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