Ch17Bonds MBA2

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Investment Analysis & Portfolio Theory

MBA-IISpring Semester 2011

Lahore School of Economics

Salaar Farooq ² Assistant Professor



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Chapter 17

Fixed Income Securities:

Bonds



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Chapter 17

Bond Yields & Prices

Objectives

Bonds & Valuation

Interest rates

Bond Yields

Bond Prices

Bond Price Changes Duration, Modified Duration & Convexity



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Bonds

What is a Bond?



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BondsWhat is a Bond

Debt instrument to raise money (Loan)

Issued by Corporations & Govts

Interest Only LoanHas a standard Face Value of $1000 (Corp)

Coupons are paid semi-annually



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Fixed-Income Securities

BONDS

Capital Markets:

Where debt & equity capital is raised. Market f or long-term securities (stocks &bonds)

Fixed-Income Securities:Securities with specified payment dates and amounts, primarily Bonds

Bonds:

Future stream of cash flows are known at issue.

Principal is paid at maturity.IF, bond is sold bef ore maturity, price will reflectcurrent interest rates.



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Bond Characteristics

Par Value:

Term Bond:

Term to Maturity:

Coupon:

Coupon Rate:

Zer o Coupon:

Bond Prices: add zer o



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Bond Characteristics Par Value: Face Value of most Bonds is $1000 paid at maturity

Term Bond: Bonds typically mature on a specified date

Term to Maturity: how much longer the bond will exist

Coupon: Periodic interest paid by issuer. Typically semi-annual. Quoted APR

Coupon Rate: Is fixed at issuance & cannot vary

Zer o Coupon: Bond with no coupons sold at discount & redeemed at FaceValue

Bond Prices: Quoted as a %age of Par Value. Use 100 as conventional par

rather than 1,000. So, Price 90 = $900 (90% of 1000). Each point, or a change of 1,represents 1% of 1,000 or $10.

Easy conversion: since quoted in %age, just add an extra ZERO to get price.



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More «Bond Characteristics

Indenture:

Registered Form:

Bearer Form:

Security:

Debenture:

Sinking Fund:



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More «Bond Characteristics

Indenture:Written agreement b/w corp & creditor

Registered Form: Owner is named w/registrar at corp

Bearer Form: Owner is not named w/registrar

Security: collateral (financial) or mortgage (real) used as backing

Debenture: Unsecured debt (mat >10yrs)

Sinking Fund: account managed by bond trustees f or early bondpymts





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Fixed-Income SecuritiesBond Examples

A 10% coupon Bond has what dollar coupon?:

$100 coupon & $50 semi-annual coupon payments.

A Bond¶s quoted Px is 101 3/8. What¶s the $ price?

Represents 101.375% of 1,000, theref ore $ price is $1013.75

Price above PAR Premium (& discount)? Interest Rates declined, so priceincreased.

Interest rates are inversely related to price. More coverage later chapter.

Interest Rate & Bond Price:

Bond will be exactly worth its face value at maturity. But till then it fluctuates ar ound$1000 to adjust yield according to market interest rates.



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Call Pr ovision:

When is it attractive to the issuer?

Call Premium



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Call Pr ovision:

Gives the issuer the right to call in a security & retire it by paying off theobligation.

When is it attractive to the issuer? When interest rates dr op in the markets enough to save the issuer money.

additional cost f or the issuer called CALL PREMIUM

Issuer will CALL & then issue new ones at lower cost.

Most Corp Bonds are callable

Call Premium

Often equals one years interest (if called within a year), then decreases



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TYPES OF BONDS4 Major Types

Govt Treasuries: Since Govt can print money to pay off, it is consideredrisk free (no default risk). Size is 5k,10k,100k,500k & million

Notes = Maturities b/w 1 year & 10 years (Bills = < 1year)

Bonds = Maturities b/w 10 & 30 years

Govt Agencies: To help specific sectors of the economy, loans areguaranteed by the Govt thr ough agencies.

Municipals: Typically States & Cities. Also called Serial Bonds. Taxexempt, so yield is lower compared with taxable.

Corporates: Securities issued by Corporations to finance their operations.Typically 20-40yrs maturity, pays semi-annual, callable, par value $1000.



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Characteristics of Corporates

Senior Securities:

Convertible Bonds:

Rating Agencies:

Bond Ratings:



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Characteristics of Corporates

Senior Securities:Corp Bonds are senior to preferred stock, common stock in case of bankruptcy.

Convertible Bonds:

Bonds which are convertible (at holders option) into shares of common stock.

Rating Agencies: 3 agencies S&P, Moody¶s, Fitch pr ovide Bond ratingsrepresenting current opinions on relative credit quality of the firm. (PACRA)

Bond Ratings: Letters of the alphabet assigned to Bonds to expresspr obability of default.



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Bond Ratings

AAA:Best Quality & lowest default risk

AA:

More common: Very Good Quality.

D:

Lowest rating: means debt is in default.

BBB:

Investment Grade ± Medium grade. Adequate capacity to pay.



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Bond Prices

Valuation Principle

Bond Valuation

R1



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Bond Prices

Valuation Principle

Price of a security is based on estimated values based on

expectations. This is also called the Intrinsic Value.

It is the PV of all EXPECTED Cash Flows fr om thatasset



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Bond Prices

Cash Flows fr om a security can be in any f orm:

Example??



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Bond Prices

Cash Flows fr om a security can be in any f orm:

Dividends

Interest Payments

Coupons

Redemption value



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Bond Prices

Since C/F¶s are in future, they must be

Discounted

And converted to PV.

Sum of all the PV¶s of C/f¶s = Estimated Intrinsic Value



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Bond Prices

Formula f or any asset:

Value = E ??



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Bond Prices

Formula f or any asset:

Value = E C/F¶s / (1+i)^n



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Bond Valuation

Bond has TWO types of Cash Flows:?



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Bond Valuation

Bond has TWO types of Cash Flows:

1. Coupons

2. Face Value

Both of these are known in advance



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Bond Prices

Formula f or Bonds:

Value = E C/F¶s / (1+i)^t + FV/(1+i)n

Note: Coupons are ?

t =

n =

i =



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Bond Prices

Formula f or Bonds:

Value = E C/F¶s / (1+i)^t + FV/(1+i)n

Note: Coupons are semi-annual

t = period on semi-annual basis

n = periods also semi-annual

i = semi-annual rate

John Burr Williams published this eq. in 1938



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Bond Prices

Formula Breakdown:

Value = E C/F¶s / (1+i)^t + FV/(1+i)^n

3 stages:

1) Coupons??

2) FV is single C/F: Find simple PV

3) Add all PV¶s together



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Bond Prices

Formula Breakdown:

Value = E C/F¶s / (1+i)^t + FV/(1+i)^n

3 stages:

1) Coupons make it an annuity: Find Annuity PV

2) FV is single C/F: Find simple PV

3) Add all PV¶s together



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M55

Calculating Bond Price

Example bond AA new 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual is issued.

What is the Price of the Bond?



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Example bond AA new 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual is issued.

What is the Price of the Bond? Ordinary annuity of 40

APV = 40 x ((1- 1/1.04^20) / .04

= 40 *( 1-0.4563)/ .04 = 40*13.59

= 543.61 (coupons PV)

FV PV = (1000) / (1.04)^20 = 1000*.45638 = 456.38

Bond PV = 543.61 + 456.38 = 1000 (same as Face value)



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Example bond A ±

NOW INTEREST RATES GO UP

Same 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual is issued. But market interest rates are at 10%.

What is the NEW Price of the Bond?



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Example bond A ± NOW INTEREST RATES GO UPSame 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual isissued. But market interest rates are at 10%.

What is the NEW Price of the Bond? Ordinary annuity of 40

APV = 40 x ((1- 1/1.05^20) / .05

= 40 *( 1-0.3768)/ .05 = 40*12.46

= 498.48 (coupons PV)FV PV = (1000) / (1.05)^20 = 1000*.3768 = 376.89

Bond PV = 498.48 + 376.89 = 875.36 (Now at discount since i is up)



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Bond Price over time

FV = 1000 at issue FV = 1000 at end

Interest rate DROPSPrice RisesCos Discounting at lower rate

Interest rate RISESPrice DROPSCos discounting & higher rate

Bond Px

Converge to Par Value over time



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Bond Price Changes

Why is this happening?



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Bond Price Changes

Why is this happening?

1. Coupons are the same

2. Face Value is the same

3. If interest rates in the market change,ONLY the PV can & must change

to reflect the interest rates



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Bond Price Changes

What does price change do?

1. Theref ore,

Capital Gain (LOSS) is built in to compensatethe investor f or the change in interest rates inthe market.

Discount = gain Premium = Loss



S l Fi



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Example bond A ± NOW INTEREST RATES GO DOWN

Same 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual isissued. But market interest rates are at 6%.

What is the NEW Price of the Bond? Ordinary annuity of 40

APV = 40 x ((1- 1/1.03^20) / .03

= 40 *( 1-0.55366)/ .03 = 40*14.87

= 595.099 (coupons PV)FV PV = (1000) / (1.03)^20 = 1000*.55366 = 553.66

Bond PV = 595.099 + 553.669 = 1148.775 (Now at premium since i is down)

S l Fi



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R elationship Bond Px & Interest R ates

Bond Px is«?

S l Fi



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R elationship Bond Px & Interest R ates

Bond Px is«

INVERSELY Related TO Interest Rates

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Measuring Bond Yields

There are 4 measures

Current Yield

Yield to Maturity

Yield to Call

Realized Compound Yield

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There are 4 measures

Current Yield (CY)

A Bond¶s annual coupon divided by current market price.

Yield to Maturity (YTM)

Pr omised Compound rate of return at the current market price till mat.

Yield to Call (YC)Pr omised return on a Bond fr om present till date it is likely to be called

Realized Compound Yield (RCY) Yield earned based on actual reinvestment rates-Historical

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Current Yield (CY)

A Bond¶s annual coupon divided by current market price.

Ratio of coupon interest to current Mkt px.

e.g.A 3 year 10% coupon Bond with interest payments occurring exactly 6mths fr om now & so on. Current price of the Bond is $1052.42

What¶s the current yield?

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A 3 year 10% coupon Bond with interest payments occurring exactly 6mths fr om now & so on. Current price of the Bond is $1052.42

What¶s the current yield?

C/Px = 100 / 1052.10 = 9.5%

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Yield to Maturity (YTM) ± semiannual rate


The rate most often quoted

Return based on fixed assumptions«???

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Yield to Maturity (YTM) ± semiannual rate


The rate most often quotedReturn based on fixed assumptions

1) The Bond is held to Maturity

2) Coupons are re-invested at the YTM

Means: Compounded return giving PV of Bond as its current price

Same as IRR f or the Bond

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Yield to Maturity (YTM)


Small letters = semi-annual

Capital letters = annual

PV = E c/(1+ytm)^t + FV/ (1+ytm)^n

FV = 1000



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YTM

Example on zer o-coupon

A zer o coupon bond has 12 yrs to maturity & is selling f or $300.

Calculate the ytm & BEY?

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YTM

Example on zer o-couponA zer o coupon bond has 12 yrs to maturity & is selling f or $300.

Calculate the ytm & BEY.

PV = FV / (1+ytm)^n

300 = 1000 / (1+ytm)^24

1+ytm^24 = 1000/300

1+ ytm = 3.33^(1/24)

ytm = 3.33^.04166 = 1.0514 ± 1 = 5.14% ytm

5.14x2 = 10.28 BEY

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YTM

Example on regular coupon bondA 10% coupon bond has 3 yrs to maturity & is selling f or 1052.42


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YTM

Example on regular coupon bondA 10% coupon bond has 3 yrs to maturity & is selling f or 1052.42


PV = E 50/(1+ytm)^t + 1000/(1+ytm)^6

1052.42 = 50 x 5.242 (APVF) + 1000x 0.790 (PVF)

Trial & err or ± will cover after basic theory

Ytm = 4% semi annual

BEY = 4x2 = 8% annual

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Yield to Call (YTC)

Pr omised rate of return on a bond fr om present till called.

Steps to calculate: USE SAME FORMULA & MODIFY

a) find n till call date

b) Call Price substituted f or Face Value

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Realized Compound Yield (RCY)-historical

Yield earned based on actual re-investment rates (IRR)

Semi-annual realized compound yield is:

RCY = ( total wealth=FV / Pur. Px=PV ) ^ 1/n - 1

NOTE: comes from the FV=PV(1+i)^n, reshuffling for i

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R C Y (realized c yield)

Example on regular coupon bondAn investor had $1000 to invest 3 yrs ago. She purchased a 10%coupon bond with a 3 year maturity at Face Value. The YTM was 10%.

Assume the investor re-invested each coupon at the semi-annual rateor ytm of 5%.

What is the total ending wealth after 3 yrs?

What is the RCY?

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R C Y

Example on regular coupon bondAn investor had $1000 to invest 3 yrs ago. She purchased a 10%coupon bond with a 3 year maturity at Face Value. The YTM was 10%.

Assume the investor re-invested each coupon at the semi-annual rateor ytm of 5%.

What is the total ending wealth after 3 yrs?

(1+.05)^6= 1.340095 * 1000 = 1340.09 (includes the initial outlay)

thus, earnings were 340.09

What is the RCY?

(1340.10 / 1000 )̂ 1/6 ± 1 = 1.05 ± 1 = .05 RCY x2 = 10% BEY



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R e-investment R isk

assumption

Part of interest rate risk resulting fr om

uncertainty abo

ut the rate at which futureinterest coupons can be re-invested

Assumption is unlikely in real life«why?

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Bonds sources of returns

Coupons

Capital Gains

Interest on interest (coupon re-investment):

largest part of the RCY

POINTS to NOTE:

a) As maturity increases, reinvestment risk increases.

b) Higher the coupon rate, the higher reinvestment risk.

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For a zer o-coupon bond«.

What is the RCY equal to?

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For a zer o-coupon bond«.

What is the RCY equal to?

RCY = YTM since there are NO COUPONS &theref ore no re-investment risk

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Illustration of re-investment portion

Example on regular coupon bondA 10% coupon bond with 20yr maturity purchased at PAR ($1000). If all coupons are re-invested at 5% on semi-annual basis, (ytm=5%),

then«.

What is the total dollar return at end 20yrs?

What is the break down of returns? FV+Coupons+interest on interest

M20

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Illustration of re-investment portion

Example on regular coupon bondA 10% coupon bond with 20yr maturity purchased at PAR ($1000). If all coupons are re-invested at 5% on semi-annual basis, (ytm=5%),

then«.

What is the total dollar return at end 20yrs? Ordinary annuity of 50

AFV = 50 x ((1+.05)^40 ± 1 ) / .05

= 50 *( 7.03-1)/ .05 = 50*120.79

= 6039.50 (includes coupons) + FV (1000) = $7040

What is the break down of returns? FV+Coupons+interest on interest

1000+2000+4040 = 7040 Note: 6040-2000 coupons = 4040

POINT: 4040/7040 = 57% comes fr om re-investment!!

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W hat is one advantage of

Zer o-coupon bond«.in terms of risk?

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W hat is one advantage of

Zer o-coupon bond«.in terms of risk?

NO Re-investment risk, since no coupons

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Horizon R eturn

All yield measures have pr oblems.

How do investors solve this pblm?

1. They make future assumptions about the re-investmentrates.

2. Calculating the return on bonds based on futureassumptions is called, Horizon Return

NOTE: Yield curve moves based on this.

Salaar Salaar --FinanceFinance



PART II

BONDS & INTEREST RATES

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Bonds

What is a Bond?

Skip

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BondsWhat is a Bond

Debt instrument to raise money (Loan)

Issued by Corporations & Govts

Interest Only Loan

Has a standard Face Value of $1000 (Corp)

Coupons are paid semi-annually

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Bond Yields

2 Components of Interest rates

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Bond Yields

Components of Interest rates

Real Risk Free rate:

opportunity cost of f oregoing consumption (given no inflation)

Nominal Interest Rates:

Real rate PLUS inflation adjustment

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Bond Yields

For T- Bills:

The nominal rate is a function of real rate &expected inflationary premium.

RF =(appr ox) RR + EI

Where: RF = T-bill rate

RR = real risk free rate

EI = expected inflation rate

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Bond Yields

Equation RF =(appr ox) RR + EI

Is called

Fisher Hypothesis (irving fisher)

Means:Nominal rate on ST, RF securities rises point f or point withexpected inflation, with RR unaffected (consumption opp cost)

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Fish

er Effect

The nominal rate R is a function of real rate r

& expected inflatio

nary premium h.

Equation is«?

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Fish

er Effect

The nominal rate R is a function of real rate r

& expected inflatio

nary premium h.

1+ R = (1+r) x (1+h)

Where: R = N

ominal Rate

r = real risk free rate

h = inflation rate

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M

EASUR

INGR

ETUR

NSFormula ± Real Returns

Real Return (inflation adjusted)

TRia (r) = ( (1+ R (nominal) ) / (1+h) ) ± 1

TRia = total return inflation adjusted

h = inflation rate

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PROBLEM Finding inflation adjusted returns

Suppose the nominal return on a stock is 28.5731%and the inflation rate is 1.6119%.

a) W h

at is th

e real rate?b) W hat is the inflation adjusted return?

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PROBL

EM

Finding inflation adjusted returns

Suppose the nominal return on a stock is 28.5731%

and the inflation rate is 1.6119%.

a) W hat is the real rate?

1.2857/1.0161 = 1.2653 ² 1 = 26.53%

b) Same as a!!

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PO

INT TO

NO

TE

Most financial rates are quoted in NOMINAL.

We will use the symbol R f or this nominal rate

START

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D

eterminants of B

ond Yields

Term Structure of Interest Rates

1) Real Rate (opportunity cost)

2) Expected Inflation (investors require higher nominal rates)

3) Interest Rate risk premium (coupon bonds) ± yield curve

Definition of TSIR

Nominal rate on default-free, pure discount bonds of all maturities

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D

eterminants of B

ond Yields

Inflation Premium

Portion of Nominal rate representing compensation f or Exp. Inflation

Interest Rate risk premium

Compensation f or taking on interest rate risk (coupons)

Longer maturity has higher interest risk, theref ore positive slope



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M5

Yield Curve

A chart of the yields of T-bills & Bonds w/maturity

Almost same as the term structure.

Difference:1) Yield curve plotted based on coupon bonds (interest rate component)

2) Term structure based on pure discount bonds (no interest rate risk)

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Corp Bonds have additional

Premiums

Default risk premium

Compensation f or possibility of default

Taxability PremiumCompensation f or unfavorable tax implications

Liquidity PremiumCompensation f or lack of liquidity (ability to sell)

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Bond Determinants summary

Real RateAdjusted f or inflation

Expected Inflation

Compensation f or inflation expectations

Interest Rate riskCompensation f or future changes in interest rates

Default risk premium

Compensation f or possibility of default

Taxability PremiumCompensation f or unfavorable tax implications

Liquidity Premium

Compensation f or lack of liquidity (ability to sell)

Govt. Bonds

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W hat affects Bond Prices

Effects of Maturity? +ve

Effects of coupon size? -ve

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W hat affects Bond Prices

Effects of Maturity?

For a given change in rates, price of longer term

bonds will change more than shorter term bonds

Effects of coupon size?

Bond Px fluctuations (volatility) are inversely related to coupon rates.

The larger the coupon, f or same maturity, the lower the volatility

m40 Bond Px sensitivity ( 100bp chg on 10% & 100% coupons effect on %px chg!)



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F

orB

onds

Longer the maturity, higher the IR risk

WHY?

Cos Longer term has a greater discounting effect(compounding curve) on the Face Value receivedat maturity!

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For Bonds

Maturity exposure

Longer the maturity, higher the IR risk

Theref ore

Since the Principal is at end, Bond¶s Px ismore sensitive to time (maturity)

PV on 30yrs is more sensitive to PV on 1 year!! (f or Face Value)

NOTE: But it increases (term risk) at a decreasing rate

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For Bonds

The other factor

Coupon Rate risk

?

WHY?

M20

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For Bonds

The other factor

Coupon Rate risk

Lower Coupons have higher risk.

WHY?

M20

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For Bonds

The other factor

Coupon Rate risk

Lower Coupons have higher risk.

Lower coupons make Bond Px more dependant on FV!

(Risk = % changes in Bond px f or given chg in interest rates)

WHY?

Since C/F¶s are paid towards the back end of time line &

relative to Face value are smaller in size! (affected moreby discounting)

NOTE: compare PV of zer o-coupon vs regular

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Measuring Volatility:

Duration

Duration

A measure of a bond¶s lifetime which accounts f or the

entire pattern of cash flows.Measures the weighted average maturity of C/F¶s on a PVbasis.

To solve which pr oblem?

Changes in interest rates result in different % changes inBond Px¶s. Duration combines the coupon & maturity

(size & timing) of C/F effects into one yardstick.

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Measuring Volatility:

Duration Example

Duration of 4.054 (on 5 yr bond) means:

The TVM weighted average number of years

needed to

reco

ver the co

sto

f thisBo

nd is 4.054.

Although the bond has 5 yrs to maturity, interest paymentsare received in the first 4 yrs

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Calculating Duration

Duration (stated in yrs)

Co

nvert TIME to

a weighted time perio

d.

Concept

All time periods are weighted & summed. The result is

duration.

PV of each cash flow as a %age of the current price serves asthe weights, which are then applied to time periods. SUM of these equals 1.0



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Exercise



Coupon 10% 5% 50

Maturity 5 10 coupon $

Price 1000

n t C/F PV Factor PV of CF PV /Px t x (PV /Px)

1 0.5 50 0.9524 47.6190 0.0476 0.0238

2 1.0 50 0.9070 45.3515 0.0454 0.0454

3 1.5 50 0.8638 43.1919 0.0432 0.0648

4 2.0 50 0.8227 41.1351 0.0411 0.0823

5 2.5 50 0.7835 39.1763 0.0392 0.0979

6 3.0 50 0.7462 37.3108 0.0373 0.1119

7 3.5 50 0.7107 35.5341 0.0355 0.1244

8 4.0 50 0.6768 33.8420 0.0338 0.1354

9 4.5 50 0.6446 32.2304 0.0322 0.1450

10 5.0 1050 0.6139 644.6089 0.6446 3.2230

TO TALS 1000 1.00 4.0539

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Coupon 10% 5% 50



Maturity 5 10 coupon $

Price 1000

n t C/F PV Factor PV of CF PV /Px t x (PV /Px)

1 0.5 50 0.9524 47.6190 0.0476 0.0238

2 1.0 50 0.9070 45.3515 0.0454 0.04543 1.5 50 0.8638 43.1919 0.0432 0.0648

4 2.0 50 0.8227 41.1351 0.0411 0.0823

5 2.5 50 0.7835 39.1763 0.0392 0.0979

6 3.0 50 0.7462 37.3108 0.0373 0.1119

7 3.5 50 0.7107 35.5341 0.0355 0.1244

8 4.0 50 0.6768 33.8420 0.0338 0.1354

9 4.5 50 0.6446 32.2304 0.0322 0.1450

10 5.0 1050 0.6139 644.6089 0.6446 3.2230

TO TALS 1000 1.00 4.0539

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Duration

Shorter Formula :

Duration D = (1+ytm)/ytm . ( 1- (1/ytm^n))

Use semi-annual rateDouble the n

Divide answer by 2 to convert to annual basisM10

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Understanding Duration

Duration depends on 3 factors:

Final Maturity of Bond

Duration expands with time to maturity: directly related:For zer o coupon, D=Maturity

Coupon Payments

Coupon size is inversely related to duration.

YTM

YTM is inversely related to duration

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What does Duration tell us?

Difference b/w effective LIVES of bonds

Allows us to compare bonds on this basis

D is a measure of bond-price sensitivity to interest rate movements.

It measures interest rate risk

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d d




Example

Given a 10% coupon bond, ytm of 10%..

If maturity is 5yrs, D = 4.054 (effective life!)

If maturity is 10yrs, D = 6.76

If maturity is 20yrs, D = 9.36

If maturity is 50yrs, D = 10.91 yrs

Reason is, C/F¶s in distant future result in smaller PV¶s

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Modified Duration

Defined

Mod Duration is Duration divided by (1+ytm)

Mod D* = D / (1+ytm)

Ytm = semi-annual

Mod D, is used to calculate % price change f or agiven change in interest rates (usually 100bp)

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M difi d D i



Modified Duration

EXAMPLE

Using Duration of 4.054 yrs & YTM of 10%,

What is the Modified Duration?

Mod D* = D / (1+ytm)

Ytm = semi-annual

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M difi d D i



Modified Duration

EXAMPLE

Using Duration of 4.054 yrs & YTM of 10%,

What is the Modified Duration?

D* = 4.054 / (1+.05) = 3.861

Now we use this to calculate change in Px f or a given interest rate change

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M difi d D i



Modified Duration

EXAMPLE

To calculate %change in Price, use«

% Chg in Px = -D* x yield change

So, what¶s the change in Px of example, if yield

changes by + 20 bp?

M5

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M difi d D i



Modified Duration

EXAMPLE

To calculate %change in Price, use«

% Chg in Px = -D* x yield change

So, what¶s the change in Px of example, if yieldchanges by + 20 bp?

-3.861 x (+0.002) x 100 = -0.772%100 is to convert to % basis.



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C it



Convexity

rice ro i a ion in ra ion

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PRICE EST. PRICE

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Finance Environment?

OVERVIEW

COVER HERE & Ref Charts

GDP (Equation, components, growth bands, & targets)

Inflation (result of growth)

Interest Rates (tool of central banks, fed funds & principle)

Yield Curve (fisher equation, expected inflation back end)

MARKET DASH BOARD!

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Chapter 17



p

Bond Yields & Prices

What we have learnt

Bonds & Valuation

Interest rates

Bond Yields

Bond Prices

Bond Price Changes

Duration, Modified Duration & Convexity

Documents

Ch17Bonds MBA2