Review of Bond Pricing

Fixed Income Securities

3

What are FI securities?

Financial claims issued by governments, government agencies, state governments, municipalities, corporations, banks, and other financial intermediaries

They represent contract obligations of the issuers.

E.g., bonds – principals, coupons Also called debt securities

4

FI Markets

FI securities are issued, traded, and invested in markets that are called fixed-income markets

Representations: Issuers Financial intermediaries investors

5

FI Markets

Issuers: Governments and their

agencies Corporations State and

municipalities Special purpose

vehicles (SPV) Foreign institutions

Intermediaries Primary dealers Other dealers Investment banks Credit agencies Credit and liquidity

enhancers

6

FI Markets: Investors

Governments Pension funds Insurance companies Mutual funds Commercial banks Foreign institutions House holders

7

FI: Terminology

Coupon, Principals, Time to maturity/term Bid/Ask spread Bullet security

With fixed maturity date and fixed coupons Without options embedded

Debts with options Callable / putable convertible

8

Risks

Interest rate risk Price of FI securities critically depends on the

overall interest – price uncertain when one has to sell the debt security

Market risk Reinvestment risk

Coupons are assumed to be reinvested and earn interest

Greater for longer holding periods

9

Risks: cont’d

Credit risk FI securities are contract obligation for the

issuer Issuers may fail to pay the promised cashflow,

which will lead to default What will happen when default occur? Important for low grade corporate bonds (or high

yields/junk bonds)

10

Risks: cont’d

Inflation risk Purchasing power risk Reflected in the relative size of coupon rate and

inflation rate For some FIS, coupon is indexed to some

consumer index Timing risk

When it has callable feature Very important for MBS

11

Risks: cont’d

Liquidity risk / Marketability risk Reflected through bid/ask spread Less important if plan to hold the security to

maturity FX risk Volatility risk Risk-risk

12

Classifications of FI Securities US Treasury Securities

On-the-run vs off-the-run T-bills, T-notes, T-bonds Regularly issue 3m, 6m, 1y, 2y, 3y, 5y, 10y, and

30y bonds Issued through auction

Discriminating prices Uniform prices

13

Government bonds

Canadian government debt Maturities range from 2 to 25 years Issued through auction on yields

Gilt Index linked gilt – both face value and coupons Convertible gilts Irredeemable gilts, or perpetual Single price auction

14

Government bonds

JGB 2, 5, and 10 years maturities Callable Part of the issue is underwritten by banks,

insurance company and security firms. Remaining is by auction

15

Agency Securities

They are sponsored and backed by the government, but are usually privately owned

Examples FHLB Federal National Mortgage Association (Fannie

Mae) Federal Home Loan Mortgage Corp. (Freddie Mac)

Debentures and MBS Student Loan Marketing Association (Sally Mae)

floaters

16

Corporate Debt Securities

Debt issued to raise capitals Actively traded Present credit risk

What will happen if default? Usually rated by rating agencies

Usually carry coupons Mandatory sinking funds May have call features, or convertible features High yields / junk bonds

17

Securitized Assets -- MBS

Originator1

Originator1

Originator1

Pool of Mortgages

SPV

MBS

Intermediaries

Institutional and retail investors

Credit rating agencies

Credit and liquidity Enhancements(guarantees)

18

MBS Types

Pass-through securities IOs POs High timing risk

19

Municipal Issues

General obligation bonds Revenue bonds Have default risks Tax plays important roles in this type of

bonds

Face or par value Coupon rate

Zero coupon bond Compounding and payments Indenture Issuers

Bond Characteristics

Secured or unsecured Registered or bearer bonds (Canada) Call provision Convertible provision Retractable and extendible (puttable)

bonds Floating rate bond

Provisions of Bonds

Bond Yields

04/19/23 Wulin Suo 23

Discounting

Simple rate: Time (expressed in years) is less than or equal

to one year Subject to different day count convention in

different market Annual rate:

y is called the annual rate N is the number of years

0FV= (1 )P y

0FV= (1 )NP y

04/19/23 Wulin Suo 24

Compounding

Annual compounding Semi-annual comp.: Compounding m time a year

Continuous compounding:

FV vs PV

0FV= (1 )NP y2

0FV= (1 )2NyP

0FV= (1 )mNyP m

m

0FV= yNPe

04/19/23 Wulin Suo 25

Compounding …

m=1 m=2 m=4 m=12 m=365 continuous1% 1.010000$ 1.010025$ 1.010038$ 1.010046$ 1.010050$ 1.010050$ 5% 1.050000$ 1.050625$ 1.050945$ 1.051162$ 1.051267$ 1.051271$

10% 1.100000$ 1.102500$ 1.103813$ 1.104713$ 1.105156$ 1.105171$ 20% 1.200000$ 1.210000$ 1.215506$ 1.219391$ 1.221336$ 1.221403$

100% 2.000000$ 2.250000$ 2.441406$ 2.613035$ 2.714567$ 2.718282$

04/19/23 Wulin Suo 26

Rate Conversion

1 1 c

n myn my ye

n m

ln 1 nc

yy n

n

/ 1cy nny n e

/

1 1m n

mn

yy n

m

04/19/23 Wulin Suo 27

Annuity

Pays same amount C in N consecutive periods FV:

PV

PV for perpetuity

(1 ) 1NCy

y

11

(1 )NC

y y

C

y

04/19/23 Wulin Suo 28

Yields (1)

Yield is the internal rate of return; also known as yield to maturity

Annual compounding: for a bond pays $C per year for N years, yield is defined as the y such that

2( 100)

1 (1 ) (1 )

1 1001

(1 ) (1 )

N

N N

CC CP y y y

C

y y y

04/19/23 Wulin Suo 29

Yields (2)

Write the coupon as a percentage of face value: C=100c,

P <, =, or >100 if c<y, c=y, or c>y, respectively

Current yield:

100 100(1 / )

(1 )Nc c y

Py y

/cy C P

04/19/23 Wulin Suo 30

Yields: Semi-Annual Notation:

C: total annual coupon y: semi-annual compounding yield P: price of the bond N: number of coupons remaining

Premium, par, and discount Also called bond equivalent yield

1 1001

(1 / 2) (1 / 2)N N

CP

y y y

04/19/23 Wulin Suo 31

Yields: m-Compounding

Compounding m-times a year, with N years remaining:

Continuous yield:

1 1001

(1 / ) (1 / )mN mN

CP

y y m y m

m

0

100

100

NyN yt

yN

P e Ce dt

C Ce

y y

04/19/23 Wulin Suo 32

Yields in Other forms

Yield to call yield to 1st call, yield to 2nd call, etc yield to put

Yield to par Yield to worst: compute the yield to maturity,

yield to call, and yield to put

04/19/23 Wulin Suo 33

Cash Flow Yield

Amortizing securities: cash flow includes interest + principle prepayments (e.g., MBS/ABS)

CF yield: yield such that the PV of the projected cash flows equal to par

04/19/23 Wulin Suo 34

Yield Measure for Floating Rate Securities Coupon rate may change according to some

reference rate impossible to determine the future cash flow

Effective margin: a measure estimates the average spread or margin over the reference rate that the investor can expect to earn over the life of the security

04/19/23 Wulin Suo 35

Floating Rate Securities …

How to compute the effective margin: Step 1: Determine the CF assuming the reference

rate does not change Step 2: Select a margin (spread) Step 3: Discount the CF in Step 1 by the current

value of the reference rate plus the margin in Step 2

Step 4: Compare PV in Step 4 with the price.

04/19/23 Wulin Suo 36

Price for Treasuries Day cont convention:

T-bill: Actual/360 T-bill:

d is called the discount rate, or the quoted price

Quote Date Mat Date Bid

d

Ask

d

Yield

12/04/90 05/23/90 6.78 6.76 7.06

04/19/23 Wulin Suo 37

T-Bills d: annualized dollar return provided by the T-bill

expressed as a percentage of the face value Cash price P: or

for the example, ask price

Discount rate is different from rate of return:

For the example, rate of return is 7.0877%

360 1(100 )

100d P

n

100 1360

nP d

100 [1 160 0.0678 / 360] 96.98667P

100 365P

P n

04/19/23 Wulin Suo 38

Price T-Notes/Bonds

Price paid to buy the note/bond is different from those prices quoted:

Invoice price = Quoted Price + Accrued Int Invoice price is also called cash price, or dirty price Quoted price: clean price Day count convention for calculating accrued interest: Act/Act Day Count convention

Actual/Actual (in period): Treasury Bonds 30/360: Corporate Bonds Actual/360: Money Market Instruments

Example

Quote Date Mat Date Coupon Bid d Ask d Yield

12/17/91 15/11/21 8% 102.29 102.31 7.74

04/19/23 Wulin Suo 39

Price T-Notes/Bonds Compute accrued interest:

LCD: 15/05/91 NCD: 15/11/91 Total days between the coupon dates: 182 Days between quote date and LCD: 32 Accrued int: (8/2)x(32/182)=$0.7033 Quoted bid price: 102.29=$102+29/32=102.9062 Cash price (bid): 103.6720

04/19/23 Wulin Suo 40

Yields for T-Bills For maturity n<182 days:

discount rate

relationship

example

100 365BEY=

P

P n

100 360

100

Pd

n

365BEY

360

d

dn

04/19/23 Wulin Suo 41

Yields for T-Bills … For n>182 days

Assume a coupon is paid in 6 month’s time and interest is reinvested

at maturity:P

(1 / 2)P y 100

365 / 21 1 100

2 365

y nP y

22 365 2 ( 365) (2 365 1)(1 100 )

2 365 1

n n n Py

n

04/19/23 Wulin Suo 42

Yields for T-Notes/Bonds If in the last coupon period: short government

accrued interest: cash price is calculated as before yield can be annualized as in practice, yield is quoted as price

LCD MATS.D.

x

z

2

C x z

x

100 2 365C P

P z

100 2 2C P x

P z

100 2

12

CP

zy

x

04/19/23 Wulin Suo 43

Yields for T-Notes/Bonds … With more than one coupon left

if P is the invoice price, then yield is defined by

N is the number of remaining coupons

1/(1 2)z x

PP

y

1

1 10

2 100

(1 2) (1 2)

N

j Nj

CP

y y

04/19/23 Wulin Suo 44

Yield Curve Graphical depiction of the relationship between the yields of the

same credit and different maturities Treasury yield curve is the benchmark

very liquid Should not be used to price a bond

sometime bonds with similar maturities carry very different coupons

not appropriate to discount all cash flows by the same rate should treat the bond’s each cash flow separately

Risks

Rating companies Moody’s Investor Service Standard & Poor’s

Canadian Bond Rating Service (CBRS)

Rating Categories Investment grade Speculative grade

Default Risk and Ratings

Coverage ratios Leverage ratio Liquidity ratios Profitability ratios Cash flow to debt

Factors Used by Rating Companies

Financial Ratios by Rating Class

US Industrial LT Debt,

1997-1999 Medians

AAA A BBB B

EBIT interest coverage 17.5 6.8 3.9 1.0

EBITDA interest coverage 21.8 9.6 6.1 2.0

Funds flow/total debt (%) 105.8 46.1 30.5 9.4

Free operating CF/debt (%) 55.4 15.6 6.6 (4.6)

Return on capital (%) 28.2 19.9 14.0 7.2

Operating income/sales (%) 29.2 18.3 15.3 11.2

LT debt/capital (%) 15.2 32.5 41.0 70.7

Total debt/capital (%) 26.9 40.1 47.4 74.6

Sinking funds Subordination of future debt Dividend restrictions Collateral

Protection Against Default

Managing interest rate risk

Bond price risk Coupon reinvestment rate risk Matching maturities to needs The concept of duration Duration-based strategies Controlling interest rate risk with derivatives

Inverse relationship between price and yield An increase in a bond’s yield to maturity results in a smaller price

decline than the gain associated with a decrease in yield Long-term bonds tend to be more price sensitive than short-term bonds As maturity increases, price sensitivity increases at a decreasing rate Price sensitivity is inversely related to a bond’s coupon rate Price sensitivity is inversely related to the yield to maturity at which

the bond is selling

Bond Pricing Relationships

52

Price risk Price change relative to yield changes

In practice, it is scaled by 100 and treat it as a price change relative to a change of 1% in yield:

2 1

1 1001

(1 2) 2 (1 2)N N

P C N C y

y y y y

1

100P

P

y

( 1%) ( ) PP y P y

53

Price Value of a Basis Point (PVBP) What is the price change caused by the

change of 1bp in yield?

Sometimes it is multiplied by $1m In practice, PVBP for T-bills

( 1 ) ( ) 1

/100P

PP y bp P y bp

y

$1 1 100360 360

z zPVBP m bp

54

Duration Duration is defined as the price elasticity

with respect to yield: the percentage change in price in response to 1% change in yield: Macaulay duration:

Modified duration:

Similar to PVBP:

1P yD

y P

1PMD

y P

(1 )PVBP yD

P

PVBPMD

P

55

Duration … As weighted cash flow:

As a measure of sensitivity:

2

2

100

1 (1 ) (1 )

1 1 1 1 1001 2

1 (1 ) (1 )

N

N

C C CP

y y y

P y C C CD N

y P P y P y P y

1

PMD y

PP D

yP y

56

Price Yield and Duration For semi-annual compounding:

As time weighted discounted cash flow:

1 2P yD

y P

1P

MDy P

1

1

1 1

2 (1 / 2)

1

2

Ni

ii

N

ii

CD i

P y

i x

57

Properties of Duration

Depends on three variables maturity, coupon, and yield to maturity

Increase coupon: reduce duration Increase yield: reduce coupon Increase maturity:

increase duration if the bond is trading at a premium

trading at a discount: increase, and then decrease

58

Duration of a Bond Through Time Duration jumps periodically through its life

Usually decreases through the coupon period Jumps immediately after a coupon payment

Jumps more on a longer maturity bond The risk of the bond?

59

Portfolio Risk Measures Consider a portfolio consists of m bonds

n is the number of that bonds in the portfolio For each bond:

Portfolio can be written as

1 1 2 2P m mV n P n P n P

1 (1 / 2)

jTjl

j ll j

CP

y

1 1 (1 / 2)

jTmj jl

P lj l j

n CV

y

60

Portfolio … Treat the portfolio as a new bond: yield is

defined as y such that

Treating the yield curve as flat. PVBP:

Again, assuming yield is the same for all bonds, and changing in yields are the same --- parallel shift in the yield curve

1 1 (1 / 2)

jTmj jl

P lj l

n CV

y

1 1P m mPVBP n PVBP n PVBP

61

Portfolio … Duration: assuming the same yield for all the

bonds in the portfolio:

It can be written as

or

1 / 2( ) P

P

V yD P

y V

1 1 1

1

1 / 2 1 / 2( ) m m m

P m P

n P Py n P P yD P

P V y P V y

1 2( ) (1) (2) ( )mD P x D x D x D m

j jj

P

n Px

V

62

Application to Hedging Suppose one has n bonds P1 with a long

maturity, and he has a bond P2 is available to hedging his position What do we mean by hedging?

eliminate risk, or uncertainty Suppose he needs n2 of P2: If the yields changes are the same for both

bonds, then

1 1 2 2PV n P n P

1 1 2 2PPVBP n PVBP n PVBP

1 12

2

n PVBPn

PVBP

63

Hedging … If the yield changes are not the same for the

two bonds Use historically date for similar bonds and estimate

the correlation of the yield changes, say Value change of the portfolio:

The variance of the portfolio:1 1 1 2 2 2PV n PVBP y n PVBP y

2 2 2 21 1 1 2 2 2

1 2 1 2 1 2

( ) ( ) ( )

2 ( , )PVar V n PVBP Var y n PVBP Var y

n n PVBP PVBP Cov y y

64

Hedging … To minimize the risk, or variance:

Similarly, one could use modified duration

1 1 1 22

2 2

( , )

( )

n PVBP Cov y yn

PVBP Var y

1 1 1 1 22

2 2 2

( , )

( )

n P MD Cov y yn

P MD Var y

65

Convexity Remember

Bond’s value change is approximated as a linear function of yield change

However, the value of a bond as a function of yield is nonlinear (see diagram)

2nd order approximation:

PMD y

P

22 2

2

2 2

1 1 1( )

2

( )x

P dP d Py y o y

P P dy P dy

D y C y o y

66

actual price

P

yy

67

Convexity … Where

is called the convexity of the bond larger convexity means more curvature

2

2

1 1

2x

d PC

P dy

2x

PD y C y

P

xC

68

Convexity … For semi-annual coupon bonds

If there are exactly N coupons left (in full)

1

2

2 21

21

(1 / 2)

( 1)1

4(1 / 2) (1 / 2)

( 1)1

8(1 / 2) (1 / 2)

Ni

ii

Nii

i

Ni

x ii

CP

y

i i CP

y y y

i i CC

y y

2

2 2 2 1 3

( 1)(100 / ) 2 1

4(1 / 2) (1 / 2) (1 / 2)N N N

P N N C y CN C

y y y y y y

69

Properties of Convexity Option free (or bullet bonds ) has the

following properties Positive convexity: As the required yield increases

(decreases), the convexity of a bond decreases (increases).

For a given yield and maturity, the lower the coupon, the greater the convexity of a bond

For a given yield and duration, the lower the coupon, the smaller the convexity

Duration and Convexity of Callable Bonds

0 Interest Rate

Call Price

Region of positive convexity

Region of negative convexityPrice-yield curve is below tangent

5% 10%