presentation bond pricing and risk evaluation

8/2/2019 Presentation Bond Pricing and Risk Evaluation

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2

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, if

applicable

Sinking fund provisions

Call provisions Details of protective covenants


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8-3

Types and terms of bonds

Callable bond: the issuer has right to retire the bondbefore maturity, at a predetermined price that isalways specified in the bond contract. Almost all corporate bonds are callable. If interest rates

then fall in the future, firms can retire these existing bondsand replace them with new lower rate bonds.

Callable bonds will command a higher interest rate or yield(lower price) than a comparable risk non-callable bond.

Mortgage bond: bond is secured or collateralized by

some physical asset in case the issuer defaults. Commonly used in the transportation industry.


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8-4

Types and terms of bonds, continued

Convertible bond: bond can be converted into apredetermined number of shares of common stock.Investors are willing to accept a lower yield on suchbonds. The right to convert may become veryvaluable. A convertible bond thus has the opportunity to become an

exciting investment if the firm does unexpectedly well.

Debenturebond: bond is backed by the issuers

ability to generate future cash flow to make thepromised payments. There is no collateral.


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8-5

Types and terms of bonds, continued

Subordinatedbonds: the bonds claim on the issueris junior to one or more senior bond issues. Themore senior bonds have the higher priority inbankruptcy and/or liquidation.

Sinking fundprovision: issuer may be required toretire a certain amount of an issue each year. Forexample, having to retire 10% of a 20 year bond issueeach year from year 11 to year 20.


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6

Bond Characteristics and Required

Returns

The coupon rate depends on the risk characteristics

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


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8-7

Evaluating default risk:

Bond ratings

Bond ratings are designed to reflect the probability of a

bond issue going into default. The lower the rating (the

higher the default risk), the higher the required yield.

AAA or Aaa bonds have the highest rating.

Depository institutions, e.g., commercial banks and Savings &

Loans may only own Investment Grade bonds.

Investment Grade Junk BondsMoodys Aaa Aa A Baa Ba B Caa C

S & P AAA AA A BBB BB B CCC D


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8

Interest Rate Risk

Price Risk Change in price due to changes in interest rates

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

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

bonds

Reinvestment Rate Risk

Uncertainty concerning rates at which cash flows can be reinvested

Short-term bonds have more reinvestment rate risk than long-term

bonds

High coupon rate bonds have more reinvestment rate risk than lowcoupon rate bonds


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9

Term Structure and the Interest Rate Risk


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10

Upward-Sloping Yield Curve


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11

Downward-Sloping Yield Curve


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12

Bond Pricing Theorems

Bonds of similar risk (and maturity) will be priced to

yield about the same return, regardless of the

coupon rate

If you know the price of one bond, you can estimateits YTM and use that to find the price of the second

bond

This is a useful concept that can be transferred to

valuing assets other than bonds


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13

The Bond Pricing Equation

t

t

r)(1

F

r

r)(1

1-1

CValueBond


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14

Clean vs. Dirty Prices Clean price: quoted price

Dirty price: price actually paid = quoted price plus accruedinterest

Example: Consider a bond with 8% coupon and mature atNovember 15, 2021, Face Value $100,000.

Assume today is July 15, 2007 Number of days since last coupon = 61 (from May 16 till July 15)

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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15

Inflation and Interest Rates

Real rate of interest change in purchasing power

Nominal rate of interest quoted rate of interest,

change in purchasing power, and inflation

The ex ante nominal rate of interest includes ourdesired real rate of return plus an adjustment for

expected inflation


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The Fisher Effect

The Fisher Effect defines the relationship between

real rates, nominal rates, and inflation

(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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Factors Affecting Bond Yields

Default risk premium remember bond ratings Taxability premium remember municipal versus

taxable

Liquidity premium bonds that have more frequenttrading will generally have lower required returns

Anything else that affects the risk of the cash flowsto the bondholders will affect the required returns


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Elements of a Bond Valuation Model

The potential benchmark interest rates that can beused in bond valuation are those in the Treasury

market, a specific bond sector with a given credit

rating, or a specific issuer.

Benchmark interest rates can be based on either an

estimated yield curve or an estimated spot rate

curve.


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Relative Valuation Measures

Yield spread measures are used in assessing therelative value of securities.

Relative value analysis is used to identify securities asbeing overpriced (rich), underpriced (cheap), or

fairly priced relative to benchmark interest rates.


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Traditional Yield Measures

Nominal spread

Static (z) spread

These measures do not consider the effect ofembedded options (reinvestment or call

features).

Yield to worst, i.e., smallest of:

Yield to maturity

All yields to calls and puts


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Traditional Yield Measures

The interpretation of a spread measuredepends on the benchmark used.


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Nominal Yield Spread

The nominal spread is the difference between a non-Treasury bonds yield and the YTM for a benchmarkTreasury coupon security.

The nominal yield spread measures the compensation forthe additional credit risk, option risk, and liquidity risk aninvestor is exposed to by investing in a non-Treasurysecurity with the same maturity.


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Zero-Volatility Spread The zero-volatility or Z- spread is a measure of the spread the

investor would realize over the entire Treasury spot rate curve ifthe bond is held to maturity.

It is not the spread off of one point on the Treasury yield curve (nominalspread), it is an average over all spot rates.

The Z-spread is also called a static spread and is calculated asthe spread which will make the present value of the cash flowsfrom the non-Treasury bond, when discounted at the Treasuryspot rate plus the spread, equal to the non-Treasury bonds

price. Trial and error is used to determine the Z-spread.

C l l i f P i f 25 Y 8 8% C B d


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Calculation of Price of a 25-Year 8.8% Coupon Bond

Using Treasury Spot Rates

Period Cash FlowTreasury Spot

Rate (%)Present Value

1 4.4 7.00000 4.2512

2 4.4 7.04999 4.1055

3 4.4 7.09998 3.9628

4 4.4 7.12498 3.8251

5 4.4 7.13998 3.69226 4.4 7.16665 3.5622

. . . .

46 4.4 10.10000 0.4563

47 4.4 10.30000 0.4154

48 4.4 10.50000 0.3774

49 4.4 10.60000 0.3503

50 104.4 10.80000 7.5278

Theoretical price 96.6134

C l l ti f th St ti S d f 25 Y 8 8% C


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Calculation of the Static Spread for a 25-Year 8.8% Coupon

Corporate Bond

Present Value if Spread Used Is:

Period Cash Flow

Treasury Spot

Rate (%) 100 BP 110 BP 120 BP

1 4.4 7.00000 4.2308 4.2287 4.2267

2 4.4 7.04999 4.0661 4.0622 4.0583

3 4.4 7.09998 3.9059 3.9003 3.8947

4 4.4 7.12498 3.7521 3.7449 3.7377

5 4.4 7.13998 3.6043 3.5957 3.5871

. . . . . .

46 4.4 10.10000 0.3668 0.3588 0.3511

47 4.4 10.30000 0.3323 0.3250 0.3179

48 4.4 10.50000 0.3006 0.2939 0.2873

49 4.4 10.60000 0.2778 0.2714 0.2652

50 104.4 10.80000 5.9416 5.8030 5.6677

Total present value 88.5474 87.8029 87.0796


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Comparison of Traditional Yield Spread and Static Spread for

Various Bondsa

Spread (basis points)

Bond Price Yield toMaturity (%) Traditional Static Difference

25-year 8.8% Coupon Bond

Treasury 96.6133 9.15

A 88.5473 10.06 91 100 9

B 87.8031 10.15 100 110 10C 87.0798 10.24 109 120 11

. . . . . .

5-year 8.8% Coupon Bond

Treasury 105.9555 7.36

J 101.7919 8.35 99 100 1

K 101.3867 8.45 109 110 1

L 100.9836 8.55 119 120 1aAssuming the same Treasury spot rate curve given in the previous example


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Static Spread Vs Yield Spread

The magnitude of the difference between the traditional yield spread and

the static spread also depends on the shape of the yield curve. The steeper the yield curve, the more the difference for a given coupon and maturity.

We also find that the shorter the maturity of the bond, the less the static

spread will differ from the traditional yield spread.

Differences are lower when the corporate bond makes a bullet payment at

maturity.

Similarly, the difference will be considerably greater for sinking fund

bonds and mortgage-backed securities in a steep yield curve environment.


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What is the best spread?

Option Adjusted Spread (OAS)

The Z-spread, which looks at measuring the spread over a spotrate curve, has a problem in that it fails to take future interestrate volatility into consideration which could change the cashflows for bonds with embedded options.

The option-adjusted spread (OAS) was developed to take thedollar difference between the fair valuation and the marketprice and convert it to a yield spread measure.

The OAS is used to reconcile the fair price (value) and the market price

by finding a return (spread) that will equate the two. The spread is measured in basis points.


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Option Adjusted Spread (OAS)

The option-adjusted spread is a measure of option risk.

Depending on the benchmark interest rates used to generatethe interest rate tree, the option-adjusted spread may or may

not capture credit risk.

The option-adjusted spread is not a spread off of one maturityof the benchmark interest rates; rather, it is a spread over theforward rates in the interest rate tree that were constructed

from the benchmark interest rates.

ll bl d d h i


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Callable Bonds and Their Investment

Characteristics

The presence of a call option results in two disadvantagesto the bondholder:

i. callable bonds expose bondholders to reinvestment risk

ii. price appreciation potential for a callable bond in a declining

interest-rate environment is limitedo This phenomenon for a callable bond is referred to asprice

compression.

If the investor receives sufficient potential compensation

in the form of a higher potential yield, an investor wouldbe willing to accept call risk.


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Characteristics (continued)

Traditional Valuation Methodology for Callable Bondso When a bond is callable, the practice has been to calculate a

yield to worst, which is the smallest of the yield to maturity

and the yield to call for all possible call dates.

o Theyield to call (like the yield to maturity) assumes that all

cash flows can be reinvested at the computed yieldin thiscase the yield to calluntil the assumed call date.

o Moreover, the yield to call assumes that

i. the investor will hold the bond to the assumed call date

ii. the issuer will call the bond on that date.o Often, these underlying assumptions about the yield to call are

unrealistic because they do not take into account how an

investor will reinvest the proceeds if the issue is called.

ll bl d d h


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Characteristics(continued)

Price-Yield Relationship for a Callable Bond

o The priceyield relationship for an option-free bond is convex.o The figure on the next slide shows the priceyield relationship for both

a noncallable bond and the same bond if it is callable.

o The convex curve aa' is the priceyield relationship for the

noncallable (option-free) bond.

o The unusual shaped curve denoted by ab is the priceyieldrelationship for the callable bond.

o The reason for the shape of the priceyield relationship for the callable

bond is as follows. When the prevailing market yield for comparable bonds is higher than the

coupon interest, it is unlikely that the issuer will call the bond.o If a callable bond is unlikely to be called, it will have the same convex

priceyield relationship as a noncallable bond when yields are greater

thany*.


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Price-Yield Relationship for a Noncallable and Callable Bond

Price

Yieldy*

b

Noncallable Bonda- a

a

a

Callable

Bond

a - b

ll bl d d h


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Price-Yield Relationship for a Callable Bondo As yields in the market decline, the likelihood that yields will decline

further so that the issuer will benefit from calling the bond increases.

o The exact yield level at which investors begin to view the issue likely

to be called may not be known, but we do know that there is some

level, sayy*.o At yield levels belowy*, the price-yield relationship for the callable

bond departs from the price-yield relationship for the noncallable

bond.

o For a range of yields belowy*, there is price compressionthat is,

there is limited price appreciation as yields decline.o The portion of the callable bond price-yield relationship belowy* is

said to be negatively convex.

ll bl d d h i


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Price-Yield Relationship for a Callable Bondo Negative convexity means that the price appreciation will be

less than the price depreciation for change in yield of a

given number of basis points.

For a bond that is option-free and displays positive convexity,

the price appreciation will be greater than the price

depreciation for a change in yield of a given number of basis

points.

o It is important to understand that a bond can still trade

above its call price even if it is highly likely to be called,because of administrative costs of calling the bond.


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Price Volatility Implications of Positive and

Negative Convexity

Absolute Value of Percentage Price Change

Change in Interest Rates Positive Convexity Negative Convexity

-100 basis points greater than X% Less than Y%+100 basis points X% Y%

Components of a Bond ith an Embedded


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Components of a Bond with an Embedded

Option

To develop a framework for analyzing a bond with an embedded option, it

is necessary to decompose a bond into its component parts.i. buys a noncallable bond from the issuer for which she pays some priceii. sells the issuer a call option for which she receives the option price

A callable bond is equal to the price of the two components parts; that is,

callable bond price =noncallable bond price call option price

The call option price is subtracted from the price of the noncallable bondbecause when the bondholder sells a call option, she receives the option

price.

The difference between the price of the noncallable bond and the callable

bond at any given yield is the price of the embedded call option.


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Decomposition of a Price of a Callable Bond

Price

Yieldy**

b

Noncallable Bonda- a

a

a

Callable

Bonda - b

y*

PNCB

PCB

Note: At y** yield level: PNCB = noncallable bond pricePCB = callable bond price

PNCB - PCB = call option price

f


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Components of a Bond with an Embedded

Option(continued)

The logic applied to callable bonds can be similarly applied

toputable bonds.

In the case of a putable bond, the bondholder has the right to

sell the bond to the issuer at a designated price and time.

A putable bond can be broken into two separate transactions.

i. The investor buys a noncallable bond.

ii. The investor buys an option from the issuer that allows the

investor to sell the bond to the issuer.

The price of a putable bond is then

putable bond price =non-putable bond price + put option price


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Valuation of Bond with Embedded Option

The bond valuation process requiresthat we use the theoretical spot rate todiscount cash flows.

For an embedded option the valuationprocess also requires that we take intoconsideration how interest-rate

volatility affects the value of a bondthrough its effects on the embeddedoptions.


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Binomial Model A single factor interest rate model that, given an

assumed level of volatility, suggests that interestrates have an equal probability of taking on one of

two possible values in the next period.

The set of possible interest rate paths that areused to value bonds with a binomial model is

called a binomial interest rate tree

Th l i hi h f i


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The relationship among the set of rates is a

function of the interest rate volatility assumption

of the model being employed to generate the tree

(e.g. )

For this tree, it is assumed that the interest rate

tree should generate arbitrage-free values for on-

the-run issues of the benchmark security (i.e. the

value of these on-the-run issues produced by the

interest rate tree must equal their market prices)

Construction of Arbitrage Free Interest


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Construction of Arbitrage-Free Interest

Rate Tree Step 1: Use the yield on the current 1-year on-the-run Treasury security

issue fori0. Suppose, for example, that i0 = 4.5749%. Step 2: Make an assumption about the volatility of interest rates. Suppose,

for example, we assume = 15%.

Step 3: Given the coupon rate and market value of the 2-year on-the-runissue, provide a guess of i1,L, compute i1,U = i1,Le

2, and use the resultinginterest rate tree to compute the value of the on-the-run issue. Suppose,for example, that the coupon rate and market price of the 2-year on-the-run treasury security issue are 7% and $102.999, respectively.

Step 4: If the value from the model is higher than the market price,increase the guess ofi1,L, recompute i1,U, and compute the new value ofthe on-the-run issue. If the model value is too low, decrease the interest

rates in the tree. Step 5: Repeat this iterative process until the value generated by the

model is equal to the market price. Suppose, for example, we determinethat if i1,L = 5.321%, and i1,U = 5.321% e

2(0.15) = 7.1826%, then the valuefrom the model is equal to the market price of $102.999. Then theinterest rate tree (as shown in the next example) is arbitrage-free.


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Assuming that the probabilities of an up move anda down move are both 50%


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Valuation of a Callable Bond Valuation of callable bond is similar to non-

callable bond, however, the value used at anynode corresponding to the call date and beyond

must be either the price at which the issuer will

call the bond at that date or the computed valueif the bond is not called, whichever is less.

Assuming that the 2-year bond can be called in

one year at 100. The issuer will call the bond if thecomputed bond price exceeds 100 one year from

today


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Vcall = VnoncallableVcallable Vcall = $102.999 $102.238 = $0.761

Vputable = Vnonputable +Vput


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Incorporating Default Risk

o The basic binomial model explained above can be

extended to incorporate default risk.

o The extension involves adjusting the expected cash

flows for the probability of a payment default and the

expected amount of cash that will be recovered when a

default occurs. Modeling Risk

o The user of any valuation model is exposed to

modeling risk.

o This is the risk that the output of the model is incorrectbecause the assumptions upon which it is based are

incorrect.

O i dj d S d


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Option-Adjusted Spread

Translating OAS to Theoretical Value

o Although the product of a valuation model is the OAS, theprocess can be worked in reverse.

o For a specified OAS, the valuation model can determine the

theoretical value of the security that is consistent with that

OAS.

o As with the theoretical value, the OAS is affected by theassumed interest rate volatility.

o The higher (lower) the expected interest rate volatility, the

lower (higher) the OAS.

Determining the Option Value in Spread Termso The option value in spread terms is determined as follows:

option value (in basis points) = static spreadOAS


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Effective Duration and Convexity

There is a duration measure that is more appropriate

for bonds with embedded options than the modifiedduration measure.

In general, the duration for any bond can be

approximated as follows:

P_ =price if yield is decreased by x basis points

P+ =price if yield is increased by x basis pointsP0 = initial price (per $100 of par value)

y (ordy) = change in rate used to calculate price (xbasis

points in decimal form)

0

P_ Pduration

2 P dy


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Effective Duration and Convexity(continued)

When the approximate duration formula is applied to a

bond with an embedded option, the new prices at thehigher and lower yield levels should reflect the value from

the valuation model.

Duration calculated in this way is called effective duration

or option-adjusted duration. The standard convexity measure may be inappropriate for

a bond with embedded options because it does not

consider the effect of a change in interest rates on the

bonds cash flow.


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Modified Duration Versus Effective Duration

Modified DurationDuration measure in which it is assumed

that yield changes do not change

the expected cash flow

Effective DurationDuration measure in which recognition

is given to the fact that yield changes may

change the expected cash flow

DurationInterpretation: Generic description of the sensitivity of a bonds price

(as a percent of initial price) to a parallel shift in the yield curve


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Valuing a Floating-Rate Note

To value a floating-rate note that has a cap, the coupon ateach node of the tree is adjusted by determining whether ornot the cap is reached at a node; if the rate at a node doesexceed the cap, the rate at the node is the capped rate ratherthan the rate determined by the floaters coupon formula.

For a floating-rate note, the binomial method must beadjusted to account for the fact that a floater pays in arrears;that is, the coupon payment is determined in a period but not

paid until the next period.


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Convertible Securities

Convertible and exchangeable securities can be

converted into shares of common stock.

The conversion ratio is the number of common stockshares for which a convertible security may be

converted.

Almost all convertible securities are callable and

some are putable.


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The conversion value is the value of the convertible bond if it isimmediately converted into the common stock.

The market conversion price is the price that an investor effectively paysfor the common stock if the convertible security is purchased and thenconverted into the common stock.

The premium paid for the common stock is measured by the marketconversion premium per share and market conversion premium ratio.

The straight value or investment value of a convertible security is its value

if there was no conversion feature.

The minimum value of a convertible security is the greater of theconversion value and the straight value.


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A fixed income equivalent (or a busted convertible) refers tothe situation where the straight value is considerably higherthan the conversion value so that the security will trade muchlike a straight security.

A common stock equivalent refers to the situation where theconversion value is considerably higher than the straight valueso that the convertible security trades as if it were an equityinstrument.

A hybrid equivalent refers to the situation where theconvertible security trades with characteristics of both a fixedincome security and a common stock instrument.


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While the downside risk of a convertible security usually isestimated by calculating the premium over straight value, thelimitation of this measure is that the straight value (the floor)changes as interest rates change.

An advantage of buying the convertible rather than thecommon stock is the reduction in downside risk.

The disadvantage of a convertible relative to the straightpurchase of the common stock is the upside potential give-upbecause a premium per share must be paid.


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An option-based valuation model is a moreappropriate approach to value convertible securitiesthan the traditional approach because it can handlemultiple embedded options.

There are various option-based valuation models:one-factor and multiple-factor models.

The most common convertible bond valuation modelis the one-factor model in which the one factor is thestock price movement.

presentation bond pricing and risk evaluation

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