Financial engineering principles phần 6 pdf

In tranche 2, investors would be ond in line to receive coupon cash flows generated by the underlying mort-gages.. This com-ment helps to reinforce the idea that tranche creation does no

bundle that comprises the MBS Accordingly, if some homeowners happen todefault on their mortgages, the excess supply of mortgages in the bundle willhelp to cover that event Another way that MBS products are able to secure

a triple-A rating is by virtue of their being supported by federal agencies Thethree major agencies of the United States involved with supporting mortgagesinclude Ginnie Mae, Fannie Mae, and Freddie Mac.6The key purpose of thesegovernmental organizations is to provide assurance and confidence in the mar-ket for MBSs and other mortgage products

Table 4.3 summarizes key differences between an MBS and a callablebond

The most dramatic differences between MBSs and callable bonds are thatthe options embedded with the former are continuous while the single optionembedded in the latter tends to be discrete, and the multiple options within

an MBS can be triggered by many more variables

Figure 4.15 shows how an MBS’s cash flows might look; none of thecash flow boxes is solid because none of them can be relied on with 100percent certainty While less-than-100% certainty might be due partly to thevagaries of what precisely is meant by saying that the federal agencies issu-ing these debt types are “supported by” the federal government, more of theuncertainty stems from the embedded optionality Although it may very well

be unlikely, it is theoretically possible that an investor holding a backed security could receive some portion of a principal payment in one

mortgage-of the very first cash flows that is paid out This would happen if a

home-6 Ginnie Mae pass-thrus are guaranteed directly by the U.S government regarding timely payment of interest and principal Fannie Mae and Freddie Mac pass-thrus carry the guarantee of their respective agency only; however, both agencies can borrow from the Treasury, and it is not considered to be likely that the U.S government would allow any of these agencies to default.

TABLE 4.3 MBS versus Callable Bond Optionality

Callability Continuous Discrete (sometimes continuous)

Call period Immediately Eligible after the passage of some time

Call trigger Level of yields Level of yields, other cost considerations

Homeowner defaults Homeowner sells property for any reason

Property is destroyed as

by natural disaster

owner decides or is forced to sell the home almost immediately after chase and pays off the full principal of the loan In line with what we wouldgenerally expect, principal payments will likely make their way more mean-ingfully into the mix of principal-coupon cash flows after some time passes(or, in the jargon of the marketplace, with some seasoning).

pur-How can probabilities be assigned to the mortgage product’s cash flowsover time? While we can take the view that we adopted for our callabledebenture—that at the start of the game every uncertain cash flow has a50/50 chance of being paid—this type of evenly split tactic may not be verypractical or realistic for mortgage products For example, a typical homemortgage is a 30-year fixed-rate product This type of product has beenaround for some time, and some useful data have been collected to allowfor the evaluation of its cash flows over a variety of interest rate and eco-nomic environments In short, various patterns can and do emerge with thenature of the cash flows Indeed, a small cottage industry has grown up forthe creation and maintenance of models that attempt to divine insight intothe expected nature of mortgage product cash flows It is sufficient heremerely to note that no model produces a series of expected cash flows fromyear 1 to year 30 with a 50/50 likelihood attached to each and every pay-out Happily, this conforms to what we would expect from more of an intu-itive or common sense approach

Given the importance of prepayment rates when valuing an MBS, eral models have been developed to forecast prepayment patterns Clearly,investors with a superior prepayment model are better equipped to identifyfair market value

sev-In an attempt to impose a homogeneity across prepayment assumptions,certain market conventions have been adopted These conventions facilitatetrades in MBSs since respective buyers and sellers know exactly whatassumptions are being used to value various securities

One commonly used method to proxy prepayment speeds is the constant

prepayment rate (CPR) A CPR is the ratio of the amount of mortgages

pre-paid in a given period to the total amount of mortgages in the pool at thebeginning of the period That is, the CPR is the percentage of the principaloutstanding at the beginning of a period that will prepay over the follow-ing period For example, if the CPR for a given security in a particular month

is 10.5, then the annualized percentage of principal outstanding at the ning of the month that will repay during the month is 10.5 percent As thename implies, CPR assumes that prepayment rates are constant over the life

begin-of the MBS

To move beyond the rather limiting assumption imposed by a CPR—that prepayments are made at a constant rate over the life of an MBS—the

industry proposed an alternative measure, the Public Securities Association

(PSA) model The PSA model posits that any given MBS will prepay at an

annualized rate of 0.2 percent in the first month that an MBS is ing, and prepayments will increase by 0.2 percent per month until month

outstand-30 After month 30, it is assumed that prepayments occur at a rate of 6 cent per year for all succeeding months

per-Generally speaking, the PSA model provides a good description of payment patterns for the first several years in the life of an MBS and hasproven to be a standard for comparing various MBSs Figure 4.16 showstheoretical principal and coupon cash flows for a 9 percent Ginnie Mae MBS

pre-at 100 percent PSA When an MBS is quoted pre-at 100 percent PSA, this meansthat prepayment assumptions are right in line with the PSA model, above

An MBS quoted at 200 percent PSA assumes prepayment speeds that aretwice the PSA model, and an MBS quoted at 50 percent PSA assumes aslower prepayment pattern

140 120

100 80 40 20

$1,000s

60 120 180 240 300 360

Month

Interest Principal

9% 30-year Ginnie Mae, 100% PSA

FIGURE 4.16 The relationship between pay-down of interest and principal for a

pass-thru.

“Probability-weighted coupon” means the statistical likelihood of

receiv-ing a full coupon payment (equivalent to 100 percent of F times C) As

prin-cipal is paid down from par, the reference amount of coupon paymentdeclines as well (so that when principal is fully paid down, a coupon pay-

ment is equal to zero percent of F times C, or zero).

“Probability-weighted principal” means the statistical likelihood ofreceiving some portion of principal’s payment

As is the case with a callable debenture, the initial price of an MBS is

par, and Y C However, unlike our callable debenture, there is no formal

lockout period with an MBS While we might informally postulate that

prob-ability values for F should be quite small in the early stages of an MBS’s life

(where maturities can run as long as 15 or 30 years), this is merely an cated guess The same would be true for postulating that probability values

edu-for C should be quite large in the early stages of an MBS’s life Because an

MBS is comprised of an entire portfolio of short call options (with each onelinked to an individual mortgage), in contrast with the single short option

embedded in a callable debenture, the modeling process for C and F is more

complex; hence the existence and application of simplifying benchmark els, as with the CPR approach

mod-At this stage we have pretty much defined the two extremes of ality with fixed income products in the U.S marketplace However, thereare gradations of product within these two extremes For example, there arePACs, or planned amortization class securities

option-Much like a Thanksgiving turkey, an MBS can be carved up in a ety of ways At Thanksgiving, some people like drumsticks and others pre-fer the thigh or breast With bonds, some people like predictable cash flowswhile others like a higher yield that comes with products that behave in lesspredictable ways To satisfy a variety of investor appetites, MBS pass-thrus

vari-can be sliced in a variety of ways For example, classes of MBS vari-can be

cre-ated Investors holding a Class A security might be given assurances that theywill be given cash distributions that conform more to a debenture than apass-thru Investors in a Class B security would have slightly weaker assur-ances, those in a Class C security would have even weaker assurances, and

so forth As a trade-off to these levels of assurances, the class yield levelswould be progressively higher

A PAC is a prime example of a security type created from a pool of gages What happens is that the cash flows of an MBS pool are combinedsuch that separate bundles of securities are created What essentially distin-guishes one bundle from another is the priority given for one bundle to beassured of receiving full and timely cash flows versus another bundle.For simplicity, let us assume a scenario where a pool of mortgages isassembled so as to create three tranches of cash flow types In tranche 1,investors would be assured of being first in line to receive coupon cash flows

mort-generated by the underlying mortgages In tranche 2, investors would be ond in line to receive coupon cash flows generated by the underlying mort-gages If homeowners with mortgages in this pool decide to pay off theirmortgage for whatever reason, then over time tranche 2 investors would notexpect to receive the same complete flow of payouts relative to tranche 1investors If only for this reason, the tranche 2 investors should not expect

sec-to pay the same up-front price for their investment relative sec-to what is paid

by tranche 1 investors They should pay less Why? Because tranche 2investors do not enjoy the same peace of mind as tranche 1 investors of beingkept whole (or at least “more whole”) over the investment horizon Andfinally, we have tranche 3, which can be thought of as a “residual” or “clean-up” tranche The tranche 3 investors would stand last in line to receive cashflows, only after tranche 1 and tranche 2 investors were paid And consis-tent with the logic presented above for tranche 2, tranche 3 investors shouldnot expect to pay the same up-front price for their investment as tranche 1

or 2 investors; they should pay less

Note that tranche 1 investors are not by any means guaranteed of ing all cash flows in a complete and timely matter; they only are the first inline as laying priority to complete and timely cash flows In the unlikely eventthat every mortgage within the pool were to be paid off at precisely the sametime, then each of the three tranches would simply cease to exist This com-ment helps to reinforce the idea that tranche creation does not create new cashflows where none existed previously; tranche creation simply reallocatesexisting cash flows in such a way that at one end of a continuum is a securitytype that at least initially looks and feels like a more typical bond while at theother end is a security type that exhibits a price volatility in keeping with itsmore uncertain place in the pecking order of all-important cash flow receipts.This illustration is a fairly simplified version of the many different ways

receiv-in which products can be created out of mortgage pools Generally ing, PAC-type products are consistent with the tranche 1 scenario presented.Readers can refer to a variety of texts to explore this kind of product cre-ation methodology in considerable detail From PACs to TACs to A, B, C,and Z tranches (and much, much more), there is much to keep an avid mort-gage investor occupied

speak-Figure 4.18 applies the PAC discussion to our cash flow diagram.Notice that the cash flow boxes in the early part of the PAC’s life aredrawn in with solid lines PACs typically come with preannounced lockout

periods Here, lockout refers to that period of time when the PAC is

pro-tected from not receiving complete and timely cash flow payments owing tooption-related phenomena The term of lockouts varies, though is generally

5 to 10 years Again, the PAC is protected in this lockout period because itstands first in line to receive cash flows out of the mortgage pool Many times

a PAC is specified as being protected only within certain bandwidths of

option-related activity Typically the activity of homeowners paying off their

mortgages is referred to as prepayment speed (or speed) Moreover, a

con-vention exists for how these speeds are quoted Accordingly, often PAC widths define an upper and lower bound within which speeds may varywithout having any detrimental effect on the PAC’s cash flows The widerthe bandwidth, the pricier the PAC compared to PACs with narrower bands.Once a particular PAC has experienced a prepayment speed that falls out-side of its band, it is referred to as a “busted PAC.” A PAC also is “busted”once its lockout period has passed Not surprisingly, once “busted,” a PAC’svalue tends to cheapen

band-As perhaps the next logical step from a PAC, we have DUS, or delegatedunderwriting and servicing security In brief, a DUS carries significant pre-payment penalties, so borrowers do not have a great incentive to prepay theirloans Accordingly, a DUS can be thought of as having significant lockoutprotection

The formula for a PAC or DUS or a variety of other products createdfrom pass-thru might very well look like our last price formula, and it isrepeated below What would clearly differ, however, are the values we insertfor probability While large bond fund investors might perform a variety ofcomplex analyses to calibrate precise probability values across cash flows,other investors might simply observe whether respective yield levels appear

to be in line with one another That is, we would expect a 10-noncall-five

to trade at a yield below a 10-year DUS, a 10-year DUS to trade at a yieldbelow a 10-year PAC with a lockout of five years, and so forth

FIGURE 4.18 Applying the PAC discussion to the cash flow diagram.

Figure 4.19 summarizes the yield relationship to the different callablebond structures presented in this section Each successive layer represents adifferent and higher-yielding callable product.

For another perspective on the relationships among products, cashflows, and credit, consider Figure 4.20, which plots the price volatility of atriple-A-rated pass-thru against four 10-year final maturity bonds One of thebonds is a bullet, while the other three are different types of callables Each

Prepayment penalties are comparable with PACs, but there are no

bands to limit exposure to changes in prepayment activity, and these

uncertain changes contribute to the uncertainty in timing of both

coupon and principal payments.

Planned Amortization Class Security

Prepayment penalties are not as severe as with DUS, and although there are bands intended to limit exposure to changes

in prepayment activity, these changes are nonetheless uncertain and thus contribute to the uncertainty in timing of both coupon

and principal payments.

DUS

Although relatively severe penalties exist for early prepayments, there is uncertainty associated with the timing of both coupon and principal payments.

Callable Non-Treasury Coupon-Bearing Bond

After an initial lockout period, there is uncertainty

of timing of final coupon and principal.

Non-Treasury Coupon-Bearing Bond Credit risk

Coupon-Bearing Treasury Bond Market risk

Layers of increasing option-related risks (on top of credit risk and market risk)

FIGURE 4.19 Summary of the yield relationship to callable bond structures.

of the callables is a 10-noncall-2, but each has a different status with regard

to the relationship between F and K Namely, one has F  K, the other has

F much greater than K (deep in-the-money), and the last has F much less than

K (deep out-of-the-money) The price volatility of an at-the-money

10-non-call-2 is the same as that for a generic double-A-rated corporate security.Accordingly, with all the shortcomings and limitations that a mappingprocess represents, it would appear that such a process might be used to findconnectors between things like credit profiles and cash flow compositions.The particular relationship highlighted in the figure might be of special inter-est to an investor looking for an additional and creative way to identify valueacross various financial considerations inclusive of credit and structure types.Parenthetically, a financing market exists for MBS securities as well An

exchange of an MBS for a loan of cash is referred to as a dollar roll A

dol-lar roll works very much like the securities lending example described lier in this chapter, though obviously there are special accommodations forthe unique coupon and price risk inherent in an MBS as opposed to a genericTreasury Bond

ear-A preferred stock is a security that combines characteristics of bothbonds and equities (see Figure 4.21) Like bonds, a preferred stock usuallyhas a predetermined maturity date, pays regular dividends, and does notconvey voting rights Like an equity, a preferred stock ranks rather low in

BAA

AA A

AAA

Credit ratings Cash flow types

Price volatility

2 1 3

4

The intersection of the price volatility of

a triple-A rated noncall-2 and a double-A rated 10- year bullet bond.

10-1 Deep in-the-money

2 At-the-money

3 Deep out-of-the-money

4 10-year bullet bond

FIGURE 4.20 Mapping process.

priority in the event of a default, but typically it ranks above common stock.The hybrid nature of preferred stock is supported by the fact that whilesome investment banks and investors warehouse these securities in theirfixed income business, others manage them in their equity business.

One special type of preferred stock is known as a convertible As the

name suggests, the security can be converted from a preferred stock uct into something else at the choice of the investor The “something else”

prod-is usually shares of stock in the company that originally prod-issued the preferredstock A convertible typically is structured such that it is convertible at anytime, the conversion right is held by the investor in the convertible, and theconvertible sells at a premium to the underlying security Investors acceptthe premium since convertibles tend to pay coupons that are much higherthan the dividends of the underlying common stock

Generally speaking, as the underlying common stock of a convertibledeclines, the convertible will trade more like a bond than an equity That is,the price of the convertible will be more sensitive to changes in interest ratesthan to changes in the price of the underlying common stock However, asthe underlying stock price appreciates, the convertible will increasinglytrade much more in-line with the price behavior of the underlying equity thanwith changes in interest rates

Figure 4.22 shows a preferred stock’s potential evolution from more of

a bond product into more of an equity product

A convertible’s increasingly equitylike behavior is entirely consistent withthe way a standard equity option would trade That is, as the option tradesmore and more in-the-money, the more its price behavior moves into lock-step with the price behavior of the underlying equity’s forward or spot price.Parenthetically, an option that can be exercised at any time is called an

American option, while an option that can be exercised only at expiration is

called a European option In the case of a European option type structure, if

the underlying equity price is above the convertible-equity conversion price

as the convertible comes to maturity, then the conversion should be made;the option to receive equity in exchange for the convertible ought to be exer-cised But if the underlying equity price is below the applicable conversionprice, conversion should not be made; an investor is better off with takingthe redemption value of the convertible.

To add another twist to this scenario, convertibles also can be issuedwith callable features A callable feature entitles the issuer to force a givensecurity into an early redemption Thus, depending on its precise charac-teristics, the correct valuation of a convertible can be a complex undertak-ing

The cash flow triangle shows how the price behavior of an in-the-moneypreferred stock can be seen as more spot- or forward-like, as well as moreequity- or bond-like (see Figure 4.23)

The answer to the question of “What really is a convertible?” can verymuch depend on the particular time in the life of the convertible when thequestion is being posed An understanding and appreciation of the factorsdriving the convertible around the triangle (pun very much intended) willgreatly facilitate an investor’s assessment of relative value and opportunity.There are a few different ways to creatively influence the credit quality

of a bond as illustrated by Figures 4.24 and 4.25 Within the world of fixedincome, there are bonds with short call options embedded in them (callables)and bonds with long puts embedded in them (putables) Chapter 2 explainedthat a put option is generally thought of as providing downside price pro-tection; as price falls, the value of a put option rises Concomitantly, a creditcall option suggests there is downside protection against the risks typicallyassociated with a deteriorating credit story These risks might include price-related dynamics as the market adjusts itself to a less favorable credit envi-ronment Being long a credit call option will not prevent a credit ratingagency from placing a company on credit watch or downgrading a companyoutright, but being long a credit call option might help to ameliorate theadverse price consequences typically associated with negative credit events

Underlying stock price moves higher

Underlying stock price moves lower

More like an equity More like an bond

Gray Area Convertible

FIGURE 4.22 Transformation scenarios for a convertible bond.

As previously stated, a putable bond is composed of a bond and a longput option The put option is most commonly viewed as being a put option

on price; that is, if interest rates rise, causing price to fall, the put optionpresumably takes on value since it provides a support or floor level for prices

Forward Equity Spot bond

Spot Equity

• A convertible preferred security is a combination of a bond and an embedded long call option on an equity.

• A convertible that trades increasingly in-the-money (above its conversion value) and is immediately exercisable (American style) is increasingly likely to mirror the price behavior of the underlying equity’s spot price.

• A convertible that trades increasingly in-the-money and is not immediately exercisable (European style) is more likely to mirror the price behavior of the underlying equity’s relevant forward price.

• For convertibles that may embody more than one optionlike feature (as with a callable provision along the lines of the previous section), a more detailed evaluation of respective option contributions would be appropriate.

• A convertible that trades increasingly out-of-the-money (below its conversion value) is increasingly likely to mirror the price behavior of a debt instrument of the

underlying issuer (and as such be designated as a busted convert).

FIGURE 4.23 Cash flow triangle.

A putable bond differs from a callable bond in at least two fundamentalrespects.

1 With a putable bond, the embedded put is a long embedded put, andwith a callable bond, the embedded call is a short embedded call

2 As a direct consequence of number 1, the combination of being long abond and long a put option, as with a putable bond, results in a payoffprofile that much resembles a synthetic long call, while the combination

of being long a bond and short a call option, as with a callable bond,results in a payoff profile that much resembles a synthetic short put The combination of a long call and a short put results in a payoff profileresembling a simple long position in a forward The diagrams in Figure 4.26show these various relationships

All else being equal, except for being defensive on the market, put-callparity and efficient markets would suggest that we would be indifferentbetween the callable and the putable That is, being short an embedded call

or being long an embedded put are defensive or bearish strategies However,

if all else were not equal, and if credit risk were a particular matter of cern, then the put bond would take on a greater value relative to the callable.Since the putable bond has a payoff profile of a synthetic long call, down-side price risk is limited while upside price potential is unlimited If anadverse credit event were to occur, the putable bond still would provide pricesupport on the downside

con-The rationale for this downside support is simply that covenants forputable bonds (indeed, all covenants that this author is aware of) tend not

to make any stipulations about the price support features of the put ing segmenting market-related phenomena (as with changes in interest rates)versus any other phenomena (as with changes in credit risk) Accordingly,the put option embedded in a putable bond de facto provides a level of pricesupport for any event that might otherwise push the price of a bond lower.This contrasts with a callable bond, where with its synthetic short put pro-

regard-= Credit-enhanced bond Spot

Bond

Forward

Currency swap

FIGURE 4.25 Use of spot and forwards to create a credit-enhanced bond.

file, the price of the bond clearly does not receive any price support on thedownside and indeed has its price appreciation limited on the upside.

In sum, while callable and putable bonds may be viewed primarily asinterest rate risk bond products, they also can be viewed as being importantcredit products In the case of a putable, downside credit risk protection (aswith a downgrade) exists, and favorable credit-related appreciation (as with

an upgrade) is limited With a callable, favorable credit-related appreciation

is also limited, as is downside credit risk protection

A propitious choice of currency denomination also can have a favorablecredit impact for a financial product For example, it is no mere happen-stance that the so-called Brady bonds of the 1990s were explicitly intended

to assist Latin American countries with servicing their debt obligations, yetwere denominated in U.S dollars rather than pesos, or sucres, or colons, and

so forth Aside from any public relations benefit from having the bondsdenominated in dollars, U.S Treasury zero-coupon bonds and other high-grade instruments collateralize the principal and certain interest cash flows

of these bonds In sum, the involvement of the United States, including theinternational cachet of the U.S dollar, greatly enhanced the real and per-ceived credit benefits of Brady bonds

Short put option Long call option Synthetic long forward

FIGURE 4.26 Use of a long call and a short put to create a synthetic long forward.

Portfolio construction

Obviously, a portfolio is an amalgamation of products, cash flows, and creditrisks There are hundreds of thousands of portfolios and investment funds

in the world, typically managed with an orientation to a particular ment style For example, funds occasionally are described as being either rel-

invest-ative or absolute return oriented A relinvest-ative return fund, as the name

suggests, is a fund whose performance is evaluated relative to a benchmark

or index For example, a relative return equity fund might be evaluated ative to the S&P 500 equity index Accordingly, if a portfolio managerreturns 20 percent in a given year, this may or may not be an impressive feat

rel-If the S&P 500 returned 33 percent, then the portfolio manager’s mance would not be very impressive at all But if the S&P 500 returned 3percent, then a 20 percent portfolio return would be very impressive indeed.Generally speaking, larger institutional fund managers manage relativereturn funds

perfor-Conversely, an absolute return fund typically is managed without

ref-erence to a particular benchmark or index; the objective is not so much toprovide a return that is impressive relative to a market benchmark (thoughthat may be welcomed) as much as it is to provide an attractive return on

an absolute basis To achieve such a goal, it is expected that an absolutereturn fund would experience more return volatility relative to a relativereturn fund, but with larger longer-run aggregated returns in exchange forthe higher year-over-year risk being taken Generally speaking, smaller fund

managers manage absolute return funds, as with hedge funds (special funds

that are subject to special privileges and restrictions relative to more mon investment funds)

com-Because of their more aggressive objectives, absolute return funds erally are more likely to bias their investments toward relatively more riskyproduct, cash flow, and credit profiles in relation to relative return funds That

gen-is, absolute return funds are more likely to invest in equity than bonds, morelikely to invest in futures and options than spot, and more likely to dip intolower credit quality investments than higher credit quality securities

ABSOLUTE RETURN INVESTING

The following list of fund-types are all, broadly speaking, absolute oriented styles

return-Aggressive Growth

These funds typically invest in equities expected to experience acceleration

in growth of earnings per share, have generally high P/E ratios and low or

no dividends, and often are smaller-cap stocks This category also includessector specialist funds such as technology, banking, or biotechnology There

is a general bias toward being long the market

Distressed Securities

These funds buy equity, debt, or trade claims at deep discounts of nies in or facing bankruptcy or reorganization Profits are realized from themarket’s underappreciation of the true worth of these securities and frombargain prices precipitated by selling by institutional investors who cannotown below-investment-grade securities

compa-Emerging Markets

These funds invest in equity or debt of emerging (less mature) markets thattend to have high inflation and volatile growth

Funds of Hedge Funds

These funds invest in a mix of hedge funds and other pooled investment cles This blending of different funds aims to provide a more stable long-term investment return than any individual funds Capital preservation isoften an important consideration

vehi-Income

These funds have a primary focus on yield or current income rather than oncapital gains These funds may use leverage buying bonds and perhaps othertypes of fixed income derivatives

Macro

These funds seek to profit from changes in global economies, many timesbrought about by shifts in government policy that impact interest rates, cur-rencies, stocks, and bond markets; though the funds may not be invested inall of these markets at the same time Leverage and derivatives may be used

to maximize the impact of market moves

Market Neutral—Arbitrage

These funds attempt to hedge most market risk by taking offsetting tions, often in different securities of the same issuer The funds may be longconvertible bonds and short the underlying issuers equity, and may focus onobtaining returns with low or no correlation to both the equity and bondmarkets These relative value strategies include fixed income arbitrage,mortgage-backed securities, capital structure arbitrage, and closed-end fundarbitrage