Part1 Bond Valuation What is a bond? –Different issuers –Different maturities –Different...

Part1 Bond ValuationPart1 Bond Valuation

What is a bond?What is a bond?– Different issuersDifferent issuers– Different maturitiesDifferent maturities– Different stylesDifferent styles– Bond ratingBond rating– TerminologyTerminology

Bond pricingBond pricing– Zero coupon bondZero coupon bond– Ordinary coupon bondOrdinary coupon bond– ExamplesExamples– Converse questionConverse question

Some conceptsSome concepts– Yield to maturityYield to maturity– Term structureTerm structure

Future spot rateFuture spot rate Forward rateForward rate

What is a bond?What is a bond?

1.1. Bonds are loans that investors make to corporatiBonds are loans that investors make to corporations and governments. The borrowers get the caons and governments. The borrowers get the cash they need while the lenders earn interest. (Thsh they need while the lenders earn interest. (The Wall Street Journal – Guide to understanding e Wall Street Journal – Guide to understanding money & investing)money & investing)

a.a. Different issuers – Government (Treasury securities: Different issuers – Government (Treasury securities: T-bills, T-notes, T-bonds, Foreign-targeted T-notes), stT-bills, T-notes, T-bonds, Foreign-targeted T-notes), states, cities, counties, and towns (Municipal bonds), coates, cities, counties, and towns (Municipal bonds), corporations (corporate bonds) rporations (corporate bonds)


b.b. Different maturities – short term (less than 1 yDifferent maturities – short term (less than 1 years, e.g. T-bills) vs. long term (T-notes, T-bonears, e.g. T-bills) vs. long term (T-notes, T-bonds, etc)ds, etc)

c.c. Different styles – Coupon bonds (most of corpDifferent styles – Coupon bonds (most of corporate bonds) vs. zero-coupon bonds (e.g. STRorate bonds) vs. zero-coupon bonds (e.g. STRIPS – Separate Trading of Registered Interest IPS – Separate Trading of Registered Interest and Principal of Securities)and Principal of Securities)

→ → Why are STRIPS popular especially for an agent Why are STRIPS popular especially for an agent who can defer tax? who can defer tax?


d. Bond rating – Hard to evaluate the default risk in a bond. Bond rating – Hard to evaluate the default risk in a bondd

1.1. Agencies: S&P (Standard and Poor’s) and Moody’sAgencies: S&P (Standard and Poor’s) and Moody’s

2.2. Investment grade: At least BBB (S&P) or Baa(Moody’s)Investment grade: At least BBB (S&P) or Baa(Moody’s)

3.3. Junk bonds: At most CC (S&P) or Ca (Moody’s)Junk bonds: At most CC (S&P) or Ca (Moody’s)

What is a bond?What is a bond? Moody’s Long-Term Rating DefinitionsMoody’s Long-Term Rating Definitions - Example - Example

AaaAaaObligations rated Aaa are judged to be of the highest quality, with miniObligations rated Aaa are judged to be of the highest quality, with minimal credit risk.mal credit risk.

AaAaObligations rated Aa are judged to be of high quality and are subject to Obligations rated Aa are judged to be of high quality and are subject to very low credit risk.very low credit risk.

AAObligations rated A are considered upper-medium grade and are subjeObligations rated A are considered upper-medium grade and are subject to low credit risk.ct to low credit risk.

BaaBaaObligations rated Baa are subject to moderate credit risk. They are conObligations rated Baa are subject to moderate credit risk. They are considered medium-grade and as such may possess certain speculative csidered medium-grade and as such may possess certain speculative characteristics.haracteristics.

What is a bond?What is a bond? BaBa

Obligations rated Ba are judged to have speculative elements and are subject tObligations rated Ba are judged to have speculative elements and are subject to substantial credit risk.o substantial credit risk.

BBObligations rated B are considered speculative and are subject to high credit risObligations rated B are considered speculative and are subject to high credit risk.k.

CaaCaaObligations rated Caa are judged to be of poor standing and are subject to very Obligations rated Caa are judged to be of poor standing and are subject to very high credit risk.high credit risk.

CaCaObligations rated Ca are highly speculative and are likely in, or very near, defaObligations rated Ca are highly speculative and are likely in, or very near, default, with some prospect of recovery of principal and interest.ult, with some prospect of recovery of principal and interest.

CCObligations rated C are the lowest rated class of bonds and are typically in defaObligations rated C are the lowest rated class of bonds and are typically in default, with little prospect for recovery of principal or interest.ult, with little prospect for recovery of principal or interest.


e. Terminologye. Terminology

1.1. Par value (face value):Par value (face value): Dollar amount of the Dollar amount of the bond at the time it is issued. It appears several bond at the time it is issued. It appears several times on the face of bond. Usually, multiplies of times on the face of bond. Usually, multiplies of $1,000.$1,000.

2.2. Interest rate (coupon rate):Interest rate (coupon rate): Percentage of par Percentage of par value that is paid to the bond-holder on a regular value that is paid to the bond-holder on a regular

basisbasis..

Bond pricingBond pricing

a.a.Zero coupon bondZero coupon bond

Price = PV (Face value) =Price = PV (Face value) =

b.b.Ordinary coupon bondOrdinary coupon bond

Price = PV (Coupons) + PV (Face value)Price = PV (Coupons) + PV (Face value)

==

(1 )T

FV

r

1 2 3(1 ) (1 ) (1 ) (1 )T

C C C FV C

r r r r


c.c. ExamplesExamples Suppose you now own a T-bond that has a coupon rate of 5Suppose you now own a T-bond that has a coupon rate of 5

%, face value of $1,000, and maturity of 5 years. A compara%, face value of $1,000, and maturity of 5 years. A comparable investment has rate of return of 6%. ble investment has rate of return of 6%.

Assume annual couponsAssume annual coupons

2 5 5

50 50 50 1000

1.06 1.06 1.06 1.06$210.62 $747.26 $957.88

PV


Assume semi-annual couponsAssume semi-annual coupons

Why is there difference of bonds prices between annuaWhy is there difference of bonds prices between annual coupons and semi-annual coupons?l coupons and semi-annual coupons?

2 conflicting factors: 2 conflicting factors: More immediate paymentMore immediate payment and and higher discount ratehigher discount rate. → . → cannot be determined simply.cannot be determined simply.

1 2 3 10 10

25 25 25 25 1000

1.03 1.03 1.03 1.03 1.03$213.26 $744.09 $957.35

PV


d.d. Converse question Converse questionSince the market decides the bond price, then we can bSince the market decides the bond price, then we can back out “r”. So, this “r” and the price of a bond have onack out “r”. So, this “r” and the price of a bond have one-to-one relationship.e-to-one relationship.

Example: Suppose a bond with face value of $1,000, coupon Example: Suppose a bond with face value of $1,000, coupon rate of 11.5%, and maturity of 5 years is being trate of 11.5%, and maturity of 5 years is being t

raded raded at $1161.84. Then, what is “r” for this bond?at $1161.84. Then, what is “r” for this bond?

1161.841161.84 = 115/(1+r) + 115/(1+r) = 115/(1+r) + 115/(1+r)22 + … + …

+ 115/(1+r)+ 115/(1+r)55 + 1000/(1+r) + 1000/(1+r)55..r = 0.075, but usually through trial and error method.r = 0.075, but usually through trial and error method.

Some conceptSome concept

a. Yield to maturity (YTM) – “r” in our notation.a. Yield to maturity (YTM) – “r” in our notation.IRRIRR for a bond. Financial newspapers report thi for a bond. Financial newspapers report this number.s number.

IRR: Internal rate of returnIRR: Internal rate of return


Example: WSJ 5/15/02Example: WSJ 5/15/02

Maturity ASKMaturity ASK

RATE MO/YR BID ASKED CHG YLDRATE MO/YR BID ASKED CHG YLD

6.500 May 05 107:6 107:7 6 3.90 6.500 May 05 107:6 107:7 6 3.90


““6.500” means 6.500” means coupon rate (%)coupon rate (%)

““May 05” means May 05” means maturitymaturity

““107:6” is bid price and it is not $107.6, but 107:6” is bid price and it is not $107.6, but (107 and 6/32) (107 and 6/32) %% of FV (usually, $1,000) → $1,071.875 of FV (usually, $1,000) → $1,071.875

““107:7” is ask price → 107:7” is ask price → (107 and 7/32) %(107 and 7/32) % of FV, $1,072.187 of FV, $1,072.18755

Notice: Spread is really small, 1/32 (about 0.03) % oNotice: Spread is really small, 1/32 (about 0.03) % of f FV, Spread of stock price – around FV, Spread of stock price – around 1-2%1-2%


““6”means that the bond price is increased by 6”means that the bond price is increased by $6/32$6/32 from th from the previous day.e previous day.

“ “3.90” is YTM: This is backed out by using3.90” is YTM: This is backed out by using1072.19 = 65/(1+r) + 65/(1+r)1072.19 = 65/(1+r) + 65/(1+r)22

+ 65/(1+r)+ 65/(1+r)33 + 1000/(1+r) + 1000/(1+r)33

→ → r r ≈≈ 0.039 0.039

Problems of YTM – The assumption of Problems of YTM – The assumption of the same discount rthe same discount rateate from t=1 to T. It is very unusual. from t=1 to T. It is very unusual. → → Think the job of Mr. Greenspan. Think the job of Mr. Greenspan.


b. Change of interest rate and bond priceb. Change of interest rate and bond priceIn our example (price is $1072.19), if interest rate (not In our example (price is $1072.19), if interest rate (not coupon rate) is increased into 15%, then the bond price coupon rate) is increased into 15%, then the bond price will be decreased.will be decreased.

PV at 15% = 65/(1.15) + 65/(1.15)PV at 15% = 65/(1.15) + 65/(1.15)22

+ 65/(1.15)+ 65/(1.15)33 + 1000/(1.15) + 1000/(1.15)33

= = 805.93805.93

So, price is So, price is decreaseddecreased with higher interest rate. with higher interest rate.


c. Rate of return c. Rate of return What is the difference between rate of return and What is the difference between rate of return and

YTM for a bond?YTM for a bond?

From the above example, after 1 year, the bond price From the above example, after 1 year, the bond price is 1077.10. Then, what would be rate of return and is 1077.10. Then, what would be rate of return and YTM?YTM?

First, rate of return is First, rate of return is ($65 + $1,077.10 - ($65 + $1,077.10 - $1,072.19)/$1,072.19$1,072.19)/$1,072.19 = 6.52% = 6.52%


For YTM, For YTM, 1077.10 (PV at year 1)1077.10 (PV at year 1) = 65/(1+r) + 65/(1+ = 65/(1+r) + 65/(1+r)r)22 + 1000/(1+r) + 1000/(1+r)22

Then, YTM = 2.5%Then, YTM = 2.5%

What is the relationship between YTM and the rate of What is the relationship between YTM and the rate of return?return?

→ → If interest rates do not change, the bond price changes witIf interest rates do not change, the bond price changes with time so that h time so that the total return on the bond is equal to the YTthe total return on the bond is equal to the YTMM..


d. Spot rate and forward rated. Spot rate and forward rate

1 Year

r1 = 0f1

r1 = 0f1

r2 = 0r2 represented by 1 year rate

1f2 at t=0


i. Future spot ratei. Future spot rateThe actual interest rate in the future, e.g. the actual The actual interest rate in the future, e.g. the actual

interest rate from year 1 to year 2 at year 1.interest rate from year 1 to year 2 at year 1.

YTM on zero coupon bond.YTM on zero coupon bond.

ii. Forward rateii. Forward rateSince a future spot rate is unknown now, we have to Since a future spot rate is unknown now, we have to

infer it from various bonds prices. infer it from various bonds prices.

In general, (1 + In general, (1 + t-1t-1fftt)(1 + )(1 + 11r r t-1t-1))(t-1)(t-1) = (1 + = (1 + 11rrtt))tt


Example for spot rate and forward rate: ConsideExample for spot rate and forward rate: Consider following 4 bonds with face value of $100 and r following 4 bonds with face value of $100 and annual coupons.annual coupons.

Maturity Maturity Coupon Coupon Price Price YTM YTM 11 00 94.3494.34 6.006.00

22 6.56.5 100.56100.56 6.196.19

33 6.06.0 98.9898.98 6.386.38

33 11.011.0 112.28112.28 6.376.37


What is 2 year spot rate (r2)?What is 2 year spot rate (r2)?

100.56 = 100.56 = 6.5/(1.06) + 6.5/(1+r6.5/(1.06) + 6.5/(1+r22))22

+ 100/(1+r+ 100/(1+r22))22

So, r2 = 6.2%So, r2 = 6.2%


Then, rThen, r33??

98.98 = 98.98 = 6.0/(1.06) + 6.0/(1+0.062)6.0/(1.06) + 6.0/(1+0.062)22 + 106/(1+ + 106/(1+ rr33))33

So, rSo, r33 = 6.4% = 6.4%


What are What are 11ff22??

(1+r(1+r22))22 = (1+r = (1+r11)*(1+)*(1+11ff22))

In the above problem, we know that rIn the above problem, we know that r22 is is 6.2% and r6.2% and r11 is 6%. is 6%.

So, So, 11ff22 is 6.4% is 6.4% And And 22ff33??

With the same logic, With the same logic, 22ff33 = 6.8% = 6.8%


e. Term structuree. Term structure The relationship between short- and long-term interest rates at The relationship between short- and long-term interest rates at

a particular time on bonds that are a particular time on bonds that are fundamentallyfundamentally similar similar except maturity. The relationship can be presented graphically except maturity. The relationship can be presented graphically as a yield curve. Usually, the yield curve is as a yield curve. Usually, the yield curve is upward slopingupward sloping. In . In our example, rour example, r11 < r < r22 < r < r33..

Advanced points: Term structure can be represented by the Advanced points: Term structure can be represented by the relation between spot rate (e.g. rrelation between spot rate (e.g. r22) and forward rate () and forward rate (11ff22). If r). If r22 > >

rr11, then , then 11ff22 > r > r22, and if r, and if r22 < r < r11, then , then 11ff22 < r < r22. .

Part2 BOND RISKPart2 BOND RISK

Default RiskDefault Risk Call RiskCall Risk Market RiskMarket Risk

Default Risk (1)Default Risk (1)

Default RiskDefault Risk: Uncertainty that the realized r: Uncertainty that the realized return will deviate from the expected return beturn will deviate from the expected return because the issuer will fail to meet the contraecause the issuer will fail to meet the contractual obligations specified in the indenture. ctual obligations specified in the indenture.

The major concern is failure to meet interest The major concern is failure to meet interest and principal payments.and principal payments.

Default Risk(2)Default Risk(2)

Most investors do not directly access a bonMost investors do not directly access a bond’s default risk, but instead use the quality rd’s default risk, but instead use the quality ratings provided by Moody’s, S&P, and Fetcatings provided by Moody’s, S&P, and Fetch to evaluate the degree of risk.h to evaluate the degree of risk.– AAA, AA, A, BBB, ……, C, ……AAA, AA, A, BBB, ……, C, ……

Default Rates:Default Rates:– .12% per year for all bonds since WWII..12% per year for all bonds since WWII.– 1980s: 3.27% per year on junk bonds.1980s: 3.27% per year on junk bonds.– 1990-1991: 9% on junk bonds 1990-1991: 9% on junk bonds


Empirical studies that looked at the relation bEmpirical studies that looked at the relation between default risk premium (RP) and the statetween default risk premium (RP) and the state of the economy.e of the economy.

.

.

.

C ARP YTM YTM

C Genericlowqualitybond

A Generichighqualitybond


Study looked at the yield curves for different quality bonds.Study looked at the yield curves for different quality bonds. Found that the YC for lower quality bonds tended on average to be negativeFound that the YC for lower quality bonds tended on average to be negative

ly sloped.ly sloped. Reason: Greater concern over the repayment of principal on low quality bonReason: Greater concern over the repayment of principal on low quality bon

ds.ds.

MC

YTM


Applied the Evans and Archer methodology to bond portfolios grouped Applied the Evans and Archer methodology to bond portfolios grouped in terms of their quality ratings.in terms of their quality ratings.

Found that lower quality bond portfolios had less risk because of their Found that lower quality bond portfolios had less risk because of their lower correlationslower correlations

n

AAA

n

B

Call Risk (1)Call Risk (1)

Call RiskCall Risk: Uncertainty that the realized return will : Uncertainty that the realized return will deviate from the expected return because the deviate from the expected return because the issuer calls the bond, forcing the investor to issuer calls the bond, forcing the investor to reinvest in a market with lower rates.reinvest in a market with lower rates.

Note: When a bond is called the holder receives Note: When a bond is called the holder receives the call price (CP). Since the CP usually exceeds the call price (CP). Since the CP usually exceeds the principal, the return the investor receives over the principal, the return the investor receives over the call period is often greater than the initial YTM. the call period is often greater than the initial YTM. The investor, though, usually has to reinvest in a The investor, though, usually has to reinvest in a market with lower rates which often causes his market with lower rates which often causes his return for the investment period to be less than the return for the investment period to be less than the initial YTM. initial YTM.


Example:Example: Compare the ARR for the call period Compare the ARR for the call period with the ARR for the investment period for a bond with the ARR for the investment period for a bond that is called. Buy:that is called. Buy:– 10-year, 10% annual coupon bond at par ($1000); 10-year, 10% annual coupon bond at par ($1000); – callable at 110: CP = $1100.callable at 110: CP = $1100.

Assume:Assume:– HD = 10 years.HD = 10 years.– Flat YC at 10%.Flat YC at 10%.– YC stays at 10% until the end of year 3.YC stays at 10% until the end of year 3.– Year 3, the YC shifts down to a flat 8% and the bond is Year 3, the YC shifts down to a flat 8% and the bond is

called.called.– Investor reinvest at 8% for the next 7 years.Investor reinvest at 8% for the next 7 years.


0

3

2

1

$14311 .1268

$1000

100(1.10) 100(1.10) 100 1100 1431

CC B

C

CallDateValueARR

P

ARR

CallDateValue

0

7

10

3 710

1

$1431(1.08)1 .0938

$1000

(1.1268) (1.08) 1 .0938

HDHD B

HD

HD

HDValueARR

P

ARR

ARR

9.38% 10%HDARR YTM


Call Risk PremiumCall Risk Premium

Price CompressionPrice Compression: Call features put limitations : Call features put limitations on the price-yield curve. At the rate where the on the price-yield curve. At the rate where the bond could be called (YTM*), the YC flattens, wibond could be called (YTM*), the YC flattens, with the price equal to the CP. th the price equal to the CP.

Call NCRP YTM YTM

RPgreater in higher interest rate periods

R RP

Call Risk (5)Call Risk (5) Points:Points:

Need for a valuation model for callable bonds different from the PV model. Need for a valuation model for callable bonds different from the PV model. – A 10-year callable bond may be more like a 3-year bond.A 10-year callable bond may be more like a 3-year bond.– Binomial Tree Model or option pricing model.Binomial Tree Model or option pricing model.

YTM

PB0

A

A

YTM *

A'CP

AA NonCallable Bond

AA Callable Bond'

Market Risk (1)Market Risk (1)

Market RiskMarket Risk: Uncertainty that the realized : Uncertainty that the realized return will deviate from the expected return return will deviate from the expected return because interest rates change. because interest rates change.

Recall, the return on a bond comes from Recall, the return on a bond comes from coupons, the interest earned from coupons, the interest earned from reinvesting coupons (interest on interest), reinvesting coupons (interest on interest), and capital gains.and capital gains.

A change in rates affects interest on interest A change in rates affects interest on interest and capital gains or losses.and capital gains or losses.


Note: One way to eliminate market risk is to Note: One way to eliminate market risk is to buy a PDB with M = HD.buy a PDB with M = HD.

r

( )INT on INT

PtB


Example:Example: HD = 3.5 yearsHD = 3.5 years Buy Noncallable 10-year, 10% annual bond Buy Noncallable 10-year, 10% annual bond

at par (F=$1000) at YTM of 10%.at par (F=$1000) at YTM of 10%. YC flat at 10%.YC flat at 10%.

If there is no change in the YC, your ARIf there is no change in the YC, your ARR for 3.5 years will be 10%:R for 3.5 years will be 10%:

2.5 1.5 .53.5

3.5 .5

6

4 61

1/3.5

3.5

100(1.10) 100(1.10) 100(1.10) 1395.97

1000 100: 1048.81

(1.10)

100 10001000

(1.10) (1.10)

1395.971 .10

1000

B

B

Bt

t

HDValue P

where P

P

ARR


Example:Example: If the YC shifted up to a flat 12% shortly after you bought the bond and remained there uIf the YC shifted up to a flat 12% shortly after you bought the bond and remained there u

ntil your HD, then your ARR for 3.5 years would be 8.23%:ntil your HD, then your ARR for 3.5 years would be 8.23%:

2.5 1.5 .53.5

3.5 .5

6

4 61

1/3.5

3.5

100(1.12) 100(1.12) 100(1.12) 1319

917.77 100: 961.70

(1.12)

100 1000917.77

(1.12) (1.12)

13191 .0823

1000

B

B

Bt

t

HDValue P

where P

P

ARR


Example:Example: If the YC shifted down to a flat 8% shortly after you bought the bond If the YC shifted down to a flat 8% shortly after you bought the bond

and remained there until your HD, then your ARR for 3.5 years wouand remained there until your HD, then your ARR for 3.5 years would be 11.95%:ld be 11.95%:

2.5 1.5 .53.5

3.5 .5

6

4 61

1/3.5

3.5

100(1.08) 100(1.08) 100(1.08) 1482.82

1092.46 100: 1147.44

(1.08)

100 10001092.46

(1.08) (1.08)

1484.821 .1195

1000

B

B

Bt

t

HDValue P

where P

P

ARR


Relation:Relation:

Rate change have two opposite effects: Rate change have two opposite effects: – Inverse price effectInverse price effect– Direct interest on interest effectDirect interest on interest effect

Whether the ARR varies directly or inversely to rate changes Whether the ARR varies directly or inversely to rate changes depends on which effect dominates. In this case (10-year, depends on which effect dominates. In this case (10-year, 10% coupon bond with HD = 3.5 yrs), the inverse price effect 10% coupon bond with HD = 3.5 yrs), the inverse price effect dominates. This causes the ARR to vary inversely with rate dominates. This causes the ARR to vary inversely with rate changes.changes.

3.5

8% 11.95%

10% 10.00%

12% 8.23%

R ARR


With an HD of 3.5 years, a 4 year, 20% bond has an interest on interest effect With an HD of 3.5 years, a 4 year, 20% bond has an interest on interest effect which dominates the price effect. This bond’s ARR would vary directly with ratwhich dominates the price effect. This bond’s ARR would vary directly with rate changes.e changes.

3.5

8% 9.845%

10% 10.00%

12% 10.16%

R ARR .52.5 1.53.5

3.5 .5

1/3.5

3.5

4

0 41

200(1.12) 200(1.12) 200 1.12 1848

1000 200: 1134

(1.12)

18481 .1016

1317

200 1000: 1317

(1.10) (1.10)

B

B

Bt

t

HDValue P

where P

ARR

Where P


Note: It is possible to buy a bond in which the price and Note: It is possible to buy a bond in which the price and interest on interest effect exactly offset each other. In this interest on interest effect exactly offset each other. In this case, the ARR will be invariant to changes in rates.case, the ARR will be invariant to changes in rates.

For example, a 4-year, 9% coupon bond purchased when For example, a 4-year, 9% coupon bond purchased when the YC is at 10% (Po = 968.30) would yield an HD value at the YC is at 10% (Po = 968.30) would yield an HD value at year 3.5 of 1351 and an ARR of 10%, regardless of rates.year 3.5 of 1351 and an ARR of 10%, regardless of rates.

3.5

8% 10%

10% 10%

12% 10%

R ARR


Bond Immunization:Bond Immunization: What is distinctive about a 4-year, 9% coupon bond when What is distinctive about a 4-year, 9% coupon bond when

the YTM is 10% is that it has a duration (D) equal to 3.5 the YTM is 10% is that it has a duration (D) equal to 3.5 years -- duration = HD.years -- duration = HD.

Duration can be defined as the weighted average of the Duration can be defined as the weighted average of the time periods.time periods.

Bond immunization is bond strategy of minimizing market Bond immunization is bond strategy of minimizing market risk by buying a bond or a portfolio of bonds with a risk by buying a bond or a portfolio of bonds with a duration equal to the HD.duration equal to the HD.

1 0

( )Mt

Bt

PV CFD t

P

p i iD w D


Duration of 4-year, 9% coupon Bond:Duration of 4-year, 9% coupon Bond:

D tPV CF

Pt

Bt

M

( )

01

t CFCF PV CF

P t Weight

P D

tt

tt

B

B

( . )( )

( )( )

. . .

. . .

. . .

. . .

. .

110

1 90 81818 0 084496 0 084496

2 90 74 348 0 076815 0153630

3 90 67 618 0 069832 0 209496

4 1090 744 485 0 768857 3 075428

968 30 352

0

0

Duration (1)Duration (1)

Duration is also a measure of a bond’s price sensDuration is also a measure of a bond’s price sensitivity to interest rate changes.itivity to interest rate changes.

This measure of duration is known as the This measure of duration is known as the modifiemodified durationd duration; the measure of duration as a weighted ; the measure of duration as a weighted average of the time periods is known as average of the time periods is known as MacauleMacauley’s durationy’s duration..

0 0% %

%

B BP PD

YTM YTM


0

1 0

% ( )1

(1 )

B Mt

Bt

P PV CFD t

YTM YTM P

ModifiedDuration Macaulay sDuration

:

1(3.5) 3.18

(1.10)

Example

ModifiedDuration


A modified duration measure for a bond which A modified duration measure for a bond which pays coupons each period and its principal at pays coupons each period and its principal at maturity (note: let y = YTM):maturity (note: let y = YTM):

2 10

0

1 [ ( / )]1

(1 ) (1 )% M MB

B

C M F C yy y yP

y P


Annualized DurationAnnualized Duration: Duration is defined in terms : Duration is defined in terms of the length of the period between payments. If tof the length of the period between payments. If the CFs are distributed annually, then duration is ihe CFs are distributed annually, then duration is in years; if CFs are semi-annual, then duration is n years; if CFs are semi-annual, then duration is measured in half years. The convention is to exprmeasured in half years. The convention is to expressed duration as an annual measure. The annuessed duration as an annual measure. The annualized duration is obtained by dividing duration by alized duration is obtained by dividing duration by the number of payments per year (n):the number of payments per year (n):

Durationforn periodsAnnualized Duration

n


Example: The duration in half-years for a 10-yExample: The duration in half-years for a 10-year, 9% coupon bond selling at par (F= 100) anear, 9% coupon bond selling at par (F= 100) and paying coupons semiannually is -13 and its ad paying coupons semiannually is -13 and its annualized duration is -6.5:nnualized duration is -6.5:

2 20 210

4.5 1 20[100 (4.5 / .045)]1

.045 (1.045) (1.045)%13

100

136.5

2

BP

y

AnnualizedDuration


Descriptive ParameterDescriptive Parameter: Measure of a bond’s : Measure of a bond’s price sensitivity to interest rate changes -- a price sensitivity to interest rate changes -- a measure of a bond’s volatility. Note, duratiomeasure of a bond’s volatility. Note, duration is consistent with our bond price relationn is consistent with our bond price relation


Define StrategiesDefine Strategies: Use duration to define : Use duration to define active (speculative) and passive bond active (speculative) and passive bond strategies. Examples:strategies. Examples:– Rate-Anticipation SwapRate-Anticipation Swap: Rates expected to : Rates expected to

decrease across all maturities, go long in high decrease across all maturities, go long in high duration bonds. Rates expected to increase duration bonds. Rates expected to increase across all maturities, change bond portfolio across all maturities, change bond portfolio composition so that it has lower duration bonds.composition so that it has lower duration bonds.

– Bond Immunization StrategyBond Immunization Strategy..


Estimate the percentage change in a bond’s price for a given change iEstimate the percentage change in a bond’s price for a given change in rates:n rates:

For 10-year, 9% bond, an increase in the annualized yield by 10BP (.0For 10-year, 9% bond, an increase in the annualized yield by 10BP (.09 to .0910) would lead to a .65% decrease in price (the actual is .64769 to .0910) would lead to a .65% decrease in price (the actual is .6476%). A 200BP increase (9% to 11%) would lead to an estimated price d%). A 200BP increase (9% to 11%) would lead to an estimated price decrease of 13%; the actual decrease, though, is only 12%.ecrease of 13%; the actual decrease, though, is only 12%.

%[ ]

( )

% [ ] ( )

B

B

PModifiedDuration

YTM

P ModifiedDuration YTM

% [ 6.5] (.0910 .09) .0065BP

% [ 6.5] (.11 .09) .13BP

Convexity (1)Convexity (1)

Duration is a measure of the slope of the Duration is a measure of the slope of the price-yield curve at a given point -- price-yield curve at a given point -- first-first-orderorder derivativederivative..

Convexity Convexity is a measure of the change in the is a measure of the change in the slope of the price-yield curve -- slope of the price-yield curve -- second-second-order derivativeorder derivative..

Convexity measures how bowed-shaped Convexity measures how bowed-shaped the price-yield curve is.the price-yield curve is.


Property:Property: The greater a bond’s convexity, th The greater a bond’s convexity, the greater its capital gains and the smaller its e greater its capital gains and the smaller its capital losses for given absolute changes in capital losses for given absolute changes in yields.yields.

YTM

0BP

0y

0BP

YTM1y 2y

{KLoss{

0y1y 2y

KGain{{

0 1 2 0y y y y


Measure:Measure:

02

10

/ ( 1) ( )1

(1 )

B Mt

B tt

Slope P t t PV CFConvexity

y P y

2

ConvexityfornlengthAnnualizedConvexity

n


The convexity measure for a bond which pays cThe convexity measure for a bond which pays coupons each period and its principal at maturity:oupons each period and its principal at maturity:

3 2 1 2

0

2 1 2 ( 1)[ ( / )]1

(1 ) (1 ) (1 )M M M

B

C CM M M F C yy y y y y

ConvexityP


ExampleExample: The convexity in half-years for a 10-yea: The convexity in half-years for a 10-year, 9% coupon bond selling at par (F= 100) and par, 9% coupon bond selling at par (F= 100) and paying coupons semiannually is 225.43 and its annuying coupons semiannually is 225.43 and its annualized convexity is 56.36:alized convexity is 56.36:

3 20 2 21 22

2

2(4.5) 1 2(4.5)(20) (20)(21)[100 (4.5 / .045)]1

.045 (1.045) (.045) (1.045) (1.045)225.43

100225.43

56.362

Convexity

AnnualizedConvexity


Descriptive ParameterDescriptive Parameter: Greater k-gains and small: Greater k-gains and smaller k-losses the greater a bond’s convexity.er k-losses the greater a bond’s convexity.

Estimation ofEstimation of : Using Taylor Expan : Using Taylor Expansion, a better estimate of price changes to discretsion, a better estimate of price changes to discrete changes in yield than the duration measure can e changes in yield than the duration measure can be obtained by combining duration and convexity be obtained by combining duration and convexity measures. measures.

% / %BP y


Taylor Expansion:Taylor Expansion:

For 10-year, 9% bond, an increase in the annualized yield For 10-year, 9% bond, an increase in the annualized yield by 200 BP (9% to 11%) would lead to an estimated 11.87by 200 BP (9% to 11%) would lead to an estimated 11.87% decrease in price using Taylor Expansion (the actual is 1% decrease in price using Taylor Expansion (the actual is 12%):2%):

21% [ ] [ ]( )

2BP ModifiedDuration y Convexity y

21% [ 6.5] (.02) [56.36](.02) .1187

2BP


Note: Using Taylor Expansion the percentage increases in Note: Using Taylor Expansion the percentage increases in price are not symmetrical with the percentage decreases foprice are not symmetrical with the percentage decreases for given absolute changes in yields.r given absolute changes in yields.

2

9% 11% :

1% [ 6.5] (.02) [56.36](.02) .1187

2B

y from to

P

2

9% 7% :

1% [ 6.5] ( .02) [56.36]( .02) .1413

2B

y from to

P

Derivation of Duration and Convexity (1)Derivation of Duration and Convexity (1)

Duration:Duration:

Take derivative with respect to y:Take derivative with respect to y:

01 1

1 20 1 2

(1 )(1 )

(1 ) (1 ) (1 )

M MB tt

ttt t

B MM

CFP CF y

y

P CF y CF y CF y

2 3 ( 1)1 2( 1) (1 ) ( 2) (1 ) ( ) (1 )

BM

M

dPCF y CF y M CF y

dy


2 3 ( 1)

1 2( 1) (1 ) ( 2) (1 ) ( ) (1 )B

MM

dPCF y CF y M CF y

dy

1 21 2

1 21 2

1 2

1(1) (1 ) (2) (1 ) ( ) (1 )

(1 )

1(1) (2) ( )

(1 ) (1 ) (1 ) (1 )

1(1) ( ) (2) ( ) ( ) ( )

(1 )

BM

M

BM

M

B

M

dPCF y CF y M CF y

dy y

CF CF CFdPM

dy y y y y

dPPV CF PV CF M PV CF

dy y


1 2

0 0 0

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

(1 )

BM

B B B

PV CF PV CF PV CFdP dP PM

dy P dy y P P P

1 0

( )/ 1

(1 )

Mt

Bt

PV CFdP PModifiedDuration t

dy y P


Convexity: Convexity: Take the derivative ofTake the derivative of

2 3 ( 1)1 2( 1) (1 ) ( 2) (1 ) ( ) (1 )

BM

M

dPCF y CF y M CF y

dy

23 4 ( 2)

1 22

0

2 210 1

2 20 0 0

2 (1 ) 6 (1 ) ( 1) (1 )

exp :

( 1)( 1) ( )

/1 1(1 )(1 )

BM

M

B

M Mt

BB ttt t

B B B t

d PCF y CF y M M CF y

dy

DividethroughbyP and ressasasummation

t t CFt t PV CF

d Pd P yConvexity

dy P dy P P y

Part1 Bond Valuation What is a bond? –Different issuers –Different maturities –Different...

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Transcript of Part1 Bond Valuation What is a bond? –Different issuers –Different maturities –Different...