25.2.1 Hedging with Credit Derivatives

Chapter 26 discusses simple credit derivatives. Although simple credit derivatives, such as credit default swaps (CDS), are not usually referred to as structured products, they often serve similar roles. Credit default swaps allow for the cost-effective transfer of default risk.

Consider an investor who holds the bonds of XYZ Corporation but wishes to hedge the risk that the bonds of XYZ might default. The investor enters a CDS on the debt of XYZ Corporation with a major bank. In effect, the bank sells credit protection to the investor that functions much like an insurance contract. The CDS transfers the financial risk of XYZ's default from the credit protection buyer (the investor) to the credit protection seller (the bank). Now the investor is hedged. If the XYZ bonds that the investor holds experience default, the investor is made whole through the CDS. Of course, the bank receives compensation from the investor for providing the credit protection. CDSs help organizations manage their credit risk.

25.2.2 Structuring with Tranches

CDOs are structures that partition the risk of a portfolio into ownership claims called tranches, which differ in seniority. More senior tranches tend to be the first to receive cash flows and the last to bear losses. The key point of a CDO, therefore, is to engineer the risk of a portfolio into a spectrum of risks tailored to meet the needs, preferences, or market views of various investors. The tranching of CDOs performs a function quite similar to the capital structure of an operating corporation.

For example, the sponsor of a highly simplified CDO structure might buy bonds of XYZ Corporation and place them into the portfolio of a CDO structure (typically along with other corporate bonds). The CDO structure has various tranches with claims to receiving the coupons and principal payments from those bonds. Investors can select a tranche that best meets their preferences for risk and return.

25.2.3 Creating Structured Products

The term structured products can be used as an umbrella term to describe a spectrum of innovative financial instruments, or it can be used more specifically to refer to specially tailored securities that are financially engineered to provide specific attributes, such as risk, that meet the preferences of one or more investors. An example of a structured product based on the equity of XYZ Corporation would be a security that paid an investor greater amounts of money if the value of XYZ equity performed poorly and lower amounts of money if XYZ performed well, but had a minimum value to the payout. The structured product might be ideal for an investor with a very large position in XYZ stock who is trying to avoid selling that position due to the potential tax liabilities from a sale. The investor desires downside protection while retaining some upside potential, so a major bank structures a product tailored to meet the investor's precise preferences with regard to size, timing, and payoff profiles.

25.3.1 Completing Markets as an Economic Role

One of the most central motivations to structured products is market completion. A complete market is a financial market in which enough different types of distinct securities exist to meet the needs and preferences of all participants.

For example, consider a world without any risk, uncertainty, taxes, or transaction costs. In such a world, the only difference between securities would be the timing of their cash flows. A complete market in this idealized example would exist when investors could assemble a portfolio that offered exactly the cash flows they desired on every possible date. Thus, a pension fund obligated to disperse cash on the first day of every month would be able to establish long and short positions in existing securities that generated cash on exactly those days that the cash was needed (i.e., the first day of every month).

In the United States, investors seeking riskless investments (in terms of U.S. dollars) tend to invest in U.S. Treasury bills, notes, and bonds. The market for Treasury securities contains many securities across a wide spectrum of maturity dates. But even ignoring the risk of a U.S. Treasury default, the Treasury market could not be described as being perfectly complete. The longest ordinary Treasury security is the 30-year Treasury bond, with an initial maturity of 30 years. What should an investor such as a pension fund do with liabilities requiring cash flows in perhaps 40 or 50 years? And there was a four-year period (2002 to 2006) when even the 30-year Treasury bond was no longer being issued. The U.S. Treasury began issuing the 30-year bond again based in part on the very function being discussed here—the benefits of completing a market by creating investment products that meet the needs of investors (in this case, mostly financial institutions with long-term time horizons).

In the idealized world of a complete market, individual investors could manage their wealth optimally because sufficient distinct securities would exist to allow any desired investment exposure. It should be noted that the financial market will never be fully completed. The term completing the market simply means that the market is being brought one step closer to completion by offering investors unique opportunities with which to manage their finances.

25.3.2 States of the World within Structured Products

In the real world of uncertainty and asymmetric information, markets are highly incomplete. Incomplete markets are understood in the context of “states of the world.” A state of the world, or state of nature (or state), is a precisely defined and comprehensive description of an outcome of the economy that specifies the realized values of all economically important variables. For example, a particular state of the world might be briefly summarized as being when an equity market index closes at $X, a bond market index closes at $Y, the gross domestic product (GDP) of a particular nation reaches $Z, and so on. The concept is theoretical since it is impossible to fully describe the entire world or all states that could occur. However, the concept provides valuable insight into why many structured products exist.

To demonstrate, let's examine a highly simplified example in which an investor defines the states of the world on only three outcomes: her job, the level of the equity market, and the level of the debt market. One of the many states in this example might be an outcome in which global stock markets rise, interest rates fall, but the investor gets fired from her job. How can this investor prepare for this potential state of the world? One answer would be to purchase unemployment insurance—although it might be very expensive or impossible to get large amounts of insurance against the economic consequences of being fired. The reason, of course, is that the insurance company would be concerned about moral hazard: the possibility that the insured would intentionally perform poorly at work in order to collect insurance. The point is that markets will always be in a condition of having substantial and important incompleteness.

25.3.3 Structured Products as Market Completers

Although markets can never be complete, the primary role of structured products is to move them toward being more complete. For example, most investors would define the states of the world as including the condition of their physical properties. How can investors prepare for the potential that fire might destroy their real estate? The answer, of course, is to purchase fire insurance. Centuries ago, in a world with very incomplete markets, investors might not have been able to purchase fire insurance and so would have had to bear the very undesirable and highly diversifiable risk of losing substantial wealth due to fire. But in a complete market, investors could purchase fire insurance, a “security” that pays a substantial payoff in states in which the real estate burns and pays nothing in other states. This example illustrates that the primary economic role of insurance companies is to make the market more complete.

To summarize, people and organizations can be viewed as analyzing future scenarios of the world (i.e., states of the world) and estimating their probabilities. For risk management purposes, investors typically seek products that offer high payoffs in those states in which the investor's wealth would otherwise be low. For return enhancement purposes, investors might seek products that offer high payouts in states that the investor believes are unusually likely to occur. In both cases, the structuring of products serves the economic role of meeting the needs and preferences of these investors by completing the market. In other words, the structured products offer an otherwise unavailable combination of payoffs in various states that enables the investor to better manage risk and return.

In the context of alternative investments, financial institutions strive to meet the preferences of various investors by creating securities or products that move the market toward being more complete. As shown, insurance companies are an excellent example of a type of financial institution that addresses the deficiencies of incomplete markets. Major banks, insurance companies, and other financial institutions offer structured products that are tailored to the needs of individuals and institutions for risk management or risk enhancement purposes.

It should be noted that many simple financial derivatives, such as call options and put options, trade in the financial markets and can be used by market participants to manage basic risks of traditional assets, such as indices and individual securities. But when a market participant desires a product that is peculiar to individualized circumstances or preferences, structured products may be the solution that can be engineered to tailor a solution.

25.4.1 Prioritization of Claims within CMOs

The key distinguishing feature between CMOs and other investment pools, such as mutual funds or the mortgage-backed securities discussed in Chapter 14, is that CMOs use extensive structuring. Specifically, CMOs are financed with security classes or tranches that have substantially varied characteristics. A tranche is a distinct claim on assets that differs substantially from other claims in such aspects as seniority, risk, and maturity. Each tranche is typically tradable in units that may differ in size.

A CMO issuer structures these tranches to have different seniorities to the cash flows from the underlying mortgages. Exhibit 25.2 illustrates a stylized CMO structure for insured mortgages with only three tranches. In practice, CMOs usually have numerous tranches.

The assets on the left side of Exhibit 25.2 are often referred to as the collateral pool. The assets generate the cash inflows that are structured and distributed to the various tranches. In the case of insured residential mortgages, the structuring of the cash flows focuses on maturity and cash flow timing, because lenders bear little or no risk of principal losses due to mortgage defaults. In the case of commercial mortgages and subprime residential mortgages, the focus of the structuring of the cash flows from the mortgage pool is more on the allocation of default losses, since these loans are generally not insured.

The issuer of the CMO receives the monthly mortgage payments (principal and interest payments) from the collateral pool, and after collecting its fees, the issuer passes the payments on to the various tranches, following the procedures and priorities defined in the CMO prospectus. Each tranche has a coupon that it is promised and a prespecified priority in receiving distributions of principal payments.

25.4.2 Structuring of Sequential-Pay CMOs

The sequential-pay collateralized mortgage obligation is the simplest form of CMO. In a sequential-pay CMO, each tranche receives a prespecified share of the interest payments based on each tranche's coupon and principal amount. Each tranche also potentially receives principal. When there is no default risk, it is the seniority to principal payments that is the focus of CMOs.

In the case of a sequential-pay CMO, the first-pay tranche (labeled as the senior tranche in Exhibit 25.2) receives all principal repayments until the tranche's face value has been fully repaid. As a tranche's principal is paid down, its receipt of coupon payments is proportionately reduced. A tranche matures once it has received repayment of its entire principal value. The next senior tranche then receives the entire principal payments until it, in turn, matures. There is a final tranche, typically called the Z-tranche, that receives any residual cash flows.

The purpose to the structuring offered by a CMO is that it provides investors with a spectrum of risk and return opportunities. For example, an investor seeking short-term, low-risk securities may purchase a highly senior tranche, while a longer-term investor might seek a tranche with a longer maturity, higher yield, and greater uncertainty of cash flow timing.

The structuring of the cash flows from the underlying mortgage collateral pools divides the prepayment risks (and, in other cases, default risks) of the pool into tranches that have low risk and tranches that have high risk. The higher-risk tranches can have extreme sensitivity to unexpected changes in prepayment rates (and, in some cases, default rates). Accordingly, the analysis and modeling of prepayment risks and default risks becomes even more crucial in the case of highly structured products.

In the case of insured residential mortgages, the exposure of each tranche to prepayment risk depends on the seniority of that tranche. The most senior tranches are virtually certain to mature quickly, regardless of prepayment rates. The tranches with the lowest seniority for receiving principal payments can have maturities that are extremely sensitive to prepayment rates. If interest rates increase, then prepayments by homeowners are likely to fall.

25.4.3 Longevity Characteristics of CMO Tranches

Fluctuations in interest rates and other factors that drive mortgage prepayments cause a phenomenon known as extension risk. Extension risk is dispersion in economic outcomes caused by uncertainty in the longevity—especially increased longevity—of cash flow streams. For example, when interest rates rise, prepayment rates usually fall, and the life of most tranches, especially the more junior tranches, is extended, thereby increasing or extending the expected life of the tranche further than originally expected. In most CMO tranches, extension lowers the value. This reflects the general tendency of fixed-income instruments to fall in value when interest rates rise. However, some tranches can benefit from extension. These types of tranches might fall in value due to contraction when anticipated longevity declines. Contraction risk is dispersion in economic outcomes caused by uncertainty in the longevity—especially decreased longevity—of cash flow streams.

Consider a CMO with an underlying collateral pool of mortgages that generates $1,620,000 of cash flow in its first month (after fees): $1,500,000 in interest fees (9% annualized) and $120,000 in principal repayments. A stylized sequential-pay two-tranche CMO structure is presented in Exhibit 25.3. Payments are made first to Tranche A and then to Tranche B.

Exhibit 25.3 illustrates that, in month 1, both Tranche A and Tranche B receive their corresponding interest payments of $1,062,500 ($150,000,000 × 8.5%/12) for Tranche A and $437,500 ($50,000,000 × 10.5%/12) for Tranche B, for a total of $1,500,000 in interest payments. In this simplified example, the interest payments received equal the interest payments owed to the two tranches, and there is no residual tranche. The remaining cash flow of $120,000 is a principal repayment received from the underlying collateral, and it is used only to pay principal to Tranche A. The reason is that this is a sequential-pay two-tranche CMO, in which principal payments are made to Tranche A until the principal of Tranche A has been fully paid off, after which payments are made to Tranche B. Therefore, at the end of month 1, the principal balance for Tranche A is reduced to $149,880,000 ($150,000,000 – $120,000), and the principal balance of Tranche B remains at the initial $50,000,000, since Tranche B received no principal payment.

The mechanics of the payments in the following months will be similar. In the case of Tranche A, however, the interest payments due in the second month will decline because the total principal is decreasing with each principal repayment. For example, in the second month, Tranche A would be entitled to interest payments of only $1,061,650 ($149,880,000 principal at 8.5%/12 interest).

The principal for Tranche B will start to be paid off only after the principal for Tranche A has been fully paid. If the prepayment rates of the mortgages underlying the CMO increase, Tranche A would be paid off faster and Tranche B would start to be amortized earlier. Conversely, if unscheduled principal repayments slow, the anticipated longevity of Tranche A will extend, but perhaps only modestly, as scheduled principal payments would presumably continue (i.e., extension risk would likely be minimal). However, the anticipated term of Tranche B could extend substantially. In this simplified example, Tranche B would continue until the last underlying mortgage made its final payment, and therefore, depending on prevailing rates and anticipated reinvestment opportunities, would likely be positively exposed to extension risk.

In actual CMO structures, there is typically an accrual tranche, or Z-bond, that receives no promised interest or coupon payments. Rather, the tranche serves as a residual, equity-like claimant, with rights to cash flows that remain after all fixed-income tranches have been satisfied.

25.4.4 Other CMO Structures and Tranches

Numerous variations can be structured within a CMO issue other than the sequential-pay structure introduced in the previous section. This section highlights an important aspect of structuring: There is often an evolution that occurs in structured products wherein relatively simply structured products, if successful, evolve into increasingly complex and sophisticated products.

Here are several of the more popular types of CMOs.

PLANNED AMORTIZATION CLASS TRANCHES: Planned amortization class (PAC) tranches receive principal payments in a more complex manner than do sequential-pay CMOs. Investors in some PAC tranches have high priority for receiving principal payments as long as the prepayment rates are within a prespecified range (the planned prepayment levels). When prepayments diverge from what was originally projected, the relative priorities of tranches can shift. In a sequential-pay structure, the relationship between tranche longevity and prepayment rates is somewhat linear, meaning that each tranche's longevity to changes in prepayment speeds is somewhat stable at various levels of prepayment. But with PAC tranches, it is possible that a tranche will contract in longevity as prepayment rates accelerate to a certain point but then extend in longevity beyond that point. In other words, a tranche might have high priority to receiving principal payments in one range of prepayment speed and low priority if other prepayment speeds occur. Thus, PAC tranches can be riskier and more complex to analyze.

TARGETED AMORTIZATION CLASS TRANCHES: Targeted amortization class (TAC) tranches receive principal payments in a manner similar to PAC tranches but generally with an even narrower and more complex set of ranges. The amortization procedures tend to identify narrower ranges of prepayment speeds within which tranches have particular priorities for receiving principal payments and interest. These prepayment ranges can be viewed more as targeted outcomes than as planned outcomes. TAC tranches can be especially complex and risky. A sensitive TAC tranche can quickly switch from being quickly paid off to receiving no principal payments (and vice versa), even when prepayment speeds change by only a small amount.

PRINCIPAL-ONLY TRANCHES AND INTEREST-ONLY TRANCHES: Principal-only (PO) tranches receive only principal payments from the collateral pool, whereas interest-only (IO) tranches receive only interest payments from the collateral pool. Both tranches are therefore created by dividing cash flows from the mortgage collateral into the portion that is principal repayment and the portion that is interest. The principal repayment cash flows are distributed to one bond, the PO, and the interest cash flows are distributed to a second bond, the IO.

Investors in PO bonds are ultimately paid the face value of their bonds as borrowers eventually make the principal payments on their mortgages. The logic behind a PO is that investors buy these bonds at a discount from face value and eventually receive the face value through the scheduled principal repayments and prepayments received from the mortgages. PO tranches are positively exposed to extension risk in that their values decline when prepayments slow, since they receive no coupons.

An IO bond has a notional principal used to compute each interest payment. The cash flows received by investors in IOs decline as the principal is paid down. IO tranches are positively exposed to contraction risk in that their values decline when prepayments accelerate, since their payments are only interest because notional principal is not repaid.

Prepayment sensitivity tends to be severe for POs and IOs, with one generally profiting when the other suffers. For example, in the case of POs on fixed-rate mortgages, when interest rates decline, the speed of prepayments typically accelerates. This contraction in longevity reduces the life of both the IO and the PO. PO tranches benefit from quicker receipt of their only cash flows: principal repayments in the fixed amount of the PO's face value. IO tranches suffer from principal reductions, since their only cash flows (interest payments) are proportionately reduced. On the other hand, when interest rates increase, the speed of prepayments declines, and the PO investor is paid the face value further in the future, lowering its effective return, while the IO receives a longer annuity of interest payments. Both tranches can be issued with adjustable- and fixed-rate underlying mortgages.

FLOATING-RATE TRANCHES: Floating-rate tranches earn interest rates that are linked to an interest rate index, such as the London Interbank Offered Rate (LIBOR), and are usually used to finance collateral pools of adjustable-rate mortgages. A collateral pool of adjustable-rate mortgages provides a stream of variable interest rate payments that can flow through to floating-rate tranches, which also have floating coupons. Floating-rate tranches can be structured to have rates that move more than the underlying index (e.g., twice the floating rate) or even in the opposite direction, which is known as an inverse floater tranche. An inverse floater tranche offers a coupon that increases when interest rates fall and decreases when interest rates rise. Floating-rate tranches can have specified upper and lower limits to their adjustable coupons.

25.4.5 Motivations of Structured Mortgage Products

The primary motivations driving the demand for CMOs in the case of insured mortgages are summarized in Exhibit 25.4. Mortgages offer up to 30 years of coupons and principal payments, with high uncertainty regarding the level of unscheduled prepayments. Some investors prefer to take slices from this maturity range rather than invest in the entire range. Tranches permit investors to select securities that match their preferred exposures in terms of longevity and sensitivity to unscheduled prepayments.

As depicted in Exhibit 25.4, these preferences are driven by two primary motivations: risk and return. Investors can lower their risk by selecting tranches with durations that match the duration of their liability stream. Cash flow matching is one of many risk-reducing strategies that can be facilitated by the structuring of claims.

Some investors have a market view of future interest rates or prepayment speeds. An investor can select tranches of CMOs that offer enhanced returns to the extent that the investor's market view is superior.

In the cases of both motivations in Exhibit 25.4, the overall role being served by the CMOs is completion of the market. CMOs create numerous otherwise unavailable investment opportunities (i.e., tranches) from a pool of previously existing collateral. When the structuring is formulated in response to market demand, the enhanced set of opportunities facilitates improved portfolios from the perspective of the market participants.

25.4.6 Valuing Default-Free CMOs

Chapter 14 discusses unscheduled mortgage principal payments (i.e., prepayments). Prepayment decisions are made by property owners based on idiosyncratic events to the homeowners (e.g., job-related relocations) and macroeconomic factors, such as interest rates and housing prices. Chapter 14 discusses the role of these unscheduled prepayment rates in driving the risks, returns, and values of residential mortgage pools.

The effect of prepayment speeds on the valuation of CMO tranches can be even more critical than the effects on the overall mortgage pools. The reason is that structuring creates tranches with varying risks. Suppose that overall mortgage values drop by 1% due to increased interest rates. Whereas collateral pools will tend to drop by 1%, the losses to various tranches will vary based on the sensitivity of each tranche to interest rates. Long-term, highly sensitive tranches might drop by 5% or more, while very short-term tranches may be virtually unaffected. Some tranches, such as IO tranches, might even gain in value.

TAC, IO, and PO tranches can be especially sensitive to interest rates and prepayment speeds. Valuation of tranches requires careful and sophisticated analysis using advanced models of interest rates and prepayment speeds. The complexity of many CMOs creates both opportunities and threats. The sophistication of the models used to evaluate tranches creates the potential for analysts with superior skills to locate tranches of CMOs that are mispriced. However, the complexity of the products and models also carries the danger that analysts with inferior skills will be induced into consistently making trading decisions that generate negative net present values. Highly complex tranches with innovative characteristics can appear to be attractively priced when they are actually overpriced; thus, the importance of due diligence cannot be overstated.

25.4.7 Systemic Risk and the History of Structured Mortgage Products

There was a U.S. financial crisis involving CMOs on insured residential mortgages in 1994. Interest rates rose dramatically, causing most CMO tranches to extend in maturity as prepayment rates fell. The combination of extended maturities and higher interest rates caused market values of most tranches to fall, some quite severely. As investors and institutions suffered large losses, market liquidity eroded, and CMO tranches began to trade at prices reflecting even more conservative prepayment rate forecasts, further exacerbating the crisis.

Perhaps the worst case involved inverse-floating TAC tranches. Some of these tranches offered high coupons and were expected to mature within months at the end of 1993, due to high prepayment rates in the underlying mortgage pool and low interest rates. Therefore, these high-coupon and presumably short-term tranches traded at premium prices to their principal values and appeared to have little risk exposure to small or moderate interest rate changes. However, as part of a TAC structure, the tranches could experience dramatic shifts in seniority to principal payments if prepayment rates deviated from target ranges. In early 1994, prepayment speeds dropped such that the anticipated maturities of some of the TAC tranches extended from several months to many years, and switched from being the most senior to being the least senior tranches. Further, in the case of inverse floaters, the coupons on the tranches fell from high coupons to zero coupons as interest rates such as LIBOR skyrocketed.

The result was that some of the tranches, previously viewed by some market participants as having very low risk, fell from trading at premiums to trading at as little as 20% of face value by the summer of 1994. This incredible drop in value occurred on tranche securities even though there was never a doubt that the principal value of the tranche would ultimately be recovered, since the underlying mortgage pool was insured by U.S. government agencies. Many institutions suffered huge losses, some firms collapsed, and the crisis widened until interest rates reversed their climb in the fall of 1994.

The prevalence and power of the structuring of mortgage products is often cited as causing or exacerbating both the financial crisis of 1994 and the financial crisis that began in 2007. In the most recent financial crisis, the financial losses in mortgage-backed structured products centered on default risk. In both cases, structured products contributed to increased systemic risk, substantially harming or even bankrupting major financial institutions and increasing uncertainty throughout financial markets.

Despite the past problems with structured mortgage products, the power of structured products has generated tremendous benefits. The long maturity and substantial prepayment risk of insured residential mortgages make unstructured ownership of mortgages undesirable to most market participants. Mortgages offer cash flows that range in maturity from 1 month to 360 months and offer uncertainty as to the size of each cash flow due to the prepayment options held by the borrowers. Relatively few market participants find mortgages to be attractive direct investments because of their range of scheduled cash flows and their exposure to prepayment rates and interest rates.

However, structured mortgage products allow market participants to select longevities and risk exposures that more closely align with their preferences. Thus, shorter-term fixed-income money managers can purchase short-term senior tranches, and longer-term managers, such as pension funds, can focus on longer-term tranches of insured mortgages. The emergence of structured mortgage products in the past several decades coincides with substantially reduced mortgage rate spreads, suggesting that structured products have enabled hundreds of millions of homeowners to enjoy substantially lower financing costs.

25.4.8 Commercial CMOs and Default Risk

The previous sections discussed prepayment risk and extension risk. For CMOs with underlying portfolios of commercial mortgages or subprime residential mortgages, the primary risk is usually default risk. In CMO structures with substantial default risk, the primary result of the structuring of the cash flows is to vary the level of exposure of each tranche to default risk. The most senior tranches have the first right to scheduled and unscheduled principal payments and are last to bear losses from defaults. Conversely, the more junior tranches are highly subject to default losses from the underlying mortgage portfolio. The exposure of most CMO tranches to default risk is indicated by credit ratings assigned to each tranche.

As would be expected, credit ratings tend to differ quite considerably between the different tranches of a commercial mortgage-backed security (CMBS) because each tranche has different risk profiles, maturities, and subordination. Due to their subordination, more junior tranches have lower credit ratings. Conversely, senior tranches are often rated AAA because they have a high-priority claim on the cash flows and enjoy the extra security embedded by having initial default losses absorbed by the junior tranches.

The most junior tranches, often referred to as first-loss tranches, are often rated at non-investment-grade levels. This dispersion in credit risk exposure and credit ratings has the advantage of broadening the pool of appropriate investors. The senior, investment-grade-rated tranches are generally viewed as fixed-income securities, since they have limited expected exposure to default risk and are therefore primarily analyzed in the context of interest rate risk. In contrast, the most junior tranches are generally viewed and analyzed as risky securities substantially influenced by the risks of the underlying real estate rather than being influenced primarily by interest rate risks. Even a single large default can have a considerable impact on the performance of these junior securities. Therefore, in the case of CMBSs, junior tranches generally have higher coupons than senior tranches in the same structure. Particular attention is placed on the credit quality and other risk characteristics of the underlying mortgage pool, which is the collateral for the structure.

Default-risk CMO models focus on the expected rates of default, the correlation between defaults, and the losses on each defaulted issue. The idea is to forecast the probabilities of various cash flow streams from the underlying mortgage pool and to project the likelihood of payoffs to each of the tranches.

25.5.1 The Intuition of Merton's Structural Model

Robert Merton pioneered the understanding of the option-like aspects of capital structure.¹ The key to Merton's approach is to recognize the option-like characteristics of structured cash flows, especially the option-like characteristics of credit risk that are inherent in the simplified capital structure of a traditional operating firm.

For simplicity, assume that a levered operating firm has only two securities: a single issue of zero-coupon debt and a single class of equity. Perhaps the most intuitive way of seeing the option-like nature of traditional corporate securities is based on call options. The call option view of capital structure views the equity of a levered firm as a call option on the assets of the firm. The call option implicit in equity has a strike price equal to the face value of the debt and an expiration date equal to the maturity date of the debt. If the firm does well, the firm pays its debt holders fully when the debt matures, and the assets of the firm belong to the shareholders. If the firm does poorly, the shareholders can declare bankruptcy and walk away from the firm, leaving the assets to the debt holders. The situation is highly analogous to a traditional call option, in which the owner of the call either pays the strike price of the option to claim the underlying asset or lets the option expire worthless. Equity holders are like the owner of a call option who enjoys unlimited upside potential from gains in the underlying asset but has limited loss exposure to declines in the underlying asset, since the option can be allowed to expire. This situation is depicted in Equation 25.1:

(25.1)

The view of the equity of a corporation as a call option also leads to an option-based view of the corporation's debt. Specifically, if the value of the assets of the firm equals the sum of the liabilities plus equity, and if equity is a call option, then owning debt is equivalent to owning the assets and writing a call option. In other words, owning debt is equivalent to owning a covered call, meaning being long assets and short a call option on those assets.

An analogous application of options theory can be performed using put options rather than call options. Note that due to put-call parity (see Chapter 6), a call option can be viewed as a long position in a put option and the underlying assets financed with a riskless bond. By inserting these positions in place of the call options from the call option view of capital structure, the relationship is changed to the put option view of capital structure. The put option view of capital structure views the equity holders of a levered firm as owning the firm's assets through riskless financing and having a put option to deliver those assets to the debt holders. As depicted in Equation 25.2, the risky debt of a levered firm can be viewed as being equivalent to owning a riskless bond and writing a put option that allows the stockholders to put the assets of the firm to the debt holders without further liability (i.e., in exchange for the debt).

(25.2)

In Equation 25.2, the put option reflects the ability of equity owners to declare bankruptcy and enjoy limited liability. If the assets fall sufficiently, the debt holders suffer losses because they must pay a strike price to the stockholders that equals the face value of the riskless bond. In default, debt holders receive only the depleted value of the underlying assets rather than the face value of their debt, a risk that is captured by the short put position that debt holders have in the put option view of capital structure (part of which is shown in Equation 25.2).

Within either the call option view or the put option view of the levered firm, the value of the securities of a firm can be viewed in terms of the values of the underlying assets and the options on those assets. Accordingly, arbitrage-free option pricing models such as the Black-Scholes option pricing model (discussed in Chapter 6) may be used to analyze credit instruments. The analyst implementing the structural approach examines market prices to find reasonable values of the model's parameters, such as asset volatility and interest rate levels, and inserts those parameters into the structural model to generate prices for assets with credit risk.

25.5.2 The Conflict of Interest Regarding Risk in Structuring

There is an inherent conflict between the stockholders and the bondholders with regard to the optimal level of risk for a firm's assets. The equity holders, with their long position in a call option, prefer higher levels of risk, especially when the value of the firm's assets is near or below the face value of the debt. This is because the value of the equity at the maturity of the debt is the maximum of zero and the difference between the value of the firm's assets and the face value of the debt. As long as there is time before the debt matures and volatility in the value of the underlying assets, the implicit call option of the equity has time value. Importantly, the time value of the equity as a call option monotonically increases with higher asset volatilities (everything else being equal). Especially when the credit risk of the debt is high, equity holders may have a strong incentive to encourage managers to invest in risky projects, because if the projects fail, the bondholders are the losers, whereas the shareholders gain more when the projects succeed. Conversely, bondholders prefer safer projects and reduced asset volatility, as seen through their short position in a put option. The conflict of interest may be viewed as a zero-sum game in which managers can transfer wealth from bondholders to stockholders by increasing the risk of the firm's assets (or vice versa).

The conflict of interest between stockholders and bondholders in the capital structure of a firm is similar to the case of structured products with multiple tranches. The manager of the collateral pool can cause wealth transfers between tranches by altering the risk of the assets. In most structures, high levels of asset risk benefit junior tranche holders at the expense of senior tranche holders.

25.5.3 The Mechanics of Merton's Structural Model

This section takes a more precise look at Merton's application of option theory to credit risk. Throughout this discussion, it is assumed that the firm has a simple capital structure consisting of a single issue of debt in the form of a zero-coupon bond and a single issue of equity. The structural model view of the firm's capital structure expresses the firm's debt and equity in terms of a hypothetical call option and put option on the firm's assets, with a strike price equal to the face value of the zero-coupon bond and an expiration date equal to the maturity of the bond.

Inserting the call option view of the equity of a firm and the put option view of the debt into the fundamental relationship that the value of the firm equals the sum of the value of the equity and the debt produces the relationship in Equation 25.3:

(25.3)

The term in the first bracket on the right-hand side of Equation 25.3 represents the equity, and the terms in the second bracket represent the firm's risky debt. Note that the value of the risky debt is equal to the value of an otherwise identical riskless bond reduced by the value of the put. The reduction in the value of the debt by the value of the put option is the market's price for bearing the credit risk of the firm.

Equation 25.3 illustrates the conflict of interest between stockholders and bondholders. Consider a change in the anticipated volatility of the firm's assets that leaves the current value of the firm's assets unchanged. Perhaps the firm embarked on a risky venture with a net present value of zero. Equation 25.3 indicates that the equity is a long position in a call option on the underlying assets of a levered firm. Thus, the value of the equity, like any call option, will rise when the volatility of the underlying assets increases (everything else being equal). Equation 25.3 indicates that for every dollar that the equity increases in value, the firm's risky debt must fall in value by $1. The decline in the value of the firm's debt is captured in Equation 25.3 as an increase in the value of the put option. Equity's increase when volatility increases is due to its long vega exposure, while the decline in the value of the debt is due to its short vega exposure.

25.5.4 Valuing Risky Debt with Black-Scholes Option Pricing

The Black-Scholes option pricing model can be used along with Equation 25.3 to derive estimates of the value of debt that contains credit risk. In other words, fixed-income analysts can value risky debt using option pricing models. For example, a credit analyst wishes to value the risk of Firm XYZ's only issue of debt. The analyst follows a four-step process, which involves estimating the volatility of the firm's assets and using the estimated volatility to price the debt:

1. Estimating the volatility of Firm XYZ's equity: This estimate may be derived through analysis of XYZ's historical stock volatility, through the implied volatility of options on XYZ's stock, or through a combination of the two approaches.
2. Unlevering XYZ's estimated equity volatility (from step 1) based on XYZ's capital structure: XYZ's estimated asset volatility, σ_assets, can be approximated as XYZ's estimated equity volatility, σ_equity, times the ratio of the value of XYZ's equity to the value of the firm's assets, as illustrated in Equation 25.4) assuming that the debt is riskless for simplicity).
   (25.4)
3. Solving for the price of a call and put on the firm's assets: The estimated asset volatility can be inserted into the Black-Scholes option pricing model along with observable parameters to generate call and put prices.
4. Using the call price as the value of XYZ's stock, and subtracting the put price from the price of a riskless bond to value XYZ's debt.

Note that the accuracy of estimated option values may be reduced to the extent that the assumptions of the model are violated. Three assumptions are particularly troublesome: (1) that the percentage changes in the values of the firm's underlying assets through time are lognormally distributed, (2) that the volatility of the firm's assets can be accurately estimated, and (3) that there is a single issue of debt with no coupon. Nevertheless, option pricing models can be especially useful in providing normative guidance of relative yields within the same firm or between similar firms.

25.5.5 Binomial Trees and Structured Product Valuation

The application of the structural model is not limited to use of the Black-Scholes option pricing model. Chapters 10 and 15 discussed the application of binomial option pricing to real options—that is, options regarding real assets. As introduced in Chapter 6, binomial tree models are extremely flexible and valuable tools for analyzing assets with embedded options. In the case of credit instruments, binomial tree models allow analysts to estimate prices based on volatilities and observable parameters using the principles of risk-neutral pricing.

For example, the value of credit-risky securities in a capital structure or a structured product can often be well estimated using two underlying binomial trees: one for the value of the assets, and one for interest rates. The analyst simply estimates future cash flows contingent on the asset values and then prices the securities through backward induction. Whereas the Black-Scholes option pricing model is often used in simple option analysis, it is the binomial tree approach that serves as the primary valuation tool in the case of most structured products with complex optionalities.

25.5.6 Advantages and Disadvantages of Structural Model Applications

Merton's structural model and its extensions have two major potential advantages:

1. The structural approach tends to rely on data from equity markets, such as observed stock price volatilities or implied stock price volatilities backed out of option prices. Since equity markets are generally more liquid and transparent than corporate bond markets, some argue that equity markets provide more reliable information than credit markets provide.
2. Structural models are well suited for handling different securities of the same issuer, including bonds of various seniorities and convertible bonds. The different securities or tranches rely on the same assets with the same asset parameters.

The structural model has three major disadvantages as well:

1. If equity prices are highly unreliable, then estimates of asset volatility and values are also highly unreliable. For example, private equity or real estate equity valuations may be unreliable sources to the extent that the valuations are not based on liquid markets.
2. Current data on a firm's or structure's liabilities may be unreliable and, in the case of sovereign issuers, may be unworkable.
3. The valuations generated by simple structural models are sometimes unreasonable, especially for short-term, very high-quality debt and for debt that is very near default.

25.6.1 A Stylized CDO

Exhibit 25.5 illustrates the concept of a CDO that is being used to structure the cash flows and default risk from a portfolio of high-yield bonds. There are $100 million of high-yield bonds on the left-hand side of Exhibit 25.5 that serve as the assets or collateral portfolio (or pool) for the structure. These bonds generate cash flows in the form of coupon payments and principal payments. The bond portfolio can also generate losses from events such as defaults in the bonds and from profits or losses from trading activity. On the right-hand side of the structure are the various classes of securities (or tranches) that provided the financing for the portfolio and that have claims of varying seniorities to receive cash inflows.

The cash flows from the collateral pool of assets are distributed using a waterfall approach somewhat analogous to that discussed in Chapter 3 for limited partnerships. Without any defaults, the assets should generate $7 million in coupon income, found as the product of the asset size ($100 million) and the average coupon of the assets (7%). The first priority of the cash flows is to meet direct expenses and fees for the operation and management of the CDO. For simplicity, this example ignores the expenses and fees.

After expenses and fees, the cash flows are distributed to the various tranches in order of their seniority. The senior tranche is a tranche with the first or highest priority to cash flows in the structured product. In this case, the senior tranche is owed the first $2.1 million per year in coupon income, found as the product of its size ($70 million) and the coupon rate (3%). A mezzanine tranche is a tranche with a moderate priority to cash flows in the structured product and with lower priority than the senior tranche. In this case, the mezzanine tranche is owed $1.2 million per year after the senior tranche has been paid. The equity tranche has lowest priority and serves as the residual claimant. In this case, the equity tranche has claim to the remaining $3.7 million.

The previous numerical example ignored defaults in the bond portfolio. However, this CDO contains bonds subject to substantial credit risk, since they are below investment grade (i.e., are rated BB or lower), and therefore it is reasonable to expect defaults. When collateral assets default, the order of the waterfall reverses relative to the order for receiving cash flows, such that the lowest seniority tranches experience the losses first. The losses from asset defaults are posted against the lowest remaining tranche until that tranche is wiped out, at which point the losses are posted against the next lowest seniority tranche. For example, if $11 million of assets completely defaulted (i.e., experienced no recovery), the equity tranche would be wiped out, and the notional principal of the mezzanine tranche would be reduced from $20 million to $19 million. If $11 million of assets defaulted with 60% recovery, the assets would drop by only $4.4 million, causing loss only to the equity tranche.

Note that defaults in the CDO's collateral portfolio reduce both the left and right sides of Exhibit 25.5. The assets are reduced in size and in annual income. The aggregated tranches are reduced in size by reducing the tranches, starting with the lowest-seniority tranche. As the debt tranches are reduced in notional principal, their claim to coupon income is reduced. Thus, if the mezzanine tranche in the example is reduced from $20 million to $19 million, the annual coupons owed to the mezzanine tranche holders would fall from $1.20 million to $1.14 million.

25.6.2 Attachment Points, Detachment Points, Calls, and Puts

Note that the 20% of the structure in Exhibit 25.5 that is represented by the mezzanine debt tranche lies between the 70% financed by senior debt and the 10% financed by the most junior tranche (the equity tranche). As losses to the collateral pool are experienced due to defaults in the portfolio of bonds, the first 10% of the losses are applied against the equity tranche, and the last 70% are applied against the senior tranche. Thus, the losses to the mezzanine tranche begin when 10% of the collateral assets have been lost to default and end when 30% of the collateral assets have been lost to default and the mezzanine tranche is eliminated.

The first percentage loss in the collateral pool that begins to cause reduction in a tranche is known as the lower attachment point, or simply the attachment point. The higher percentage loss point at which the given tranche is completely wiped out is known as the upper attachment point, or the detachment point. Thus, the mezzanine tranche in the simplified example has a lower attachment point of 10% and an upper attachment point of 30%. Each tranche is often identified using these points, such that the mezzanine tranche in the example might be described as being a 10%/30% mezzanine tranche.

The risks and payoffs to the most senior and most junior tranches in a CDO can be viewed using positions in either a call or a put. Similarly, the structural credit risk model discussed earlier in this chapter expressed the positions of equity holders and debt holders in the traditional capital structure of an operating firm as being synonymous with call options or put options. The senior debt tranche in the example illustrated in Exhibit 25.5 may be viewed in the structural model as either a covered call or a riskless bond with a short put position on the assets. Similarly, the equity tranche can be viewed in the structural model as either a long position in a call option or a financed long position in the assets with a long put position.

25.6.3 Three Option Strategies Similar to a Mezzanine Tranche

The economic essence of the 10%/30% mezzanine tranche in the previous section was that the mezzanine tranche benefits from investment success in the collateral pool within the range of the assets retaining 70% of their value to 90% of their value. If the assets fall below 70% of their original value, the mezzanine tranche is wiped out, and the senior tranche begins bearing losses. If the assets retain 90% or more of their value, the value in excess of 90% benefits the equity tranche. Whereas the most senior and most junior tranches can be viewed with single positions in options, mezzanine tranches can be viewed with option strategies involving two options. There are three theoretically equivalent option strategies that mimic a mezzanine tranche, each involving two options: a collar position, a bull call spread, and a bull put spread.

Let's begin with viewing a mezzanine tranche as a collar position. As detailed in Chapter 6, a collar combines a long position in an asset with a short position in a call option and a long position in a put option. In this example, the long position in the asset is 70% financed. Both options are on the same asset and have the same expiration date, but the call option has a higher strike price than the put option. Owning a mezzanine tranche is like owning the collateral asset, owning a put option that places a floor on losses when the assets are below a particular amount (70% in the example), and writing a call option that places a cap on profits when the assets remain at or above a particular amount (90% in the example).

Thus, the mezzanine tranche in Exhibit 25.5 may be described as a collar with a financed position in the collateral pool, a long position in a put at the lower attachment point, and a short position in a call at the upper attachment point. The mezzanine tranche is net long between $70 million and $90 million in assets, with profits limited at and above $90 million in assets, and losses limited at and below $70 million in assets.

As noted in Chapter 6, a collar position has the same payout as a bull option spread; thus, a mezzanine tranche may be mimicked with a bull option spread. As discussed in Chapter 6, a bull spread combines a long and short position in either calls or puts, which differ only with regard to strike prices. A bull spread involves a long position in the option with the lower strike price and a short position in the option with the higher strike price. The bull spread offers positively correlated performance between the strike prices, places a cap on profits at the higher strike price, and places a floor on losses at the lower strike price. The top left panel in Exhibit 6.5 illustrates the profit and loss diagram of a bull spread (as well as a collar position).

Bull spreads may be formed with two calls or two puts. A bull call spread has two calls that differ only by strike price, in which the long position is in the lower strike price and the short position is in the higher strike price. A bull put spread has two puts that differ only by strike price, in which the long position is in the lower strike price and the short position is in the higher strike price. Bear spreads are the mirror positions. The underlying assets to the options are the collateral pool.

A bull call spread that mimics the 10%/30% mezzanine tranche in the example contains a long call option with a strike price of $70 million and a short call option with a strike price of $90 million. The bull call spread, like the mezzanine tranche, benefits from increases in the collateral pool between $70 million and $90 million. The analogous bull spread with put options (i.e., a bear call spread) is long a put option with a strike price of $70 million and short a put option with a strike price of $90 million. In summary, a mezzanine tranche can be viewed as a bull call spread or a bull put spread on the CDO's portfolio.

It may appear counterintuitive that a bull option spread has a long position in the option with the lower strike price and a short position in the option with the higher strike price regardless of whether the spread uses calls or puts. However, note that in the previous example, when the structure's assets (i.e., the collateral pool) are worth $80 million, the bull call spread has only one option that is in-the-money (a long call), and the bull put spread has only one option that is in-the-money (a short put). Both a long call and a short put are bullish positions.