Lecture 4(Ch6 Bonds) NCBA&E

7/31/2019 Lecture 4(Ch6 Bonds) NCBA&E

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Interest Rates and BondValuations

Winter Semester 2011

NCBA&E

Instructor

Jamal Nasir Khan


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Lecture 3


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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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Bonds

What is a Bond

Debt instrument to raise money (Loan)

Issued by Corporations & GovernmentsInterest Only Loan

Has a standard Face Value of $1000 (Corp)

Coupons are paid semi-annually


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Fixed-Income SecuritiesBONDS

Capital Markets:

Where debt & equity capital is raised. Market for 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 before maturity, price will reflectcurrent interest rates.


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Fixed-Income Securities

Bond Characteristics

Par Value:

Term Bond: Term to Maturity:

Coupon:

Coupon Rate:

Zero Coupon: Bond Prices: add zero


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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

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

Bond Prices: Quoted as a %age of Par Value. Use 100 as conventional parrather 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 for early bondpayments


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Bond Examples

A 10% coupon Bond has what dollar coupon?:

A Bonds quoted Px is 101 3/8. Whats the $ price?

Interest Rate & Bond Price:


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Bond Examples

A 10% coupon Bond has what dollar coupon?:

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

A Bonds quoted Px is 101 3/8. Whats the $ price?

Represents 101.375% of 1,000, therefore $ price is $1013.75

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

Interest rates are inversely related to price. Morecoverage later chapter.

Interest Rate & Bond Price: Bond will be exactly worth its face value at maturity. But till then it fluctuates around

$1000 to adjust yield according to market interest rates.


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Call Provision:

When is it attractive to the issuer?

Call Premium


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Call Provision:

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 drop in the markets enough to save the issuer money.

Additional cost for 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 BONDS

4 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 through 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 ofbankruptcy.

Convertible Bonds:

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

Rating Agencies: 3 agencies S&P, Moodys, Fitch provide Bond ratingsrepresenting current opinions on relative credit quality of the firm. (PACRA inPakistan)

Bond Ratings: Letters of the alphabet assigned to Bonds to expressprobability 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 Yields

2 Components of Interest rates


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Bond Yields

Components of Interest rates

Real Risk Free rate:

opportunity cost of foregoing consumption (given no inflation)

Nominal Interest Rates:

Real rate PLUS inflation adjustment


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Bond Yields

Example of inflation effects

If real rate is 10% & inflation is 12%.

Whats the value of $100 investment today?


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Bond Yields



Whats the value of$110 today?


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Bond Yields



Whats the value of $110 today?110 / (1.12) = 98.21 < 100 originally invested!

Therefore, an expected inflation premium is needed to adjust

purchasing power.


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Bond Yields

For T- Bills:

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

RF = RR + EI

Where: RF = T-bill rate

RR = real risk free rate

EI = expected inflation rate


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Bond Yields

Equation RF = RR + EI

Is called

Fisher Hypothesis (Irving Fisher)

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


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Fisher Effect

The nominal rate R is a function of real rate r

& expected inflationary premium h.

Equation is?


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Fisher Effect

The nominal rate R is a function of real rate r

& expected inflationary premium h.

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

Where: R = Nominal Rate

r = real risk free rate

h = inflation rate


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MEASURING RETURNS

Formula 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) What is the real rate?

b) What is the inflation adjusted return?


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PROBLEM Finding inflation adjusted returnsSuppose the nominal return on a stock is28.5731% and the inflation rate is 1.6119%.

a) What is the real rate?

1.2857/1.0161 = 1.2653 1 = 26.53%

b) Same as a!!


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POINT TO NOTE

Most financial rates are quoted in NOMINAL.

We will use the symbol R for this nominalrate


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Determinants of Bond 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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Determinants of Bond Yields

Inflation Premium

Portion of Nominal rate representing compensation for Exp. Inflation

Interest Rate Risk Premium

Compensation for taking on interest rate risk

Longer maturity has higher interest risk, therefore positive slope


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Term Structure (based on pure discount bonds)

Real Rate

Inflation Premium

Interest Rate risk premium

InterestRate

Maturity


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M5

Yield Curve (plotted against coupon bonds)

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 for possibility of default

Taxability PremiumCompensation for unfavorable tax implications

Liquidity PremiumCompensation for lack of liquidity (ability to sell)


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Bond Determinants Summary

Real RateAdjusted for inflation

Expected Inflation

Compensation for inflation expectations

Interest Rate riskCompensation for future changes in interest rates

Default risk premium

Compensation for possibility of default

Taxability PremiumCompensation for unfavorable tax implications

Liquidity Premium

Compensation for lack of liquidity (ability to sell)


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Bond Prices

Valuation Principle

Bond Valuation


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Bond Prices

Valuation PrinciplePrice 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 from thatasset


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Bond Prices

Cash Flows from a security can be in any

form:

Example??


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Bond Prices

Cash Flows from a security can be in any

form:

Dividends

Interest Payments

Coupons

Redemption value


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Bond Prices

Since C/Fs are in future, they must be

Discounted

And converted to PV.

Sum of all the PVs of C/fs = Estimated Intrinsic Value


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Bond Prices

Formula for any asset:

Value = E ??


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Bond Prices

Formula for any asset:

Value = E C/Fs / (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 for Bonds:

Value = E C/Fs / (1+i)^t + FV/(1+i)n

Note: Coupons are ?t =

n =

i =


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Bond Prices

Formula for Bonds:


Note: Coupons are semi-annualt = period on semi-annual basis

n = periods also semi-annual

i = semi-annual rate


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Bond Prices

Formula for Bonds:


Note: Coupons are semi-annualt = 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/Fs / (1+i)^t + FV/(1+i)^n

3 stages:1) Coupons??

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

3) Add all PVs together


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Bond Prices

Formula Breakdown:

Value = E C/Fs / (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 PVs 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. Therefore,

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

Discount = gain Premium = Loss


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Checking the discount gain logic

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

Whats the PV of $20(100-80) coupon not paid by thisBond?


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Checking the discount gain logic


Whats the PV of $20(100-80) coupon not paid by thisBond?

APV = 10x ((1- 1/1.05^20) / .05

= 10 *( 1-0.3768)/ .05 = 10* .62/.05 = 10*12.464

= 124.64 (missed coupons PV)

NOTICE: Bond Discount was 1000-875.36 = 124.64

& PV OF $20 MISSED ANNUITY IS ALSO 124.64 TO COMPENSATE

FOR PREMIUM: INVESTOR WILL HAVE TO PAY EXTRA


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Example bond ANOW INTEREST RATES GO DOWNSame 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual isissued. But market interest rates are at 6%.



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Example bond ANOW INTEREST RATES GO DOWNSame 8% coupon bond with 10 year maturity, PAR of 1000, semi-annual isissued. But market interest rates are at 6%.


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)


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Checking the premium loss logic Example bond A

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

Whats the PV of $20(80-60) coupon paidextra by thisBond?


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Calculating Bond PriceChecking the premium loss logic


Whats the PV of $20(80-60) coupon paidextra by thisBond?

APV = 10x ((1- 1/1.03^20) / .03

= 10 *( 1-0.553676)/ .03 = 10* .4463/.03 = 10*14.877

= 148.77 (extra coupons paid PV)

NOTICE: Bond Premium was 1148.775-1000 = 148.77

& PV OF $20 Extra ANNUITY IS ALSO 148.77 (offset by charging more)


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Relationship Bond Px & Interest Rates

Bond Px is?


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Relationship Bond Px & Interest Rates

Bond Px is

INVERSELY Related TO Interest Rates


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Bond Yields

Looking at Risk Now

All rates are affected by 2 variables from theRisk free rate:

a) Maturityb)Risk Premium

c) Coupon Rate RiskKeep this in mind!


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Time Factor

(maturity differentials & i )

All Interest Rates are affected byTIME FACTOR.

Therefore,

Longer Term maturities yield more thanshorter term

Thus, Bonds i > Notes i > Bills (Typical relationship)


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For Bonds

Longer the maturity, higher the Interest RateRisk

WHY?


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For Bonds

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

Therefore

Since the Principal is at end, Bonds Px is

more sensitive to time (maturity)

PV on 30yrs is more sensitive to PV on 1 year!! (for 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 for given chg in interest rates)

WHY?

Since C/Fs are paid towards the back end of time line &

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

NOTE: compare PV of zero-coupon vs regular


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Third Factor

All interest rates other than risk less areaffected by a THIRD Factor

Risk Premium

Also called yield spread OR differential


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What affects Bond Prices

Effects of Maturity? +ve

Effects of Coupon Size? -ve


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What 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 tocoupon rates.

The larger the coupon, for same maturity, the lower the volatility

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


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Bond Yields

Components of Interest rates Measuring Bond Yields

Current yield (CY)

Yield to maturity (YTM)

Yield to call (YC) Realized Compound yield (RCY)

Re-investment Risk


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Measuring Bond Yields


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Measuring Bond YieldsThere 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 Bonds annual coupon divided by current market price.

Yield to Maturity (YTM)

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

Yield to Call (YC)

Promised return on a Bond from 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 Bonds 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 from now & so on. Current price of the Bond is $1052.42

Whats the current yield?


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

Whats the current yield?

C/Px = 100 / 1052.10 = 9.5%


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Yield to Maturity (YTM) semiannual rate

Promised Compound rate of return at the current market price till mat. The rate most often quoted

Return based on fixed assumptions???


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Yield to Maturity (YTM) semiannual rate

Promised Compound rate of return at the current market price till mat. 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 for the Bond


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Small letters = semi-annual

Capital letters = annual

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

FV = 1000

N= no. of semiannual periods & t=payment period


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Bond Equivalent Yield

Yield on an annual basis derived by doubling the semi-annual yield.

eg. 5.1% semi-annual ytm is 5.1x2 = 10.2% BEY


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YTM

Example on zero-couponA zero coupon bond has 12 yrs to maturity & is selling for $300.

Calculate the ytm & BEY?


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YTM

Example on zero-couponA zero coupon bond has 12 yrs to maturity & is selling for $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 bond has 3 yrs to maturity & is selling for 1052.42



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YTM

Example on regular coupon bondA bond has 3 yrs to maturity & is selling for 1052.42


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

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

Trial & errorwill cover after basic theoryYtm = 4% semi annual

BEY = 4x2 = 8% annual


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Yield to Call (YTC)

Promised rate of return on a bond from present till called.

Steps to calculate: USE SAME FORMULA & MODIFY

a) find n till call date

b) Call Price substituted for 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 theFV=PV(1+i)^n, reshuffling for i


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RCY (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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RCY

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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Points to Note

Coupons are seldom re-invested at same rates

YTM is totally dependant on some assumptions RCY is the actual rate earned at the end.

Assumptions:a) Bond is held to maturity

b) Coupons are re-invested at same rates as YTM

YTM = IRR


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Re-investment Riskassumption

Part of interest rate risk resulting fromuncertainty about the rate at which future

interest coupons can be re-invested

Assumption is unlikely in real lifewhy?


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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 zero-coupon bond.

What is the RCY equal to?


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For a zero-coupon bond.

What is the RCY equal to?

RCY = YTM since there are NO COUPONS &therefore 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). Ifall 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). Ifall 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 from re-investment!!


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What is one advantage of

Zero-coupon bond.in terms of risk?


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What is one advantage of

Zero-coupon bond.in terms of risk?

NO Re-investment risk, since no coupons


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Horizon Return

All yield measures have problems.

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.

Measuring Volatility:


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Duration

Duration?



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Duration

Duration

A measure of a bonds lifetime which accounts for theentire pattern of cash flows.

Measures the weighted average maturity of C/Fs on a PVbasis.

To solve which problem?

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

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



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Duration Example

Duration of 4.054 (on 5 yr bond) means:

The TVM weighted average number of yearsneeded to recover the cost of this Bond is 4.054.

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

Calculating Duration


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Calculating Duration

Duration (stated in yrs)

Convert TIME to a weighted time period.

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 ofthese equals 1.0


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Duration

Formula :

Duration D = E (PV (CFt) / Mkt Px ) x t

PV of CF is found at YTM discount

Mkt Px is current PV of bond

Coupon 10% 5% 50

Exercise


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p

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

TOTALS 1000 1.00 4.0539

Coupon 10% 5% 50



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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 0.145010 5.0 1050 0.6139 644.6089 0.6446 3.2230

TOTALS 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

Understanding Duration


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g

Duration depends on 3 factors:

Final Maturity of Bond

Duration expands with time to maturity: directly related:For zero 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 tointerest rate movements.

It measures interest rate risk



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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/Fs in distant future result in smaller PVs

Modified Duration


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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 for agiven change in interest rates (usually 100bp)

Modified Duration


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EXAMPLE

Using Duration of 4.054 yrs & YTM of 10%,

What is the Modified Duration?

Mod D* = D / (1+ytm)

Ytm = semi-annual

Modified Duration


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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 fora given interest rate change

Modified Duration


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EXAMPLE

To calculate %change in Price, use

% Chg in Px = -D* x yield change

So, whats the change in Px of example, if yield

changes by + 20 bp?

M5

Modified Duration


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EXAMPLE

To calculate %change in Price, use

% Chg in Px = -D* x yield change

So, whats 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.

C i C


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Convexity Concept

Mod. D is an approximation because, therelationship of Mod D to actual price changesis convex.

Mod D itself is a linear relationship.

Mod D is a tangent to the actual relationship &therefore is accurate for smaller changes, butdiverges for larger changes.


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Convexity

Price Approximation Using Duration

55

65

75

85

95

105

115

125

135

145

155

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Yield

Price

PRICE EST. PRICE

Chapter 17

B d Yi ld & P i


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Bond Yields & Prices

What we have learnt

Bonds & Valuation

Interest rates

Bond Yields

Bond Prices

Bond Price Changes

Duration, Modified Duration & Convexity

Documents

Lecture 4(Ch6 Bonds) NCBA&E