rate of return of y through
time T The risk that coupons cannot be reinvented at a rate greater than or
equal to y due to market conditions is called “reinvestment risk”
The yield to maturity is the investor’s expected return on investment and is thus the issuer’s rate cost It’s the issuer’s cost of debt, kD, for the bond
The yield reflects both the time value of money and the credit worthiness of the borrower The expected variance in the cash flow is reflected in the yield
8
Bond Price
The discount rate y is the yield to maturity or simply the yield on a coupon bond
It’s an 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 side
M
1it
ii)y(1
CFP
M
1ii
i
my
1
FCP
9
y is the nominal annual yield to maturity in this formula with integer periods
y is effective annual yield to maturity in this formula with discrete real time periods
For a fractional initial coupon period: t1 < ∆t
Fractional 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+Dt tM= T
C = coupon payment F = face or par value
10
Zero Coupon Bonds Again
A bond dealer can split a coupon bond into ZCBs one for the principal and one for each coupon 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
11
Bond Equation Applications
Find the yield-to-maturity, y, from a known market price, P Solve for y (nominal, y, or effective, y ‘bar’)
Solve for the roots of a nonlinear equation In this course use Excel Goal Seek
Example: Compute both the effective and nominal yield for a bond with $1000 face value, current market price of $800, coupon rate of 7% paid semiannually, and 4.5 years to maturity.
M
1it
ii)y(1
CFP
M
1ii
i
my
1
FCP
12
Bond Equation Applications
$1,000 F7.00% c nominal
13.434% y effective t CF DF DCF
0 $0 $0.000.5 $35 0.939 $32.86
1 $35 0.882 $30.851.5 $35 0.828 $28.97
2 $35 0.777 $27.202.5 $35 0.730 $25.54
3 $35 0.685 $23.983.5 $35 0.643 $22.51
4 $35 0.604 $21.144.5 $1,035 0.567 $586.94
Sum $1,315 P $800.00
13.011% y nominalt i CF DF DCF
0 0 $0 $0.000.5 1 $35 0.939 $32.86
1 2 $35 0.882 $30.851.5 3 $35 0.828 $28.97
2 4 $35 0.777 $27.202.5 5 $35 0.730 $25.54
3 6 $35 0.685 $23.983.5 7 $35 0.643 $22.51
4 8 $35 0.604 $21.144.5 9 $1,035 0.567 $586.94
Sum $1,315 P $800.00
M
1it
ii)y(1
CFP
M
1ii
i
my
1
FCP
13
Bond Equation Applications
Convert the nominal yield to the effective yield
Find market price from a known yield For the bond in the last example, what is the price?
Given an effective annual yield of 12% or A nominal annual yield of 12%
12y
1y
12
%011.131%434.13
2
2
14
Bond Equation Applications
$1,000 F7.00% c nominal
12.000% y effective t CF DF DCF
0 $0 $0.000.5 $35 0.945 $33.07
1 $35 0.893 $31.251.5 $35 0.844 $29.53
2 $35 0.797 $27.902.5 $35 0.753 $26.36
3 $35 0.712 $24.913.5 $35 0.673 $23.54
4 $35 0.636 $22.244.5 $1,035 0.601 $621.53
Sum $1,315 P $840.34
M
1it
ii)y(1
CFP
12.000% y nominalt i CF DF DCF
0 0 $0 $0.000.5 1 $35 0.943 $33.02
1 2 $35 0.890 $31.151.5 3 $35 0.840 $29.39
2 4 $35 0.792 $27.722.5 5 $35 0.747 $26.15
3 6 $35 0.705 $24.673.5 7 $35 0.665 $23.28
4 8 $35 0.627 $21.964.5 9 $1,035 0.592 $612.61
Sum $1,315 P $829.96
M
1ii
i
my
1
FCP
15
Bond Equation Applications
For the bond with a 12% effective yield and price $840.34 at time 0, here’s a plot of price as time progress from 0 to 4.5 years assuming a constant yield of 12%
$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
Pric
e16
Corporate Credit Rating
From Investopedia
17
AAA companies
Reinvestment Risk
Consider a $1000 bond with a coupon rate of 10% paid annually for 10 years. Initially, the yield is 11%, the price is $941.11, and the yield curve is flat. Prior to the payment of the next coupon, we consider three scenarios1. the yield curve shifts parallel down to 9%2. the yield curve remains flat at 11%3. the yield curve shifts parallel up to 12%What are the actual yields?
$1,000 F10.00% c nominal Year CF DF DCF 9% 11% 12%11.00% y nominal 1 100$ 0.9009 90.09$ 217.19$ 255.80$ 277.31$
2 100$ 0.8116 81.16$ 199.26$ 230.45$ 247.60$ 3 100$ 0.7312 73.12$ 182.80$ 207.62$ 221.07$ 4 100$ 0.6587 65.87$ 167.71$ 187.04$ 197.38$ 5 100$ 0.5935 59.35$ 153.86$ 168.51$ 176.23$ 6 100$ 0.5346 53.46$ 141.16$ 151.81$ 157.35$ 7 100$ 0.4817 48.17$ 129.50$ 136.76$ 140.49$ 8 100$ 0.4339 43.39$ 118.81$ 123.21$ 125.44$ 9 100$ 0.3909 39.09$ 109.00$ 111.00$ 112.00$
10 1,100$ 0.3522 387.40$ 1,100.00$ 1,100.00$ 1,100.00$ Sum 941.11$ 2,519.29$ 2,672.20$ 2,754.87$
Yield To Maturity 10.35% 11.00% 11.34%
Bond Price Calculation Future Value of Coupon Reinvestment
18
Plot price v. yields to maturity
$700
$800
$900
$1,000
$1,100
$1,200
$1,300
0% 2% 4% 6% 8% 10% 12% 14% 16%
Yield
Pric
e
F=$1000c=7% semiannualT=4.5 yrs
Bond “price – yield” or P-y curve
Illustrates how price changes as yield-to-maturity changes for a particular bond ( c, m, M, and F are constant)
Each point represents a DCF calculation
M
1it
ii)y(1
CFP
19
Home Mortgage Calculation
Given the nominal interest rate, m=12, P, and N, what is the monthly payment, C?
C : monthly payment Includes principal repayment and interest –
there is no return of principal “F” N : number of years m : number of compounding periods per year (12 for home loans) y : nominal fixed interest rate for the loan P : loan principal (the mortgage amount) Solve for C using Excel Goal Seek
Find the value of C that equates the left and right hand sides
M
1ii
i
my
1
CP
20
Mortgage Example
You wish to borrow $300,000 at 6.5% fixed for 30 years. The following excel table shows the calculations for the first
12 months and the last 5 months. The monthly payment of $1896 is determined using goal seek
to force the sum of the last column to $300,000. Note that you will pay out $682,633 in principal and interest
$300,000 in principal $382,633 in interest
21
Mortgage Example
t i CF DF DCF0.000 0 -$ -$ 0.083 1 1,896$ 0.995 1,886$ 0.167 2 1,896$ 0.989 1,876$ 0.250 3 1,896$ 0.984 1,866$ 0.333 4 1,896$ 0.979 1,856$ 0.417 5 1,896$ 0.973 1,846$ 0.500 6 1,896$ 0.968 1,836$ 0.583 7 1,896$ 0.963 1,826$ 0.667 8 1,896$ 0.958 1,816$ 0.750 9 1,896$ 0.953 1,806$ 0.833 10 1,896$ 0.947 1,796$ 0.917 11 1,896$ 0.942 1,787$ 1.000 12 1,896$ 0.937 1,777$
29.667 356 1,896$ 0.146 277$ 29.750 357 1,896$ 0.145 276$ 29.833 358 1,896$ 0.145 274$ 29.917 359 1,896$ 0.144 273$ 30.000 360 1,896$ 0.143 271$
Sum 682,633$ P 300,000$
M
1ii
i
my
1
CP
$300,000 P6.500% y nominal
12 m6.697% y annual effective0.542% y monthly effective
22
Perpetuity 23
Now in the case that M=∞C is constant
and of course y < 1
This is a perpetuity
myC
P
If a nominal annual rate, y, is used then
P
C
i
Example: How much money do you need to invest, P, to pay out $1 per year forever if the pay out rate is 10% (effective) per year?
24
Annuity
Now how much money do you need to invest at 10% to receive a $1 / year payout for M years ?
That’s an annuity (a perpetuity would pay out forever) P
C
i M M+1
Annuity: Present Value
Annuity: Payout
25
Annuity
Now how much money do you need to invest at 10% to receive a $1 / year payout for M years ?
That’s an annuity (a perpetuity would pay out forever)
M=20 yearsC=$1Y=10%P=$8.51
26
Annuities
Closed Form Formulas
Annuity Home mortgage annuity formula example
Bonds Annuity for coupon payment plus the discounted face
value
20.1896$1)%542.0(1
0.542%)(1 0.542%$300,000C 360
360
MM
my
1
F
my
1my
1
my1
CP
27
Closed Form Formulas
Bonds Example of bond w/ F=$1000, c=7% semi-annual,
T=4.5yrs, y annual nominal = 13.011%
Bond with fractional initial period
00.800$
2y
1
$1000
213.011%
12
13.011%
1
213.011%
135$P 99
deMM
my
1
1
my
1
F
my
1my
1
my1
1CP
28
Closed Form Formulas
.825 .175
last coupon
next coupon
e=64 days d = 365 dayse/d=.175
8/15/08 8/15/09 8/15/10 8/15/11 8/15/12 8/15/13 8/15/14
6/12/09
F=$100y=4% annualc=5% annual
y & c are effective & nominal
Clean and Dirty Price example (p. 7.10) using closed form
$100
$101
$102
$103
$104
$105
$106
$107
$108
$109
$110
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Time
Pric
e
29
70.108$)%4(1
1)%4(1
$100)%44%(1
14%1
15$P365
6455