The Choice of Optimal Risk Retention Forms from the Perspective of Asymmetric Information

Abstract

The risk retention rule requires issuers to retain part of their securities and share the interests of investors. The different forms of risk retention chosen lead to different financing effects of enterprises. In order to explore the optimal choice of risk retention forms in different environments, according to asset pricing theory and asset securitization practice combined with risk retention rules, we obtain issuer payoff models under three forms of risk retention. Through numerical simulation and economic meaning analysis, we draw the following conclusions: hybrid retention can not only alleviate the side effects of horizontal retention but also reduce the proportion of vertical retention, which can improve the issuer payoffs; horizontal retention is more suitable for situations where asset pool losses are small or asset volatility is large, while vertical retention is the opposite. Therefore, we suggest that securities issuers should consider the assets of enterprises and macroeconomic situations to choose the optimal form of risk retention.

1. Introduction

In the aftermath of the global financial crisis, regulators around the world have realized the contradiction between risk transfer and information asymmetry in securitization. Many countries (e.g., the United States and European Union) have introduced regulatory requirements for risk retention in financial reforms, such as the European Union’s CRD IV, which stipulates that sponsors must retain at least 5% of risk. The goal of these policies is to increase market transparency while reducing systemic risk. Therefore, the study of the optimal form of risk retention in different information environments not only contributes to theoretical research but also has important guiding significance for policy formulation in practice.

Risk retention is an effective means to solve the problem of incentive incompatibility and agency in the process of asset securitization, which realizes the consistency of interests between sponsors and investors by requiring sponsors to retain some securitization products and promotes the monitoring and control of loans by banks and other sponsors [1,2,3,4]. The mode choice of risk retention form has always been the focus of academic and industry attention. Because subprime securities have the highest risk and the greatest possibility of loss, Kiff and Kisser [5] believe that subprime retention will minimize the problem of information asymmetry. After examining the optimal loan screening effect, the size of initial lenders and securitization institutions, Kiff and Kisser [6] believe that only the initial lenders keeping subprime files can maximize the promotion of bank due diligence. Malekan and Dionne [7] studied the optimal allocation scheme of risk between sponsors and investors from the perspective of the product structure design of asset securitization and concluded that the optimal risk retention contract must include some subprime products. The retention of intermediate and priority files is also worthy of attention. By comparing different forms of retention, Fender and Mitchell [8] found that if the sponsor faces a large systemic risk, the effect of retention in the middle or priority is better. From the perspective of retention cost, Admati et al. [9] believe that if the retention cost of a subprime product is too high, banks may turn to the middle of retention.

Different transaction structures choose different forms of retention. Guo and Wu [10] present a model with mandated vertical risk retention only. Wang [11] believes that there are two forms of risk retention, horizontal retention and vertical retention, and the adoption of vertical retention has a positive effect on improving the enthusiasm of bank asset securitization, while Flynn et al. [12] found that the adoption of horizontal retention can indicate a higher collateral value in the asset pool. Gürtler [13] believes that horizontal retention prevails in equilibria with high effort levels, whereas vertical retention arises rather in equilibria with lower effort levels. Liu [14] believes that hybrid retention can give full play to the ratio of risk and return by comparing it with the current method of risk retention. Therefore, in different economic environments, the most effective retention mechanism is not exactly the same, and risk retention policies should be formulated accordingly based on the macroeconomic situation and the type and quality of underlying assets.

This paper takes the securities issuer as the subject of risk retention obligation and discusses the form of risk retention from the perspective of asset pricing [15,16,17,18,19]. In the process of securitization financing, securities issuers package the assets of enterprises in the asset pool and design them as asset-backed securities with different risk levels; they then sell them to external investors to complete a financing. However, there is asymmetric information between issuers and investors; investors do not know about the quality of asset pools, but issuers can use the form of risk retention to convey the signal of asset pool quality. Therefore, the financing results are not only related to the quality of the asset pool, but also related to the form of risk retention, so what impact will these two have on the issuer payoffs? In different market environments, how should issuers choose the optimal form of risk retention? The exploration of the above problems aims to reveal the economic mechanism of the form of risk retention on the financing of enterprise asset securitization and to provide a possible theoretical explanation for the actual financial phenomenon.

Therefore, on the premise of stable return flow of an asset pool under enterprise asset securitization, we assume that multi-enterprise assets obey geometric Brownian motion and use asset pricing theory to obtain analytical solutions for the pricing of various securities. Based on this, we combine the risk retention rule to obtain the issuer payoff models of vertical retention, horizontal retention and hybrid retention. We draw some conclusions. Firstly, hybrid retention is conducive to improving the issuer payoffs. This conclusion verifies that hybrid retention can not only alleviate the side effects of horizontal retention but also reduce the proportion of vertical retention. Secondly, horizontal retention is more suitable for situations where asset pool losses are small or asset volatility is large. Thirdly, vertical retention is more suitable for situations where asset pool losses are large or asset volatility is small. The above two conclusions show that different forms of risk retention have different applicability, and issuers should make choices according to the actual situation. Fourthly, issuer payoffs are inversely U-shaped with the coupon and negatively correlated with the correlation between assets, asset volatility and the discount rate of subordinated securities. This conclusion provides a theoretical basis for issuers to screen asset pool assets, price securitization products and choose the form of risk retention.

This article is most closely related to research by Skarabot [16] and Flynn et al. [12], but there are differences. Firstly, in order to comprehensively consider the macroeconomic situation and enterprise assets, we issue low-risk debt-type notes as senior securities and high-risk equity-type notes as subordinated securities so as to combine asset pricing with risk retention forms, and we comprehensively analyze their impact on enterprise asset securitization financing, but the above literature does not involve it. Secondly, Flynn et al. [12] assume that investors’ expected losses to asset pools obey the same uniform distribution and give a risk retention model by separating equilibrium conditions. However, our model assumes that investors’ expected losses to asset pools obey different normal distribution, and their expectations are given in the form of risk retention, which enables us to obtain a risk retention model based on asset pricing. Thirdly, Flynn et al. [12] set the discount rate through the form of risk retention, while we set the discount rate through the type of securities, which has a better explanation for practical financial problems. In addition, in order to expand and enrich the theoretical model, we give vertical retention, horizontal retention and hybrid retention models, as well as their adaptability in different environments, and make economic explanations in theory.

In the remaining sections of the paper, Section 2 will introduce the model setup of asset securitization, Section 3 will present the issuer risk retention model, Section 4 will give the experiments and results, and Section 5 will summarize the work of this paper.

2. Model Setup

Effective investment timing is one of the standards that reflect enterprise quality. Generally, the earlier the investment timing, the better the enterprise quality [20,21,22,23]. We assume that the securities issuer can select $n$ high-quality enterprises by investment timing and organize their corporate assets into an asset pool. In addition, we assume that the uncertainty in the model is presented by the random process for the value of the assets $V(V_1,V_2,\cdots ,V_n)$. Process $V(V_1,V_2,\cdots ,V_n)$ follows n-dimensional Ito’s process in the following sense. Given a probability space $(\Omega ,F,P)$, we assume that each process $V_i$ solves a stochastic differential equation of the form

$$d{V}_i={\mu }_i{V}_{i}dt+{\sigma }_i{V}_{i}d{z}_i,$$

(1)

where ${\mu }_i$ and ${\sigma }_i$ denote the drift parameter and volatility parameter of the ith asset, and each $z_i$ is a standard one-dimensional Brownian motion. Motions $z_i$ and $z_j$ are correlated, and $\mathrm{cov}(d{z}_i,d{z}_j)\equiv {\rho }_{ij}dt$, ${\rho }_{ii}={\rho }_{jj}=1$, ${\rho }_{ij}={\rho }_{ji}$, $i,\hspace{0.17em}j=1,\hspace{0.17em}2,\cdots ,n$.

We assume that there is a risk-free asset in the market, and the interest rate is constant $r$. Generally, the value of a securities product based on enterprise assets is a function of the current asset value and time, recorded as $F(V,t)$, and its payment rate is $C(V,t)$. Using multi-dimensional Ito’s lemma in [24] and Merton’s Corporate Securities Pricing Theory in [25], it can be derived that $F(V,t)$ satisfies the n-dimensional version of the Black–Scholes equation:

$${\displaystyle \frac{\partial F}{\partial t}}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{\rho }_{ij}{\sigma }_i{\sigma }_j{V}_i{V}_j\frac{{\partial }^{2}F}{\partial V_i\partial V_j}}}+{\displaystyle \sum _{i=1}^{n}r{V}_i\frac{\partial F}{\partial V_i}}-rF+C(V,t)=0,$$

(2)

where $V\in \mathrm{B}\subset R_{++}^n$ and $0\le t\le T$, and region $\mathrm{B}$ is determined by the specific settlement provisions of securities products.

Following Leland [15], we assume that SPV will issue two types of securitization products: one is debt-type notes paying a fixed coupon, recorded as $D(V,C,t)$, and the other is equity-type notes, recorded as $E(V,t)$. Part of the future asset payoffs of the asset pool will be given to creditors with coupons, and the rest will be given to equity holders. Determining the optimal capital structure is to split the total payoffs of the asset pool.

Consider a debt claim $D(V_1,V_2,\cdots ,V_n)$ that continuously pays a coupon $C$. We assume that debt promises a perpetual coupon payment so that the debt claim has no explicit time dependence. Valuation Equation (2) can be rewritten as

$${\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{\rho }_{ij}{\sigma }_i{\sigma }_j{V}_i{V}_j\frac{{\partial }^{2}D}{\partial V_i\partial V_j}}}+{\displaystyle \sum _{i=1}^{n}r{V}_i\frac{\partial D}{\partial V_i}}-rD+C=0,$$

(3)

where $V$ is in the solvency range $V\in \mathrm{B}\subset R_{++}^n$ and Equation (3) has appropriate boundary conditions. Naturally, if the value of assets $V$ grows large, then the firm is far away from bankruptcy and the value of debt $D(V_1,V_2,\cdots ,V_n)$ should be close to the value of risk-free perpetual debt:

$$V_1+V_2+\cdots +V_n\to \infty \Rightarrow D(V_1,V_2,\cdots ,V_n)\to {\displaystyle \frac{C}{r}}.$$

(4)

If the firm enters bankruptcy, that means $V$ is at the boundary $\partial \mathrm{B}$ of the solvency region $\mathrm{B}$, then

$$(V_1,V_2,\cdots ,V_n)\in \partial \mathrm{B}\Rightarrow D(V_1,V_2,\cdots ,V_n)=(1-\alpha )(V_1+V_2+\cdots +V_n),$$

(5)

with $0<\alpha \le 1$. Here, we assume that if bankruptcy occurs, a fraction $\alpha$ of the total value of the assets will be lost due to bankruptcy costs. In addition, the boundary of the solvency region $\partial \mathrm{B}$ is endogenized. Problem (3) becomes a free-boundary problem with an additional smooth-pasting condition at the default boundary.

However, there exists a problem that is preventing us from simply rewriting Leland’s model [15] to a multi-asset setting. The value of the firm’s assets is represented by the sum of $n$ geometric Brownian motions. Because the sum of lognormal processes is not a lognormal process, that represents a serious problem in the case that we are searching for some variation of closed-form solutions for Equation (3). Following Skarabot [16], we replace the sum of multi-asset values $V_1+V_2+\cdots +V_n$ with $n\sqrt[n]{V_1{V}_2\cdots V_n}$. According to the inequality property of arithmetic geometric mean, if and only if $V_1=V_2=\cdots =V_n$, the two are completely equal.

We assume that $V_1^{*}\ge V_2^{*}\ge \cdots \ge V_n^{*}$, $k^{\prime }{V}_1^{*}\le V_n^{*}$, $0<k^{\prime }<1$ and $k^{\prime }$ is a constant. The value of descending assets is $(V_1,V_2,\cdots ,V_n)$, where $V_i^{*}$ is the ith asset value. The approximate error of the sum of multi-asset values after replacement is

$$(V_1+V_2+\cdots +V_n)-n\sqrt[n]{V_1{V}_2\cdots V_n}\le V_1^{*}\left(1+(n-1)k^{\prime }-n{k^{\prime }}^{{\displaystyle \frac{n}{n-1}}}\right),$$

(6)

Inequality (6) shows that if the approximation error is to be minimized, the value of the assets $(V_1,V_2,\cdots ,V_n)$ should be approximately equal, and $k$ should approach 1. Once the approximation error is fairly small, the boundary conditions (4) and (5) are rewritten as

$$n\sqrt[n]{V_1{V}_2\cdots V_n}\to \infty \Rightarrow D(V_1,V_2,\cdots ,V_n)\to {\displaystyle \frac{C}{r}},$$

(7)

$$(V_1,V_2,\cdots ,V_n)\in \partial \mathrm{B}\Rightarrow D(V_1,V_2,\cdots ,V_n)=(1-\alpha )(n\sqrt[n]{V_1{V}_2\cdots V_n}).$$

(8)

Therefore, the value of various contingent claims can be given by the following proposition:

Proposition 1.

Values of debt, bankruptcy costs, tax benefits, equity, and the total value of the firm are given as (proofs are relegated to Appendix A)

$$D(V_1,V_2,\cdots ,V_n)={\displaystyle \frac{C}{r}}-\left({\displaystyle \frac{C}{r}}-(1-\alpha )V_B\right){\left({\displaystyle \frac{V_B}{n\sqrt[n]{V_1{V}_2\cdots V_n}}}\right)}^{n{x}_m},$$

(9)

$$BC(V_1,V_2,\cdots ,V_n)=\alpha V_B{\left({\displaystyle \frac{V_B}{n\sqrt[n]{V_1{V}_2\cdots V_n}}}\right)}^{n{x}_m},$$

(10)

$$TB(V_1,V_2,\cdots ,V_n)={\displaystyle \frac{\tau C}{r}}-\left({\displaystyle \frac{\tau C}{r}}\right){\left({\displaystyle \frac{V_B}{n\sqrt[n]{V_1{V}_2\cdots V_n}}}\right)}^{n{x}_m},$$

(11)

$$E(V_1,V_2,\cdots ,V_n)=n\sqrt[n]{V_1{V}_2\cdots V_n}+{\displaystyle \frac{(1-\tau )C}{r}}+\left({\displaystyle \frac{(1-\tau )C}{r}}-V_B\right){\left({\displaystyle \frac{V_B}{n\sqrt[n]{V_1{V}_2\cdots V_n}}}\right)}^{n{x}_m},$$

(12)

$$v(V_1,V_2,\cdots ,V_n)=n\sqrt[n]{V_1{V}_2\cdots V_n}+{\displaystyle \frac{\tau C}{r}}-\left({\displaystyle \frac{\tau C}{r}}+\alpha V_B\right){\left({\displaystyle \frac{V_B}{n\sqrt[n]{V_1{V}_2\cdots V_n}}}\right)}^{n{x}_m}.$$

(13)

Proposition 2.

The bankruptcy value and optimal coupon are given as

$$V_B={\displaystyle \frac{(1-\tau )C{x}_m}{r(x_m+1/n)}},$$

(14)

$$C^{*}=n\sqrt[n]{V_1{V}_2\cdots V_n}{\left({\displaystyle \frac{1}{h(1+n{x}_m)}}\right)}^{{\displaystyle \frac{1}{n{x}_m}}},$$

(15)

where $\tau$ is the corporate tax rate and

$$x_m={\displaystyle \frac{\left(nr-\frac{1}{2}{\displaystyle \sum _{i=1}^n{\sigma }_i^2}\right)+\sqrt{{\left(nr-\frac{1}{2}{\displaystyle \sum _{i=1}^n{\sigma }_i^2}\right)}^2+2\left({\displaystyle \sum _{i=1}^n{\sigma }_i^2+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j\ne i}^n{\rho }_{ij}{\sigma }_i{\sigma }_j}}}\right)r}}{{\displaystyle \sum _{i=1}^n{\sigma }_i^2+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1,j\ne i}^n{\rho }_{ij}{\sigma }_i{\sigma }_j}}}}}.$$

Thus, we derive the optimal values of debt $D^{*}(V_1,V_2,\cdots ,V_n)$ (abbreviated as $D^{*}$), the total value of the firm $v^{*}(V_1,V_2,\cdots ,V_n)$ (abbreviated as $v^{*}$), and the optimal values of equal $E^{*}(V_1,V_2,\cdots ,V_n)$(abbreviated as $E^{*}$). See Appendix B for certification and results.

3. Issuer Risk Retention Model

The issuer transmits the quality signal of the asset pool in the form of risk retention, so the issuer payoff is not only related to the quality of enterprise assets in the asset pool but also related to the form of risk retention. Based on the asset pricing model in Section 2 and the risk retention rules, we design a risk retention model and obtain the issuer’s total payoff model under three forms of risk retention.

3.1. Model and Assumptions

We assume that investors and issuers in the market are risk-neutral, and there is asymmetric information between them. Issuers issue two types of securities, debt-type notes $D^{*}$, with lower risk as senior securities, and equity-type notes $E^{*}$, with higher risk as subordinated securities, which are used to absorb the first loss of the asset pool. The asset pool generates future cash flow given by

$$A=(1-\beta -\epsilon )v^{*},$$

(16)

where $\beta$ and $\epsilon$ are random variables. The total realized losses of the asset pool is $(\beta +\epsilon )v^{*}$. Part $\beta v^{*}$ denotes losses that can be predicted based on some soft information about the securitized loans, which is available to the issuer but not to the investors or credit rating agencies. Investors can only estimate $\beta$ according to the form of risk retention. Part $\epsilon v^{*}$ of the losses is caused by factors not related to asymmetric information. Both the issuer and investors believe that $\epsilon$ is distributed according to the probability density function $f_{\epsilon }$ with support $[0,\overline{\epsilon }]$, and the expected values is ${\epsilon }_0$.

In our setting, the issuer chooses the security design (vertical type $v$, horizontal type $h$ or hybrid type $l$) after it learns private information about the asset pool, as shown in Figure 1. As a result, the risk retention type is a signal of the issuer’s private information. In vertical retention, the issuer retains a fraction $\lambda$ of each type of securities in the deal, with $\lambda \in (0,1)$. In horizontal retention, the issuer retains only a fraction ${\phi }_h$ of the subordinated securities, with ${\phi }_h\in [0,1]$. In order to satisfy a fair market value assessment requirement, the horizontal tranche is equal to at least $\lambda$ of the fair market value of all securities issued. Hybrid retention is a combination of vertical retention and horizontal retention. In hybrid retention, the issuer horizontally retains a fraction ${\phi }_l$ of all subordinated securities, with ${\phi }_l\in [0,{\phi }_h]$, and vertically retains a fraction ${\lambda }_l$ of other securities, with ${\lambda }_l\in [0,\lambda ]$. Similarly, hybrid retention must satisfy a fair market value assessment requirement. Therefore, the hybrid tranche is also equal to at least $\lambda$ percent of the fair market value of all securities issued.

We assume that investors believe that $\beta$ follows the normal distribution with support $[0,\overline{\beta }]$, and the expected value $E[\beta |k^{″}]={\beta }_{k^{″}}$ by type $k^{″}=v,\hspace{0.17em}h,\hspace{0.17em}l$. Following Flynn et al. [12], under the same risk retention requirements, the more subordinated securities the issuer retains, the better the quality of the asset pool. Then, investors think that the losses of the asset pool are smaller, that is, $0<{\beta }_h\le {\beta }_l\le {\beta }_v<1$. We emphasize that investors will judge ${\beta }_l$ according to the subordinated securities included in the hybrid retention, and the larger (smaller) ${\lambda }_l$ is, the larger (smaller) ${\beta }_l$ is. In particular, ${\beta }_l={\beta }_v$ when ${\lambda }_l=\lambda$, and ${\beta }_l={\beta }_h$ when ${\lambda }_l=0$. We suppose that the issuer discounts future cash flows at a rate higher than the market rate. For notational convenience, we normalize the market discount factor to one. Then, there exist discount factors ${\delta }_d$ and ${\delta }_e$ that denote the fractional value to the issuer of senior and subordinated securities such that ${\delta }_e<{\delta }_d<1$. A lower discount factor ${\delta }_e$ reflects a higher cost for the issuer to hold the subordinated securities. We assume that the size of subordinated securities retained horizontally is bigger than possible realized losses; that is, ${\phi }_h{E}^{*}>(\overline{\beta }+\overline{\epsilon })v^{*}$. To sum up, we give the following three types of risk retention models.

3.2. Vertical Retention

In the case of vertical retention, the total expected payoffs ${\pi }_v(\beta )$, discount payoffs of retained securities $S_v(\beta )$ and market payoffs of sold securities $P_v$ of the issuer are given as

$${\pi }_v(\beta )=S_v(\beta )+P_v,$$

$$S_v(\beta )={\delta }_d\lambda D^{*}+{\delta }_e\lambda (E^{*}-(\beta +{\epsilon }_0)v^{*}),$$

(17)

$$P_v=(1-\lambda )D^{*}+(1-\lambda )(E^{*}-({\beta }_v+{\epsilon }_0)v^{*}).$$

(18)

In Equation (17), ${\delta }_d\lambda D^{*}$ denotes the discount payoffs of retained senior securities, and ${\delta }_e\lambda (E^{*}-(\beta +{\epsilon }_0)v^{*})$ denotes the discount payoffs of retained subordinated securities after offsetting the losses, where $E[(\beta +\epsilon )|\beta ]v^{*}=(\beta +{\epsilon }_0)v^{*}$ is the expected total losses of the issuer on the asset pool. In Equation (18), $(1-\lambda )D^{*}$ denotes the market payoffs from selling senior securities and $(1-\lambda )(E^{*}-({\beta }_v+{\epsilon }_0)v^{*})$ denotes the market payoffs from selling subordinated securities, where $E[(\beta +\epsilon )|{\beta }_v]v^{*}=({\beta }_v+{\epsilon }_0)v^{*}$ is the expected total losses of the investor on the asset pool. We note that retention value $S_v(\beta )$ depends on $\beta$, but market value $P_v$ does not depend on $\beta$ because investors do not know the issuer’s private information. However, the payoffs from the retained securities do depend on $\beta$, and the payoffs from the selling securities do depend on the form of risk retention.

3.3. Horizontal Retention

In the case of horizontal retention, the total expected payoffs ${\pi }_h(\beta )$, discount payoffs of retained securities $S_h(\beta )$ and market payoffs of sold securities $P_h$ of the issuer are given as

$${\pi }_h(\beta )=S_h(\beta )+P_h,$$

$$S_h(\beta )={\delta }_e({\varphi }_h{E}^{*}-(\beta +{\epsilon }_0)v^{*}),$$

(19)

$$P_h=D^{*}+(1-{\varphi }_h)E^{*}.$$

(20)

Equation (19) denotes the discount payoffs of retained subordinated securities ${\phi }_h{E}^{*}$ after deducting the losses $(\beta +{\epsilon }_0)v^{*}$. Equation (20) denotes the market payoffs of sold senior securities $D^{*}$ and other subordinated securities $(1-{\phi }_h)E^{*}$. In order to satisfy a fair market value assessment requiring the market value of horizontal retention securities, that is, $E[{\phi }_h{E}^{*}-(\beta +\epsilon )v^{*}|{\beta }_h]={\phi }_h{E}^{*}-({\beta }_h+{\epsilon }_0)v^{*}$, $\lambda$ of the fair market value of all securities issued, that is, $\lambda E[A|{\beta }_h]=\lambda (1-{\beta }_h-{\epsilon }_0)v^{*}$, the two must be equal, so ${\phi }_h$ must make ${\varphi }_h{E}^{*}-({\beta }_h+{\epsilon }_0)v^{*}=\lambda (1-{\beta }_h-{\epsilon }_0)v^{*}$.

We obtain the fraction of subordinated securities retained horizontally as

$${\varphi }_h={\displaystyle \frac{\lambda v^{*}+(1-\lambda )({\beta }_h+{\epsilon }_0)v^{*}}{E^{*}}}.$$

(21)

3.4. Hybrid Retention

In the case of hybrid retention, the total expected payoffs ${\pi }_l(\beta )$, discount payoffs of retained securities $S_l(\beta )$ and market payoffs of sold securities $P_l$ of the issuer are given as

$${\pi }_l(\beta )=S_l(\beta )+P_l.$$

On the one hand, for the part retained by the issuer, when the size of the subordinated securities retained horizontally is bigger than the expected total losses of the issuer, that is, ${\phi }_l{E}^{*}>(\beta +{\epsilon }_0)v^{*}$, only the horizontal retention part suffers losses, so

$$S_l(\beta )={\delta }_d{\lambda }_l{D}^{*}+{\delta }_e({\lambda }_l(1-{\varphi }_l)E^{*}+{\varphi }_l{E}^{*}-(\beta +{\epsilon }_0)v^{*}),$$

(22)

where ${\delta }_d{\lambda }_l{D}^{*}$ and ${\delta }_e{\lambda }_l(1-{\phi }_l)E^{*}$ denote the discount payoffs of senior securities and subordinated securities in the vertical retention portion and ${\delta }_e{\phi }_l{E}^{*}$ denotes the discount payoffs of subordinated securities in the horizontal retention portion. In order to satisfy a fair market value assessment requirement, ${\phi }_l$ must make

$${\lambda }_l{D}^{*}+{\lambda }_l(1-{\varphi }_l)E^{*}+{\varphi }_l{E}^{*}-({\beta }_l+{\epsilon }_0)v^{*}=\lambda (1-{\beta }_l-{\epsilon }_0)v^{*}.$$

We obtain the fraction of subordinated securities retained horizontally as

$${\varphi }_l={\displaystyle \frac{(\lambda -{\lambda }_l)v^{*}+(1-\lambda )({\beta }_l+{\epsilon }_0)v^{*}}{(1-{\lambda }_l)E^{*}}}.$$

(23)

Accordingly, we can also obtain the fraction of vertical retention as

$${\lambda }_l={\displaystyle \frac{\lambda v^{*}-{\varphi }_l{E}^{*}+(1-\lambda )({\beta }_l+{\epsilon }_0)v^{*}}{v^{*}-{\varphi }_l{E}^{*}}}.$$

On the one hand, for the part retained by the issuer, when the size of subordinated securities retained horizontally is smaller than expected total losses of the issuer, that is, ${\phi }_l{E}^{*}<(\beta +{\epsilon }_0)v^{*}$, both horizontal retention and vertical retention suffered losses, so

$$S_l(\beta )={\delta }_d{\lambda }_l{D}^{*}+{\delta }_e{\lambda }_l(E^{*}-(\beta +{\epsilon }_0)v^{*}),$$

(24)

where ${\delta }_e{\lambda }_l(E^{*}-(\beta +{\epsilon }_0)v^{*})$ denotes the discount payoffs of subordinated securities in hybrid retention after deducting losses.

On the other hand, for the part sold by the issuer, when the size of subordinated securities retained horizontally is bigger than the expected total losses of the investor, that is, ${\phi }_l{E}^{*}>({\beta }_l+{\epsilon }_0)v^{*}$, investors believe that the purchased securities will not suffer any losses, so

$$P_l=(1-{\lambda }_l)(D^{*}+(1-{\varphi }_l)E^{*}),$$

(25)

where $(1-{\lambda }_l)D^{*}$ and $(1-{\lambda }_l)(1-{\phi }_l)E^{*}$ denote the market payoffs of sold senior and subordinated securities. When the size of subordinated securities retained horizontally is smaller than the expected total losses of the investor, that is, ${\phi }_l{E}^{*}<({\beta }_l+{\epsilon }_0)v^{*}$, investors believe that the purchased securities will suffer losses, so

$$P_l=(1-{\lambda }_l)D^{*}+(1-{\lambda }_l)(E^{*}-({\beta }_l+{\epsilon }_0)v^{*}),$$

(26)

where $(1-\lambda )(E^{*}-({\beta }_v+{\epsilon }_0)v^{*})$ denotes the market payoffs from selling subordinated securities and $E[(\beta +\epsilon )|{\beta }_l]v^{*}=({\beta }_l+{\epsilon }_0)v^{*}$ is the expected total losses of the investor on the asset pool. From (21) and (23), we can conclude that ${\lambda }_l=0$ when ${\phi }_l={\phi }_h$, and hybrid retention is equivalent to horizontal retention. Equation (24) is the same as (17) when ${\lambda }_l=\lambda$, so when ${\phi }_l=0$, hybrid retention is equivalent to vertical retention.

4. Comparative Analysis

How can the best form of risk retention be chosen? This is the issue of most concern to the issuer. For simplicity, we present a numerical simulation analysis based on two assets and an optimal coupon. With reference to Skarabot [16] and Flynn et al. [12], our parameter settings are as follows: $V_1=100$, $V_2=100$, $\alpha =0.4$, $\tau =0.1$, $\rho =0.5$, $r=0.02$, ${\sigma }_1={\sigma }_2=0.3$, $\lambda =0.05$, ${\lambda }_l=0.04$, ${\epsilon }_0=0.05$, ${\delta }_d=0.98$, ${\delta }_e=0.6$, $\beta =0.005$, ${\beta }_v=0.007$, ${\beta }_h=0.003$, ${\beta }_l=0.006$ and $\overline{\beta }=0.02$.

4.1. Impact of Loss Ratio on Issuer Payoffs

The loss ratio $\beta$ represents the asset pool loss determined by the issuer. The larger (smaller) the loss ratio, the lower (higher) the asset pool income. By changing the loss ratio $\beta$ and keeping other parameters unchanged, the change in the issuer payoffs ${\pi }_{k^{″}}(\beta )$ can be analyzed. In particular, a sensitivity analysis of the ratio of vertical retention ${\lambda }_l$ in hybrid retention to the issuer payoffs ${\pi }_l(\beta )$ is given.

As shown in Figure 2a, there is a negative correlation between the issuer payoffs ${\pi }_{k^{″}}(\beta )$ and the loss rate $\beta$. When the loss ratio is small, the hybrid retention payoff is better. When the loss ratio is large, the vertical retention payoff is better. The economic intuition behind it is as follows: first, the proportion of asset pool losses increases, the consumption of subordinated securities used by issuers to offset losses increases and retained earnings decrease. Second, the more senior securities, the stronger the ability to resist risks. Hybrid retention has more senior securities than horizontal retention, so it has an advantage when the loss rate is small. The size of the senior securities in vertical retention is the largest, so when the proportion of losses is large, it has more advantages than other forms of retention. That is to say, hybrid retention can not only alleviate the side effects of horizontal retention but also reduce the proportion of vertical retention. Because issuers sell more subordinated securities, which improves their enthusiasm for the product issuance, and keep more senior securities to ensure the quality of securitization products, it can improve the total payoffs of issuers.

As shown in Figure 2b, in hybrid retention, there is a positive correlation between the issuer payoffs ${\pi }_l(\beta )$ and the vertical retention rate ${\lambda }_l$. This conclusion is valid when the loss rate ${\beta }_l$ of the asset pool remains unchanged. That is to say, if investors are not sensitive to the retention ratio of senior securities and subordinated securities, it is more beneficial for issuers to retain more senior securities to improve their own returns.

4.2. Impact of Discount Rate and Correlation Coefficient on Issuer Payoffs

The larger (smaller) the discount rate ${\delta }_e$ is, the more (less) a unit of subordinated securities is converted into cash. The correlation coefficient is positive (negative), indicating that the returns of the two assets change in a positive (negative) way. By changing the discount rate ${\delta }_e$ and correlation coefficient $\rho$, and keeping other parameters unchanged, the change in the issuer payoffs ${\pi }_{k^{″}}(\beta )$ can be observed.

As shown in Figure 3a, there is a positive correlation between the issuer payoffs ${\pi }_{k^{″}}(\beta )$ and the discount rate ${\delta }_e$. Its economic intuition is that with the increase in the discount rate ${\delta }_e$, the more cash is converted from subordinated securities, and the holding cost of the issuer decreases and the income increases. In addition, due to the largest number of subordinated securities in the horizontal retention, the issuer payoffs ${\pi }_{k^{″}}(\beta )$ increased fastest.

As shown in Figure 3b, there is a negative correlation between the issuer payoffs ${\pi }_{k^{″}}(\beta )$ and the correlation coefficient $\rho$. This is because a low correlation between different assets can reduce the risk of the overall portfolio, because when one asset underperforms, the others may not be as affected. For example, if the housing market is depressed and auto loans are performing well, the overall portfolio loss is reduced. In addition, as the correlation between assets decreases, the total value of securitized products may increase. This is because investors are generally willing to pay a premium for lower-risk investments. High-quality securitized products can attract more investors, thus increasing the total income of the issuer. Understanding this relationship is a key element for both investors and issuers in developing strategies and assessing risk.

4.3. Impact of Volatility on Issuer Payoffs

Next, we will explore the differences of different risk retention methods by changing volatility ${\sigma }_1$ and ${\sigma }_2$ and keeping other parameters unchanged. Due to the status equivalence of assets in the asset pool, we will fix the volatility ${\sigma }_1$, change the volatility ${\sigma }_2$, and draw the change figure of issuer payoffs ${\pi }_{k^{″}}(\beta )$ and volatility ${\sigma }_2$. The simultaneous changes of two asset volatilities ${\sigma }_1$ and ${\sigma }_2$ are shown in Figure 4b–d.

As shown in Figure 4a, there is a negative correlation between the issuer payoffs ${\pi }_{k^{″}}(\beta )$ and the volatility ${\sigma }_2$. The economic explanation is that with the increase in asset risk, the value of the debt of low-risk assets decreases significantly, while the value of the equity of high-risk assets increases slightly, so the issuer payoffs decrease. In addition, due to the largest number of debt notes in the vertical retention, the issuer payoffs ${\pi }_v(\beta )$ decreased fastest.

As shown in Figure 4b–d, if the volatility ${\sigma }_1$ and ${\sigma }_2$ increase at the same time, the issuer payoffs ${\pi }_{k^{″}}(\beta )$ will show a downward trend. If one volatility is fixed and another volatility is changed, the issuer payoffs also decrease with the increase in volatility, which is consistent with the conclusion reflected in Figure 4a.

4.4. Impact of Coupon on Issuer Payoffs

Finally, we change the coupon $C$ and keep the other parameters unchanged to observe the change in the issuer payoffs ${\pi }_{k^{″}}(\beta )$. On this basis, if volatility ${\sigma }_1$ and ${\sigma }_2$ change at the same time, the impact is shown in Figure 5b–d.

As shown in Figure 5, there is an inverted U-shaped relationship between the issuer payoffs ${\pi }_{k^{″}}(\beta )$ and the coupon $C$. Specifically, as the coupon increases, the issuer payoffs first rise and then fall. The reason for this relationship can be explained by two effects: the tax avoidance effect and the bankruptcy effect. In the low coupon phase, the tax avoidance effect dominates. Interest payments on debt notes can be deducted before taxes, thus reducing the issuer’s tax burden. This tax benefit increases the net income of the issuer so that the level retention strategy is more advantageous when the coupon is low. As the coupon increases, the bankruptcy effect gradually becomes dominant. A high coupon means higher interest payment obligations, which increases the financial burden of issuers and raises the risk of bankruptcy. When interest expenses become too onerous, it can lead to a cash flow crunch or even bankruptcy. In this case, the vertical retention strategy is more appropriate.

The value of debt notes and equity notes changes with the coupon. As the coupon rises, the face value of debt notes increases because they generate a higher fixed income for creditors. In contrast, the residual income space of shareholders is squeezed, resulting in a decline in the value of equity notes. This variation implies that the issuer is able to effectively use the tax benefit through the horizontal retention strategy when the coupon is small, while the vertical retention strategy helps the issuer manage the risk of bankruptcy when the coupon is high.

The inverted U-shaped relationship between issuer payoffs and coupons reveals how issuers optimize their financial strategies under different economic environments and coupon levels. Understanding the dominant stages of the tax avoidance effect and the bankruptcy effect, as well as the changes in the value of debt and equity bills, helps issuers make more informed choices of asset allocation and retention strategies. This analysis not only has guiding significance for the financial management of issuers but also provides important clues for investors about the financial health of issuers.

4.5. Sensitivity Analysis

In order to test the robustness of the main conclusions, we take the bankruptcy loss ratio $\alpha$ as an example and analyze the impact of key parameters on the issuer payoffs by adjusting its size. Specifically,

Table 1. Sensitivity analysis of key parameters for vertical issuer payoffs.

Key Parameters	Vertical Issuer Payoffs
	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$
$\beta =0.005$	189.827	187.522	186.505
$\beta =0.010$	189.796	187.492	186.475
$\beta =0.015$	189.766	187.462	186.445
${\delta }_e=0.5$	189.256	186.794	185.699
${\delta }_e=0.7$	190.398	188.251	187.311
${\delta }_e=0.9$	191.540	189.709	188.922
$\rho =-0.5$	196.521	194.144	193.023
$\rho =0$	196.099	193.730	192.655
$\rho =0.5$	195.818	193.461	192.420
${\delta }_2=0.2$	196.836	194.459	193.311
${\delta }_2=0.4$	195.052	192.755	191.836
${\delta }_2=0.6$	194.126	191.977	191.261
$C=0$	190.720	190.720	190.720
$C=2$	195.602	193.347	191.091
$C=4$	195.261	188.953	182.644

shows the impact of the loss ratio $\beta$, the discount rate ${\delta }_e$, the correlation coefficient $\rho$, volatility ${\sigma }_2$ and coupon $C$ on the vertical issuer payoffs ${\pi }_v(\beta )$.

Table 2. Sensitivity analysis of key parameters for horizontal issuer payoffs.

Key Parameters	Horizontal Issuer Payoffs
	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$
$\beta =0.005$	188.880	187.203	186.489
$\beta =0.010$	188.269	186.597	185.886
$\beta =0.015$	187.658	185.991	185.282
${\delta }_e=0.5$	187.956	186.287	185.577
${\delta }_e=0.7$	189.804	188.119	187.402
${\delta }_e=0.9$	191.653	189.951	189.227
$\rho =-0.5$	195.461	193.683	192.868
$\rho =0$	195.107	193.353	192.583
$\rho =0.5$	194.872	193.141	192.405
${\delta }_2=0.2$	195.728	193.937	193.094
${\delta }_2=0.4$	194.237	192.595	191.970
${\delta }_2=0.6$	193.486	192.019	191.567
$C=0$	191.252	191.252	191.252
$C=2$	194.819	192.601	190.383
$C=4$	193.607	187.404	181.201

shows the impact of these key parameters on the horizontal issuer payoffs ${\pi }_h(\beta )$.

Table 3. Sensitivity analysis of key parameters for hybrid issuer payoffs.

Key Parameters	Hybrid Issuer Payoffs
	$\mathit{\alpha }=0.2$	$\mathit{\alpha }=0.4$	$\mathit{\alpha }=0.6$
$\beta =0.005$	189.839	187.658	186.701
$\beta =0.010$	189.228	187.052	186.097
$\beta =0.015$	188.616	186.446	185.493
${\delta }_e=0.5$	189.170	186.865	185.847
${\delta }_e=0.7$	190.508	188.451	187.554
${\delta }_e=0.9$	191.845	190.037	189.262
$\rho =-0.5$	196.511	194.252	193.192
$\rho =0$	196.102	193.855	192.839
$\rho =0.5$	195.830	193.596	192.616
${\delta }_2=0.2$	196.817	194.555	193.467
${\delta }_2=0.4$	195.089	192.922	192.061
${\delta }_2=0.6$	194.198	192.184	191.520
$C=0$	191.024	191.024	191.024
$C=2$	195.647	193.397	191.146
$C=4$	195.131	188.836	182.542

shows the impact of these key parameters on the hybrid issuer payoffs ${\pi }_l(\beta )$. According to the results of Table 1, Table 2 and Table 3, we find that under different bankruptcy loss rates $\alpha$, the issuer payoffs ${\pi }_{k^{″}}(\beta )$ are negatively correlated with the loss ratio $\beta$, the correlation coefficient $\rho$ and volatility ${\sigma }_2$, are positively correlated with the discount rate ${\delta }_e$, and are inversely U-shaped with coupon $C$. To sum up, the numerical results obtained in Table 1, Table 2 and Table 3 are consistent with the above, and the relationship between key parameters and issuer payoffs under different risk retention structures is robust.

5. Conclusions

In this study, we used asset pricing theory and risk retention rules to build vertical retention, horizontal retention and hybrid retention models. Through numerical simulation and economic implication analysis, we discussed the impact of asset pool assets and risk retention forms on the issuer payoffs. We found that hybrid retention can not only alleviate the side effects of horizontal retention but also reduce the proportion of vertical retention, which can effectively improve the issuer payoffs. Horizontal retention is applicable to situations where asset pool losses are small or asset volatility is high, while vertical retention is the opposite. Issuer payoffs are inversely U-shaped with coupons and negatively related to the correlation between assets, asset volatility and the discount rate of subordinated securities. Therefore, the issuer should choose the form of risk retention that maximizes the payoffs so as to improve the size of enterprise financing. It is worth emphasizing that in the face of the complexity of the real world, issuers should also combine other methods such as empirical research and expert judgment to obtain more comprehensive and reliable research conclusions. In addition, in order to improve the effectiveness of risk retention strategies, issuers should also comprehensively analyze and dynamically adjust risk retention strategies by combining dynamic factors such as market conditions, investor behavior and macroeconomic variables.

Based on the content of this paper, there are several extended issues that can be further explored. Firstly, this paper assumes a geometric Brownian motion as the price generating process, thus obtaining analytical solutions. However, for extreme risk scenarios, the model will not be applicable, so this work needs to be extended to integrate price processes with jumps, an extension that will help issuers better capture the sharp fluctuations in financial markets. Secondly, this paper assumes a fixed correlation coefficient between the two assets, which may oversimplify the dynamics of asset returns. In financial markets, asset correlations tend to vary with market volatility, liquidity changes and investor sentiment fluctuations. Therefore, it will be our next step to investigate how dynamic correlations can be integrated into existing asset pricing and risk management frameworks to improve the applicability of the model under different market conditions. In addition, the use of a static approach in risk retention modeling in this paper leads the model to ignore dynamic factors, such as changes in market conditions, investor behavior or macroeconomic variables, which can significantly affect the effectiveness of retention strategies over time. Therefore, how to extend the static model to the dynamic model is worthy of further study.