STATIC CASH FLOW YIELD METHODOLOGY Learning Objectives After reading this chapter you will understand: Il the cash flow yield methodology for analyzing mortgage-backed securities § the
Trang 1364 CHAPTER l4 Analysis of Bonds with Embedded Options
e Demonstrate that if o is assumed to be 10%, the lower one-year forward rate
one year from now is 6.944%
f Demonstrate that if o is assumed to be 10%, the lower one-year forward rate
two years from now is approximately 6.437%
g Show the binomial interest-rate tree that should be used to value any bond of
this issuer
h Determine the value of an 8.5% coupon option-free bond for this issuer using
the binomial interest-rate tree given in part g :
i Determine the value of an 8.5% coupon bond that is callable at par (100) as-
suming that the issue will be called if the price exceeds par
Explain how an increase in expected interest-rate volatility can decrease the value
of a callable bond
How should an interest-rate volatility of 15% be interpreted if the prevailing in-
terest rate is 7%?
a What is meant by the option-adjusted spread?
b What is the spread relative to?
“The option-adjusted spread measures the yield spread over the Treasury on-the-
run yield curve.” Explain why you agree or disagree with this statement
What is the effect of greater expected interest-rate volatility on the option-
adjusted spread of a security?
The following excerpt is taken from an article titled “Call Provisions Drop Off”
that appeared in the January 27, 1992, issue of BondWeek, p 2:
Issuance of callable long-term bonds dropped off further last year as in-
terest rates fell, removing the incentive for many issuers to pay extra for
the provision, said Street capital market officials
The shift toward noncallable issues, which began in the late 1980s, re-
flects the secular trend of investors unwilling to bear prepayment risk and
possibly the cyclical trend that corporations believe that interest rates
have hit all time lows
a What “incentive” is this article referring to in the first sentence of the excerpt?
b Why would issuers not be willing to pay for this incentive if they feel that inter-
est rates will continue to decline?
The following excerpt is taken from an article titled “Eagle Eyes High-Coupon
Callable Corporates” that appeared in the January 20, 1992, issue of
BondWeek, p 7:
If the bond market rallies further, Eagle Asset Management may take
profits, trading $8 million of seven- to 10-year Treasuries for high-
coupon single-A industrials that are callable in two to four years accord-
ing to Joseph Blanton, senior v.p He thinks a further rally is unlikely,
however
The corporates have a 95% chance of being called in two to four years
and are treated as two- to four-year paper in calculating the duration of
*- the portfolio, Blanton said
a Why is modified duration an inappropriate ‘peasure for a high-coupon
callable bond?
b What would be a better measure than modified duration?
c Why would the replacement of 10-year Treasuries with high-coupon callable
uration be a good approximation for the effective durati ; i
22 The following information was re i ported in William M Boyce, Web i liam M Boyee, Webster tucker Peter S A Niculescu, and Michael Wald cu, » “The Implied Volatility “ ied Volatility of Freed of Fi Income Markets,” in Frank J Fabozzi (ed), › _F ed.), Advances and Innovati : ovations th whe i Bond and Mortgage Markets (Chicago: Probus Publishing, 1989) for two ities af
Effective
b Why Is it necessary to make an interest-rate volatility assumption to calculate
23 a Explain how effective duration is i calculated using the binomial i i mod
b What is assumed about the OAS in calculating effective duration? “
CHAPTER 14 Analysis of Bonds with Embedded Options 365
Trang 2
STATIC CASH FLOW YIELD METHODOLOGY Learning Objectives
After reading this chapter you will understand:
Il the cash flow yield methodology for analyzing mortgage-backed securities
§ the limitations of the cash flow yield methodology
lM how the effective duration and convexity are calculated for the cash flow yield
methodology
Ml why the Monte Carlo simulation methodology is used to value mortgage-backed
securities
@ how interest-rate paths are simulated in a Monte Carlo simulation methodology
@ how the Monte Carlo simulation methodology can be used to determine the theo-
retical value of a mortgage-backed security
M how the option-adjusted spread, effective duration, and effective convexity are
computed using the Monte Carlo simulation methodology
i the complexities of modeling collateralized mortgage obligations
Mi the limitations of option-adjusted spread
M@ modeling risk and how it gan be stress tested
Mf how the total return is calculated for a mortgage-backed security
Mi the difficulties of applying the total return framework to mortgage-backed
®
There are two approaches to the analysis of mortgage-backed securities (in-
cludirtg pass-throughs, collateralized mortgage obligations, and stripped more
gage-backed securities): (1) the static cash flow yield methodology, and (2) the
Tự
*This chapt adapted from Chapters 9 and 10 of Frank J, Fabozzi, ñ E a
Collatenttized Morigage Obligations: Structures and Analysis (Buckingham, PA: Frank J Fabozzi Associ:
The static cash flow yield methodology is the simplest to use, although we shall see that it offers little'insight into the relative value of a mortgage-backed security It be- gins with the computation of the cash flow yield measure that we described for pass- throughs in Chapter 11 The cash flow yield is based on some prepayment assumption
To illustrate the cash flow yield, we’ll use one of the CMO structures we devel- oped in Chapter 12, FJF-06 This structure is summarized in Exhibit 12-15 Exhibit 15-1 summarizes cash flow yields according to various PSA prepayment assumptions for the four tranches assuming different purchase prices Notice that the greater the
discount assumed to be paid for the tranche, the more a tranche will benefit from
faster prepayments The converse is true for a tranche for which a premium is paid
The faster the prepayments, the lower the cash flow yield
Vector Analysis One practice that market participants use to overcome the drawback of the PSA benchmark is to assume that the PSA speed can change over time This technique is referred to as vector analysis A vector is simply a set of numbers.-In the case of pre- payments, it is a vector of prepayment speeds Vector analysis is particularly useful for CMO tranches that are dramatically affected by the initial slowing down of prepay- ments, and then speeding up of prepayments, or vice versa
Exhibit 15-2 reports the cash flow yield using vector analysis for the four tranches and collateral of FJF-06 The top panel shows the cash flow yield assuming 165 PSA
Nine vectors are then shown assuming that the PSA is constant from months 1 to 36, and then changes for months 37 through 138, and again changes for months 139 through 357,
Limitations of the Cash Flow Yield
As we have noted several times already, the yield to maturity has two shortcomings as
a measure of a bond’s potential return: (1) It is assumed that the coupon payments can be reinvested at a rate equal to the yield to maturity, and (2) it is assumed that the bond is held to maturity These shortcomings are equally present in application of the cash flow yield measure: (1) the projected cash flows are assumed to be reinvested at the cash flow yield, and (2) the mortgage-backed security is assumed to be held until the final payout based on some prepayment assumption The importance of reinvest- ment risk, the risk that the cash flow will be reinvested.at a rate less than the cash flow yield, is particularly important for many mortgage-backed securities because pay- ments come as frequently as every month The cash flow yield, moreover, is depen- dent on realization of the projected cash flow according to some prepayment rate If
Trang 3Yield Spread to Treasuries
It should be clear that at the time of purchase it is not possible to determine an exact
yield for a mortgage-backed security; the yield will depend on the actual prepaymen
experience of the mortgages in the pool Nevertheless, the convention in all fixe -in-
come markets is to méasure the yield on a non-Treasury security to that of a “compa-
_Ý—— of principal over time makes it inappropriate to compare the yield
of a mortgage-backed security to a Treasury of a stated maturity Instead, mar : pa i
ticipants have used two measures: Macaulay duration (as explained in Chapter 4) a
average Tife (as explained in Chapter 11)
5 »
As explained in Chapter 14, the practice of spreading the yield to the average life n
the interpolated Treasury yield curve is improper for an amortizing bond even in t
absence of interest-rate volatility What should be done instead is to calculate what 1s
90-31 7.85 8.12 8.49 8.95 9.69 10.13 10.89 11.83 91-31 769 = 7,93 8.25 8.66 9.31 9.70
2.89 2.35 1.90 Mod duration: 7.23 5.92 4.78 3.84 2.92 2.56
2.11 1.74 Exp maturity: 10.90 8.40 6.49 4.99 3.65
3.15 2.49 1.99
called the static spread This is the yield spread in a static scenario (i.e., no volatility of interest rates) of the bond over the entire theoretical Treasury spot rate curve, not a single point on the Treasury yield curve
As explained in Chapter 14, the magnitude of the difference between the tradi- tional yield spread and the static yield spread depends on the steepness of the yield curve: The steeper the curve, the greater the difference between the two values In a relatively flat interest-rate environment, the difference between the traditional yield spread and the static spread will be small
There are two ways to compute the static spread for mortgage-backed securities
One way is to use today’s yield curve to discount future cash flows and keep the mort- gage refinancing rate fixed at today’s mortgage rate Because the mortgage refinanc- ing rate is fixed, the investor can usually specify a reasonable prepayment rate for the life of the security, Using this prepayment rate, the bond’s future cash flow can be es- timated Use of this approach to calculate the static spread recognizes different prices today of dollars to be delivered at future dates This results in the proper discounting
of cash flows while keeping the mortgage rate fixed Effectively, today’s prices indi-
cate what the future discount rates will be, but the best estimates of future rates are
today’s rates
ties 369
Trang 4
384 CHAPTER I5 Analysis of Mortgage-Backed Securities
Now let’s look at the sensitivity to the interest-rate volatility assumption, 12% in the
base case Two experiments are performed: reducing the volatility assumption to 8% and
increasing it to 16% These results are reported in the third panel of Exhibit 15-9
Reducing the volatility to 8% increases the dollar price of the collateral by $1 and
increases the OAS from 70 in the base case to 92 This $1 increase in the price of the
collateral is not equally distributed, however, among the four tranches Most of the in-
crease in value is realized by the longer tranches The OAS gain for each of the
tranches follows more or less the OAS durations of those tranches This makes sense,
because the longer the duration, the greater the risk, and when volatility declines, the
reward is greater for the risk accepted
At the higher level of assumed interest-rate volatility of 16%, the collateral is af-
fected severely The collateral’s loss is distributed among the tranches in the expected
manner: The longer’ the duration, the greater the loss In this case tranche D and the
residual are the least affected
Using the OAS from the Monte Carlo simulation methodology, a fair conclusion
can be made about this simple plain vanilla structure: What you see is what you get
The only surprise in this structure seems to be tranches B and C In general, however,
a money manager willing to extend duration gets paid for that risk
TOTAL RETURN ANALYSIS
Neither the static cash flow methodology nor the Monte Carlo simulation methodol-
ogy will tell a money manager whether investment objectives can be satisfied The
performance evaluation of an individual mortgage-backed security requires specifica-
tion of an investment horizon, whose length for most financial institutions is dictated
by the nature of its liabilities
The measure that should be used to assess the performance of a security or a port-
folio over some investment horizon is the total return that we discussed in Chapter 3
The total dollars received from investing in a mortgage-backed security consist of
1, The projected cash flow from the projected interest payments and the pro-
jected principal repayment (scheduled plus prepayments)
2 The interest earned on reinvestment of the projected interest payments and
the projected principal prepayments
3 The projected price of the mortgage-backed security at the end of the invest-
To obtain the cash flow, a prepayment rate over the investment horizon must be
assumed The second step requires assumption of a reinvestment rate Finally, either
of the methodologies described in this chapter—cash flow yield or Monte Carlo simu-
lation—can be used to calculate the price at the end of the investment horizon under a
particular set of assumptions Either approach requires assumption of the prepayment
rate and the Treasury rates (i.e., the yield curve) at the end,of the investment horizon
The cash flow yield methodology uses an assumed spread to a comparable Treasury to
determine the required cash flow yield, which is then used to compute the projected
price The Monte Carlo simulation methodology requires an assumed OAS at the in-
CHAPTER I5 Analysis of Mortgage-Backed Securities 385
vestment hor 1zon F Tom this assumption, the OAS methodolo:
can pr oduce the hor +1- 8y
Horizon Price for CMO Tranches The most difficult part of estimating total return is projecting the price at the horizon date In the case of a CMO tranche the price depends on the characteristics of the tranche and the spread to Treasuries at the termination date The key determinants are the “quality” of the tranche, its average life (or duration), and its convexity
prepayments could become a sequential-pay tranche As another example, suppose that a PAC bond is the longest-average-life tranche in a reverse PAC structure Pro
jected prepayments in this case might occur in an amount to change the class from a long-average-life PAC tranche to a support tranche The converse is that the qualit
of a tranche may improve as well as deteriorate For example, the effective collar for 4 PAC tranche could widen at the horizon date when prepayment circumstances in- crease the par amount of Support tranches outstanding as a proportion of the deal
OAS-—Total Return
The total return and OAS frameworks can be combined to determine the projected
ly Em at we porizon date At the end of the investment horizon, it is necessary to spec-
”
Mong Cả Sere sted to change The horizon price can be “backed out” of the Assumptions about the OAS value at the investment horizon reflect the expecta- tions of the money manager It is common to assume that the OAS at the horizon date will be the same as the OAS at the time of purchase A total return calculated usin this assumption is sometimes teferred to as a constant-OAS total return Alterna, tively, active total return managers will make bets on how the OAS will change—
either widening or tightening The total return framework can be used to assess how Sensitive the performance of a mortgage-backed security is to changes in the OAS
Trang 5
386 CHAPTER !5 Analysis of Mortgage-Backed Securities
yield, (2) the mortgage-backed security is held to the maturity date, and (3) the pre-
payment speed used to project the cash flow will be realized In addition, the cash flow
yield methodology fails to recognize that future interest-rate changes will affect the
cash flow
Modified duration is not a good measure of price volatility for mortgage-backed
securities because it assumes that the cash flow does not change as yield changes Ef-
fective duration does take into consideration how yield changes will affect prepay-
A mortgage-backed security is a security whose cash flow is path dependent This
means that cash flow received in one period is determined not only by the current and
future interest-rate levels, but also by the path that interest rates took to get to the
current level A methodology used to analyze path dependent cash flow securities is
Monte Carlo simulation This methodology involves randomly generating many sce-
narios of future interest-rate paths, where the interest-rate paths are generated based :
on some volatility assumption for interest rates
The random paths of interest rates should be generated from an arbitrage-free
model of the future term structure of interest rates The Monte Carlo simulation
methodology applied to mortgage-backed securities involves randomly generating a
set of cash flows based on simulated future mortgage refinancing rates The theoreti-
cal value of a security on any interest-rate path is the present value of the cash flow on
that path, where the spot rates are those on the corresponding interest-rate path The
theoretical value of a security is the average of the theoretical values over all the in-
terest-rate paths Information about the distribution of the path values is useful in un-
derstanding the variability around the theoretical value The average life reported is
the average of the average lives from all the interest-rate paths and information about
the distribution of the average life is useful
In the Monte Carlo simulation methodology, the option-adjusted spread is the
spread that when added to all the spot rates on all interest-rate paths will make the av-
erage present value of the paths equal to the observed market price (plus accrued in-
terest) The effective duration and effective convexity are calculated using the Monte
Carlo simulation methodology by holding the OAS constant and shifting the term
structure up and down
Total return is the Correct measure for assessing the potential performance of
CMO tranches over a specified investment horizon The static cash flow yield or
Monte Carlo simulation methodology can be incorporated into a total return frame-
work to calculate the mortgage-backed security’s price at the horizon date Scenario
analysis is one waysto evaluate the risk associated with investing in a mortgage-backed
security
1 Suppose you are told that the cash flow yield of a pass-through security is 9% and
that you are seeking to invest in a security with a yield greater than 8.8%
a What additional information would you need té.know before you might inves!
in this pass-through security?
b What are the limitations of the cash flow yield for assessing the potential retur
from investing in a mortgage-backed security?
Treasury securit : y How H methodolo By, a Spread j 387
at Is vector a n VIS a Comparab] e T: 1s calculat ed ove
of effective durat; 2 uration :
“Fidelity E the Janua yes $250 Million
TY 27, 1992, issue
Federal Home Loan Mong
Pass-throughs, Th
K mainly at
Corp and Feq-
ese have higher toughs, or similar
e onvexit ye paid Wolfson, “Younes compensat
- > Re said He does not feel pra in yiel
Payme
Said He exp
slow later in _ The highe higher prepa
ts will
b What doce W $ what Prepayments wil] aecelee, Strips offi
Path present vale : ac Structure, wh at would he theoretical
An avo structure? average lives to be compared oct the distribution of th
o Wing assuming 12%, volatility using the Monte Carlo method j Stru : ond fro m the
Trang 6(basis points) (basis points)
nmen — 8 m Analysis of Convertible Bonds
a Calculate the option cost for cach tranche
: What would happen to the static spread for each tranche if a 15% volatility is
d What would happen to the OAS for each tranche if a 15% volatility is assumed?
After reading this chapter you will understand:
@ what a convertible bond is
M@ what an exchangeable bond is M@ the basic features of a convertible security
9 tion-adjusted spread vary across dealer firms? ¬
6 Bavlain how the Thumber of interest-rate paths used in the Monte Carlo simula- a conversion value, market conversion price, conversion premium per share, conver- tion methodology is determined i i i ing statement: “When the hen th sion premium ratio, and premium over straight value of a convertible bond
a Manto Carlo simulation methodology taod ta value A mortgage-backed secu- onte : r the investment features of a convertible security
ion is employed for all interest-rate paths ñ i what the minimum value of a convertible bond is
18 What acsumptionis made about the OAS in calculating the effective duration and
What are the limitations of the option-adjusted sprea -
20 What assumptions are required to assess the potential total return of a mortgage:
21 What are the ‘complications of assessing the potential total return of a CMO
tranched using the total return framework?
the pros and cons of investing in a convertible bond
Mi the options approach to valuing a convertible bond
a why an option pricing approach is needed to value convertible securities properly
methodologies for analyzing them, beginning with a review of the basic provi- sions of convertible bonds
%
CONVERTIBLE BOND PROVISIONS The conversion provision in a corporate bond issue grants the bondholder the right to convert the bond into a predetermined number of shares of common stock of the is- suer A convertible bond is therefore a corporate bond with a call option to buy the common stock of the issuer
Exchangeable bonds grant the bondholder the right to exchange the bonds for the
common stock of a firm other than the issuer of the bond For example, Dart & Kraft has
an outstanding bond issue that is exchangeable for the common stock of Minnesota Min- ing and Manufacturing (Dart & Kraft obtained the 3M common stock in exchange for
389
Trang 7the sale of Riker Laboratories.) Some Ford Motor Credit bonds are exchangeable for the common stock of the parent company, Ford Motor Company A few issues are exe ange- able into more than one security General Cinema, for example, has an oust cing i sue that is convertible into the common stock of R.J Reynolds and Sea- a Tp rat n
The number of shares of common stock that the bondholder lệ receive from oe ercising the call option of a convertible bond or an exchangeable | on 5 called © conversion ratio The conversion privilege may extend for all or on y some portion of
the bond’s life, and the stated conversion ratio may fall over time It is always adjuste
tionately for stock splits and stock dividends
are the time of issuance of a convertible bond, the issuer has effectively granted the bondholder the right to purchase the common stock at a price equal to
par value of, convertible bond conversion ratio Along with the conversion privilege granted to the bondholder, most cone
bonds are callable at the option of the issuer Some convertible bonds are pu ‘ oP
options can be classified as hard puts and soft puts A hard put is one m w rc the oe vertible security must be redeemed by the issuer only for cash int © cas of soft put, the issuer has the option to redeem the convertible security or cast A\ mmon
stock, subordinated notes, or a combination of the three In our initial analysis, g
nore any call or put options
To illustrate how to analyze a convertible bond, we use the same hypothetical bon throughout the chapter, XYZ bond:
maturity = 10 years coupon rate = 10%
conversion ratio = 50 ~ par value = $1,000 current market price of XYZ bond = $950 current market price of XYZ common stock = $17
s dividends per share = $1 The conversion price for XYZ bond is
conversion price = $1,000 = $20 ,
%
MINIMUM VALUE OF A CONVERTIBLE BOND
The conversion value of a convertible bond is the value of the bond if it is converte
conversion value = market price of common stock X conversion ratio
tụ
¡ ically i 1 be- iti i jue given here is theoretically incorrec!
, i the standard textbook definition of conversion va € tot the cae ae bondholders convert, the price of the stock wiil decline The theoretically correct definition conversion value is that it is the product of the conversion ratio and the stock price after co -
The minimum price of a convertible bond is the greater of
1, Its conversion value, or
2 Its value as a corporate bond without the conversion option—that is, based
on the convertible bond’s cash flows if not converted (ie., a plain vanilla bond) This value is called its straight value
To estimate the straight value, we must determine the required yield on a noncon- vertible bond with the same quality rating and similar investment characteristics Given this estimated required yield, the straight value is then the present value of the bond’s cash flows using this yield to discount the cash flows
If the convertible bond does not sell for the greater of these two values, arbi- trage profits could be realized For example, suppose that the conversion value is greater than the straight value and the bond trades at its straight value An investor can buy the convertible bond at the straight value and convert it By doing so, the investor realizes a gain equal to the difference between the conversion value and the straight value Suppose, instead, that the Straight value is greater than the con- version value, and the bond trades at its conversion value By buying the convertible
at the conversion value, the investor will realize a higher yield than a comparable
Hlustration For the XYZ convertible bond, the conversion value is equal to
conversion value = $17 x 50 = $850
To determine the straight value, it is necessary to determine what comparable bonds are trading for in the market Suppose that comparable bonds are trading to yield 14% The straight value is then the price of a 10% 10-year bond selling to yield 14% The price for such a bond would be $788
Given a conversion value of $850 and a straight value of $788, the minimum price for the XYZ bond is $850 To see this, note that if the bond is selling at its straight
value rather than its conversion value, an investor could buy the bond for $788 and si-
multaneously sell 50 shares of XYZ stock at $17 per share When the short sale of the stock is covered when the bond is converted, the transaction would produce an arbi- trage profit of $62 per XYZ bond purchased The only way to eliminate this arbitrage profit is for the XYZ bond to sell for $850, its conversion value
Suppose, instead, that comparable nonconvertible bonds are trading to yield 11.8% Then the straight value of XYZ bond would be $896 The minimum price for
the XYZ bond must be its straight value in this case, because that is a value higher
than the conversion value of $850 To see this, suppose that the market price of the XYZ bond is $850 At this price, the yield would be about 12.7%, 90 basis points greater than comparable nonconvertible bonds Investors would find the bond at- tractive As investors buy the bond, they will bid up its price to where the new yield 1s 11.8%
?Actually, it is $788.10, but $788 will be used in our illustrations.
Trang 8
The price that an investor effectively pays for the common stock if the convertible
bond is purchased and then converted into the common stock is called the market
conversion price It is found as follows:
: market price of convertible bond market conversion price =
conversion ratio
The market conversion price is a useful benchmark because when the actual market
price of the stock rises above the market conversion price, any further stock price in- crease is certain to increase the value of the convertible bond by at least the same per- centage Therefore, the market conversion price can be viewed as a break-even point,
An investor who purchases a convertible bond rather than the underlying stock typically pays a premium over the current market price of the stock This premium per share is equal to the difference between the market conversion price and the cur- rent market price of the common stock That is,
market conversion premium per share = market conversion price ~ current market price The market conversion premium per share is usually expressed as a percentage of the
current market price as follows:
market conversion premium ratio = :
market price of common stock Why would someone be willing to pay a premium to buy this stock? Recall that the minimum price of a convertible bond is the greater of its conversion value or its
straight value Thus, as the stock price declines, the price of the convertible bond will
not fall below its straight value The straight value therefore acts as a floor for the con-
vertible bond price
Viewed in this context, the market conversion premium per share can be seen as the price of a call option As explained in Chapter 22, the buyer of a call option limits
the downside risk to the option price In the case of a convertible bond, for a pre-
mium, the bondholder limits the downside risk to the straight value of the bond The
difference between the buyer of a call option and the buyer of a convertible bond is
that the former knows precisely the dollar amount of the downside risk, whereas the
latter knows only that the most that can be lost is the difference between the convert-
ible bond price and thestraight value The straight value at some future date, how-
ever, is not known; the value will change as the interest rate changes
Hiustration
At a market price of $950, a stock price of $17, and a conversion ratio of 50, the mar-
ket conversion price, market conversion premium per share, and market conversion
premium ratio of the XYZ convertible bond are calculated ag follows:
*The market conversion price is also called the conversion parity price
- HN, NINH OM
MARKET CONVERSION PRICE
CHAPTER 16 Analysis of Convertible Bonds
market conversion price = “ms = $19 market conversion premium per share = $19 — $17 = $2 market conversion premium ratio = 32 = 0.118 or 11.8%
CURRENT INCOME OF CONVERTIBLE BOND VERSUS STOCK
As an offset to the market conversion premium per share, investing in the convertible bond rather than buying the stock directly generally means that the investor realizes higher current income from the coupon interest paid on the convertible bond than would be received as dividends paid on the number of shares equal to the conversion ratio Analysts evaluating a convertible bond typically compute the time it takes to re- cover the premium per share by computing the premium payback period (which is also known as the break-even time) This is computed as follows:
market conversion premium per share favorable income differential per share
where the favorable income differential per share is equal to‘
coupon interest from bond — (conversion ratio X dividend per share)
conversion ratio Notice that the premium payback period does not take into account the time value
Hlustration For the XYZ convertible bond, the market conversion premium per share is $2 The favorable income differential per share is found as follows:
coupon interest from bond = 0.10 X $1,000 = $100 conversion ratio X dividend per share = 50 x $1 = $50
Therefore,
$100 — $50 _ 50 $1
favorable income differential per share =
and
premium payback period = ze = 2 years
Without considering the time value of money, the investor would recover the market conversion premium per share in two years
*A more precise methodology for calculating the favorable income from holding the convertible is recom-
mended in Luke Knecht and Mike McCowin, “Valuing Convertible Securities,” in Frank J Fabozzi (ed.), Advances and Innovations in Bond and Mortgage Markets (Chicago: Probus Publishing, 1989) In most
cases the conventional formula presented in the text is sufficient.
Trang 9
—-
DOWNSIDE RISK WITH A CONVERTIBLE BOND
Investors usually use the straight value of the bond as a measure of the downside risk
of a convertible bond, because the price of the convertible bond cannot fall below this
value Thus, the straight value acts as the current floor for the price of the convertible
bond The downside risk is measured as a percentage of the straight value and com
puted as follows:
market price of the convertible bond _ 1 premium over straight value = straight value
The higher the premium over straight value, all other factors constant, the less attrac
tive the convertible bond
Despite its use in practice, this measure of downside risk is flawed because the straight value (the floor) changes as interest rates change If interest rates rise (fall), the straight value falls (rises) making the fioor fall (rise) Therefore, the downside risk changes as interest rates change
Illustration Earlier we said that if comparable nonconvertible bonds are trading to yield 14%, the straight value of the XYZ bond would be $788 The premium over straight value is then
premium over straight value = vn —1=0.21or21%
If the yield on a comparable nonconvertible bond is 11.8% instead of 14%, the
straight value would be $896 and the premium over straight value would be
premium over straight value = ae — 1 = 0.06 or 6%
INVESTMENT CHARACTERISTICS OF A CONVERTIBLE BOND
The investment characteristics of a convertible bond depend on the stock price If the price of the stock is low, so that the straight value is considerably higher than the con- version value, the bond will tradé much like a straight bond The convertible bond in such instances is referred to as a bond equivalent or a busted convertible
When the price of the stock is such that the conversion value is considerably higher than the straight value, the convertible bond will trade as if it were an equity instrument; in this case it is said to be an equity equivalent In such cases the market
Between these two cases, bond equivalent and equity equivalent, the convertible bond trades as a hybrid security, having the characteristics of both a bond and an ¢q- uity instrument
PROS AND CONS OF INVESTING IN A CONVERTIBLE BOND _
So far we have presented several measures that can be used to analyze convertible bonds Let’s use the XYZ convertible bond to drive home the pros and cons of invest:
ing in a convertible bond
Đo
CHAPTER 16 Analysis of Convertible Bonds 395 Suppose that an investor is considering purchase of a stock or a convertible bond
The stock can be purchased in the market for $17 By buying the convertible bond, the investor is effectively purchasing the stock for $19 (the market conversion price per share)
Look at the outcome one month from now, assuming that XYZ stock rises to $34
An investor buying the stock would realize a gain of $17 ($34 — $17) on a $17 invest- ment, or a 100% return In contrast, the conversion value for the bond would be
$1,700 ($34 x 50) Because the price of XYZ bond is $950, the investor would realize
a return of about 79% The return would in fact probably be slightly higher because the convertible bond would trade at a slight premium to its conversion value The rea- son for the lower return by buying the convertible bond rather than the stock directly
is that the investor,has effectively paid $2 per share more for the stock Thus, the in- vestor realizes a gain based on a stock price of $19 rather than $17
So far, we’ve illustrated the advantage of owning the stock rather than the bond when the price of the stock rises Let’s look at the situation where the stock declines
in value to $7 The investor who buys the stock now realizes a loss of $10 per share for
a return of —59% The conversion value of the XYZ bond likewise drops, to $350 ($7 X 50) Its price, however, will not fall to that level Recall from our earlier discus- sion that the minimum price of a convertible bond will be the greater of its conversion value or its straight value Assuming that the straight value is $788, and it does not change over the one-month period, the value of XYZ bond will fall to only $788 This means that the investor realizes a loss of only 17% The loss would be even less in fact because the convertible bond would trade at a premium to its straight value
The critical assumption in this analysis is that the straight value does not change, although it can change for any of the reasons cited in Chapter 2 More specifically, if interest rates rise in the economy, the straight value will decline Even if interest rates
do not rise, the perceived creditworthiness of the issuer may deteriorate, causing in- vestors to demand a higher yield In fact, the stock price and the yield required by in- vestors are not independent When the price of the stock drops precipitously, as in our
$17 to $7 illustration, the perceived creditworthiness of the issuer may decline, causing
a decline in the straight value In any event, although the straight value may decline, it still is a floor (albeit a moving floor) for the convertible bond price In our illustration, the straight value would have to fall about $390 (59% loss on $950), to equal the loss
on the stock purchase
The illustration clearly demonstrates that there are benefits and drawbacks of in- vesting in convertible bonds The disadvantage is the upside potential given up be- cause a premium per share must be paid An advantage is the reduction in downside risk (as determined by the straight value), with the opportunity to recoup the pre- mium per share through the higher current income from owning the convertible bond
Call Risk
Convertible issues are callable by the issuer This is a valuable feature for issuers, who
deem the current market price of their stock undervalued enough so that selling stock directly would dilute the equity of current stockholders The firm would prefer to raise equity funds over incurring debt, so it issues a convertible setting the conversion ratio on the basis of a stock price it regards as acceptable When the market price reaches the conversion point, the firm will want to see the conversion happen in view
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mm ~ $9 CHAPTER | nalysis of Convertible Bonds
of the risk that the price may drop in the future This gives the firm an interest in fOrc- ing conversion, even though this is not in the interest of the owners of the security whose price is likely to be adversely affected by the call
Takeover Risk Corporate takeovers represent another risk to investing in convertible bonds n an is suer is acquired by another company or by its own management (as in ne c se or a
management-led leveraged buyout), the stock price may not appreciate lệ y or
the holders of the convertible bond to benefit from the conversion feature Ss “s och
of the acquired company may no longer trade after a takeover, the investor can e left with a bond that pays a lower coupon rate than comparable-risk corporate bonds
OPTIONS APPROACH
In our discussion of convertible bonds, we did not address the following questions:
1, What is a fair value for the conversion premium per share? oo
2 How do we handle convertible bonds with call and/or put options?
3 How does a change in interest rates affect the stock price?
The option pricing approach to valuation described in Chapter 14 can TP as ane swer these questions Consider first a noncallable/nonputable convertible bon h e investor who purchases this bond would be entering Into two separate vansac ve (1) buying a noncallable/nonputable straight bond, and (2) buying a ca oP ion or
warrant) on the stock, where the number of shares that can be purchased wi e cai
ion i the conversion ratio
Pr he question is: What is the fair value for the call option? The fair value cepencs
on the factors (discussed in Chapter 22) that affect the price of a call option ne key factor is the expected price volatility of the stock: the more the expected price vo at ity, the greater the value of the call option The theoretical value of a ca option cam
be valued using the Black-Scholes option pricing mode? or the binomia option Pe ing model.‘ As a first approximation to the value of a convertible bond, the form
convertible bond value = straight value + price of the call option on the stock The price of the call option is added to the straight value because the investor has pur-
TU md in đcommon feature of a convertible bond: the issuer’s right to cal the bond The issuer can force conversion by calling the bond For SN hưng that the call price is $1,030 per $1,000 par and the conversion value is $1,70 h ne suer calls the bonds, the optimal strategy for the investor is to convert the bon
— ~
SBischer Black and Myron Scholes, “The Pricing of Corporate Liabilities,” Journal of Political Economy,
Sohn C Con, Stephen A Ross, and Mark Rubinstein cei Pricing: AeSimpied Approach, cao St tof
i i ics, September 1979, pp 229-263; Richard J Rendlem: 1d | : > ng Coton Pricing.” Journal of Finance, December 1979, pp 1093-1110; and William F Sharpe, Investme!
(Englewood Cliffs, NJ: Prentice Hall, 1981), Chap 16
receive shares worth $1,700.’ The investor, however, loses any premium over the con-
version value that is reflected in the market price Therefore, the analysis of convert-
ible bonds must take into account the value of the issuer’s right to call the bond This depends, in turn, on (1) future interest rate volatility, and (2) economic factors that determine whether it is optimal for the issuer to call the bond
The Black-Scholes option pricing model cannot handie this situation Instead, the binomial option pricing model can be used simultaneously to value the bondholder’s call option on the stock and the issuer's right to call the bonds, The bondholder’s; put option can also be accommodated To link interest rates and stock prices together \ (the third question we raised previously), statistical analysis of historical movements
of these two variables must be estimated and incorporated into the model
The option pricing approach offers a great deal of promise and models have been proposed as far back as 1977.5 In the experience of the author, the most complicated model employed by practitioners uses the Black-Scholes model and tests the sensitiv- ity of the factors that affect any other embedded options.°
In this chapter we have discussed the basic provisions of convertible bonds and ex- plored a framework for evaluating these bonds Analysis of a convertible bond re- quires calculation of the conversion value, straight, value, market conversion price, market conversion premium ratio, and premium payback period
The downside risk of a convertible bond usually is estimated by calculating the premium over straight value The limitation of this measure is that the straight value (the floor) changes as interest rates change Convertible bond investors are also sub- ject to call risk and takeover risk
The option pricing approach can be used to determine the fair value of the em- bedded call option The value of the call option following this approach is estimated using some equity option pricing model such as the Black—Scholes model
Questions
1 In the October 26, 1992, prospectus summary of the Staples 5% convertible sub- ordinated debentures due 1999, the offering stated: “Convertible into Common Stock at a conversion price-of $45 per share .” If the par value is $1,000, what is the conversion ratio?
7Actually, the conversion value would be less than $1,700, because the per share value after conversion would decline 4See, for example, Michael Brennan and Eduardo Schwartz, “Convertible Bonds: Valuation and Optimal’
Strategies for Call and Conversion,” Journal of Finance, December 1977, pp 1699-1715; Jonathan Ingersoll,
“A Contingent-Claims Valuation of Convertible Securities,” Journal of Financial Economics, May 1977,
pp 289-322; Michael Brennan and Eduardo Schwartz, “Analyzing Convertible Bonds,” Journal of Financial
and Quantitative Analysis, November 1980 pp 907-929; and George Constantinides, “Warrant Exercise and
Bond Conversion in Competitive Markets,” Journal of Financial Economics, September 1984, pp 371-398,
See, for example, Mihir Bhattacharya and Yu Zhu, “Valuation and Analysis of Convertible Securities,” Chapter 36 in Frank J Fabozzi and T Dessa Fabozzi (eds.), The Handbook of Fixed Income Securities
(Burr Ridge, IL: Irwin Professional Publishing, 1995); and Frank J Fabozzi, Valuation of Fixed Income Se- curities and Derivatives (New Hope, PA: Frank J Fabozzi Associates, 1995), Chapter 9
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