A Novel Framework for Mortgage-Backed Securities

A Novel Framework for Mortgage-Backed Securities

Chris Wijnbergen

Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Prof. dr. P.A. Bekker

A Novel Framework for Mortgage-Backed Securities

Chris Wijnbergen

February 25

, 2018

Abstract

Contents

1 Introduction 4

2 Background 7

2.1 Mortgage-backed securities . . . 7

2.2 Risks involving mortgage payments . . . 9

2.3 Basic affine jump diffusion processes . . . 14

2.3.1 Intensity processes . . . 14

2.3.2 Risks at individual debt contract level . . . 15

2.4 Risks at debt portfolio level . . . 18

2.4.1 Including correlation . . . 18

2.5 Setting parameter values: an analytical solution . . . 19

3 Methodology: introducing prepayment processes 22 3.1 Justification . . . 22

3.2 Prepayment processes . . . 25

4 Simulating intensity processes 27 4.1 Motivation for parameter values . . . 27

4.2 Simulation of portfolio cash flows . . . 30

5 Mortgage-backed securities 34 5.1 Plain vanilla collateralized mortgage obligations . . . 35

5.2 Proposing dynamic mortgage-backed securities . . . 39

5.2.2 Dynamic mortgage-backed security with threshold ledger . . . 46

6 Results: evaluating mortgage-backed securities 48

6.1 Evaluation . . . 49

7 Discussion 57

Appendices 62

1

Introduction

Residential mortgages have become an increasingly appealing investment opportunity for financial institutions since yields on investment grade bonds are historically low. However, whereas the scheduled payments from an investment grade bond occur with a very high probability, the same cannot be said for residential mortgage payments. This thesis shows the shortcomings of conventional financial products with residential mortgages, the so called mortgage-backed securities (MBSs) and offers a solution to improve the certainty in scheduled payments. It proposes a framework of a new type of mortgage-backed security of which the scheduled payments occur with a high probability. The framework of the proposed mortgage-backed security gives insight how loss frequency and loss severity of payments of financial products with residential mortgages could be mitigated.

In order to present this new type of mortgage-backed security, a novel framework is required such that uncertainty in mortgage payments can be assessed and controlled in a transparent manner. Uncertainty in mortgage payments arises primarily from prepayment and default. Both processes are extremely complex and depend not merely on macro-economic factors such as interest rates. Regulation plays a major role in prepayment and default behavior as well. A wide variety of research has tried to capture multiple relevant factors for these behaviors, for example, how the lack of liquidity causes default behavior by Elul et al. (2010). However, one can imagine that it is practically impossible to capture all of them.

risk. We merely assume that certain drivers exist, without pretending to know to which extent. The method that allows for this agnosticism is based on basic affine jump diffu-sion processes (BAJD), previously used by Duffie and Garleanu (2001) to model default processes in a specific financial security, namely collateralized debt obligations.

Modelling both default and prepayment processes according to a BAJD process, results in several advantages. First of all, the setup of the BAJD process involves a parameter setting that allows for an intuitive interpretation. Therefore, from a supervisory and regulatory point of view, it is relatively easy to simulate “what-if” scenarios which would be difficult using other models. Second, it offers an appealing solution by considering prepayment and default risk as independent, otherwise often assumed to be negatively correlated. We show that this independence assumption functions as a lower bound of expected mortgage payments. This results in a conservative estimate of payments originating from the mortgage portfolio, which can be seen as an appealing characteristic from a risk-management point-of-view. This method provides the basis to evaluate various types of mortgage-backed securities.

high payment certainty can be met.

This brings us to our research question: to what extent can the loss frequency and loss severity of scheduled payments of a conventional mortgage-backed security be improved upon?

2

Background

This section starts with a general outline of mortgage portfolio characteristics and related financial products such as mortgage-backed securities. Furthermore, insight in the mod-elling of several risks involving mortgage payments is necessary. Therefore, an outline is given on these topics in which previous literature is discussed.

2.1

Mortgage-backed securities

Mortgage portfolios can be considered as a relatively safe investment since defaults and foreclosures of Dutch mortgages are relatively rare, for example, the annual default rate was only around 0.07% in 2012 according to Dutch Banking Association (2014). This safety partially originates from the diversification benefits of having many mortgages (and therefore borrowers) in a portfolio. This is one of the reasons that the securitization1 _of

mortgage portfolios is a popular method to sell a financial product that has exposure to the mortgage market. The product that results from the securitized mortgage portfolio is known as a mortgage-backed security (MBS). The first MBSs were relatively simple securities that paid out the cash flows from the mortgage portfolio on a per share basis, as in Figure 1. This structure is also known as a pass-through MBS. Since the introduction of this product, financial institutions could gain exposure to the mortgage market without having to look after their own mortgage portfolio.

However, more complex securities soon followed after it was found that large institu-tional investors demanded more certainty in MBS payments. This resulted in the intro-1_{Securitization is the sale of portions of the mortgage portfolio’s cash flows (securities) to third party}

duction of a payment priority scheme which could not be obtained using a pass-through MBS. A structure with different classes or tranches2 _{was created in which a priority in}

payments was explicitly formulated. As a consequence, financial products with varying low or high risk profiles could be created for institutional investors or hedge funds, re-spectively. This structure is known as a collateralized mortgage obligation3 _{(CMO) and}

is depicted in Figure 2. It shows how different tranches bear different amounts of risk. Huang et al. (2007) showed how a ledger of a CMO could be optimally designed such that more certainty was added to its payments. A ledger can be seen as a bank account or reserve such that future expected payments occur with a higher probability. They made use of two metrics that were set out against the efficiency of the ledger size. That is, a smaller ledger would be beneficial and therefore more efficient. The metrics they used were the loss frequency and the loss severity of scheduled payments. However, since they developed an optimization method only to set the optimal amount in the ledger, they failed to create insight in how the ledger should be constructed. Furthermore, they failed to take default risk into account.

2.2

Risks involving mortgage payments

An assessment of the risks involving the mortgage portfolio’s cash flows has to be made be-fore the financial products (securities) from Figure 1 and 2 can be evaluated. A multitude of risk factors have an effect on the uncertainty of future mortgage payments. Whereas the risk in conventional bonds originates from default behavior, mortgages face an addi-tional major risk as well. This risk is called prepayment risk and arises from the option

2_{Throughout the paper, classes, tranches or notes are used interchangeably.}

of borrowers prematurely paying the face value of the contract. Furthermore, changing interest rates could change the payment behavior from a borrower. However, in contrast to this interest rate risk, default and prepayment risk directly affect the expected value of future mortgage payments and can be taken into account relatively simple.

Prepayment and default risks have been modelled using a wide variety of methods. Research was at first mainly focused on modelling these risks separately. Dunn and Mc-Connell (1981) were one of the the first to present a model to evaluate prepayment risk in mortgage-backed securities. They drew parallels between callable bonds and default-free mortgages that include the option to prepay. Consequently, they modelled the prepay-ment option similar to a financial option and calculated the prepayprepay-ment value in a method comparable to how Black and Scholes (1973) valued options. The concept boils down to that a borrower decides to prepay and refinance its contract at a lower rate whenever it is profitable to do so.

Schwartz and Torous (1989) built further on this by proposing a method to value MBSs by incorporating a prepayment function that depends on time, and long-term and short term interest rates. Unlike Dunn and McConnell (1981), they did not merely assume that a borrower deterministically prepays when it is profitable to do so. They assumed that prepayments occur with some probability. However, the assumption was made that MBSs were not subject to default risk since at that time, many mortgages were assumed to be either directly or indirectly backed by the U.S. government. Since the sub-prime mortgage crisis we have learned that this assumption is not always valid and that default risk should be taken into account explicitly.

mortgage-backed securities, namely CMOs with several types of payment structures. Furthermore, they added the assumption from Stanton (1995)4 _{to their framework which implies that}

a background level of prepayments exists following a time-dependent hazard rate.

Similar to modelling prepayment processes, a common approach of representing default processes is using option valuation methods. Foster and Van Order (1984) used a similar method to model default processes as Dunn and McConnell (1981) modelled prepayment processes. However, whereas the prepayment of a mortgage can be compared to a borrower exercising a call option, default on a mortgage can be seen as a borrower exercising a put option. Kau et al. (1994) extended Foster and Van Order (1984) by calculating default probabilities of mortgages using predictions of interest rate term structures and house prices. They took the approach that the mortgage value versus the house value is the main driver of default risk. In this setting, a mortgagor will default if negative equity occurs, that is, if the value of the mortgage is higher than the value of the house (or, the put option is “in the money”).

This is in contrast to the approach that defaults are caused by a homeowner’s lack of liquidity. Ambrose et al. (1997) went further and showed that when lenders have the ability to impose transaction costs on a defaulting contract, this drastically impacts a mortgagor’s defaulting behavior. Even further, Ghent and Kudlyak (2011) showed that recourse5 _{affects default behavior by decreasing the sensitivity to negative equity. In}

short, it follows from previous research that many factors contribute to the complexity of default behavior.

4_{In 1993, Stanton (1995) was an unpublished manuscript.}

5_{Recourse indicates whether the lender can obtain other assets than the borrower’s house (collateral)}

Taking interest rate levels into account appears to be a proper approach since default and prepayment risk depend on interest rate levels. That is, when interest rates are high, it is more likely for a borrower to default. Even further, with high interest rates, it does not make sense to prepay the mortgage and apply for a new mortgage at a higher rate, hence prepayment would be low. Because of this negative relation between default and prepayment risk, Deng et al. (2000) used an approach where default and prepayment risk are modelled jointly. However, this approach has its flaws since they fail to take a lack of borrower’s liquidity into account. Elul et al. (2010) found that both a lack of liquidity and negative equity significantly affect default behavior. In short, no definitive method exists to model mortgage payments.

given in Section 2.3.

2.3

Basic affine jump diffusion processes

This section is dedicated to give an extensive explanation of modelling default processes as in Duffie and Garleanu (2001). They applied the general framework from Duffie et al. (2000) to model default behavior within a specific type of financial security, namely a collateralized debt obligation. Duffie et al. (2000) set out examples how basic affine jump diffusion (BAJD) processes could be used for modelling several econometric problems. Furthermore, they proposed how a wide variety of securities and derivatives could be priced. Duffie and Garleanu (2001) showed an extension and used this BAJD concept to model default behavior within debt contracts such that collateralized debt obligations could be priced. Below, general concepts of this approach to model default behavior will be described. Finally, it will be explained how an analytical solution can be obtained using this framework.

2.3.1 Intensity processes

In order to calculate default probabilities for debt contracts, Duffie and Garleanu (2001) used basic affine jump diffusion processes in which default arrives following a default intensity. For the sake of introducing the concept of intensities, these can be compared to the well known hazard function h used in survival analysis that is defined by

h(t) = lim

dt→0

P (t ≤ X < t + dt|X ≥ t)

dt .

employs a stochastic intensity process λ that leads to a stopping time τ at which an event of interest occurs, that is, the default of a debt contract. Intuitively, this can be seen as

P (τ < t + ∆t|Ft) ≈ λ(t)∆t, for small ∆t > 0,

where Ft denotes all information available up to time t. A more formal definition follows

from the probability of τ preceding the future period t + s, with s > 0, at the current time t < τ that is given by

P (τ < t + s|Ft) = 1 − P (τ > t + s|Ft) = 1 − E  exp Z t+s t −λ(u)du  |Ft  . (1)

Duffie and Garleanu (2001) assumed that intensity λ is a basic affine jump diffusion process, this process will be set out below.

2.3.2 Risks at individual debt contract level

Before the default processes of all debt contracts in an entire debt portfolio can be con-sidered, the default process of a single contract has to be considered first. Duffie and Garleanu (2001) assumed the existence of stochastic intensity processes λ such that all performing debt contracts default with probabilities satisfying (1). Let the default inten-sity rate λ at time t be the solution to the following affine jump diffusion process with initial value λ(0),

dλ(t) = κDθD− λ(t) dt + σDpλ(t)dWD(t) + ∆JD(t), (2)

where κD_{, θ}D _{and σ}D _{denote respectively, the mean reversion rate, mean and volatility}

process. As an addition to the CIR process, the basic affine jump diffusion process allows for shocks that can be seen as jumps in the intensity. This shock component is denoted by ∆JD and indicates a pure jump process that is independent of WD. The jumps from

∆JD _{arrive following an independent Poisson process with arrival rate ι}D _{and its jump}

size follows an exponential distribution with mean µD_.7 _{Furthermore, the long-run mean}

of the intensity in (2) is given by θD+ (ιDµD)/κD.

Basic affine jump diffusion processes can be used in simulation studies, however an important benefit is that an exact analytical solution exists as well. Duffie and Garleanu (2001) offered an exact analytical solution for calculating default probabilities. Following this framework, the default probability between periods t and t + s as in (1) of a debt contract following a default intensity process as in (2) can be calculated by

P (τ < t + s|Ft) = 1 − E  exp Z t+s t −λ(u)du  |Ft  , (3) = 1 − exp {α(s) + β(s)λ(t)} ,

where α(s) and β(s) are solutions of

α0(s) = mβ(s) + ιD µ D_β(s) 1 − µD_β(s), (4) β0(s) = nβ(s) +1 2pβ(s) 2 + q, (5)

with m = κD_θD_{, n = −κ}D_{, p = (σ}D₎2_{, q = −1, and boundary conditions α(0) = 0 and}

7_{As Duffie et al. (2000) show, the jump size of a basic affine jump diffusion process is not restricted}

β(0) = 0. The solution to the ordinary differential equations (4) and (5) is given8 _by α(s) =m(a1c1− d1) b1c1d1 log c1+ d1e b1s c1+ d1  + m c1  s +ι D_(a 2c2− d2) b2c2d2 log c2+ d2e b2s c2+ d2  + ι D c2 − ιD  s, (6) β(s) =1 + a1e b1s c1+ d1eb1s , (7) where c1 = −n +pn2_{− 2pq} 2q , d1 = n +pn2_{− 2pq} 2q , a1 = −1, b1 = d1(n + 2qc1) + a1(nc1+ p) a1c1− d1 , a2 = d1 c1 , b2 = b1, c2 = 1 − µD c1 , (8) d2 = d1− µDa1 c1 , (9)

with n = −κD, p = (σD)2, q = −1, and m = κDθD, for the specific case of a survival

function for a single debt contract as in (3). A generalization is provided in section 2.5 where multiple contracts will be considered which involve different values for q, ιD _and

m.

8_{Duffie and Garleanu (2001) erroneously formulated the solution to the ordinary differential equations}

2.4

Risks at debt portfolio level

An advantage of the basic affine jump diffusion process in (2) is that the parameter setup allows for a relatively large amount of flexibility in the construction of default intensities. Up to now the framework of the stochastic components has been formed such that default probabilities of a single debt contract can be modelled. However, another important advantage of the framework is that affine jump diffusion processes allow for correlation in default intensities between debt contracts within a debt portfolio.

2.4.1 Including correlation

An advantage that follows from a result in Duffie and Garleanu (2001) allows to include correlation in default intensity between different debt contracts. The result is set out as follows. Let there exist two independent basic affine jump diffusion processes Y and Z with parameters (κ, θY, σ, µ, ιY) and (κ, θZ, σ, µ, ιZ). The result holds that X = Y + Z

is a basic affine jump diffusion process as well. The parameters of X are then given by (κ, θY + θZ, σ, µ, ιY + ιZ).

When this result is applied to the current framework, the following default intensity for the ith debt contract at time t is obtained by

λi(t) = λS(t) + λ∗i(t), (10)

where λS and λ∗i are basic affine processes with parameters (κD, θSD, σD, µD, ιDS) and

(κD_{, θ}∗D_{, σ}D_{, µ}D_{, ι}∗D_{), respectively. Consequently, following the aforementioned result}

λi can be seen as a basic affine process with parameters (κD, θD, σD, µD, ιD) with θD =

θD S + θ

∗D _{and ι}D _{= ι}D S + ι

∗D_.9 _{A correlation between two debt default processes now}

9_{All idiosyncratic processes λ}∗

follows from considering λS as a systemic risk factor that is equal for every contract in

the portfolio. Therefore this correlation10 _{can be defined as}

ρD = ι D S ιD = θD S θD. (11)

Furthermore, λ∗_i is the stochastic process that accommodates unsystematic or

idiosyn-cratic risk for every ith debt contract.

2.5

Setting parameter values: an analytical solution

A key aspect of properly evaluating debt payment risks is to set the parameters of the intensity processes in (2) correctly. The intensity process from Duffie and Garleanu (2001) offers an insightful parameterization such that several key features of the default process can be simulated comprehensively. Among these key features are primarily the long-run means of the processes. However, the basic affine jump diffusion processes allow for the application of shocks, which is interesting from a risk management point of view. When default shocks can be set to levels similar as in financial crises, this helps to understand the risks of debt securities. The method from Duffie and Garleanu (2001) allows for a clear a priori evaluation of the default risks following an intensity as in (2) for different parameter values of ρD _{as can be seen in Figure 3.}

Debt default probabilities can be obtained analytically. Let τj denote the default time

of the jth contract, and let dj denote the event that contract j defaulted before time T ,

that is

dj = {τj < T },

furthermore, let dC_j be the complement of dj.

Figure 3: Probability of the number of defaults after 10 years for ρD equals 0.1, 0.5 and 0.9, respec-tively for a portfolio of N = 30 mortgages. The other parameters are set as κD, θD, σD, ιD, µD
= (0.5, 0.04, 0.02, 0.175, 0.12).

When the set of parameters are equal for each debt contract, then by symmetry of each jth debt contract, the probability of k defaulted contracts before time T is then given by P (M = k) =N k  ˜ q(k, N ), where ˜ q(k, N ) = P d1∩ · · · ∩ dk∩ dCk+1∩ · · · ∩ d C N
.

Then for k > 0 and using _{N −j}k
= 0 when k < N − j,

˜ q(k, N ) = N X j=1 (−1)(j+k+N +1)  k N − j  pj, pj = P (d1∪ · · · ∪ dj) ,

which is proved by induction in Duffie and Garleanu (2001).

is pj = 1 − P (min{τ1, ..., τj} > T ) , = 1 − E " exp ( − Z T 0 j X i=1 λi(t)dt )# , = 1 − exp {αS(T ) + βS(T )λS(0) + jαi(T ) + jβi(T )λ∗i(0)} ,

where αi(T ) and βi(T ) are equal to the solutions α(s) and β(s) of the ordinary differential

equations (6) and (7), respectively for ιD = ι∗D and m = κDθ∗D. Furthermore, the

systemic component follows from αS(T ) and βS(T ) which are equal to the solutions of

the same ordinary differential equations α(s) and β(s) in (6) and (7). However, this systematic component uses parameters q = −j, ιD = ιD_S and m = κDθD_S.

3

Methodology: introducing prepayment processes

Whereas Duffie and Garleanu (2001) only applied the basic affine jump diffusion pro-cess from (2) to model default behavior of general debt contracts in collateralized debt obligations, we introduce these types of processes to model default behavior in mortgage contracts. Since a mortgage contract is merely a specific type of debt, this is not a distinct addition to previous research.

However, a novelty of this paper originates from the introduction of the framework of Duffie and Garleanu (2001) to model prepayment behavior. This framework is required to answer the question to which extent conventional mortgage-backed securities could be improved upon with respect to payment certainty. First, this section will outline multiple arguments that provide a justification of modelling prepayment using this framework from Duffie and Garleanu (2001). Thereafter, an outline is given how prepayment is incorporated within this framework.

3.1

Justification

The justification of modelling the prepayment of mortgage contracts using the framework from Duffie and Garleanu (2001) has to be considered twofold. The first question is whether the prepayment intensities can be modelled using a basic affine jump diffusion process. The second question is whether the assumption that prepayment behavior and default behavior in mortgage contracts are independent is valid. Arguments in favor of both statements are provided below.

ques-tion is whether prepayment behavior can be seen as similar to default behavior within this framework. As mentioned in Section 2.3.1, default of a debt contract is a process that occurs at a particular default time τ and arrives at some intensity λ. In previous literature, Stanton (1995) already assumed the existence of a certain exogenous, or base-line, prepayment process following a time-dependent hazard rate. A borrower prepays at a certain time following this hazard rate. Conceptually, the notion of prepayment haz-ard rates is not far from the prepayment intensities that this paper proposes. The main difference is that, whereas these hazard rates proposed by Stanton (1995) are relatively simple, our assumption is that the prepayment intensity follows a more complex basic affine jump diffusion process.

Another argument can be seen in the current literature. As mentioned in Section 2.2, both processes are often modelled by a very similar concept using option theory. On the one hand, Foster and Van Order (1984) modelled default behavior as a put option, where the borrower sells the house to the lender for the remaining value of the mortgage contract. On the other hand, Dunn and McConnell (1981) assumed that borrowers exercise their call option (or decide to prepay the mortgage contract) only when refinancing the mortgage contract is cheaper than keeping the current mortgage contract. In short, a similarity between default and prepayment behavior is shown since both processes can be explained by making use of option theory.

arguments in favor of using the framework of Duffie and Garleanu (2001) to model pre-payment behavior of mortgage contracts.

The second question is whether the assumption of independence between prepayment risk and default risk can be justified in this setting. No closed form solution exists for modelling these prepayment and default processes jointly. However, it is possible to simu-late these processes independently upon first default or prepayment. It is very important to state that this approach does not try to provide a pricing model of mortgage-backed securities. It merely offers an approach to model the risks of payments of mortgage-backed securities. It can be shown that considering these processes as independent yields a con-servative estimate of what could be expected in real life, that is, the real world would yield higher cash flows. This is a very attractive property from a risk-management point of view.

Default yields an additional cash flow in the form of a (partially) recovered amount of the face value. Furthermore, prepayment yields an additional cash flow since the bor-rower decides to increase the amortization for a specific period. Default and prepayment intensities are often considered as counterparts since when default is high, prepayment is low, and vice versa. Because this negative correlation exists in real life, the independence assumption will yield a conservative estimate. This idea will be made clear by the follow-ing example. Assume that U1(t) and U2(t) are processes denoting prepayment and default

are defined by     dU1(t) dU2(t)     =     1 ρ¯ ¯ ρ 1         dW1(t) dW2(t)     ,

where W1(t) and W2(t) are Wiener processes. Furthermore, let U1(0) = U2(0) = 100. It

follows that the variance of dU (t) = dU1(t) + dU2(t) is increasing in ¯ρ. When ¯ρ < 0, that

is, when dU1(t) and dU2(t) are negatively correlated, the quantiles of U (t), denoted by

F_α−1(U (t)), are less extreme than when ¯ρ = 0. When translating this to our example,

that is, when negative correlation is assumed between default and prepayment payments as in the real world, this results in less extreme payments from the mortgage portfolio than when default and prepayment payments are assumed to be independent. This shows that assuming prepayment and default behavior to be independent, the risks involved in the cash flows originating from a mortgage portfolio are overestimated. Moreover, since cash flows from prepayment are many times larger as the recovery cash flows, the severity of this effect that follows from the independence assumption is reduced even further. As such these two questions are now answered and it is shown next how the prepayment process is incorporated in this framework.

3.2

Prepayment processes

Since arguments in favor of modelling prepayment processes have been provided, it will now be shown how prepayment can be incorporated in the proposed framework. The prepayment intensity ` can be modelled following the same method as the default process in Section 2.3. It is defined at time t by solving

where κP_{, θ}P _{and σ}P _{denote the mean reversion rate, mean and volatility parameter,}

respectively. Similarly, as in the default intensity process (2), WP _{denotes a Wiener}

process and ∆JP denotes a pure jump process with jump times according to a Poisson

arrival process with arrival rate ιP _{and an exponentially distributed jump size with mean}

µP_{. Consequently, the prepayment intensity and its corresponding probabilities can be}

calculated in a similar manner as in (3)-(9).

Similar to (10), this approach can be used to include correlation across the mortgage contracts’ prepayment intensities. The total prepayment intensity at time t for the ith mortgage contract is defined by

`i(t) = `S(t) + `∗i(t),

where `S(t) is the systemic or common component with parameters (κP, θSP, σP, µP, ιPS)

and `∗_i(t) is the idiosyncratic component for the ith mortgage contract with

parame-ters (κP_{, θ}∗P_{, σ}P_{, µ}P_{, ι}∗P_).11 _{As a result `}

i is a basic affine process with parameters

(κP_{, θ}P_{, σ}P_{, µ}P_{, ι}P_{). Furthermore, the same correlation structure θ}P

S = ρPθP and ιPS =

ρPιP holds as in (11).

Furthermore, it naturally follows that the same method to obtain an analytical solution from Section 2.5 can be applied to prepayment intensity. Since a method to model default and prepayment has been defined, the key elements to simulate mortgage payments have been outlined. This forms the basis of Section 4, where mortgage portfolio cash flows will be simulated.

11_{As is the case with the ideosyncratic and systemic default components, all idiosyncratic processes}

4

Simulating intensity processes

This section illustrates how the simulation of the default and prepayment intensity pro-cesses from Sections 2.3 and 3 results in mortgage portfolio cash flows. First of all, a motivation is provided for the choices of the parameter values that are used for the simu-lation of the mortgage portfolio. Second, an outline will be given of the framework of the mortgage portfolio cash flows. This will result in simulated cash flows of a mortgage port-folio. These cash flows are required to answer the research question to evaluate various types of mortgage-backed securities.

4.1

Motivation for parameter values

One of the problems regarding default is that it not only depends on macro-economic factors, but on regulatory factors as well. This creates a high amount of heterogeneity between mortgage portfolios under different countries with different jurisdictions. This would result in different parameters for our default intensity process in (2) as well.

Unfortunately, mortgage data is mostly proprietary and hard to obtain. However, Dutch Banking Association (2014) published data on Dutch mortgage defaults. Using this report, the default parameters are determined by visual inspection of historical default graphs and are depicted in Table 1. This results in the probability of defaults after a certain number of years, as can be seen in Figure 4.

Parameter Value κD _0.3 θD 0.04% σD _0.2% µD _0.04% ιD 0.175 ρD _0.6

Table 1: Parameter setup for the default intensity process in (2).

2 for the prepayment intensity process in (12) was set by visual inspection. This resulted in a probability of a number of prepayments after different numbers of years as shown in Figure 5. Parameter Value κP 0.5 θP _4% σP _2% µP 12% ιP _0.175 ρP _0.6

Table 2: Parameter setup for the prepayment intensity process in (12).

The difference in parameters follows mainly from the fact that the prepayment of a mortgage contract is much more common than the default of a mortgage contract. However, the downside of setting parameter values by visual inspection is that mistakes are easily made. The long-run mean of an intensity as in (2) can be easily identified, however the value of µD _{that denotes the mean of an exponentially distributed shock is}

harder to identify. Furthermore, the correlation between different contracts, denoted by ρD _{in (11) is hard to identify as well. This problem is not limited to default intensities and}

holds for prepayment intensities too. Therefore in order to make the results of this paper more robust, three alternative parameter sets will be used. Two of these alternatives are aimed at changes in the importance of systemic risk which is denoted by ρD _{and ρ}P_{. The}

in Appendix A.

Figure 5: This plot indicates the probability of the number of prepayments of a portfolio of 30 mortgages after 10, 20 and 30 years using the analytical solution provided in Section 2.5 with a parameter setup from Table 2.

To evaluate the conditional prepayment rate and default rate, the outcomes of 500 sim-ulations using the parameter setup of Tables 1 and 2 are shown in Figure 16 in Appendix B.

4.2

Simulation of portfolio cash flows

Consequently, each contract defaults and prepays following probabilities calculated from the simulated intensities.

We assume for simplicity that each contract i has a face value F Vi,t of 100 for t = 0

and payments follow a monthly linear amortization scheme for a period of 30 years. A linear amortization scheme is such that besides prepayments, amortization equals 100/360 in all 360 months until full amortization is reached. Furthermore, payments take place after each month, hence interest is paid over the remaining face value (after monthly amortization).

Most of the time, a borrower remains paying the required amount of money according to the contract. However, there are two scenarios in which a deviation from this contrac-tual payment can occur: default or prepayment. When a contract defaults, the remaining face value of the contract turns to zero.

F Vi,t =       

0 if contract i defaults before or at time t,

F Vi,t−1− Ai,t if contract i does not default before or at time t,

where Ai,t is the amortization of contract i at time t. Amortization consists out of normal

amortization payments and prepayments. Given the contract still performs, it is defined by Ai,t =       

F Vi,t−1 if contract i prepays at time t, F Vi,0

360 if contract i does not prepay at time t.

However, not all cash flows from the mortgage portfolio originate from performing contracts. When a borrower defaults, a sale of the property takes place such that an amount can be recovered where approximately a recovery rate of 70 % can be expected.12 12_{From Dutch Banking Association (2014). In practice recovery is a complex process from which}

The chosen recovery amount Ri,t is therefore Ri,t =       

min (F Vi,t−1, 0.7 · F Vi,0) if contract i defaults at time t,

0 if contract i does not default at t.

This results in a total cash flow from mortgage contract i at time t of

CFi,t =               

Ri,t if contract i defaults at t,

Ai,t+ Ii,t if contract i does not default before or on time t,

0 if contract i defaulted before t, Ii,t =

ri,t

12 · F Vi,t,

where Ii,t and ri,t denote the interest payment and interest rate of contract i at time t,

respectively. For simplicity’s sake, we assume ri,t = r = 3.25% to be the fixed for all

contracts i = 1, ..., N . The total cash flow of the entire portfolio at time t is then given by CF_tP = N X i=1 CFi,t.

The simulated mortgage portfolio cash flows differ widely across different simulations as can be seen in Figure 6. A more clear conclusion can be drawn from Figure 7 in which quantile plots of the simulated cash flows are shown. The clear upward skewness of the portfolio’s cash flows is noteworthy.

Finally, it is important to note that the total repayment of the face value of a contract is not limited to the regular amortizing payments and prepayment, but should also include the recovery amount. Therefore, the portfolio’s total amortization (including recovery) is

Figure 6: Results of 500 simulations of a mortgage portfolio of 100 mortgages following default and prepayment processes from (2) and (12), respectively. Tables 1 and 2 are used for parameter input.

denoted by AP ∗_t = N X i=1 A∗_i,t, where

A∗_i,t = Ai,t+ Ri,t.

Figure 7: Depicts 1%, 50% and 99% quantiles of 500 simulations of a portfolio of 100 mortgages for each month. In addition, one specific outcome path and a ‘normal’ payment scheme without default and prepayment are portrayed.

5

Mortgage-backed securities

The previous sections described how default and prepayment intensities could be trans-formed into probabilities such that payments of a mortgage portfolio could be simulated. However, besides offering a novel approach to simulate mortgage portfolio cash flows, it still remains to provide an answer to the research question to what extent the payment certainty of conventional mortgage backed securities could be improved upon.

5.1

Plain vanilla collateralized mortgage obligations

A plain vanilla collateralized mortgage obligation is a type of mortgage-backed security which involves a sequential payment structure. It is a conventional and relatively simple collateralized mortgage obligation (CMO) and consists out of multiple classes or tranches. McConnell and Singh (1993) discussed a conventional CMO with five subordinated classes. The same structure is discussed in Section 2.1 and visualized in Figure 2. It consists of five tranches in total with a waterfall payment structure. The first three tranches hold the first three sequential payment priorities. Secondly, an interest accrual tranche, also known as a Z-bond and thirdly an equity tranche are considered. Since the main scope of this paper is on payments that occur with a high probability, the focus will be on the most senior tranche, i.e. tranche A. It can be justified to assume that no costs are made by the special purpose vehicle, since these costs will incurred by the equity tranche and not by the tranche of interest, that is, tranche A.

𝐶𝐹𝑡𝑃

𝐶𝐹𝑡𝐴 𝐶𝐹𝑡𝐵 𝐶𝐹𝑡𝐶 𝐶𝐹𝑡𝑍 𝐶𝐹𝑡𝐸

Figure 8: The special purpose vehicle (SPV) receives payments from the mortgage portfolio and dis-tributes it according to a scheme from (13) to (14).

are set out below. The A tranche, B tranche, C tranche and Z-bond have an initial face value of F VA

0 , F V0B, F V0C, F V0Z, respectively. The payments from the mortgage portfolio,

as simulated in Section 4.2, are directed towards the three different senior tranches in the following manner

CF_tA = AA_t + I_tA, (13)

CF_tB = AB_t + I_tB,

CF_tC = AC_t + I_tC,

where CFh

t , Aht and Ith are the cash flow, amortization and interest payment for tranche

h = A, B, C, respectively, in the t-th period. Furthermore, the remaining face value of tranche A, B, and C at time t is defined by F VA

t , F VtB, F VtC, respectively. The

amorti-zation for the three tranches is defined by

F V_tA= max ( F VA− t X j=1 AP ∗_j , 0 ) , F V_tB = max ( 0, min F V₀A+ F V₀B− t X j=1 AP ∗_j , F V₀B !) , F V_tC = max ( 0, min F V₀A+ F V₀B+ F V₀C − t X j=1 AP ∗_j , F V₀C !) , AA_t = F V_t−1A − F V_tA, AB_t = F V_t−1B − F VB t , AC_t = F V_t−1C − F VC t ,

defined by I_tA = rA· F VA− t X j=1 AA_j ! , I_tB = rB· F VB− t X j=1 AB_j ! , I_tC = rC· F VC − t X j=1 AC_j ! ,

where rA, rB and rC are the interest rates for the A, B and C tranche, respectively.13

The interest accrual tranche or (Z-bond) accrues its interest payments first to the principal. It only starts to pay at time tZ when the A, B and C tranche are fully

amortized. Hence, the face value of the Z-bond at time t is defined by

F V_tZ =                F VZ t−1+ rZ 12 · F V Z t−1 if t < tZ, maxn0, F VZ t−1−  Pt j=1A P ∗ j − (F V0A+ F V0B+ F V0C) o if t = tZ, max0, F V_t−1Z − AP ∗ t if t > tZ,

where rZ denotes the interest rate on the Z-bond. The amortization of the Z-bond at

time t is defined by AZ_t =        0 if t < tZ, F V_t−1Z − F VZ t if t ≥ tZ.

After amortization of the accrued face value of the Z-bond has been initiated, no further accrual takes place. The cash flow of the Z-bond at time t ≥ tZ consists out of the

amortization and interest payment IZ

t and is defined by

CF_tZ = AZ_t + I_tZ,

I_tZ = rZ· F V_tZ.

13_{As McConnell and Singh (1993) indicate, no rate of prepayments should result in inadequate funds}

of coupon interest payments, hence rA_{, r}B_{, r}C _{< r, where r denotes the interest rate on the mortgage}

Now, the cash flows to the equity tranche remains. These payments can be seen as a residual and are as follows

CF_tE = maxCF_tP − (CFA

t + CFtB+ CFtC+ CFtZ), 0 . (14)

Figure 9 denotes the result of a mortgage portfolio simulation evaluated on this type of mortgage-backed security. However, a problem arises when certainty is demanded in the cash flow of a particular tranche. The amortization of plain vanilla CMOs occur in a sequential (waterfall) structure. Therefore, the date of the final payment is uncertain for each tranche.

This will be made clear by using the following concept of Value-at-Risk (VaR). Note that this concept will be used throughout this paper. VaR is a quantitative method to evaluate risks. It is partially because of its simplicity and intuitiveness a widely used method in regulatory guidelines such as Basel III and Solvency II. McNeil et al. (2015) stated the definition of the VaR of a random variable X ∼ FX by

VaRγ(X) = min{x ∈ R : P (X > x) ≤ 1 − γ}, (15)

= min {x ∈ R : FX(x) ≥ γ} , γ ∈ (0, 1). (16)

Intuitively, it can be seen as a similar concept as the γ% quantile of FX. Therefore, the

method for evaluating the uncertainty in mortgage-backed securities’ payments in this paper is done by assessing cash flow quantiles of the simulations in each period.

structure is not optimal in CMOs when payment certainty is demanded. Therefore, in Section 5.2 a novel MBS payment structure will be proposed.

5.2

Proposing dynamic mortgage-backed securities

The purpose of this study is to evaluate conventional mortgage-backed securities and to find out to what extent an improvement can be made with respect to certainty in scheduled MBS payments. The theoretical framework of a conventional mortgage backed security was introduced in Section 5.1, this section is dedicated to propose a novel alternative mortgage-backed security. In order to obtain a tranche of a mortgage-backed security in which the payments of one tranche occur with a very high amount of certainty, the risk has to be directed away from one tranche towards the other tranche. A novel payment structure is introduced in this section such that an improvement is made with respect to certainty in payments compared to more traditional mortgage-backed securities.

This novel payment structure only contains two tranches as shown in Figure 11. It is more simplistic than the plain vanilla collateralized mortgage obligation (CMO) from Section 5.1. Furthermore, where the plain vanilla CMO pays according to a fixed and constant coupon rate, a dynamic mortgage-backed security (DMBS) offers an entirely different approach. It begins with what could possibly happen to the portfolio and adjusts its payment structure on that.

However, insight in the uncertainty of cash flows of mortgage-backed securities has to be gained first. This insight is obtained in the concept of Value-at-Risk as in (15). This risk-based approach can be seen as:

𝐶𝐹𝑡𝑃

𝐶𝐹𝑡𝑅 𝐶𝐹𝑡𝐴

portfolio’s cash flow is determined. That is, a cash flow amount CFA∗

t that is

achieved with a high degree of certainty in this period is set as the payment for the senior tranche.

• The residual acts as where the remaining cash flows are directed to. This tranche holds in fact most of the prepayment and default risk.

We define CFA

t and CFtR as the executed cash flows to the senior A tranche and the

residual tranche, respectively for time t. That is,

CFA∗_t = VaRγ CFtP|F0

(17) CF_tA= minCF_tP, CFA∗_t ,

CF_tR= max0, CF_tP − CF_tA ,

where VaRγ CFtP|F0
denotes the a priori simulated γ%-th quantile of mortgage portfolio

cash flow CFP

t . The expected cash flow levels of the DMBS for γ is 0.5%, 1% and 5% are

displayed in Figure 12.

As can be seen in Figure 13, the cash flows of the senior note (or tranche) are relatively low and are still subject to the risk of not being fully paid. The residual receives highly fluctuating cash flows since these are mainly influenced by the amount of prepayments and defaults for each time period.

5.2.1 Dynamic mortgage-backed security with ledger

Figure 12: Expected DMBS senior note cash flow levels from (17) for γ equals 0.5%, 1% and 5%.

portfolio. It is assumed that the interest rate on this ledger is zero. This can be justified since all costs are incurred by the residual tranche that is outside the scope of this paper.

𝐶𝐹𝑡𝑃

𝐶𝐹𝑡𝐿

𝐶𝐹𝑡𝑅

𝐶𝐹𝑡𝐴

Figure 14: Structure of a dynamic mortgage-backed security with ledger. Note that CFL

t can be both

positive and negative. It is positive (negative) when the cash flow goes into (out of) the ledger.

We define the amount in the ledger as Lt for each time t (with L0 = 0) and for each

period the ledger has a maximized amount of Lt. Furthermore, CFA∗t is defined as the

expected payment to the senior note at time t and CFA∗

t is defined as the realized payment

to the senior note.

CFA∗_t = VaRγ CFtP|F0
,

CF_tA∗= minCFP

t + Lt−1, CFA∗t .

Note that CFA∗

t cannot exceed the amount that is available from the portfolio and ledger

or the γ% quantile of the simulations. Moreover, the cash flow14 from the ledger to the

special purpose vehicle at time t is defined as CFL t . CF_tL=        minLt− Lt−1, CFtP − CFA∗t if CF_tP ≥ CFA∗ t ,

− minLt−1, max Lt−1− Lt, CFA∗t − CFtP

if CFP

t < CFA∗t .

Furthermore, the maximum amount in the ledger Lt is proportional to the expected cash

flow CFA∗

t at time t, that is,

Lt = x · CFA∗t where x ∈ (0, 1),

Lt = Lt−1+ CFtL.

Finally, the residual tranche cash flow is defined by

CF_tR∗ = CF_tP − CFA∗ t − CF

L

t . (18)

However, simply adding a ledger that holds a standard amount of liquid assets cannot be categorized as efficient. The ledger acts as a reserve and no reserves should be held when no abnormal prepayments and defaults occur. Therefore, this type of ledger will be used as a benchmark and a more efficient ledger will be introduced next.

5.2.2 Dynamic mortgage-backed security with threshold ledger

Including a ledger can be seen as increasing the certainty in payments to the DMBS’s senior note. However, it could be beneficial to consider more complex rules with respect to the amount of reserves that should be held in the ledger. The following alternative only adds money to the ledger when an extremely large cash flow takes place. As in the previous section, CFA∗

t and CFtA∗ denote the expected payment and realized payment to

the senior tranche at time t, respectively. This is formulated as

CFA∗_t = VaRγ CFtP|F0
,

CF_tA∗= minCF_tP + Lt−1, CFA∗t .

CF_t¯γ = VaR¯γ CFtP|F0
, where ¯γ ∈ (0, 1).

With notation similar to previous section, the maximum ledger amount and cash flow to the ledger are respectively defined as

Lt=        Lt−1+ x · CFtP − CF ¯ γ t
if CFP t ≥ CF ¯ γ t , Lt−1· 360−t₃₆₀
if CF_tP < CF_tγ¯, CF_tL=        minLt− Lt−1, CFtP − CFA∗t if CF_tP ≥ CFA∗ t ,

− minLt−1, max Lt−1− Lt, CFA∗t − CFtP

if CFP

t < CFA∗t .

6

Results: evaluating mortgage-backed securities

In Section 4, it was shown how mortgage payments could be simulated. Consequently, Section 5 discussed the payment structures of various mortgage-backed securities. This section combines these former two sections and evaluates the payments in a variety of mortgage-backed security types. First, the evaluation metrics of the different MBS struc-tures are discussed. Finally, Section 6.1 displays the final results.

An evaluation has to be made regarding the risks of several types of MBS payment structures. Consequently, similar measures as in Huang et al. (2007) which are described in Section 2.1, are quantified:

1. The frequency of the expected payments not fully paid (loss frequency). This is defined by

P1000

n=1

P360

j=1Ijn{Payment not met}

1000 · 360 ,

where I_jn{Payment not met} is an indicator function that equals 1 when the MBS

payment at time j of the nth simulation does not meet its a priori expected payment level.

2. The average loss size given the expected payment is not met (loss severity). This is defined by P1000 n=1 P360 j=1Loss n

j | Ijn{Payment not met} = 1

P1000

n=1

P360

j=1Ijn{Payment not met}

,

whereLossn_j | In

j {Payment not met} = 1 indicates the size of the shortage below

3. The average ledger size. This is defined by P1000 n=1 P360 j=1L n j 1000 · 360 ,

where Ln_j indicates the amount of cash in the MBS ledger at time j of simulation n.

When considering different types of asset backed securities that are not backed by mort-gages, the complication of payments occurring at a certain period are generally less severe. Figure 7 shows how prepayment causes cash flows to be shifted towards the present, a problem that is specifically relevant to mortgage payments.

6.1

Evaluation

The two mortgage-backed security types: the plain vanilla CMO and the newly proposed dynamic mortgage-backed security are compared in this section with respect to their payment certainty. Furthermore, the results of the different ledger types are investigated as well. The results for the standard parameter set are shown in Tables 3-5. The results for the alternative parameter sets are shown in Tables 12-18 in Appendix C.

Moreover, when considering the 0.5% Value-at-Risk of a Tranche A CMO as an ex-pected payment and evaluate portfolio cash flows on it, the loss severity remains relatively large. That is, 7.018 and 5.599 for a CMO tranche A with a face value of 30% and 60% of the total portfolio face value, respectively. This is because the final payment date of a CMO tranche is highly uncertain. Therefore, if the expected payment is not met, it is likely no payment will be made at all.

Now that it is clear that a DMBS can decrease the loss severity and loss frequency with respect to a traditional CMO, an addition is made to the DMBS. Adding a ledger to the DMBS, as in Section 5.2.1, helps to decrease the loss frequency. Table 3 shows that a simple ledger that holds at most 10% of the expected payment can decrease the loss frequency from 0.864% to 0.570%. However, the addition of this type of ledger to the DMBS does not lead to a decrease in the loss severity. Table 3 shows an increase in the loss severity from 1.404 without a ledger to 1.921 with a ledger.

From Table 3 it follows that the DMBS with a more sophisticated ledger, that only increases after a cash flow exceeds a threshold ¯γ as in Section 5.2.2, improves the efficiency. That is, for a given average ledger size, the loss frequency and severity of a DMBS with a ledger threshold of 99.5% is less than a DMBS with the standard ledger. This result holds for DMBS payment levels γ of 1% and 5% as well, as is shown in Tables 4 and 5.

frequency. However, it appears that when the average ledger size becomes larger, the efficiency of a ledger type with a higher retention threshold decreases.

Furthermore, the ledger is more efficient for higher retention thresholds ¯γ with respect to loss severity as well, as can be seen in Figure 15. That is, a lower loss severity is observed for a ledger type that increases only when extreme portfolio payments are observed. However, whereas the loss frequency decreases for a larger average ledger size, this is not necessarily the case for the loss severity.

Tables 12-20 show similar results for the three alternative parameter sets as Tables 3-5 do for the standard parameter set with respect to tranche A CMOs. Furthermore, this holds for adding a ledger from Section 5.2.1 as well.

Figure 17 shows the results for DMBSs with threshold ledger types for parameter set 2. Parameter set 2 assumes that the systemic risk component is more important than in the standard case. This results in more extreme mortgage default and prepayment behavior. Whereas for the standard parameter set the highest retention threshold ledger is most efficient for an average ledger size of up to 8 (top-left plot of Figure 15), this is not the case for the higher systemic risk assumption. In the latter scenario, the threshold of 90% becomes more efficient at an average ledger size of approximately 4 as can be seen in the top-left plot of Figure 17.

The results of assuming a lower importance of systemic risk, as in parameter set 3, is displayed in Figure 18. These results are very similar to the base case scenario of parameter set 1. However, for lower DMBS payment levels of γ equal to 0.5% and 1%, the efficiency of high retention threshold ledger regarding loss severity is not distinctively better as is the case with parameter set 1 (Figure 15).

pa-rameter set 4. This papa-rameter set corresponds to more extreme prepayment and default shocks, whereas the overall long-run mean is the same as in parameter set 1. Similar results for this parameter set hold as for parameter set 2. That is, the benefit of a higher retention threshold ¯γ is less pronounced as was in the base parameter set.

Type Loss frequency Loss severity Average in ledger Median CMO 30% 4.934% 5.447 -Median CMO 60% 12.978% 4.981 -VaR(CMO 30%, γ = 0.5%) 0.013% 7.018 -VaR(CMO 60%, γ = 0.5%) 0.028% 5.599 -DMBS; x=0 0.864% 1.404 -DMBS; x=0.1 0.570% 1.921 0.806 DMBS; x=0.2 0.545% 1.878 1.610 DMBS; x=0.3 0.526% 1.832 2.413 DMBS; x=0.4 0.509% 1.792 3.216 DMBS; x=0.5 0.509% 1.725 4.019 DMBS (x=0.005; ¯γ=99.5%) 0.579% 1.736 0.121 DMBS (x=0.01; ¯γ=99.5%) 0.509% 1.659 0.245 DMBS (x=0.05; ¯γ=99.5%) 0.257% 1.269 1.284 DMBS (x=0.1; ¯γ=99.5%) 0.173% 0.861 2.621 DMBS (x=0.2; ¯γ=99.5%) 0.124% 0.508 5.372 DMBS (x=0.4; ¯γ=99.5%) 0.100% 0.398 10.894 DMBS (x=0.001; ¯γ=90%) 0.560% 1.937 0.362 DMBS (x=0.0025; ¯γ=90%) 0.486% 1.913 0.908 DMBS (x=0.005; ¯γ=90%) 0.395% 1.872 1.820 DMBS (x=0.01; ¯γ=90%) 0.278% 1.882 3.655 DMBS (x=0.025. ¯γ=90%) 0.125% 1.743 9.194 DMBS (x=0.0005; ¯γ=75%) 0.566% 1.962 0.488 DMBS (x=0.001; ¯γ=75%) 0.519% 1.980 0.976 DMBS (x=0.0025; ¯γ=75%) 0.410% 2.006 2.445 DMBS (x=0.005; ¯γ=75%) 0.299% 2.057 4.900 DMBS (x=0.01; ¯γ=75%) 0.187% 2.132 9.827

Type Loss frequency Loss severity Average in ledger Median CMO 30% 4.934% 5.447 -Median CMO 60% 12.978% 4.981 -VaR(CMO 30%, γ = 1%) 0.038% 5.167 -VaR(CMO 60%, γ = 1%) 0.100% 6.579 -DMBS; x=0 1.763% 1.386 -DMBS; x=0.1 1.308% 1.677 0.911 DMBS; x=0.2 1.238% 1.651 1.820 DMBS; x=0.3 1.188% 1.622 2.728 DMBS; x=0.4 1.133% 1.611 3.638 DMBS; x=0.5 1.088% 1.603 4.547 DMBS (x=0.01; ¯γ=99.5%) 0.686% 1.347 0.179 DMBS (x=0.05; ¯γ=99.5%) 0.443% 1.094 0.920 DMBS (x=0.1; ¯γ=99.5%) 0.324% 0.931 1.870 DMBS (x=0.2; ¯γ=99.5%) 0.216% 0.827 3.800 DMBS (x=0.4; ¯γ=99.5%) 0.160% 0.822 7.707 DMBS (x=0.0005; ¯γ=90%) 1.376% 1.680 0.176 DMBS (x=0.001; ¯γ=90%) 1.304% 1.682 0.353 DMBS (x=0.005; ¯γ=90%) 0.896% 1.117 1.782 DMBS (x=0.01; ¯γ=90%) 0.628% 1.755 3.593 DMBS (x=0.025; ¯γ=90%) 0.314% 1.701 9.103 DMBS (x=0.0005; ¯γ=75%) 1.317% 1.698 0.480 DMBS (x=0.001; ¯γ=75%) 1.190% 1.733 0.961 DMBS (x=0.0025; ¯γ=75%) 0.931% 1.793 2.411 DMBS (x=0.005; ¯γ=75%) 0.668% 1.877 4.846 DMBS (x=0.01; ¯γ=75%) 0.424% 1.953 9.751

Type Loss frequency Loss severity Average in ledger Median CMO 30% 4.934% 5.447 -Median CMO 60% 12.978% 4.981 -VaR(CMO 30%, γ = 5%) 0.290% 5.028 -VaR(CMO 60%, γ = 5%) 0.774% 4.783 -DMBS; x=0 6.067% 1.830 -DMBS; x=0.1 4.941% 1.973 1.149 DMBS; x=0.2 4.638% 1.931 2.306 DMBS; x=0.3 4.402% 1.900 3.468 DMBS; x=0.4 4.182% 1.889 4.634 DMBS; x=0.5 4.009% 1.872 5.801 DMBS (x=0.05; ¯γ=99.5%) 4.146% 1.712 1.022 DMBS (x=0.1; ¯γ=99.5%) 3.449% 1.600 2.146 DMBS (x=0.2; ¯γ=99.5%) 2.721% 1.443 4.513 DMBS (x=0.3; ¯γ=99.5%) 2.292% 1.347 7.026 DMBS (x=0.4; ¯γ=99.5%) 2.000% 1.264 9.608 DMBS (x=0.001; ¯γ=90%) 5.306% 1.949 0.323 DMBS (x=0.0025; ¯γ=90%) 4.830% 1.936 0.813 DMBS (x=0.005; ¯γ=90%) 4.184% 1.920 1.645 DMBS (x=0.01; ¯γ=90%) 3.288% 1.892 3.350 DMBS (x=0.025; ¯γ=90%) 1.978% 1.762 8.634 DMBS (x=0.0005; ¯γ=75%) 5.266% 1.980 0.451 DMBS (x=0.001; ¯γ=75%) 4.952% 1.984 0.905 DMBS (x=0.0025; ¯γ=75%) 4.102% 2.025 2.282 DMBS (x=0.005; ¯γ=75%) 3.178% 2.070 4.618 DMBS (x=0.01; ¯γ=75%) 2.224% 2.050 9.373

7

Discussion

In general, financial institutions prefer to receive mortgage payments as scheduled. Sev-eral products named mortgage-backed securities have been created in an attempt to meet this preference. However, conventional mortgage-backed securities such as plain vanilla collateralized mortgage obligations have a large amount of uncertainty in their final pay-ment dates. This uncertainty can be a nuisance to financial institutions. Therefore, the introduction of a novel MBS type could meet their preferences.

To develop this new MBS type, a novel agnostic simulation method was proposed in which major components in mortgage payment behavior, namely prepayment and default, could be controlled explicitly. So far, mortgage payment simulation methods mostly fo-cused on using data that relied on several different economic factors. Using our novel simulation method, the loss frequency and severity of a conventional mortgage-backed se-curity was compared to a novel type of MBS, named a dynamic mortgage-backed sese-curity. It was found that a DMBS decreases the loss severity and frequency severely.

of the study.

Finally, it was observed that results of the DMBS depend severely on which set of 1,000 simulated mortgage portfolio was used to set the a priori DMBS payment levels. The realized loss frequencies of the DMBSs should be close to their a priori set Value-at-Risk levels. However, it was found that fluctuations could occur. The best explanation can be found in that the mortgage portfolio cash flows are highly volatile such that Value-at-Risk levels of the DMBS payments are affected by this.

In order to develop this novel simulation framework that was used to evaluate the MBSs, several assumptions were required. The most important one was that default and prepayment behavior were assumed to be independent, which is in contrast to the negative relation that is empirically observed. This could result in a conservative estimate of mortgage portfolio payments with respect to what would be empirically observed. However, it remains doubtful to what extent this would change the results. Furthermore, our agnostic simulation framework which does not depend on interest rate levels, could be considered as a downside as well since both prepayment and default behavior correlate with interest rates in real-life.

on interest rates.

References

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Appendices

Appendix A: Alternative parameter sets

Parameter Value κD _0.3 θD _0.04% σD 0.2% µD _0.04% ιD _0.175 ρD 0.75

Parameter Value κP _0.5 θP 4% σP _2% µP _12% ιP 0.175 ρP _0.75

Table 7: Alternative parameter setup 2 for the prepayment intensity process in (12). With respect to the base parameter setup, the only difference is the increase in correlation from 0.6 to 0.75. Parameter Value κD _0.3 θD _0.04% σD 0.2% µD _0.04% ιD _0.175 ρD 0.45

Parameter Value κP _0.5 θP 4% σP 2% µP _12% ιP 0.175 ρP _0.45

Table 9: Alternative parameter setup 3 for the prepayment intensity process in (12). With respect to the base parameter setup, the single difference is the decrease in correlation from 0.6 to 0.45. Parameter Value κD 0.3 θD _0.03417% σD _0.2% µD 0.05% ιD _0.175 ρD _0.6

Parameter Value κP _0.5 θP 3.3% σP _2% µP _14% ιP 0.175 ρP _0.6

Table 11: Alternative parameter setup 4 for the prepayment intensity process in (12). The change with respect to the base parameter setup is that the mean of the exponentially distributed jump is increased from 12% to 14%. However, the mean of the CIR process is decreased such that the long-term mean of the default intensity remains the same.

Figure 16: Annualized CPR and Default Rates using parameter sets from Tables 1 and 2.

Type Loss frequency Loss severity Average in ledger Median CMO 30% 4.773% 10.647 -Median CMO 60% 12.128% 8.118 -VaR(CMO 30%, γ = 0.5%) 0.012% 3.220 -VaR(CMO 60%, γ = 0.5%) 0.033% 3.192 -DMBS; x=0 0.435% 1.503 -DMBS; x=0.1 0.353% 1.524 0.722 DMBS; x=0.2 0.320% 1.451 1.441 DMBS; x=0.3 0.296% 1.392 2.160 DMBS; x=0.4 0.274% 1.356 2.879 DMBS; x=0.5 0.256% 1.340 3.599 DMBS (x=0.01; ¯γ=99.5%) 0.329% 1.437 0.220 DMBS (x=0.025; ¯γ=99.5%) 0.261% 1.272 0.554 DMBS (x=0.05; ¯γ=99.5%) 0.187% 1.211 1.118 DMBS (x=0.1; ¯γ=99.5%) 0.131% 1.172 2.261 DMBS (x=0.2; ¯γ=99.5%) 0.104% 1.022 4.553 DMBS (x=0.4; ¯γ=99.5%) 0.086% 0.978 9.138 DMBS (x=0.001; ¯γ=90%) 0.356% 1.557 0.342 DMBS (x=0.005; ¯γ=90%) 0.211% 1.398 1.717 DMBS (x=0.01; ¯γ=90%) 0.116% 1.437 3.442 DMBS (x=0.02; ¯γ=90%) 0.054% 1.449 6.911 DMBS (x=0.025; ¯γ=90%) 0.043% 1.406 8.647 DMBS (x=0.0005; ¯γ=75%) 0.352% 1.580 0.521 DMBS (x=0.001; ¯γ=75%) 0.310% 1.536 1.043 DMBS (x=0.0025; ¯γ=75%) 0.204% 1.495 2.611 DMBS (x=0.005; ¯γ=75%) 0.117% 1.510 5.232 DMBS (x=0.01; ¯γ=75%) 0.050% 1.581 10.488

Type Loss frequency Loss severity Average in ledger Median CMO 30% 4.773% 10.647 -Median CMO 60% 12.128% 8.118 -VaR(CMO 30%; γ = 1%) 0.036% 5.753 -VaR(CMO 60%; γ = 1%) 0.085% 3.974 -DMBS; x=0 0.943% 1.444 -DMBS; x=0.1 0.693% 1.642 0.808 DMBS; x=0.2 0.649% 1.552 1.614 DMBS; x=0.3 0.617% 1.480 2.421 DMBS; x=0.4 0.586% 1.434 3.227 DMBS; x=0.5 0.558% 1.400 4.034 DMBS (x=0.025; ¯γ=99.5%) 0.613% 1.409 0.541 DMBS (x=0.05; ¯γ=99.5%) 0.517% 1.266 1.094 DMBS (x=0.1; ¯γ=99.5%) 0.397% 1.139 2.217 DMBS (x=0.2; ¯γ=99.5%) 0.285% 1.073 4.502 DMBS (x=0.4; ¯γ=99.5%) 0.224% 1.053 9.090 DMBS (x=0.001; ¯γ=90%) 0.728% 1.664 0.339 DMBS (x=0.0025; ¯γ=90%) 0.652% 1.574 0.849 DMBS (x=0.005; ¯γ=90%) 0.530% 1.478 1.701 DMBS (x=0.01; ¯γ=90%) 0.372% 1.352 3.415 DMBS (x=0.025; ¯γ=90%) 0.175% 1.059 8.591 DMBS (x=0.0005; ¯γ=75%) 0.722% 1.675 0.518 DMBS (x=0.001; ¯γ=75%) 0.667% 1.619 1.036 DMBS (x=0.0025; ¯γ=75%) 0.503% 1.557 2.595 DMBS (x=0.005; ¯γ=75%) 0.348% 1.461 5.203 DMBS (x=0.01; ¯γ=75%) 0.202% 1.281 10.440

Type Loss frequency Loss severity Average in ledger Median CMO 30% 4.773% 10.647 -Median CMO 60% 12.128% 8.118 -VaR(CMO 30%; γ = 5%) 0.273% 4.958 -VaR(CMO 60%; γ = 5%) 0.694% 4.688 -DMBS; x=0 5.120% 1.670 -DMBS; x=0.1 3.900% 1.867 1.058 DMBS; x=0.2 3.607% 1.821 2.125 DMBS; x=0.3 3.402% 1.775 3.196 DMBS; x=0.4 3.234% 1.739 4.270 DMBS; x=0.5 3.071% 1.721 5.346 DMBS (x=0.025; ¯γ=99.5%) 3.887% 1.741 0.457 DMBS (x=0.05; ¯γ=99.5%) 3.403% 1.685 0.950 DMBS (x=0.1; ¯γ=99.5%) 2.858% 1.598 1.989 DMBS (x=0.2; ¯γ=99.5%) 2.285% 1.502 4.153 DMBS (x=0.4; ¯γ=99.5%) 1.777% 1.414 8.630 DMBS (x=0.001; ¯γ=90%) 4.299% 1.849 0.312 DMBS (x=0.0025; ¯γ=90%) 3.905% 1.833 0.785 DMBS (x=0.005; ¯γ=90%) 3.343% 1.826 1.584 DMBS (x=0.01; ¯γ=90%) 2.552% 1.816 3.217 DMBS (x=0.025; ¯γ=90%) 1.498% 1.697 8.252 DMBS (x=0.0005; ¯γ=75%) 4.243% 1.868 0.489 DMBS (x=0.001; ¯γ=75%) 3.934% 1.871 0.981 DMBS (x=0.0025; ¯γ=75%) 3.148% 1.908 2.473 DMBS (x=0.005; ¯γ=75%) 2.299% 1.962 4.998 DMBS (x=0.01; ¯γ=75%) 1.528% 1.925 10.129

Type Loss frequency Loss severity Average in ledger Median CMO 30% 5.189% 5.399 -Median CMO 60% 12.814% 4.747 -VaR(CMO 30%; γ = 0.5%) 0.015% 2.262 -VaR(CMO 60%; γ = 0.5%) 0.036% 2.539 -DMBS; x=0 0.382% 1.216 -DMBS; x=0.1 0.266% 1.319 0.837 DMBS; x=0.2 0.233% 1.266 1.672 DMBS; x=0.3 0.214% 1.202 2.507 DMBS; x=0.4 0.199% 1.154 3.342 DMBS; x=0.5 0.186% 1.133 4.176 DMBS (x=0.01; ¯γ=99.5%) 0.251% 1.339 0.200 DMBS (x=0.025; ¯γ=99.5%) 0.193% 1.256 0.503 DMBS (x=0.05; ¯γ=99.5%) 0.149% 1.125 1.014 DMBS (x=0.1; ¯γ=99.5%) 0.108% 0.905 2.039 DMBS (x=0.2; ¯γ=99.5%) 0.066% 0.778 4.097 DMBS (x=0.001; ¯γ=90%) 0.267% 1.400 0.335 DMBS (x=0.0025; ¯γ=90%) 0.214% 1.332 0.838 DMBS (x=0.005; ¯γ=90%) 0.158% 1.236 1.680 DMBS (x=0.01; ¯γ=90%) 0.106% 1.033 3.369 DMBS (x=0.02; ¯γ=90%) 0.055% 0.825 6.753 DMBS (x=0.025; ¯γ=90%) 0.038% 0.859 8.447 DMBS (x=0.0005; ¯γ=75%) 0.276% 1.418 0.495 DMBS (x=0.001; ¯γ=75%) 0.241% 1.384 0.990 DMBS (x=0.0025; ¯γ=75%) 0.170% 1.314 2.479 DMBS (x=0.005; ¯γ=75%) 0.112% 1.241 4.965 DMBS (x=0.01; ¯γ=75%) 0.055% 1.115 9.942