# Charles Davì

## The Anatomy of Deflation

In Not So Efficient Market Hypothesis, Slight Departure on December 11, 2008 at 1:38 am

#### Slight Departure

In this article, and others to follow under the category “Slight Departure,” I will take a slight departure from the narrow topics that dominate this blog. Instead, I will focus on broader economic issues and attempt to make sense of current macro economic conditions. Enjoy!

#### One Way Ticket

Yields on U.S. Treasuries have fallen quite a bit over the last 12 months across maturities. This was probably caused in part by what has been dubbed a “flight to safety.” That is, investors have sold off “risky” positions in exchange for safer ones. But why? My personal opinion is that while risk aversion is certainly playing a part in the quick movement into Treasuries, this risk aversion was itself spurred by a rush to meet short term obligations. Please note that this is my own version of what happened and what is to come. Anyone claiming to have a conclusive answer to such a broad question is either incorrect or correct as a matter of coincidence.

#### Collective Failure

Mortgage backed securities, CDOs comprised of mortgage backed securities, and the like (collectively refereed to as “Structured Products”) were widely held and widely believed to be fairly liquid and safe. But, for reasons that will likely be debated for decades if not longer, they became illiquid as the markets in which they were traded fell apart. If you were holding one of these, you were stuck with it, unless you were willing to sell it for pennies on the dollar. But before the failure of these markets, those holding Structured Products continued to accumulate short term obligations based on the assumption that Structured Products could be sold, if necessary, to meet these short term obligations. That is, they continued to operate under the reasonable and shared assumption that at any point, Structured Products could be exchanged for an amount of cash close to the par value of the product. When this assumption failed, it caused those holding Structured Products to sell other liquid assets and if necessary draw on existing lines of credit to meet these short term obligations. In my opinion, this bout of collective selling and exhaustion of credit began the first wave of downward pressure on prices.

#### Capital Requirements

Many of the entities holding Structured Products were banks or other institutions subject to capital requirements and regulations. When the Structured Products they were holding became illiquid, devalued, and subsequently downgraded by the various ratings agencies, various capital regulations forced them to either raise equity capital or shift their asset allocation towards high rated assets. In order to do this, many such entities simply sold riskier assets and purchased low risk assets or held the cash. This collective selling put enormous downward pressure on asset prices across asset classes, with the notable exception of Treasuries, where prices soared and yields plummeted. Moreover, the most dramatic drops along the Treasury yield curve were front loaded, in short term Treasuries. This is consistent with a rush out of other asset classes and into short term treasuries. Thus, it is arguable that those entities holding Structured Products simply liquidated other assets en masse in order to meet short term obligations and comply with capital requirements.

#### The Supply Of Credit

As mentioned in the previous paragraph, many of the entities holding Structured Products were banks. And for banks, like any other business, resource allocation is a zero sum game. That is, if banks allocate more cash to one endeavor, they must allocate less to at least one other. This simple observation suggests that as banks allocated more cash to Treasuries they could have cut back on lending. The dramatic escalation in short term LIBOR rates that took place over the last year (which has since subsided) suggests that there was indeed a severe shortage of credit, particularly short term interbank lending. As the cost of credit increases, the total cost of assets purchased using credit increases, which lowers the price at which these purchasers are willing to purchase assets. So those who purchase assets using credit will become less competitive during a credit shortage when compared to those who pay for assets with cash. Thus, the demand for assets in general should be expected to shrink given a large enough contraction in the supply of credit. This exerts further downward pressure on asset prices across asset classes.

#### Joe The Plumber

A decline in asset prices across asset classes decreases the balance sheet wealth of not only institutions, but individuals as well. This could lead to a decrease in consumer spending as the perceived wealth of individuals decreases, which would exert downward pressure on non-financial asset prices. This is consistent with the recent decline in the CPI.

And in addition to the individual’s role in the so called “real economy,” the individual also plays a role in the financial system. If individuals do not believe that the assets they have in the financial system (bank deposits, mutual funds, etc.) will maintain their value, they will have an incentive to sell these assets in exchange for cash. This not only exerts downward pressure on asset prices, but it exerts upward pressure on the cost of capital, particularly for banks, since this type of behavior erodes the deposit base.

#### Rinse, Lather, Repeat

A decline in asset prices across asset classes could cause the capital structure of a bank and therefore its ability and willingness to lend to deteriorate. As mentioned above, it could also cause its cost of capital to increase. As a result, the entire process outlined above could begin anew as a result of it having occurred once, compounding the downward pressure on asset prices.

#### Tis Good To Be King

So what should we do to stop this financial recursion? I have a few ideas. Again, these are my ideas, and you should keep that in mind. These problems are maddeningly complex and gigantic in scale and so you may not agree with my ideas for any number of reasons. That said, something must be done. And despite deep reservations, I think that U.S. government spending during this crisis is the thing to do.

Right now, the U.S. Government is able to borrow at almost 0% interest at a time that the U.S. dollar is at its strongest in years against certain currencies. This is a truly odd combination, but could be explained if we accept that people are leaving capital markets all over the world and running for the hills, i.e., U.S. Treasuries and therefore U.S. dollar denominated assets. This market dynamic will change in the long run, hopefully. So, my plan to exploit this aberration would be as follows:

Flood the market with short term and medium term U.S. Treasuries and thereby borrow to the hilt. Spend this money on fixing the U.S. financial system (creating more repo facilities, buying toxic assets, etc.) and U.S. infrastructure projects (to create jobs and demand for non-financial assets). The net effect of this is to allow the private entities in the U.S. economy to borrow at near 0% interest, giving the U.S. a tremendous near term advantage over the rest of the world.

At some point, this borrowing will push the interest rates on Treasuries up and will push the dollar down. When interest rates on treasuries reach non-bargain basement levels of borrowing, the U.S. should stop borrowing. But what about the green back? As a bonus, because the U.S. dollar will eventually fall from its current heights, the U.S. debt issued until that point will depreciate in value.

Thus, the U.S. could borrow for almost no interest, pump that money into its own economy, and then get a free principle reduction at the end. As for the rest of the world, yes, my solution rests your fate on the shoulders of the U.S. economy. I’m sorry, but I just don’t have enough time to think of a way out of your mess as well.

## Synthetic CDOs, Ratings, And Super Senior Tranches: Part 3

In Politicized Economy, Systemic Counterparty Confusion on December 8, 2008 at 9:39 pm

#### Prescience and Precedent

In the previous articles (part 1 and part 2), we discussed both the modeling and rating of  CDOs and their tranches. In this article, we will discuss the rating of synthetic CDOs and those fabled “super senior” tranches. As mentioned in the previous articles, I highly recommend that you read my article on Synthetic CDOs and my article on tranches.

#### Funded And Unfunded Synthetic CDOs

As explained here, the asset underlying a synthetic CDO is a portfolio of the long positions of credit default swaps. That is, investors in synthetic CDOs have basically sold protection on various entities to the CDS market through the synthetic CDO structure. Although most CDS agreements will require collateral to be posted based on who is in the money (and may also require an upfront payment), as a matter of market practice, the protection seller does not fund the long position. That is, if A sold \$1 million worth of protection to B, A would not post the \$1 million to B or a custodian. (Note that this is a market convention and could change organically or by fiat at any moment given the current market context). Thus, B is exposed to the risk that A will not payout upon a default.

Because the long position of a CDS is usually unfunded, Synthetic CDOs can be funded, unfunded, or partially funded. If the investors post the full notional amount of protection sold by the SPV, then the transaction is called a fully funded synthetic CDO. For example, if the SPV sold \$100 million worth of protection to the swap market, the investors could put up \$100 million in cash at the outset of the synthetic CDO transaction. In this case, the investors would receive some basis rate, usually LIBOR, plus a spread. Because the market practice does not require a CDS to be funded, the investors could hang on to their cash and simply promise to payout in the event that a default occurs in one of the CDSs entered into by the SPV. This is called an unfunded synthetic CDO. In this case, the investors would receive only the spread over the basis rate. If the investors put up some amount less than the full notional amount of protection sold by the SPV, then the transaction is called a partially funded synthetic CDO. Note that the investors’ exposure to default risk does not change whether the transaction is funded or unfunded. Rather, the SPV’s counterparties are exposed to counterparty risk in the case of an unfunded transaction. That is, the investors could fail to payout upon a default and therefore the SPV would not have the money to payout on the protection it sold to the swap market. Again, this is not a risk borne by the investors, but by the SPV’s counterparties.

#### Analyzing The Risks Of Synthetic CDOs

As mentioned above, whether a synthetic CDO is funded, unfunded or partially funded does not affect the default risks that investors are exposed to. That said, investors in synthetic CDOs are exposed to counterparty risk. That is, if a counterparty fails to make a swap fee payment to the SPV, the investors will lose money. Thus, a synthetic CDO exposes investors to an added layer of risk that is not present in an ordinary CDO transaction. So, in addition to being exposed to the risk that a default will occur in any of the underlying CDSs, synthetic CDO investors are exposed to the risk that one of the SPV’s counterparties will fail to pay. Additionally, there could be correlation between these two risks. For example, the counterparty to one CDS could be a reference entity in another CDS. Although such obvious examples of correlation may not exist in a given synthetic CDO, counterparty risk and default risk could interact in much more subtle and complex ways. Full examination of this topic is beyond the scope of this article.

In a synthetic CDO, the investors are the protection sellers and the SPV’s counterparties are the protection buyers. As such, the payments owed by the SPV’s counterparties could be much smaller than the total notional amount of protection sold by the SPV. Additionally, any perceived counterparty risk could be mitigated through the use of collateral. That is, those counterparties that have or are downgraded to low credit ratings could be required to post collateral. As a result, we might choose to ignore counterparty risk altogether as a practical matter and focus only on default risk. This would allow us to more easily compare synthetic and ordinary CDOs and would allow us to use essentially the same model to rate both. Full examination of this topic is also beyond the scope of this article. For more on this topic and and others, go here.

#### Synthetic CDO Ratings And Super Senior Tranches

After we have decided upon a model and run some simulations, we will produce a chart that provides the probability that losses will exceed X. We will now compare two synthetic CDOs with identical underlying assets but different tranches. Assume that the tranches are broken down by color in the charts below. Additionally, assume that in our rating system (Joe’s Rating System), a tranche is AAA rated if the probability of full repayment of principle and interest is at least 99%.

Note that our first synthetic CDO has only 3 tranches, whereas the second has 4, since in in the second chart, we have subdivided the 99th percentile. The probability that losses will reach into the green tranche is lower than the probability that losses will reach into the yellow tranches of either chart. Because the yellow tranches are AAA rated in both charts, certain market participants refer to the green tranche as super senior. That is, the green tranche is senior to a AAA rated tranche. This is a bit of a misnomer. Credit ratings and seniority levels are distinct concepts and the term “super senior” conflates the two. A bond can be senior to all others yet have a low credit rating. For example, the most senior obligations of ABC corporation, which has been in financial turmoil since incorporation, could be junk-rated. And a bond can be subordinate to all others but still have a high credit rating. So, we must treat each concept independently. That said, there is a connection between the two concepts. At some point, subordination will erode credit quality. That is, if we took the same set of cash flows and kept subdividing and subordinating rights in that set of cash flows, eventually the lower tranches will have a credit rating that is inferior to the higher tranches. It seems that the two concepts have been commingled in the mental real estate of certain market participants as a result of this connection.

#### Blessed Are The Forgetful

So is there a difference between AAA notes subordinated to some “super senior” tranche and plain old senior AAA rated notes? Yes, there is, but that shouldn’t surprise you if you distinguish between credit ratings and seniority. You should notice that the former note is subordinated while the latter isn’t. And bells should go off in your mind once you notice this. The rating “AAA” describes the probability of full payment of interest and principle. Under Joe’s Ratings, it tells you that the probability that losses will reach the AAA tranche is less than 1%. The AAA rating makes no other statements about the notes. If losses reach the point X = L*, investors in the subordinated AAA notes (the second chart, yellow tranche) will receive nothing while investors in the senior AAA notes (the first chart, yellow tranche) will not be fully paid, but will receive a share of the remaining cash flows. This difference in behavior is due to a difference in seniority, not credit rating. If we treat these concepts as distinct, we should anticipate such differences in behavior and plan accordingly.

## Synthetic CDOs, Ratings, And Super Senior Tranches: Part 2

In Politicized Economy, Systemic Counterparty Confusion on December 4, 2008 at 11:34 pm

#### Bait And Switch

My apologies, but this is going to be a three part article.  I have come to the conclusion that each topic warrants separate treatment. In this article, I will discuss the rating of CDO tranches. In the next, I will discuss the rating of Synthetic CDOs and those illusive “Super Senior” tranches.

#### Portfolio Loss Versus Tranche Loss

In the previous article, we discussed how rating agencies model the expected losses on the portfolio of bonds underlying a CDO. The end result was a chart that plotted losses against a scale of probabilities. This chart purports to answer the question, “how likely is it that the portfolio will lose more than X?” So if our CDO has a single tranche, that is if the payment waterfall simply passes the cash flows onto investors, then this chart would presumably contain all the information we need about the default risks associated with the CDO. But payment waterfalls can be used to distribute default risk differently among different tranches. So, if our CDO has multiple tranches, then we need to know the payment priorities of each tranche before we can make any statements about the expected losses of any tranche. After we know the payment priorities, we will return to our chart and rate the tranches.

#### Subordination And Default Risk

Payment waterfalls can be used to distribute default risk among different tranches by imposing payment priorities on cash flows. But in the absence of payment priorities, cash flows are shared equally among investors. For example, if each of 10 investors had equal claims on an investment that generated \$500, each investor would receive \$50. Assuming each made the same initial investment, each would have equal gains/losses. However, by subordinating the rights of certain investors to others, we can insulate the senior investors. For example, continuing with our 10 investors, assume there are 2 tranches, A and B, where the A notes are paid only the first \$500 generated by the investment and the B notes are paid the remainder. Assume that 5 investors hold A notes and that 5 investors hold B notes. If the investment generates only \$500, the A investors will receive \$100 each while the B investors will receive nothing. If however the investment generates \$1,500 the A investors will receive \$100 each and the B investors will receive \$200 each. This is just one example. In reality, the payment waterfall can assign cash flows under any set of rules that the investors will agree to.

If the investment in the previous example is a portfolio of bonds with an expected total return of \$1,000, then the payment waterfall insulates the A investors against the first \$500 of loss. That is, even if the portfolio loses \$500, the A investors will be fully paid. So, the net effect of the payment waterfall is to shift a fixed amount of default risk to the B investors.

#### Rating CDO Tranches

As a general rule, rating agencies define their various gradations of quality according to the probability of full payment of principal and interest as promised under the bonds. Assume that Joe’s Rating Agency defines their rating system as follows:

AAA rated bonds have at least a 99% probability of full payment of principal and interest;

AA rated bonds have at least a 95% probability of full payment of principal and interest;

A rated bonds have at least a 90% probability of full payment of principal and interest; and

Any bonds with less than a 90% probability of principal and interest are “Sub Investment Grade (SIG).”

Assume that the bonds underlying our CDO collectively promise to pay a total of \$100 million in principal and interest over the life of the bonds. For simplicity’s sake, assume that the CDO investors will receive only one payment at maturity. Further, assume that we have conducted several hundred thousand simulations for our CDO and constructed the chart below:

It follows from the data in the chart that the probability that losses on the CDO will be less than or equal to: \$35 million is 90%; \$40 million is 95%; \$65 million is 99%. We define the tranches as follows: tranche A is paid the lesser of (i) \$35 million and (ii) the total return on the CDO pool (the green tranche);  tranche B is paid the lesser of (i) \$25 million and (ii) the total return on the CDO pool less any amounts paid to tranche A (the yellow tranche); tranche C is paid the lesser of (i) \$5 million and (ii) the total return on the CDO pool less any amounts paid to tranches A and B (the blue tranche); and tranche D is paid the lesser of (i) \$35 million and (ii) the total return on the CDO pool less any amounts paid to tranches A, B, and C (the red tranche).

After some thought, you should realize that, according to Joe’s Ratings, tranche A is AAA; tranche B is AA; tranche C is A; and tranche D is SIG.

## Synthetic CDOs, Ratings, And Super Senior Tranches: Part 1

In Politicized Economy, Systemic Counterparty Confusion on December 3, 2008 at 9:40 pm

#### Super Senioritis

I’ve been perusing the finance blogs lately and I’ve noticed a recent obsession with Synthetic CDOs, specifically the super senior tranches of these transactions. And so I felt it was necessary for me to chime in on the debate, by applying my usual toast-dry analysis to Synthetic CDOs for the second time. This is a huge topic that requires consideration of how Synthetic CDOs function, how they’re rated, and how tranches distribute risk among investors. As a result, I’ve decided to break the article into two parts. This first part deals with the basics of rating the assets contained in CDOs. The next will examine the application of ratings to tranches of CDOs, how that translates into the world of synthetic CDOs, and ultimately, culminate in a discussion of what are known as “super senior tranches.”

You are likely to struggle greatly with this article unless you have some familiarity with Synthetic CDOs. And because I am an unabashed self-promoter, I highly recommend you read my introductory article on Synthetic CDOs and my article on Tranches. If you’re going to read only one, then read the one on tranches.

#### Tranches And Structured Products

Payment waterfalls allow the risks of an investment to be allocated among different groups of investors, or tranches. For example, in the case of Mortgage Backed Securities, a fixed amount of prepayment risk can be allocated to one tranche by tailoring the rules in the payment waterfall to pass all prepayments of principal to that tranche. But there are risks beyond prepayment risk. One obvious example is default risk. In the MBS world, this is the risk that because of defaults in the underlying mortgages the cash flows from the mortgages backing the notes will be inadequate to make payments on those notes. Obviously, default risk will be a primary concern of any investor. The risk that you will not get paid is arguably paramount to all others. So, payment waterfalls have been developed to address this risk and tailor the distribution of default risk in a way that allows each investor to assume a desired default risk level. But before we can understand how investors distinguish between these different levels of default risk, we must understand how rating systems describe these different levels.

#### Rating Systems And Rating Agencies

You have undoubtedly heard terms such as “AAA rated” and “AA rated” thrown somewhere near names like S&P and Moody’s. It’s not necessary to become familiar with the peculiarities of each rating agency’s system to appreciate the general idea: the ranking of default risk. That is, the market wants a short-hand system that both describes the probability of default for a particular financial product and can be compared across a disparate class of financial products. So, rating agencies developed models and systems of ratings (using confusingly similar labels like “AAA,” “Baa,” etc.) that purport to do just that.

#### Part 1: Past Performance And Correlation

The models that rating agencies use to produce their ratings are backward looking. That is, the first step in the process is to accumulate data about how financial products have behaved in the past. Rating agencies, and investors, will look to the past and produce charts like this:

They will note that in the past, of all bonds that Moody’s deemed Aaa, less than 1% of such bonds defaulted within 10 years of issuance. People then assume, quite reasonably, that this data provides probabilities of default across time for the various ratings. That is, they assume that if we wish to know the probability that a B rated bond will default in year three, we simply look to the above chart and discover that it is .1977 or 19.77%. Examination of this assumption is beyond the scope of this article. But for a great article on that topic (containing the above table and more!) go here.

A CDO is in essence a portfolio of bonds. So in order to model the cash flows of the portfolio, rating agencies turn to charts like the one above and examine the past performance of bonds similar to those in the portfolio. They also look at the correlation of default between the bonds in the CDO portfolio. Correlation, in this context, is a very precise term. And it’s impossible to do justice to the concept in a few sentences. That said, in layman’s terms, when considering the correlation of default between two bonds, rating agencies are looking for a connection between the bonds defaulting. That is, if bond 1 defaults, how does that change our expectation of the probability that bond 2 will default? Exactly how this is done is also well beyond the scope of this article. For those of you who are interested, you can read all about this and more here.

#### Part 2: Scenario Analysis

So after we have all of our data, we can begin to construct a chart of how likely a given level of loss is. This is done through scenario analysis. That is, the models are run hundreds of thousands of times (and possibly more) using different inputs. In each of these simulations, some bonds might “default.” That is, the model predicts that given a particular set of inputs, certain bonds will default. After each of these simulations, an amount of loss will be calculated, which is based on the estimated recovery values for the bonds in the pool that “defaulted” during that particular simulation. We can then ask, out of all of the simulations, how many times did the loss go above X? So if we ran our simulation 500,000 times, and if the loss was greater than \$1 million in only 5,000 of these simulations, then we could say that the probability of the loss being greater than \$1 million is .01, or 1%.

## Tranches And Risk

In Politicized Economy, Systemic Counterparty Confusion on December 1, 2008 at 7:32 am

#### What Is A Tranche?

Tranche is a French word that means slice. Every investment will convey certain rights in the cash flows produced by the investment to the investors. A tranche is a slice of those rights. Quite literally, each tranche represents a unique piece of the investment pie. So the term tranche connotes a fairly accurate indication of how the term is used in finance. And after all, it’s easier to tell investors that they’re buying tranches as opposed to “pits” or “buckets.”

#### Payment Waterfalls

A payment waterfall determines who gets paid what and when. That is, each dollar produced by an investment will be “pushed through” a payment waterfall and allocated according to the rules in the payment waterfall. For example, assume that there are 3 investors, A, B and C. They collectively invest in venture X. The payment waterfall for X is defined as follows: on the first of each month, A will be paid the lesser of (i) \$100 and (ii) all of the cash flows produced by X in the previous month; B will be paid the lesser of (i) \$100 and (ii) all of the cash flows produced by X in the previous month less any amounts paid to A; and C will be paid the lesser of (i) \$100 and (ii) all of the cash flows produced by X in the previous month less any amounts paid to A and B.

Assume that in month 1, X produced \$300 in cash. On the first day of month 2, the \$300 will be pushed through the waterfall. So A will get \$100; B will get \$100; and C will get \$100. Note that in the case of C, the two choices will produce equal amounts, so the term “lessor of” isn’t technically accurate. But assume that when the choice is between equal amounts, we simply pay that amount. Now assume that X produced \$150 in month 1. On the first day of month 2, the \$150 will be pushed through the waterfall. So A will get \$100; B will get \$50; and C will get \$0. Because A is “first” to get paid, so long as X produces \$100 per month, A is fully paid. B is fully paid so long as X produces \$200 per month and C at \$300 per month. So in this case, A’s tranche is said to be the least risky of the 3 tranches, with B and C being more risky in that order. Note that I am not using my technical definition of risk.

So why would C agree to be last in the pecking order? Well, one simple explanation is that C paid the least for his tranche. In another example we could have given C the right to any amounts left over each month after all other tranches are paid. This type of right is called a residual right. It is basically an equity stake. So in that case C would bear the risk that X’s cash flows will fall short in exchange for the right to acquire any excess cash flows produced by X. As is evident, the terms of the waterfall can be anything that the parties agree to. As such, we can cater the payment priorities to meet the specific desires of investors and distribute risks accordingly.

#### Mortgage Backed Securities And Prepayment Risk

Securitization is a fairly simple process to grasp in the abstract. In reality, turning thousands of mortgages into interest bearing notes is not a simple process. However, we can at least begin to understand the process by considering how a payment waterfall can be used to streamline the payments to investors. Viewed as a bond, a mortgage is a bond where the borrower, in this case the mortgagor, has a right to call the bond at any point in time. That is, at any point in time, a mortgagor can simply repay the full amount owed and terminate the lending agreement. Additionally, even if the mortgagor doesn’t pay the full amount owed, it is free to pay more than the amount obligated under the mortgage and allocate any additional amounts to the outstanding principal on the mortgage. For example, if A has a mortgage where A is obligated to make monthly payments of \$100, A could pay \$150 in a particular month, and request that the lender allocate the additional \$50 to reduce the outstanding principal on the mortgage.

The typical practice for a mortgage is to require the mortgagor to make fixed payments over the life of the mortgage. So each payment will consist of an interest portion and a principal portion. The amount allocated to principal is predetermined and said to amortize over the life of the mortgage. And as mentioned above, any amount over the fixed amount can be allocated to principal at the option of the mortgagor. The risk that any given loan will pay an amount above the required fixed payment is called prepayment risk.

While getting your money back is usually a good thing, investors prefer to defer repayment to some future date in exchange for receiving more money than they invested. So getting all of their principal back today is not the most preferred outcome. They prefer to get their principal at maturity plus interest over the life of the agreement. For example, if all of the mortgages in a pool of mortgages that have been securitized prepay the full amount before the anticipated maturity date of the notes, then the investors will presumably be repaid, but will not receive the remaining interest payments over the anticipated life of the notes. If this prepayment en masse occurs on the second day of the life of the notes, it would defeat the purpose of the investment.

#### Prepayment Risk And Payment Waterfalls

We can use payment waterfalls to distribute prepayment risk into different tranches. In reality, this can become a mind numbingly complex endeavor. We propose one simple example to demonstrate how tranches can be used to redistribute complex risks.

Assume that our mortgage pool consists of $N$ mortgages; the remaining principal on each mortgage is $p_i$; and the total remaining principal on the pool is $P = p_1 + \cdots + p_N$. Because each mortgage payment consists of some interest and some principal, each month, there will be a scheduled reduction in the outstanding total principal on the pool. Let $S$ denote the scheduled reduction of $P$. That is, $S$ is the sum of all of the principal portions of the fixed payments to be made in the pool. If there are any prepayments in the underlying mortgages, the actual reduction in $P$ will exceed the scheduled reduction. Let $A$ denote the actual reduction in $P$. The question now becomes, what do we do with $A - S$? That is, how do we distribute the amount by which the actual reduction in total principal exceeds the scheduled reduction? The simple answer, and the one considered here, is to push the entire prepayment amount onto one tranche, and reduce the outstanding principal on that tranche by that same amount.

For example, assume that a mortgage pool contains mortgages with a total \$100 million principal outstanding and that \$100 million worth of notes were issued against that pool. Further, assume that there are two tranches of notes: the A series and B series, with \$50 million face value of each outstanding. For simplicity’s sake, assume the notes pay interest monthly. On any interest payment date, we could pay the B series the entire prepayment amount $A - S$ and reduce the face value on the B series notes by $A - S$. For example, if on the first interest payment date, $A - S =$ \$10 million, then we would pay the \$10 million to the B series note holders and reduce the face value on the B series to \$40 million. Thus, any prepayment amount less than or equal to \$50 million will be completely absorbed by the B series note holders. So the net effect is to cushion the A series against a certain amount of prepayment risk. The B series note holders will likely demand something in return for bearing this risk.