**After completing this reading you should be able to: **

- Describe common types of structured products.
- Describe tranching and the distribution of credit losses in a securitization.
- Describe a waterfall structure in a securitization.
- Identify the key participants in the securitization process and describe conflicts of interest that can arise in the process.
- Compute and evaluate one or two iterations of interim cashflows in a three-tiered securitization structure.
- Describe a simulation approach to calculating credit losses for different tranches in a securitization.
- Explain how the default probabilities and default correlations affect the credit risk in a securitization.
- Explain how default sensitivities for tranches are measured.
- Describe risk factors that impact structured products.
- Define implied correlation and describe how it can be measured.
- Identify the motivations for using structured credit products.

## What is a Structured Product?

A **structured product**, also known as a market-linked investment, is a **pre-packaged** structured finance investment built from a single security or a basket of securities. That includes bonds, options, indices, commodities, currencies, and derivatives.

Each structured product has its own benefits and risks, which often differ from those of the original assets. Let’s look at the main structured products on the market.

## The Common Types of Structured Products

### Covered Bonds

Covered bonds are debt securities issued by banks that are guaranteed by a secure cover pool consisting of mortgage loans. The cover pool – assets designated as security for the bond – stay on the balance sheet of the issuer. However, the pool is segregated from other assets in case of a winding-up. These assets cannot be used to settle claims from other stakeholders before all the claims from covered bondholders have been met.

Covered bonds are not considered fully-fledged structured products. There are two main reasons:

- The underlying assets remain on the issuer’s balance sheet. In most cases, structured products are held off the balance sheet often in special purpose entities.
- Covered bonds have dual recourse, but structured products are a nonrecourse form of financing. Principal and interest payments to bondholders are paid out of the issuer’s general cash flows rather than out of the cover pool’s direct cash flows

### Mortgage Pass-through Securities

A mortgage pass-through security is a security created when mortgages are pooled to create a product that’s marketable in form of shares – often called** participation certificates.** The holder is entitled to a pro-rata share of all principal and interest payments made on the pool of mortgage loans. A pool can consist of several thousands of mortgages.

Unlike covered bonds, mortgage pass-through securities are off-balance-sheet securitizations where investors receive cash flows based entirely on the performance of the pool. Payments are made to security holders each month from a **servicer**. These cash flows are usually reduced by the “service and other administrative fees” of the mortgages. A large percentage of pass-throughs carry implicit or explicit guarantee of performance, making default risk less of a concern to the investor. The main risk arises from prepayment, where the homeowner returns the principal earlier than scheduled. Prepayment implies that the investor loses out on interest that would be paid on that part of the principal.

### Collateralized Mortgage Obligations

A collateralized mortgage obligation (CMO) is a fixed income security that uses mortgage-backed securities as collateral. CMOs are subdivided into tranches that vary in interest and risk based on the **maturity structure** of the mortgages. These tranches can have short or long maturities, cash flows that are fixed or floating, and other conditions. The most common payment structure is the waterfall mechanism (also called sequential pay) in which the tranches are ordered, with “Class A” receiving all principal repayments from the loan until it is retired, then “Class B,” and so on. The senior tranches carry less prepayment risk than a pass-through, while the lower ones bear more.

Collateralized mortgage obligations offer investors an opportunity to profit from a diversified, and therefore risk-reduced, set of mortgage-backed securities.

### Structured Credit Products

Structured credit products work much like CMOs: they are backed by credit-risky loans or bonds and are split into tranches that have different degrees of credit risk. However, they introduce a new innovation – **sequential distribution of losses.**

The most junior tranches lie first “in the line of fire” as they bear the first losses and are most likely to be written down. More senior tranches only begin to bear credit risk losses once the junior tranches have been written down to zero. As such, the most senior tranche has the highest credit rating and the lowest probability of writedowns.

In some cases, structured credit products can combine sequential distribution of losses with sequential payment technology as applied in CMOs to form complex products.

### Collateralized Debt Obligations (CDOs)

CDO are Securitizations that repackage other securitizations. They may come in the form of a bond issued against a collateral pool consisting of ABS, MBS, or CLOs), collateralized mortgage obligations (CMOs), or collateralized bond obligations.

Structured credit products are usually set up as **special purpose entities** (SPE) or **vehicles** (SPV), also known as a trust. The SPE/SPV is intended to separate the assets and liabilities of the structured product from those of the original creditors (originators) and of the servicer. The SPV/SPE often has a predefined purpose and legal identity. If the originator goes bankrupt, the assets sold to the SPV/SPE are not at risk. Hence, the SPV/SPV is said to be **bankruptcy-remote. **

## Tranching and Distribution of Losses in a Securitization

As noted earlier, most structured products have a sequential capital structure with different tranches.

$$ \begin{array}{c|c} \textbf{Assets} & \textbf{Liabilities} \\ \hline {} & \text{Senior Debt} \\ {{\text{Underlying Debt Instruments}}} & \text{Mezzanine debt} \\ {} & \text{Equity} \\ \hline \end{array} $$

The safest tranche is also known as the **senior tranche. **It offers the lowest interest rate, but it’s the **first** to receive cash flows from the underlying asset portfolio.

The **middle (mezzanine) tranche** offers a slightly higher interest rate and ranks just below the senior tranche. It takes the **second** spot during cash flow distribution. It will absorb losses only after the equity tranche is completely written down. The boundary between the senior tranche and the mezzanine tranche, expressed as a percentage of the total of the liabilities, is called the **attachment point** of the more senior tranche and ** detachment point **of the junior (mezzanine) tranche. The equity tranche only has a detachment point, and the most senior only has an attachment point.

The most junior tranche, also called the** equity tranche**, offers the highest interest rate (or high spread if the claim is floating) but ranks **last** during cash flow distribution. It’s also the first tranche to absorb any loss that may be incurred. The amount available for distribution to the equity (junior) tranche is whatever is left from the two other tranches less management fees. These fees can range from 0.5% to 1.5% annually.

The part of the capital structure below a given tranche is called its **subordination **or **credit enhancement.** We can interpret it as the fraction of the collateral pool that must be lost before the tranche takes any loss. As such, credit enhancement is greater for more senior bonds in the structure.

In general, credit enhancement (CE) takes two forms – internal CE or external CE. Internal CE may result from overcollateralization or excess spread.

**Over-collateralization** (OC) is the provision of collateral that is worth more than enough to cover the claims from investors. In other words, the assets underlying the product have a value that’s in excess of the face value of securities issued to investors.

**Excess spread** is the difference between the cash flows collected and the payments made to all bondholders. For instance, let the interest rate received on the underlying collateral be 11%, and the coupon on the issued structured security be 10% (including fees). In this case, the excess spread is 1%. The excess spread is a** built-in margin of safety** that protects the pool (and originator) from losses. Its presence can actually improve the ratings on the structured product being assembled and make the resulting security more attractive to investors.

Investors in these tranches can protect themselves from default by purchasing credit default swaps. This CDS guarantees a pre-specified compensation in the event that a given tranche defaults. In turn, the investors must make regular payments to the credit protection seller (writer of the CDS).

Each tranche is assigned its own credit rating, except the equity tranche. For instance, the senior tranche is constructed to receive an AAA rating. Highly rated tranches are sold to investors, but the junior ranking ones may end up being held by the issuing bank. That way, the bank has an incentive to monitor the underlying loans.

### Waterfall Structure in a Securitization

The waterfall structure has much to do with the rules about how the cash flows from the collateral are distributed to the various securities in the capital structure. A typical structured credit product begins life with a certain amount of hard overcollateralization, since part of the capital structure is an equity note, and the debt tranches are less than 100 percent of the deal. Soft overcollateralization mechanisms may begin to pay down the senior debt over time with part of the collateral pool interest, or divert part of it into a reserve that provides additional credit enhancement for the senior tranches.

#### Example: Waterfall Structure

A collateralized loan obligation is comprised of 100 identical leveraged loans with a par value of $1,000,000 each, priced at par. The loans pay a fixed spread of 4% over one –month Libor. The capital structure consists of equity, and a junior and a senior bond. Determine the excess spread, assuming that there are no defaults in the collateral pool

Assumptions:

- The swap curve (“Libor”) is flat at 5%
- There are no upfront, management, or trustee fees
- The loans in the collateral pool and the liabilities are assumed to have a maturity of five years.
- All coupons and loan interest payments are annual, and occur at year-end
- There are no defaults in the collateral pool

$$ \textbf{Loan and Tranche Information} $$

$$ \begin{array}{c|c|c} \textbf{B/S component} & \textbf{Par value} & \textbf{Coupon} \\ \hline \textbf{Assets} & {$100,000,000} & \textbf{ Libor} { + 4\%} \\ \hline \textbf{Liabilities} & {} & {} \\ \hline \text{Equity note} & {$5,000,000} & {} \\ \hline \text{Junior tranche} & {$10,000,000} & \text{ Libor}{ + 500} \text{ bps} \\ \hline \text{Senior tranche} & {$85,000,000} & \text{ Libor} {+ 50} \text{ bps} \\ \end{array} $$

**Solution**

From the data, we can be able to see that the junior bond has a much wider spread than that of the senior, and much less credit enhancement; the mezzanine/ junior tranche attachment point is 5 percent, and the senior attachment point is 15 percent.

The annual cash flows will be as follows:

$$ \textbf{Tranche Interest Information} $$

$$ \begin{array}{c|c|c|c} {} & \textbf{Libor + Spread} & \textbf{Principal amount} & \textbf{Annual interest} \\ \hline \text{Collateral } & {0.05 + 0.04} & {$100,000,000} &{$9,000,000} \\ \hline \text{mezzanine} & {0.05 + 0.05} & { $10,000,000} & {($1,000,000)} \\ \hline \text{senior} & {0.05 + 0.005} & { $85,000,000} & {($4,675,000)} \\ \hline {} & {} & \text{Excess spread} & {$3,325,000} \\ \end{array} $$

Ideally, therefore, holders of the equity tranche would receive $3,325,000. However, the actual amount received is likely to be much less because of several reasons. First, cash flows to the equity tranche may be subject to an overcollateralization trigger where the maximum amount the tranche can receive is specified with any excess cash flows channeled to a trust account. Second, cash flows will also be lower if defaults occur.

To illustrate, let’s go back to our example and assume that the default rate is 5% per year. That implies that on average, about 5 loans will default over the year. In this case, the total cash inflow from collateral will be $8,550,000 [= $100,000,000 * 0.09 *(1 – 0.05)]. There will still be sufficient cash to pay senior and junior bondholders in full. However, the cash flows into the equity tranche will be reduced.

$$ \textbf{Tranche Interest Information Assuming 5% Annual Default Rate} $$

$$ \begin{array}{c|c|c|c} {} & \textbf{Libor + Spread} & \textbf{Principal amount} & \textbf{Annual interest} \\ \hline \text{Collateral } & {0.05 + 0.04} & { $95,000,000} &{$8,550,000} \\ \hline \text{mezzanine} & {0.05 + 0.05} & { $10,000,000} & {($1,000,000)} \\ \hline \text{senior} & {0.05 + 0.005} & { $85,000,000} & {($4,675,000)} \\ \hline {} & {} & \text{Excess spread} & {$2,875,000} \\ \end{array} $$

The assumption that all the loans and bonds have precisely the same maturity date is only to simplify the process. In practice, however, there can be more than a dozen tranches with different coupon rate and maturities. Each of these tranches is rated.

## Key Participants in the Securitization Process and Possible Conflicts of Interest

### Loan Originator

The loan originator is the original lender who creates the debt obligations in the collateral pool. This is often a bank, for example, when the underlying collateral consists of bank loans or credit card receivables. But it can also be a specialty finance company or mortgage lender. If most of the loans have been originated by a single intermediary, the originator may be called the sponsor or seller.

### Underwriter

The underwriter is the financial engineer behind the securitization structure and is at times called the arranger. The underwriter is often, but not always, a large financial intermediary. Typically, the underwriter aggregates the underlying loans, designs the securitization structure, including things like the coupon rates, tranche sizes, and triggers. In this capacity, the underwriter is also the issuer of the securities. You may also come across a somewhat technical legal term – the depositor – being used to describe the issuer. The underwriter bears **warehousing risk**, the risk that the deal will not be completed and the value of the accumulated collateral still on its balance sheet falls.

### Rating Agencies

Rating agencies help investors to assess the riskiness of the investment by assigning credit ratings to the various tranches engineered. Attachment points and the subordination structure are key cogs in the rating process. When companies are planning to issue a non-securitized bond, rating agencies have little influence over creditworthiness and are not involved in the structuring process. When issuing securitizations, however, rating agencies are usually more engaged in the structuring process and do more than just opining on creditworthiness.

A potential conflict of interest arises from the fact that rating agencies are compensated by the originators. The agencies may be faced with a situation where they feel obliged to provide a favorable rating even when there are serious issues regarding the creditworthiness of the structure. The fact that the agencies are actively involved in designing the securitization structure means only serves to exacerbate the potential conflict of interest. The agencies may dictate the amount of enhancement required to guarantee an investment-grade rating. Investors can cope with the potential conflict of interest by carrying out their own credit review of the deal or even demanding a wider spread.

Although the ratings under a securitization structure may be based entirely on the credit quality of the pool and the liability structure, they may also reflect any **performance guarantees** typically provided by **monoline insurance companies**. The insurer **guarantees** timely repayment of bond principal and interest in exchange for **insurance **premiums.

### Servicers and Managers

The servicer collects principal and interest from the loans in the collateral pool and disburses principal and interest to the liability holders, as well as fees to the underwriter and itself. Servicers may also be actively involved in the monitoring of the collateral pool. If a loan in the pool is in distress, for example, the servicer may have the authority to evaluate possible options, including extending the term of the loan or foreclosing. Servicers, therefore, are firmly in the mix when it comes to conflict of interest between themselves and bondholders, or, between different classes of bondholders. Equity holders are often keen to have all the loans performing to maturity because they are first in line to absorb any eventual loss. Therefore, they will be open to any arrangement that can help the borrower to see out the contract, including extending maturity. Senior bondholders, on the other hand, are less concerned with the possibility of default especially if the historical default rate is low such that there’s a very small chance that the senior tranche will make a loss.

Nonetheless, any loan extension may avoid immediate loss but increase the potential future loss, thereby increasing the riskiness of the bond as a whole and eroding the credit enhancement. At any one time, the servicer takes action that is better aligned with the interests of some bondholders than of others.

When the collateral pool is actively managed, the manager may not be so keen to monitor the financial health of the pool if there’s no incentive to do so. To mitigate conflict of interest, the structure may be designed in such a way that the originator or manager bears the first loss in the capital structure.

### Credit Scenario Analysis of a Securitization

Let’s revisit our example above where the excess spread (after paying senior and mezzanine tranche bondholders) was $3,325,000.

$$ \textbf{Tranche Interest Information} $$

$$ \begin{array}{c|c|c|c} {} & \textbf{Libor + Spread} & \textbf{Principal amount} & \textbf{Annual interest} \\ \hline \text{Collateral } & {0.05 + 0.04} & { $100,000,000} &{$9,000,000} \\ \hline \text{mezzanine} & {0.05 + 0.05} & { $10,000,000} & {($1,000,000)} \\ \hline \text{senior} & {0.05 + 0.005} & { $85,000,000} & {($4,675,000)} \\ \hline {} & {} & \text{Excess spread} & {$3,325,000} \\ \end{array} $$

Instead of channeling the entire amount into the pockets of equity holders, originators will often divert part of it to an excess spread account, also known as a trust/overcollateralization account.

The aim is to create further protection for the senior and mezzanine tranches by creating an alternative pool of funds that can be used to make payments to investors in case the interest from the collateral pool proves insufficient. In other words, the funds in the overcollateralization account will be used to pay interest on the bonds if there is not enough interest flowing from the loans in the collateral pool. Any built-up funds in the overcollateralization account at maturity will be released to equity holders. The funds in the overcollateralization account earn interest at the money market rate.

The overcollateralization account (we will denote this as OA) creates a reliable buffer especially when the default rate is low. Unless the default rate is very high, there will be some cash inflow into the account every year.

Before we the various possibilities, let’s make some two more assumptions:

- Loans in the collateral pay no interest if they have defaulted any time during the prior year. That means there is no partial interest; interest is paid at the end of the year by surviving loans only.
- In the event of default, the recovery rate is 40%, and all recovered monies are channeled into the overcollateralization account where accumulation occurs at the money market rate

Notation

N = number of loans in initial collateral pool (in our case, N = 100)

\({\text{d}}_{\text{t}}\) = number of defaults in the course of year t

\({\text{L}}_{\text{t}}\) = aggregate loan interest received by the trust at the end of year t

B = Total coupon (interest) due to both the junior and senior bonds (in our example above, B = $5,675,000 for all t)

K = maximum amount diverted from excess spread into the overcollateralization account at the end of year t

\({\text{OC}}_{\text{t}}\) = amount actually diverted from excess spread into the overcollateralization account at the end of year t

\({\text{R}}_{\text{t}}\)= recovery amount deposited into the overcollateralization account at the end of year t

r = money market rate, assumed to be constant (We will assume r = 5%)

To determine the cash flow to equity, we will ask ourselves the following question:

**Q: Is the excess spread positive?**

In notations, we are simply asking: is \({\text{L}}_{\text{t}}-\text{B}≥0\)?

__If yes:__

A further test must be performed to determine how much flows into the OA: is \({\text{L}}_{\text{t}}-\text{B} \ge {\text K}?\) If yes, then K is channeled into the OA, and \(\text{L}_{\text{t}}-\text{B}-\text{K}\), flows to equity holders: \({\text{OC}}_{\text{t}}=\text{K}\). If no, then the entire amount \({\text{L}}_{\text{t}}-\text{B}\) is channeled into the OA, and equity holders receive nothing: \({\text{OC}}_{\text{t}}\)=\({\text{L}}_{\text{t}}-\text{B}\)

__If No:__

If the excess spread is not positive, then the interest is not sufficient to pay bondholders and all \({\text{L}}_{\text{t}}\) goes to bondholders. The shortfall, therefore, is \(\text{B}-{\text{L}}_{\text{t}}\).

In this scenario, the next step is to check if the OA has accumulated an amout that can cover the shortfall. If the OA has enough funds, bondholders are paid in full. If the OA does not have enough, the bondholders suffer a writedown.

Now let’s see how a positive default rate affects cash flows. First, let’s assume that the number of defaults, \({\text{d}}_{\text{t}}\) is constant in any given year, and the recovery rate in year t is 40%. In this case, the recovery amount deposited into the overcollateralization account at the end of year t, \({\text{R}}_{\text{t}}\) is calculated as follows:

$$ {\text{R}}_{\text{t}}=0.4{\text{d}}_{\text{t}} \times \text{loan amount} $$

Thus, the total amount deposited into the trust account in year t is:

$$ {\text{R}}_{\text{t}}+ {\text{OC}}_{\text{t}} \quad \text{t}=1,…\text T-1 $$

The value of the overcollateralization account at the end of year t, including the cash flows from recovery and interest paid on the value of the account at the end of the prior year, is:

$$ { { \text{R} } }_{ { \text{t} } }+{ { \text{OC} } }_{ { \text{t} } }+\sum _{ \tau =1 }^{ \text{t}-1 }{ { \left( 1+\text{r} \right) }^{ \text{t}-\tau } } { { \text{OC} } }_{ { \text{t} } } \quad \text{t}=1,…\text{T}-1 $$

If the excess spread is negative, i.e.,\({\text{L}}_{\text{t}}-\text{B}\) < 0, the custodian must check if the OA has an amount enough to cover the shortfall. Formally, the test is as follows:

$$ { { \text{R} } }_{ { \text{t} } }+\sum _{ \tau =1 }^{ \text t-1 }{ { \left( 1+\text{r} \right) }^{ \text{t}-\tau } } { { \text{OC} } }_{ { \text{t} } }>\text{B}-{ { \text{L} } }_{ { \text{t} } } $$

Note that we do not have \({ { \text{OC} } }_{ { \text{t} } }\) to add to \({ { \text{R} } }_{ { \text{t} } }\) in the equation above because there is no excess spread in the current period.

If the test is true, then the bondholders are paid in full. If not, then the OA is literally swept clean and all of its holdings channeled to bondholders. In this case, they receive

$$ { { \text{R} } }_{ { \text{t} } }+ \sum _{ \tau =1 }^{ \text{t}-1 }{ { \left( 1+\text{r} \right) }^{ \text{t}-\tau } } { \text{OC} }_{ \text{t} } \text{ from the OA} $$

The amount to be diverted in any year t is given by:

$$ { { \text{ OC } } }_{ { \text{t} } }=\left\{ \begin{matrix} \text{min}\left( { { \text{ L } } }_{ { \text{ t } } }-\text{ B,K } \right) \\ \text{ max }\left[ { { \text{ L } } }_{ { \text{ t } } }-\text{ B }=\left( \sum _{ \tau =1 }^{ \text{ t }-1 }{ { \left( 1+\text{ r } \right) }^{ \text{ t }-\tau } } { { \text{ OC } } }_{ { \text{t} } }+{ { \text{ R } } }_{ { \text{t} } } \right) \right] \end{matrix} \right\} \text{for}\left\{ \begin{matrix} { { \text{ L } } }_{ { \text{t} } }\ge \text{ B } \\ { { \text{ L } } }_{ { \text{t} } } < \text{ B } \end{matrix} \right\} $$

Once we have established how much excess spread, if any, flows into the OA at the end of year t, we can determine how much cash flows to equity holders at the end of year t. The equity cash flow is

$$ \text{max}\left( { { \text{L} } }_{ { \text{t} } }-{\text{B}}-{ { \text{OC} } }_{ { \text{t} } },0\right) \quad \text{t}=1,…\text T-1 $$

In the terminal year, cash flows are examined separately for several reasons:

- All of the surviving loans up to that point reach maturity and principal is returned
- There is no diversion to the OA because the structure ends. All proceeds have to be shared
- Since there is no diversion to the OA, there’s no need to test overcollateralization triggers.

The equations below summarize terminal cash flows:

Loan interest=\(\left( \text{ N }-\sum _{ \text{t}=1 }^{ \text{T} }{ { \text{d} }_{ \text{t} } } \right) \times \left( \text{ Libor+spread } \right) \times \text{ par }\)

Proceeds from redemption of all surviving loans=\(\left( \text{ N }-\sum _{ \text{t}=1 }^{ \text{T} }{ { \text{d} }_{ \text{t} } } \right) \times \text{ par }\)

Recovery \(\text{R}_{ \text{t} }=0.4\text{d}_{ \text{t} }\times \text{par}\)

Residual in OA account = \(\sum _{ \tau =1 }^{ \text{t}-1 }{ { \left( 1+\text{r} \right) }^{ \text{t}-\tau } } \text{OC}\)

In the terminal year therefore, the waterfall mechanism takes full effect: If the sum of all terminal cash flows is large enough, the senior tranche is paid off. The remainder, if any, is used to pay off the junior tranche, and any remainder flows to equity. Sometimes there may be no cash flows left for equity holders after paying off the junior tranche, especially if the default rate has been high.

**Examples: Equity IRR given different default rates**

$$ \textbf{Cash flow structure Given a Constant Default Rate of 2%} $$

$$ \begin{array}{l|r} \textbf{Assumptions and Notation} & \textbf{Values} \\ \hline \text{Number of loansin the pool at t = 0} & \text{100} \\ \text{Default rate} & \text{2% (constant)} \\ \text{Bond coupon interest due, B (in millions)} & \text{\$5.675} \\ \text{Maximum amount diverted, K (in millions)} & \text{\$2.00} \\ \text{Recovery rate} & \text{40%} \\ \text{Money market rate} & \text{5%} \\ \text{Aggregate loan interest, L(t)} & \text{9% (5% Libor + 4% Spread)} \\ \end{array} $$

$$ \begin{array}{l|r|r|r|r|r|r} \textbf{t} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} \\ \hline \text{Defaults, d(t), rounded} & \text{} & \text{2} & \text{2} & \text{2} & \text{2} & \text{2} \\ \text{Cumulated Defaults} & \text{} & \text{2} & \text{4} & \text{6} & \text{8} & \text{10} \\ \text{Survived} & \text{} & \text{98} & \text{96} & \text{94} & \text{92} & \text{90} \\ \text{Loan interest, L(t)} & \text{} & \text{\$8.82} & \text{\$8.64} & \text{\$8.46} & \text{\$8.28} & \text{\$98.10} \\ \text{Excess spread, L(t) – B} & \text{} & \text{\$3.15} & \text{\$2.97} & \text{\$2.79} & \text{\$2.61} & \text{} \\ \text{Overcollateralization, OC(t)} & \text{} & \text{\$2.00} & \text{\$2.00} & \text{\$2.00} & \text{\$2.00} & \text{} \\ \text{Recovery, R(t)} & \text{} & \text{\$0.80} & \text{\$0.80} & \text{\$0.80} & \text{\$0.80} & \text{\$0.80} \\ \text{OC + Rec, OC(t) + R(t)} & \text{} & \text{\$2.80} & \text{\$2.80} & \text{\$2.80} & \text{\$2.80} & \text{} \\ \text{Equity flow} & \text{-5} & \text{\$1.15} & \text{\$0.97} & \text{\$0.79} & \text{\$0.61} & \text{\$10.91} \\ \text{Results} & \text{} & \text{Y} & \text{Y} & \text{Y} & \text{Y} & \text{} \\ \text{OC a/c} & \text{} & \text{\$2.80} & \text{\$5.74} & \text{\$8.83} & \text{\$12.07} & \text{\$12.68} \\ \text{Equity IRR} & \text{} & \text{\$1.15} & \text{\$0.97} & \text{\$0.79} & \text{\$0.61} & \text{\$10.91} \\ \end{array} $$

$$ \begin{array}{l|r|r} \text{Total terminal cash flows = L(5) + R(5) + OC a/c} & \text{\$111.58} & \text{IRR: 29.79%} \\ \hline \text{Amount owed to bond tranches = \$85 + \$10 + \$5.675} & \text{\$100.68} & \text{} \\ \hline \text{Equity terminal cash flow} & \text{\$10.91} & \text{} \\ \end{array} $$

With a constant default rate of 2%, the figure above shows that the equity tranche generates an IRR of approx. 29.79%. However, increasing the default rate to 8% as illustrated in the example below generates a negative IRR for equity. The excess spread declines over time as defaults pile up.

$$ \textbf{Cash flow structure Given a Constant Default Rate of 8%} $$

$$ \begin{array}{l|r} \textbf{Assumptions and Notation} & \textbf{Values} \\ \hline \text{Number of loansin the pool at t = 0} & \text{100} \\ \text{Default rate} & \text{8% (constant)} \\ \text{Bond coupon interest due, B (in millions)} & \text{\$5.675} \\ \text{Maximum amount diverted, K (in millions)} & \text{\$2.00} \\ \text{Recovery rate} & \text{40%} \\ \text{Money market rate} & \text{5%} \\ \text{Aggregate loan interest, L(t)} & \text{9% (5% Libor + 4% Spread)} \\ \end{array} $$

$$ \begin{array}{l|r|r|r|r|r|r} \textbf{t} & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} \\ \hline \text{Defaults, d(t), rounded} & \text{} & \text{8} & \text{7} & \text{7} & \text{6} & \text{6} \\ \text{Cumulated Defaults} & \text{} & \text{8} & \text{15} & \text{22} & \text{28} & \text{34} \\ \text{Survived} & \text{} & \text{92} & \text{85} & \text{78} & \text{72} & \text{66} \\ \text{Loan interest, L(t)} & \text{} & \text{\$8.28} & \text{\$7.65} & \text{\$7.02} & \text{\$6.48} & \text{\$71.94} \\ \text{Excess spread, L(t) – B} & \text{} & \text{\$2.61} & \text{\$1.98} & \text{\$1.35} & \text{\$0.81} & \text{} \\ \text{Overcollateralization, OC(t)} & \text{} & \text{\$2.00} & \text{\$1.98} & \text{\$1.35} & \text{\$0.81} & \text{} \\ \text{Recovery, R(t)} & \text{} & \text{\$3.20} & \text{\$2.80} & \text{\$2.80} & \text{\$2.40} & \text{\$2.40} \\ \text{OC + Rec, OC(t) + R(t)} & \text{} & \text{\$5.20} & \text{\$4.78} & \text{\$4.15} & \text{\$3.21} & \text{} \\ \text{Equity flow} & \text{-5} & \text{\$0.61} & \text{\$0.00} & \text{\$0.00} & \text{\$0.00} & \text{} \\ \text{Results} & \text{} & \text{Y} & \text{Y} & \text{Y} & \text{Y} & \text{} \\ \text{OC a/c} & \text{} & \text{\$5.20} & \text{\$10.24} & \text{\$14.90} & \text{\$18.86} & \text{\$19.8} \\ \text{Equity IRR} & \text{} & \text{\$0.61} & \text{\$0.00} & \text{\$0.00} & \text{\$0.00} & \text{\$0.00} \\ \end{array} $$

$$ \begin{array}{l|r|r} \text{Total terminal cash flows = L(5) + R(5) + OC a/c} & \text{\$94.14} & \text{IRR: -87.80%} \\ \hline \text{Amount owed to bond tranches = \$85 + \$10 + \$5.675} & \text{\$100.68} & \text{} \\ \hline \text{Equity terminal cash flow} & \text{\$0.00} & \text{} \\ \end{array} $$

As defaults pile up, the most dramatic cash flow changes occur in the equity tranche. At a low default rate, the equity continues to receive at least some cash almost throughout the life of the securitization. But as the default rate increases, interim cash flows to the equity **terminate **much earlier.

## The Simulation Procedure and the Role of Correlation

In our calculations up to this point, we have made several assumptions, notably that the default rate is constant, and there’s no correlation between loans. In reality, the default rate is a random variable, and loans show some correlation.

We can estimate credit losses for the various tranches in a securitization structure by way of simulation, which allows us to change our initial assumptions.

#### Step 1: Estimate parameters

The first step is to determine the parameters for the valuation, particularly the probability of default for each security. We also need to determine the correlation that ties the securities together.

#### Step 2: Generate default time simulations

Next, we need to simulate the default times of each security included in the collateral pool, using the estimated parameters and the copula approach. We simulate the default times of each security in the collateral pool.

#### Step 3: Compute the credit losses

Using the simulations results, (i.e., default probabilities and associated timings), we can simulate cash flows from the security pool in every period.

## Effect of Default Probabilities and Default Correlations on the Credit Risk in a Securitization

The interaction between default probability and default correlation has important implications on the credit risk under a securitization framework.

An increase in the default rate hurts all the tranches in a securitization structure. It increases bond losses and also decreases the equity IRR.

An increase in correlation, on the other hand, can bring about mixed results, depending on the level of default.

### Equity Tranche

The equity value tends to exhibit positive convexity: as the default rate increases from a very low level, the effect on equity value is extremely huge. But as the default rate increases further and further from its low values, the responsiveness of the equity value drops off. We can use a bit of biological analogy here: If someone happens to physically assault you, the first few blows tend to be the most painful. Or better yet, the human body tends to be hit really hard by pathogens (disease-causing organisms) at the first time of attack. As time goes, the body is able to adapt and minimize the impact caused by a further onslaught by the pathogens. Once you have lost most of your investment due to an increase in the default rate, you will be left with much less to lose following even more defaults in the future.

### Mezzanine Trance

When the default rate is low, an increase in correlation increases the likelihood of losses to the mezzanine tranche (similar to the senior tranche). But when the default rate is high, an increase in correlation actually decreases the expected losses to the mezzanine tranche because of increased likelihood of fewer defaults. In other words, the mezzanine bonds behave more like the senior bonds at low default rates and more like the equity tranche when the default rates are high. We can also look at this differently: at low default rates, there’s a very small chance that the attachment point for mezzanine bonds will be broken. But at high default rates, a breach of the attachment point is highly likely.

Convexity presents yet another interesting perspective when it comes to default rates. Maybe to take you back to bond convexity a little bit, a bond that has positive convexity would typically experience larger price increases as yields fall but relatively smaller price decreases when yields increase.

### Senior Tranche

At low correlations, senior bonds exhibit negative convexity: as defaults pile up, the decline in bond value increases. The mezzanine tranche is ambiguous: it exhibits negative convexity for low default rates and positive convexity for high default rates.

At high correlations, all bonds tend to shake off the convexity effect. They respond more linearly to an increase in default rates.

## Measuring Default Sensitivities for Tranches

Let’s now look at the effect of default on the various securitization tranches independently without tying everything to correlation. To do this, we will revive the concept of DV01.

default01. The default probability π is shocked up by 10 basis points and all tranches revalued. Then, the default probability is shocked down by 10 basis points and the tranches revalued yet again. The default01 is then computed as:

$$ \cfrac { 1 }{ 20 } \left[ \left( \text{mean value/loss for } \pi + 0.0010 \right) -\text{mean value/loss for } \pi – 0.0010 \right] $$

The default01 is computed for each combination of \(\pi\) and \(\rho \)

It can be shown that for all tranches, the default01 is positive, regardless of the values of \(\pi\) and correlation \(\rho \). This happens because equity and bond values decrease monotonically as the default probability rises. In addition, the default01 converges to zero for all tranches as the default rates increase. Once the losses are extremely high, the incremental impact of additional defaults is low.

### Tranche Risk

- Systematic Risk
Even when the collateral pool is well diversified, structured products tend to have quite high systematic. High systematic risk can be expressed through the presence of high default correlation. If the default correlation is high, it means there is a low but material probability of an unusually high number of defaults.

As long as the correlation is high, all the tranches have a degree of systematic risk. There’s a likelihood of a large loss in the senior bond if correlation is high.

The degree of systematic risk is highly dependent on the securities making up the collateral pool and the level of credit enhancement.

- Tranche thinness
In most securitization structures, the equity and junior tranches are relatively thin. In the examples we have used so far, the two tranches constituted just 15% of the collateral pool. This thinness manifests itself in VaR calculations. The 95% and 99% credit VaRs are quite close. This implies that if the two have been breached, there’s a high likelihood that the loss is very large.

- Granularity
The granularity of the collateral pool is a major indicator of the level of diversification. A collateral pool made up of a few large loans is more risky than another pool with many loans of smaller amounts.

### Implied Correlation

If you recall, the implied volatility of an option contract refers to the value of the volatility of the underlying instrument which, when input in an option pricing model, will return a theoretical value equal to the current market price of said option. The implied default correlation is a closely related concept: It is the value of the correlation built into the observable market prices of the various securitized tranches.

In essence, structured credit products are claims on cash flows of credit portfolios. Their prices therefore reflect an implied default correlation of those portfolios. Earlier on, we saw how to estimate the values of securitization tranches. In the same way, it is possible to reverse engineer the valuation process and use the market-observed prices of structured credit products and come up with a value of the default correlation.

The implied default correlation is a risk-neutral parameter that can always be estimated as long as there are observable prices of the portfolio credit products.

### Issuer and Investor Motivations for Structured Credit

To identify why securitizations are created incentives of the loan originators need to be identified.

### Incentives of the issuers

**Lower Cost of funding**Securitization provides a cheaper funding method for the issuer. The interest rate payable to investors in securitized assets is much lower than that which the originator would have to pay if they issued a coupon bond of the same maturity as the securitized portfolio.

**Exit Avenue**Without securitization, the originator would have to either have to retain the underlying assets on the balance sheet or sell them in the secondary market. As such, securitization offers a good “exit” route especially when the securitization costs much less than the next best alternative.

**Arbitrage CDOs**Sometimes, securitization may be designed to specifically capture the spread between the underlying loan interest and the coupon rates payable to investors.

**Balance sheet Relief**Hiving a significant amount of loans off the balance sheet can come with benefits for the originator. Banks, for example, may be subject to lower regulatory capital requirements.

**Fees**A sponsor may make more money from originating and possibly servicing loans than retaining them on the balance sheet.

### Incentives of Investors

**A unique Investment Opportunity**Securitization allows investors to have more direct legal claims on loans and portfolios of receivables. Capital market investors are able to participate in diversified loan pools in sectors that would otherwise be the preserve of banks alone. These include mortgages, credit card receivables, and automobile loans.

**Risk and Maturity Matching**Investors can easily access securities matching their risk, return, and maturity needs. For example, a pension fund with a long-term horizon can have access to long-term real-estate loans.

**Liquidity**Securitization allows for the creation of tradable securities with better liquidity

# Practice Questions

1) Which among the following is not a pattern with respect to the interaction between correlation and default probability?

- Convexity
- Increased bond losses
- Increased IRR of equity tranches
- Distribution of losses

The correct answer is **D**.

Convexity increased bond losses and increased IRR of equity tranches are all important patterns with respect to the interaction between correlation and default probability. The distribution of losses is not one of these patterns.