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8/2/2019 Bond Valuation Jitesh

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Submitted By :Jitesh TalesaraSec B : 12222

FYPGDMISB&M Pune

Bond ValuationProf . Neeraj Kapoor


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Bonds and Bond Valuation

More on Bond Features

Bond Ratings

Some Different Types of Bonds

Bond Markets

Inflation and Interest Rates

Determinants of Bond Yields


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33

Bond ValuationLearning Module


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Definitions

Par or Face Value -

The amount of money that is paid to the bondholders atmaturity. For most bonds this amount is $1,000. It alsogenerally represents the amount of money borrowed by thebond issuer.

Coupon Rate -The coupon rate, which is generally fixed, determines the

periodic coupon or interest payments. It is expressed as apercentage of the bond's face value. It also represents theinterest cost of the bond to the issuer.


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Definitions

Coupon Payments -

The coupon payments represent the periodic interestpayments from the bond issuer to the bondholder. The

annual coupon payment is calculated by multiplying thecoupon rate by the bond's face value. Since most bondspay interest semiannually, generally one half of the annualcoupon is paid to the bondholders every six months.

Maturity Date -

The maturity date represents the date on which the bondmatures, i.e., the date on which the face value is repaid.

The last coupon payment is also paid on the maturity date.


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Definitions

Original Maturity -

The time from when the bond was issued until its maturitydate.

Remaining Maturity -

The time currently remaining until the maturity date.

Call Date -

For bonds which are callable, i.e., bonds which can be

redeemed by the issuer prior to maturity, the call datere resents the earliest date at which the bond can be


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Definitions

Call Price -

The amount of money the issuer has to pay to call acallable bond (there is a premium for calling the bondearly). When a bond first becomes callable, i.e., on thecall date, the call price is often set to equal the facevalue plus one year's interest.

Required Return -

The rate of return that investors currently require on abond.


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Definitions

Yield to Maturity -

The rate of return that an investor would earn if hebought the bond at its current market price and held ituntil maturity. Alternatively, it represents the discountrate which equates the discounted value of a bond'sfuture cash flows to its current market price.

Yield to Call -

The rate of return that an investor would earn if hebought a callable bond at its current market price andheld it until the call date given that the bond was called

on the call date.


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Bond Valuation

Bonds are valued using time value ofmoney concepts.

Their coupon, or interest, payments aretreated like an equal cash flow stream(annuity).

Their face value is treated like a lumpsum.


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Example

Assume Hunter buys a 10-year bond from the KLMcorporation on January 1, 2003. The bond has aface value of $1000 and pays an annual 10%coupon. The current market rate of return is 12%.Calculate the price of this bond today.

1. Draw a timeline

$100

$100

$100 $10

0

$100$10

0

$100 $10

0

$100

$100

$1000 +

??


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Example

2. First, find the value of the coupon stream Remember to follow the same approach

you use in time value of moneycalculations.

You can find the PV of a cash flow stream PV = $100/(1+.12)1 + $100/(1+.12)2 + $100/

(1+.12)3 + $100/(1+.12)4 + $100/(1+.12)5 +$100/(1+.12)6 + $100/(1+.12)7 + $100/(1+.12)8 + $100/(1+.12)9+ $100/(1+.12)10

Or, you can find the PV of an annuity PVA = $100 * {[1-(1+.12)-10]/.12}

PV = $565.02


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Example

3. Find the PV of the face value

PV = CFt / (1+r)t

PV = $1000/ (1+.12)10

PV = $321.97

4. Add the two values together to get the total PV

$565.02 + $321.97 = $886.99

. Alternatively, on your calculator

PMT = 100FV = 1000n = 10

i = 12PV = ?


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Realized Return

Sometimes you will be asked to find the realizedrate of return for a bond.

This is the return that the investor actuallyrealized from holding a bond.

Using time value of money concepts, you are

solving for the required rate of return instead ofthe value of the bond.


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Example

Doug purchased a bond for $800 5-years ago and hesold the bond today for $1200. The bond paid anannual 10% coupon. What is his realized rate ofreturn?

n

PV = [CFt / (1+r)t]

t=0

$800 = [$100/(1+r) + $100/(1+r)2 + $100/(1+r)3 +$100/(1+r)4 + $100/(1+r)5] + [$1200/(1+r)5]

To solve, you need use a trail and error approach.You plug in numbers until you find the rate of return

that solves the equation.


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Example

This is much easier to find using a financial calculator:

n = 5

PV = -800FV = 1200PMT = 100i = ?, this is the realized rate of return on this bond

Note that if the payments had been semiannual,n=10, PV=-800, FV=1200, PMT=50, i=?=9.47%.Thus, the realized return would have been 2 * 9.47%

= 18.94%.


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Bond Valuation

Bonds are valued using time value ofmoney concepts.

Their coupon, or interest, payments aretreated like an equal cash flow stream(annuity).

Their face value is treated like a lumpsum.


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Interest Rates and Bond Valuation


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BOND VALUATION

Dr. Rana SinghAssociate Professorwww.ranasingh.org


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CONTENTS

IntroductionBond Returns

coupon rate

current yield spot interest rate

yield to maturity

yield to callBond Prices

Bond Pricing Theorems

Bond Risks


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INTRODUCTION

Bonds are Long-term fixed incomesecurities. Debentures are also long-termfixed income securities. Both of these aredebt securities.

The two major categories of bonds aregovernment bonds & corporate bonds.

There are the two main features of bondssuch as callability and convertibility.


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BOND RETURNS

COUPON RATE:-

It is the nominal rate of interest fixed

andprinted on the bond certificate. It iscalculated on

the face value of the bond. It is the rate atwhich

interest is payable by the issuing company tothe

bondholder.


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BOND RETURNS (CONT.)

CURRENT YIELD:-

The current market price of a bond in thesecondary market may differ from its face value.

The current yield relates the annualinterest receivable on a bond to its current marketprice. It can be expressed as follows:-

Where

In = Annual Interest

Po = Current market price

current yield=In/Po 100


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SPOT INTEREST RATE:-

Zero coupon bond is a special type of bondwhich does not pay annual interests.

The return on this bond is in the form of adiscount on issue of the bond.

This type of bond is also calledpure discountbond or deep discount bond.

Spot interest rate is the annual rate of return on

a bond that has only one cash inflow to the


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YIELD TO MATURITY (YTM):-

This is the most widely used measure ofreturn on bonds.

It may be defined as the compounded rate ofreturn an investor is expected to receivefrom a bond purchased at the current marketprice and held to maturity.

It is really the internal rate of return earnedfrom holding a bond till maturity.

YTM depends upon the cash outflow forpurchasing the bond, that is, the cost or


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Current market price of the bond as well asthe cash inflows from the bond, namely thefuture interest payments and the terminalprincipal repayment.

YTM is the discount rate that makes thepresent value of cash inflows from the bondequal to the cash outflow for purchasing thebond.

The relation between the cash outflow, thecash inflow and the YTM of a bond can beexpressed as:

MP = Ct TV

(1 + YTM)t (1 + YTM)n


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Where :-

MP = Current market price of the bond

Ct = Cash inflow from the bond

throughout the holding period.TV = Terminal cash inflow received at

the end of the holding period.


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BOND PRICING THEOREMS

The relation between bond prices andchanges in the market interest rates have beenstated by Burton G. Malkiel in the form of fivegeneral principles. These are known as Bondpricing theorems.

The five principles are:-

Bond prices will move inversely to market

interest changes.

Bond price variability is directly related to theterm to maturity; which means, for a given

change in the level of market interest rates,chan e in bond rices are reater for lon er-


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BOND PRICING THEOREMS(CONT..)

A bonds sensitivity to changes in market

interest rate increases at a diminishing rateas the time remaining until its maturity

increases.

The price changes resulting from equalabsolute increases in market interest ratesare not symmetrical, i.e. for any givenmaturity, a decrease in market interest ratecauses a price rise that is larger than theprice decline that results from an equal

increase in market interest rate.


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BOND PRICING THEOREMS (CONT..)

Bond price volatility is related to thecoupon rate, which implies that thepercentage change in a bonds price due toa change in the market interest rate will besmaller if its coupon rate is higher.


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BOND RISKS

Two types of risk are associated withinvestment in bonds, namely default riskandinterest rate risk.

DEFAULT RISK:-

Default risk refers to the possibilitythat a company may fail to pay the interest

or principal on the stipulated dates. Poorfinancial performance of the company leadsto such defaults.


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BOND RISKS (CONT..)

INTEREST RATE RISK:-

The risk that an investment's valuewill change due to a change in the absolutelevel of interest rates. Such changes usuallyaffect securities inversely and can bereduced by diversifying or hedging.

Interest rate risk affects the

value of bonds more directly than stocks,and it is a major risk to all bondholders. Asinterest rates rise, bond prices fall and viceversa.


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BOND DURATION

Duration is the weighted average measureof a bonds life. The various time periods inwhich the bond generates cash flows areweighted according to the relative size of

the present value of those flows.The formula for computing duration d is:-

(t) (Ct)

d= (1 + k)t

Ct

(1 + k)t


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BOND DURATION (CONT..)

Where :-

Ct = Annual cash flow including interest& repayment of principal.

n = Holding period.k= Discount rate which is the market

interest rate.

t= The time period of each cash flow.


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Definitions

Par or Face Value -

The amount of money that is paid to the bondholders atmaturity. For most bonds this amount is $1,000. It also

generally represents the amount of money borrowed by thebond issuer.

Coupon Rate -

The coupon rate, which is generally fixed, determines theperiodic coupon or interest payments. It is expressed as apercentage of the bond's face value. It also represents theinterest cost of the bond to the issuer.


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Definitions

Coupon Payments -

The coupon payments represent the periodic interestpayments from the bond issuer to the bondholder. The

annual coupon payment is calculated by multiplying thecoupon rate by the bond's face value. Since most bondspay interest semiannually, generally one half of the annualcoupon is paid to the bondholders every six months.

Maturity Date -The maturity date represents the date on which the bond

matures, i.e., the date on which the face value is repaid.The last coupon payment is also paid on the maturity date.


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Definitions

Original Maturity -

The time from when the bond was issued until its maturitydate.

Remaining Maturity -

The time currently remaining until the maturity date.

Call Date -

For bonds which are callable, i.e., bonds which can be

redeemed by the issuer prior to maturity, the call datere resents the earliest date at which the bond can be


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Definitions

Call Price -

The amount of money the issuer has to pay to call acallable bond (there is a premium for calling the bond

early). When a bond first becomes callable, i.e., on thecall date, the call price is often set to equal the facevalue plus one year's interest.

Required Return -The rate of return that investors currently require on a

bond.


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Definitions

Yield to Maturity -

The rate of return that an investor would earn if hebought the bond at its current market price and held it

until maturity. Alternatively, it represents the discountrate which equates the discounted value of a bond'sfuture cash flows to its current market price.

Yield to Call -The rate of return that an investor would earn if he

bought a callable bond at its current market price andheld it until the call date given that the bond was called

on the call date.


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Chapter Outline

Bonds and Bond ValuationMore on Bond Features

Bond Ratings

Some Different Types of Bonds

Bond Markets


Determinants of Bond Yields


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Bond Definitions

BondPar value (face value)

Coupon rate

Coupon payment

Maturity date

Yield or Yield to maturity

Present Value of Cash Flows as


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Present Value of Cash Flows asRates Change

Bond Value = PV of coupons + PV of par

Bond Value = PV of annuity + PV of lumpsum

Remember, as interest rates increasepresent values decrease

So, as interest rates increase, bond prices

decrease and vice versa

Valuing a Discount Bond with


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Valuing a Discount Bond withAnnual Coupons

Consider a bond with a coupon rate of 10% andannual coupons. The par value is $1,000 and thebond has 5 years to maturity. The yield tomaturity is 11%. What is the value of the bond?

Using the formula:

B = PV of annuity + PV of lump sum

B = 100[1 1/(1.11)5] / .11 + 1,000 / (1.11)5

B = 369.59 + 593.45 = 963.04Using the calculator:

N = 5; I/Y = 11; PMT = 100; FV = 1,000

CPT PV = -963.04

Valuing a Premium Bond with


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Valuing a Premium Bond withAnnual Coupons

Suppose you are looking at a bond that has a10% annual coupon and a face value of $1000.

There are 20 years to maturity and the yield tomaturity is 8%. What is the price of this bond?

Using the formula:

B = PV of annuity + PV of lump sum

B = 100[1 1/(1.08)20] / .08 + 1000 / (1.08)20

B = 981.81 + 214.55 = 1196.36

Using the calculator:

N = 20; I/Y = 8; PMT = 100; FV = 1000

CPT PV = -1,196.36


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Price and Yield-to-maturity(YTM)

Bond

Price

Yield-to-maturity(YTM)

Bond Prices: Relationship


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Bond Prices: RelationshipBetween Coupon and Yield

If YTM = coupon rate, then par value = bondprice

If YTM > coupon rate, then par value > bondprice

Why? The discount provides yield above coupon rate

Price below par value, called a discount bond

If YTM < coupon rate, then par value < bond

priceWhy? Higher coupon rate causes value above par

Price above par value, called a premium bond


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The Bond Pricing Equation


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Example 7.1

Find present values based on thepayment period

How many coupon payments are there?

What is the semiannual coupon payment?What is the semiannual yield?

B = 70[1 1/(1.08)14] / .08 + 1,000 / (1.08)14= 917.56

Or PMT = 70; N = 14; I/Y = 8; FV = 1,000; CPTPV = -917.56


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Interest Rate Risk

Price RiskChange in price due to changes in interest rates

Long-term bonds have more price risk than short-termbonds

Low coupon rate bonds have more price risk than highcoupon rate bonds

Reinvestment Rate Risk

Uncertainty concerning rates at which cash flows can be

reinvested

Short-term bonds have more reinvestment rate risk thanlong-term bonds

High coupon rate bonds have more reinvestment rate

risk than low coupon rate bonds


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Figure 7.2


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Computing Yield-to-maturity

Yield-to-maturity is the rate implied bythe current bond price

Finding the YTM requires trial and errorif you do not have a financial calculator

and is similar to the process for finding rwith an annuity

If you have a financial calculator, enterN, PV, PMT, and FV, remembering thesign convention (PMT and FV need tohave the same sign, PV the oppositesign)


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YTM with Annual Coupons

Consider a bond with a 10% annualcoupon rate, 15 years to maturity and apar value of $1,000. The current price is$928.09.

Will the yield be more or less than 10%?

N = 15; PV = -928.09; FV = 1,000; PMT = 100

CPT I/Y = 11%

YTM ith S i l


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YTM with SemiannualCouponsSuppose a bond with a 10% coupon rate

and semiannual coupons, has a facevalue of $1,000, 20 years to maturityand is selling for $1,197.93.

Is the YTM more or less than 10%?What is the semiannual coupon payment?

How many periods are there?

N = 40; PV = -1,197.93; PMT = 50; FV =1,000; CPT I/Y = 4% (Is this the YTM?)

YTM = 4%*2 = 8%


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Table 7.1

C rrent Yield s Yield to


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Current Yield vs. Yield toMaturityCurrent Yield = annual coupon / priceYield to maturity = current yield + capital gains

yield

Example: 10% coupon bond, with semiannual

coupons, face value of 1,000, 20 years tomaturity, $1,197.93 price

Current yield = 100 / 1,197.93 = .0835 = 8.35%

Price in one year, assuming no change in YTM =1,193.68

Capital gain yield = (1,193.68 1,197.93) / 1,197.93 =-.0035 = -.35%

YTM = 8.35 - .35 = 8%, which the same YTMcomputed earlier


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Bond Pricing Theorems

Bonds of similar risk (and maturity) willbe priced to yield about the samereturn, regardless of the coupon rate

If you know the price of one bond, youcan estimate its YTM and use that tofind the price of the second bond

This is a useful concept that can be

transferred to valuing assets other thanbonds

Bond Prices with a


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Bond Prices with aSpreadsheetThere is a specific formula for finding

bond prices on a spreadsheet

PRICE(Settlement,Maturity,Rate,Yld,Redemption,Frequency,Basis)

YIELD(Settlement,Maturity,Rate,Pr,Redemption,Frequency,Basis)

Settlement and maturity need to be actual dates

The redemption and Pr need to be input as % of par

value

Click on the Excel icon for an example

Differences Between Debt


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Differences Between Debtand Equity

DebtNot an ownershipinterest

Creditors do not havevoting rights

Interest is considered acost of doing businessand is tax deductible

Creditors have legal

recourse if interest orprincipal payments aremissed

Excess debt can lead tofinancial distress and

bankruptcy

EquityOwnership interest

Common stockholdersvote for the board ofdirectors and other

issuesDividends are not

considered a cost ofdoing business and arenot tax deductible

Dividends are not aliability of the firm andstockholders have nolegal recourse ifdividends are not paid

An all equity firm can not

go bankrupt merely dueto debt since it has no


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The Bond Indenture

Contract between the company and thebondholders that includes

The basic terms of the bonds

The total amount of bonds issued

A description of property used as security, ifapplicable

Sinking fund provisions

Call provisions

Details of protective covenants


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Bond Classifications

Registered vs. Bearer FormsSecurity

Collateral secured by financial securities

Mortgage secured by real property, normallyland or buildings

Debentures unsecured

Notes unsecured debt with original maturityless than 10 years

Seniority

Bond Characteristics and


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Bond Characteristics andRequired Returns

The coupon rate depends on the riskcharacteristics of the bond when issued

Which bonds will have the higher coupon,

all else equal?Secured debt versus a debenture

Subordinated debenture versus senior debt

A bond with a sinking fund versus one without

A callable bond versus a non-callable bond

Bond Ratings Investment


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Bond Ratings InvestmentQualityHigh Grade

Moodys Aaa and S&P AAA capacity to pay isextremely strong

Moodys Aa and S&P AA capacity to pay is very

strongMedium Grade

Moodys A and S&P A capacity to pay is strong, butmore susceptible to changes in circumstances

Moodys Baa and S&P BBB capacity to pay isadequate, adverse conditions will have more impacton the firms ability to pay


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Bond Ratings - Speculative

Low GradeMoodys Ba, B, Caa and Ca

S&P BB, B, CCC, CC

Considered speculative with respect to capacity topay. The B ratings are the lowest degree ofspeculation.

Very Low Grade

Moodys C and S&P C income bonds with nointerest being paid

Moodys D and S&P D in default with principal andinterest in arrears


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Government BondsTreasury Securities

Federal government debt

T-bills pure discount bonds with original maturity of oneyear or less

T-notes coupon debt with original maturity between one

and ten years

T-bonds coupon debt with original maturity greater thanten years

Municipal Securities

Debt of state and local governments

Varying degrees of default risk, rated similar to corporatedebt

Interest received is tax-exempt at the federal level


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Example 7.4

A taxable bond has a yield of 8% and amunicipal bond has a yield of 6%

If you are in a 40% tax bracket, which bond doyou prefer?

8%(1 - .4) = 4.8%

The after-tax return on the corporate bond is 4.8%,compared to a 6% return on the municipal

At what tax rate would you be indifferent

between the two bonds?8%(1 T) = 6%

T = 25%


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Zero Coupon Bonds

Make no periodic interest payments (couponrate = 0%)

The entire yield-to-maturity comes from thedifference between the purchase price and the

par valueCannot sell for more than par value

Sometimes called zeroes, deep discount bonds,or original issue discount bonds (OIDs)

Treasury Bills and principal-only Treasury stripsare good examples of zeroes


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Floating-Rate Bonds

Coupon rate floats depending on some indexvalue

Examples adjustable rate mortgages andinflation-linked Treasuries

There is less price risk with floating rate bonds

The coupon floats, so it is less likely to differsubstantially from the yield-to-maturity

Coupons may have a collar the rate cannotgo above a specified ceiling or below aspecified floor


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Other Bond Types

Disaster bondsIncome bonds

Convertible bonds

Put bonds

There are many other types of provisionsthat can be added to a bond and many

bonds have several provisions it isimportant to recognize how theseprovisions affect required returns


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Bond Markets

Primarily over-the-counter transactionswith dealers connected electronically

Extremely large number of bond issues,but generally low daily volume in singleissues

Makes getting up-to-date prices difficult,particularly on small company or

municipal issuesTreasury securities are an exception


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Work the Web Example

Bond quotes are available onlineOne good site is Bonds Online

Click on the web surfer to go to the site

Follow the bond search, corporate links

Choose a company, enter it under ExpressSearch Issue and see what you can find!


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Treasury Quotations

Highlighted quote in Figure 7.48 Nov 21 128:07 128:08 5 5.31

What is the coupon rate on the bond?

When does the bond mature?What is the bid price? What does this mean?

What is the ask price? What does this mean?

How much did the price change from theprevious day?

What is the yield based on the ask price?


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Clean vs. Dirty PricesClean price: quoted price

Dirty price: price actually paid = quoted price plusaccrued interest

Example: Consider T-bond in previous slide, assumetoday is July 15, 2007

Number of days since last coupon = 61

Number of days in the coupon period = 184

Accrued interest = (61/184)(.04*100,000) = 1,326.09

Prices (based on ask):Clean price = 128,250

Dirty price = 128,250 + 1,326.09 = 129,576.09

So, you would actually pay $ 129,576.09 for the

bond


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Real rate of interest change inpurchasing power

Nominal rate of interest quoted rate ofinterest, change in purchasing power, andinflation

The ex ante nominal rate of interestincludes our desired real rate of return

plus an adjustment for expected inflation


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The Fisher Effect

The Fisher Effect defines the relationshipbetween real rates, nominal rates, andinflation

(1 + R) = (1 + r)(1 + h), where

R = nominal rate

r = real rate

h = expected inflation rate

Approximation

R = r + h


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Example 7.5

If we require a 10% real return and weexpect inflation to be 8%, what is thenominal rate?

R = (1.1)(1.08) 1 = .188 = 18.8%Approximation: R = 10% + 8% = 18%

Because the real return and expected

inflation are relatively high, there issignificant difference between the actualFisher Effect and the approximation.

Term Structure of Interest


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Term Structure of InterestRatesTerm structure is the relationship between time to

maturity and yields, all else equal

It is important to recognize that we pull out theeffect of default risk, different coupons, etc.

Yield curve graphical representation of the termstructure

Normal upward-sloping, long-term yields are higher thanshort-term yields

Inverted downward-sloping, long-term yields are lowerthan short-term yields

Figure 7.6 Upward-Sloping


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g p p gYield Curve

Figure 7.6 Downward-Sloping


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77

g p gYield Curve

Fi 7 7


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Figure 7.7

F Aff i B d Yi ld


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79

Factors Affecting Bond Yields

Default risk premium remember bondratings

Taxability premium remember municipalversus taxable

Liquidity premium bonds that havemore frequent trading will generally havelower required returns

Anything else that affects the risk of thecash flows to the bondholders will affectthe required returns

Q i k Q i


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80

Quick Quiz

How do you find the value of a bond and why dobond prices change?

What is a bond indenture and what are some ofthe important features?

What are bond ratings and why are theyimportant?

How does inflation affect interest rates?

What is the term structure of interest rates?What factors determine the required return on

bonds?


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End of Chapter

C h i P bl


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Comprehensive Problem

What is the price of a $1,000 par valuebond with a 6% coupon rate paidsemiannually, if the bond is priced toyield 5% YTM, and it has 9 years to

maturity?What would be the price of the bond if

the yield rose to 7%.

What is the current yield on the bond ifthe YTM is 7%?


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CURRENT YIELD:-The current market price of a bond in

the secondary market may differ from itsface value.

The current yield relates theannual interest receivable on a bond to itscurrent market price. It can be expressed

as follows:-

current yield=In/Po 100

bond valuation jitesh

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