Bond price and yield pdf

Bonds Prices and Yields

Learning Objec-ves

¨  Types of bonds and bond parameters ¨  The concept of yield ¨  Bond price computed as discounted expected cash flow

¤  Applica-ons n  Compute bond yield from a known price n  Compute bond price from a known yield n  Graph the price v. yield n  Compute bond price when yield is not known n  Graph price v. Time to maturity n  Compute mortgage payments

¨  Bond price quotes

2

Bonds

¨  Corpora-ons and government en--es can raise capital by selling bonds ¤  Long term liability (accoun-ng) ¤  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, -me to maturity, term, or tenor: T

n  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) n  Coupon rate: c (annual, simple, nominal rate) n  Annual payment frequency: m; or period Δt

n  In the U.S. semiannual coupons is typical: m = 2 or Δt = .5

3

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

4

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 effec-ve 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 conven-on

T)z(1FP⋅+

=

Tz)(1FP+

=

5

F

P

t=0

t=T

Zero Coupon Bond Example

¨  The face value is $1000, the market price is $850, and the -me to maturity is 3.5 years. What is the annualized yield ?

¨  The face value is $1000, the market price is $975, and the -me-‐to-‐maturity is 0.5 years. What is the annualized yield?

$850= $1000(1+z)3.5

z= $1000$850

!

"#

$

%&

13.5

−1= 4.753%

$975= $1000(1+0.5⋅z)

z=2⋅ $1000$975

−1#

$%

&

'(=5.128%

6

T)z(1FP⋅+

=

Tz)(1FP+

=

Coupon Bond

P = current price

C = coupon payment F = face or par value

t=0.0 t=Δt t=2·∙Δt t=M·∙Δt=T i=0 i=1 i=2 i=M t0=0.0 t1=Δt t2=2Δt tM= M·∙Δt =T

7

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, Δt

¨  The total cash flow at -me ti, CFi, is defined as:

CFi $=$C$$$$$$$$for$$i<M

CFM$=$C$+$F

8

$8.125 C.5t1000$F

%625.1cexample

tFc C

=

=Δ

=

=

Δ⋅⋅=

%y12y1

%632.112

1.625%1

y rate, coupon Effective

2

2

=−⎟⎠⎞

⎜⎝⎛ +

=−⎟⎠⎞

⎜⎝⎛ +

T=num of years (floa-ng) N=num of years (integer) m=periods per year In this course, generally M=N�m 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

-me T n  The risk that coupons cannot be reinvented at a rate greater than or equal to y due to market condi-ons 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 -me value of money and the credit worthiness of the borrower ¤  The expected variance in the cash flow is reflected in the yield

9

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 lek hand side

∑= +

=M

1it

ii)y(1

CFP∑=

⎟⎠⎞

⎜⎝⎛ +

=M

1ii

i

my1

FCP

10

y is the nominal annual yield to maturity in this formula with integer periods

y is effec8ve annual yield to maturity in this formula with discrete real -me periods

¨  For a frac-onal ini-al coupon period: t1 < ∆t

Frac-onal Ini-al 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+Δt tM= T

C = coupon payment F = face or par value

11

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

12

Bond Equa-on Applica-ons

¨  Find the yield-‐to-‐maturity, y, from a known market price, P ¤  Solve for y (nominal, y, or effec-ve, y ‘bar’)

¤  Solve for the roots of a nonlinear equa-on n  In this course use Excel Goal Seek

¤  Example: Compute both the effec8ve 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

my1

FCP

13


$1,000 F7.00% c nominal

13.434% y effective t CF DF DCF

0 $0 $0.000.5 $35 0.939 $32.861 $35 0.882 $30.85

1.5 $35 0.828 $28.972 $35 0.777 $27.20

2.5 $35 0.730 $25.543 $35 0.685 $23.98

3.5 $35 0.643 $22.514 $35 0.604 $21.14

4.5 $1,035 0.567 $586.94Sum $1,315 P $800.00

13.011% y nominalt i CF DF DCF

0 0 $0 $0.000.5 1 $35 0.939 $32.861 2 $35 0.882 $30.85

1.5 3 $35 0.828 $28.972 4 $35 0.777 $27.20

2.5 5 $35 0.730 $25.543 6 $35 0.685 $23.98

3.5 7 $35 0.643 $22.514 8 $35 0.604 $21.14

4.5 9 $1,035 0.567 $586.94Sum $1,315 P $800.00

∑= +

=M

1it

ii)y(1

CFP ∑=

⎟⎠⎞

⎜⎝⎛ +

=M

1ii

i

my1

FCP

14


¤ Convert the nominal yield to the effec-ve yield

¨  Find market price from a known yield ¤ For the bond in the last example, what is the price?

n Given an effec8ve annual yield of 12% or n A nominal annual yield of 12%

12y1y

12

%011.131%434.13

2

2

−⎟⎠⎞

⎜⎝⎛ +=

−⎟⎠⎞

⎜⎝⎛ +=

15


$1,000 F7.00% c nominal

12.000% y effective t CF DF DCF

0 $0 $0.000.5 $35 0.945 $33.071 $35 0.893 $31.25

1.5 $35 0.844 $29.532 $35 0.797 $27.90

2.5 $35 0.753 $26.363 $35 0.712 $24.91

3.5 $35 0.673 $23.544 $35 0.636 $22.24

4.5 $1,035 0.601 $621.53Sum $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 $33.021 2 $35 0.890 $31.15

1.5 3 $35 0.840 $29.392 4 $35 0.792 $27.72

2.5 5 $35 0.747 $26.153 6 $35 0.705 $24.67

3.5 7 $35 0.665 $23.284 8 $35 0.627 $21.96

4.5 9 $1,035 0.592 $612.61Sum $1,315 P $829.96

∑=

⎟⎠⎞

⎜⎝⎛ +

=M

1ii

i

my1

FCP

16


¨  For the bond with a 12% effec-ve yield and price $840.34 at -me 0, here’s a plot of price as -me 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

Price

17

Yield Curve

¨  A graph can be developed that plots various (annualized) yields-‐to-‐maturity against -mes-‐to-‐maturity for bonds from the same issuing en-ty or from issuing en--es with the same risk

y

0 T

18

Bloomberg

Animated yield curve

Yield Curve

¨  The most useful version of the yield curve plots zero coupon bond yields against -mes-‐to-‐maturity

¨  A zero coupon yield curve depicts pure interest rates with no ambiguity in the risk associated with the yield

¨  Thus we can use ZCB yield curve to plot the interest rate with respect to -me-‐to-‐maturity, or holding period

¨  Yields on zero coupon U.S. Treasury debt are risk free ¨  The addi-onal yield for non-‐U.S. Treasury debt is due largely to credit risk

and is called a risk premium or a credit spread ¨  Moody’s and Standard & Poor’s evaluate bond credit risk via analysis of

the issuer’s financial posi-on and assign risk levels such as AA that translate into credit risk premiums

19

Corporate Credit Ra-ng

From Investopedia

20

AAA companies

Yield 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 and credit spread

21

Other Types of Risk

¨  Another type of risk known as liquidity risk ¨  Results from a bond having few buyers and sellers

¤  The issue is “illiquid”

¨  Newer U.S. Treasuries (referred to as “on the run”) can trade with a liquidity premium compared with old U.S. Treasuries (referred to as “off the run”) y = f(risk free -me value of money, credit risk, liquidity risk)

22

Reinvestment Risk

¨  Consider a $1000 bond with a coupon rate of 10% paid annually for 10 years. Ini-ally, 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 scenarios 1.  the yield curve shiks parallel down to 9% 2.  the yield curve remains flat at 11% 3.  the yield curve shiks 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

23

Bond Price Quotes

¨  Dirty and clean prices ¤ Dirty price

n  Price from DCF formula n  Transac-on price n  When the seller sells at this price she gets the prorated share (accumulated interest) of the next coupon

¤ Clean price n  Price quoted by bond dealer n  Excludes accumulated interest on next coupon payment

¤ Clean price = Dirty price – accumulated interest n  Accumulated as simple interest using applicable day count conven-on

24

Clean and Dirty Bond Prices

Bond purchased just aker its 8/15/2008 coupon Bond purchased 6/12/2009

From 8/15/08 to 8/15/09 is 365 days From 8/15/08 to 6/12/09 is 301 days or .825 yrs From 6/12/09 to 8/15/09 is 64 days or .175 yrs

Payment Date

t CF DF DCF

15-‐Aug-‐08 0 $0 1.000 $0.0015-‐Aug-‐09 1 $5 0.962 $4.8115-‐Aug-‐10 2 $5 0.925 $4.6215-‐Aug-‐11 3 $5 0.889 $4.4415-‐Aug-‐12 4 $5 0.855 $4.2715-‐Aug-‐13 5 $5 0.822 $4.1115-‐Aug-‐14 6 $105 0.790 $82.98

Sum $130.00 P $105.24

Payment Date

t CF DF DCF

12-‐Jun-‐09 0 $0 1.000 $0.0015-‐Aug-‐09 0.175 $5 0.993 $4.9715-‐Aug-‐10 1.175 $5 0.955 $4.7715-‐Aug-‐11 2.175 $5 0.918 $4.5915-‐Aug-‐12 3.175 $5 0.883 $4.4115-‐Aug-‐13 4.175 $5 0.849 $4.2415-‐Aug-‐14 5.175 $105 0.816 $85.71

Sum $130.00 P $108.70

F=$100, c=5%, y=4%, m=1

25

$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

Price

Dirty and Clean Prices

15 Aug 2008 Clean price @ 6/12/09 = Dirty price – accumulated interest

= $108.70 – $5 ·∙ 301 / 365 = $108.70 -‐ $4.12 = $104.58

15 Aug 2009

26 12 June 2009

Bond Price Quotes

¨  Bid and ask prices ¤  The clean price is quoted for bid and ask prices

n  The dealer will buy a bond at the bid price n  The dealer will sell a bond at the ask (offer) price

¨  Bond prices are quoted rela-ve to 100 regardless of actual par value ¤  Price is quoted as a percent of par

¨  Example

27

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

Price

F=$1000 c=7% semiannual T=4.5 yrs

Bond “price – yield” or P-‐y curve

Illustrates how price changes as yield-‐to-‐maturity changes for a par-cular bond ( c, m, M, and F are constant)

Each point represents a DCF calcula-on

∑= +

=M

1it

ii)y(1

CFP

28

Determine the fair price of a bond

¨  In this case c, m, T, and the relevant zero coupon yield curve are known

¨  Compute the fair value, P ∑= +

=M

1it

i

ii)z(1

CFP

zt

0 T for zero coupon bonds

ti for bond cash flows

CFt Cash flow diagram

Zero coupon bond yield curve

29

Example of pricing a bond

Price of the bond

F=$1000 c=7% semiannual T=4.5 yrs With the following zero coupon yield curve

$1,000 F t CF z DF DCF7.00% c nominal 0 $0 4.00% $0.00

0.5 $35 4.85% 0.977 $34.181 $35 5.20% 0.951 $33.271.5 $35 5.47% 0.923 $32.312 $35 5.70% 0.895 $31.332.5 $35 5.90% 0.867 $30.333 $35 6.08% 0.838 $29.323.5 $35 6.24% 0.809 $28.314 $35 6.40% 0.780 $27.314.5 $1,035 6.55% 0.752 $778.09Sum $1,315 P $1,024.46

0%

1%

2%

3%

4%

5%

6%

7%

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

t

z

30

Home Mortgage Calcula-on

¨  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 lek and right hand sides

∑=

⎟⎠⎞

⎜⎝⎛ +

=M

1ii

i

my1

CP

31

Mortgage Example

¨  You wish to borrow $300,000 at 6.5% fixed for 30 years. ¨  The following excel table shows the calcula-ons 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

32

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

my1

CP

$300,000 P6.500% y nominal

12 m6.697% y annual effective0.542% y monthly effective

33

Perpetuity 34

Now in the case that M=∞ C is constant and of course y < 1 This is a perpetuity

⎟⎠⎞

⎜⎝⎛

=

myCP

P=Ci

(1+y)ii=1

M

∑

P=C ⋅ 1(1+y)ii=1

∞

∑

P=Cy

P=Cy

P=Ci

(1+y)ii=1

M

∑

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% (effec-ve) per year?

Annuity 35

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

PM+1=Cy

P0M+1=

Cy⋅

1

1+y( )M

=Cy⋅ 1+y( )

−M

P =P0=Cy−Cy⋅ 1+y( )

−M

=Cy⋅ 1− 1+y( )

−M#

$%

&

'(

P=Cy⋅ 1& 1+y( )

&M"

#$

%

&'

C = P ⋅ y

1& 1+y( )&M

=P ⋅ y ⋅ 1+y( )

M

1+y( )M−1

P=C ⋅my

⋅ 1' 1+ ym

"

#$

%

&'

'M(

)

**

+

,

--

C=P ⋅ y

m

"

#$

%

&'⋅ 1+

ym

"

#$

%

&'

M

1+ ym

"

#$

%

&'

M

'1

M=20 years C=$1 Y=10% P=$8.51

Annui-es 36

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

my1

F

my1

my

1

my1CP

⎟⎠⎞

⎜⎝⎛ +

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛

−

⎟⎠⎞

⎜⎝⎛

⋅=

37


¨  Bonds ¤ Example of bond w/ F=$1000, c=7% semi-‐annual, T=4.5yrs, y annual nominal = 13.011%

¤ Bond with frac-onal ini-al period

00.800$

2y1

$1000

213.011%1

213.011%

1

213.011%

135$P 99 =

⎟⎠⎞

⎜⎝⎛ +

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ +⎟⎠⎞

⎜⎝⎛

−

⎟⎠⎞

⎜⎝⎛

⋅=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ +⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎟⎠⎞

⎜⎝⎛ +

+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ +⋅⎟

⎠⎞

⎜⎝⎛

−

⎟⎠⎞

⎜⎝⎛

+⋅=deMM

my1

1

my1

F

my1

my

1

my11CP

38


.825 .175

last coupon

next coupon

e=64 days d = 365 days e/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=$100 y=4% annual c=5% annual

y & c are effec-ve & 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

Price

39

70.108$)%4(1

1)%4(1

$100)%44%(1

14%115$P

3656455 =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

++⎟⎠

⎞⎜⎝

⎛+

−+⋅=

Economy & Finance

Bond price and yield pdf