Default and prepayment modelling in participating mortgages

Journal of Banking & Finance 61 (2015) 81–88

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Default and prepayment modelling in participating mortgages q Yusuf Varli a, Yildiray Yildirim b,⇑ a b

Research and Business Development Department, Borsa Istanbul, Turkey Zicklin School of Business, Baruch College, CUNY, United States

a r t i c l e

i n f o

Article history: Received 3 June 2014 Accepted 7 September 2015

JEL classification: G21 G32 R30

a b s t r a c t Since the 2008 financial crisis, the mortgage market has been renovating its tools and instruments in order to avoid a new crisis. One such innovative instrument is the participating mortgage, in which the lender gains part of the net operating income and/or future appreciation. In this paper, we establish a financing model for participating mortgages, incorporating early termination options such as default and two prepayment clauses, defeasance and prepayment penalty. Later, we illustrate a detailed sensitivity analysis of the model. The values of early termination options depend on the choice of parameters in the model, as well as the term structure of short term rates. Finally, we show that a participation rate of 11.24% results in zero mortgage interest rate using the parameters in our simulation. Ó 2015 Elsevier B.V. All rights reserved.

Keywords: Participating mortgages Credit risk Prepayment risk

1. Introduction Over the past two decades, mortgage products have become more prominent in the fixed-income market. The need for such products varies in accordance with the demand of the borrower and specific characteristics of the market. Participation mortgages (i.e. participating mortgages or PMs) allow borrowers to obtain below-market interest rates in return for a percentage of the property’s future appreciation and/or net operating income. They were first introduced mid-1980s, as an alternative to the fixed rate mortgages, when interest rates were high. However, they were unpopular, because borrowers were reluctant to share in the appreciation of the property and adjustable rate mortgages, which had lower initial rates, were also introduced around the same time. Furthermore, due to poorly written loan origination agreements coupled with the capital requirements of the Financial Institutions Reform, Recovery, and Enforcement Act of 1989 (FIRREA)1 participating mortgages were q We thank Brent Ambrose, Anthony Sanders, Shahid Ebrahim, Spencer Coopchik, and the seminar participants at the BIFEC conference in Istanbul for their helpful comments and suggestions. ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Varli), yildiray.yildirim@ baruch.cuny.edu (Y. Yildirim). 1 FIRREA chartered the Resolution Trust Corporation to manage insolvent thrifts formerly insured by the Federal Saving and Loan Insurance Corporation. It adapted a new regulation, making it difficult for saving institutions to hold certain amount of real estate loans. The total regulatory capital amount became 8% thereafter. The commercial real estate loans held by commercial banks had a 100% risk-weighted classification. Lastly, it also made banks onerous to liquidate commercial mortgages and curtailed originating them (see Hayre, 2001).

http://dx.doi.org/10.1016/j.jbankfin.2015.09.003 0378-4266/Ó 2015 Elsevier B.V. All rights reserved.

never popular. Until relatively recently, little has been written on these mortgages, and even now, literature has not addressed the effects of default and prepayment risks in pricing such mortgages. However, the recent financial crisis has proven that risk sharing may reduce the magnitude of the impact in case of the market crash. PM allows the borrower to have the ownership in the property while sharing the downside market risk with the lender. In conventional banking, the mortgage lender is interested with the refund of a given debt and does not consider the property appreciation. However, for a commercial participation mortgage the expected performance and risk of the investment determines the credit and debt positions of the lender and the borrower respectively. While the lender can receive a return higher than the market interest rate, borrowers may also have advantageous mortgage rates. Similar conditions can be transcribed for the borrower of a residential mortgage. She forgoes a ratio of the property’s rent or sale proceeds in order to get lower mortgage payments. Additionally, participation conditions for any kind of property can be adjusted in the contract depending on the agreement between the borrower and the lender. Caplin et al. (2008) argues that ‘‘development of shared appreciation mortgage (i.e. SAM) markets in the United States would moderate the impending decline in homeownership and lower the risk of future housing crashes. SAMs can increase the affordability of homeownership by reducing the amount of monthly payments and spreading risk more broadly between borrower and lender. . .”. Thus, the participation mortgage can be re-introduced to the market in addition to packaging mortgages and creating

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mortgage-backed securities to reduce the mortgage rate2 for affordable housing. Therefore, the PM can prevent the next potential financial crisis. However, the risk of default and prepayment for participation mortgage needs to be studied in order to prevent problem areas from arising. This paper examines these potential problem areas and establishes a path around them. Few of the earlier studies emphasize the general framework of participating mortgages. The rest of the literature focuses on a similar but more specific type of mortgage called a shared appreciation mortgage. For example, Alvayay et al. (2005) represents a partial equilibrium model to estimate the extent of the lender’s participation and conducts a comparative analysis of the factors affecting it. Ebrahim (1996) demonstrates that participating mortgages improve social welfare which implies that they are pareto superior to conventional mortgages. Ebrahim et al. (2011) establishes a basic framework of participating mortgages and describes a facility to the mortgage system. However, they use constant risk-free interest rate as a discount rate in their model. The definition of general participating mortgage in the paper is split up into different forms such as shared income, shared equity and shared appreciation mortgages. We extend their structure into a more realistic case incorporating default and prepayment options, adopting stochastic interest rate model. Initial studies on participating mortgages rely on the model as an attempt to reduce the levels of high interest payments in the U.S. (see Dougherty et al., 1982). Additionally, Page and Sanders (1986) and Dougherty et al. (1990) also focus on the effects of interest rate risk on the SAMs. Sanders and Slawson (2005) is one of the more comprehensive studies, which forms the mortgage pricing model for SAMs adapting the fixed rate mortgage model of Kau et al. (1992). The purpose of this paper is to contribute to the theoretical understanding of pricing participating mortgages by incorporating early termination clauses due to default and prepayment, and in particular to find the value of the options to the borrower. We employ three types of options namely default and two prepayment clauses, that are defeasance and prepayment penalty which are widely used in commercial and residential mortgages respectively. The option pricing method which is similar to Hilliard et al. (1998) is embedded into the model and Longstaff and Schwartz (2001) where the simulation method is used to calculate the option prices. Our numerical analysis documents that an increase in the participation rate for appreciation increases in prepayment and does not result in significant increases in default values. However, an increase in income shares increases both the prepayment and default values. For shared equity mortgages, the lender forgoes interest payments from the borrower by receiving a proportion of both net operating income and sale proceeds. In an example, we show that if income and appreciation participation rates are 11.24%, then the mortgage interest rate becomes zero. The remainder of the paper is as follows: the next section introduces the participation mortgage model; Section 3 includes prepayment and default risks into model; Section 4 documents the simulation results, finally, Section 5 summarizes the findings and concludes. 2. The model Following Ebrahim et al. (2011), we introduce the profit process Pt (i.e. operating income from operations by renting the property) can be defined as 2 Separating certain type of illiquid asset from the firm’s general risk will allow the company raise funds at a lower cost than if it could have raised the fund by issuing debt or equity (Pennacchi, 1988). Similarly, when mortgages are packaged and mortgage backed securities are created, it reduces the mortgage interest rates further.

dPt ¼ ð~r t  dP ÞPt dt þ rP Pt dZ t ; P

ð1Þ

where ~r t is the expected return (i.e. risk adjusted yield) and dP is the constant periodic cash yield (i.e. similar to dividend yield in stock). Additionally, rP denotes the volatility of profit process and Z Pt is a standard Brownian motion with respect to the physical measure. We define the real estate property value as Ht , which is generated from the following stochastic process

dHt ¼ ð~rt  dH ÞHt dt þ rH Ht dZ t : H

ð2Þ

Kau et al. (1992) defines dH as a service flow from using the real estate over time. Note that the borrower and the lender share the maintenance cost for the property, in proportion to their participation in the mortgage. The volatility rH indicates how the property value deviates from its mean. Z Ht is donated as the standard Brownian motion for the process. For the tractability of the calculations, we assume the expected return on both the profit and real estate property value are the same, ~rt , and follows Vasicek (1977) model (see Bakshi et al., 1997; Deng, 1997), ~r d~rt ¼ að~h  ~r t Þdt þ r~r dZ t ;

ð3Þ

where a denotes the speed of mean reversion, ~ h is the long-run mean rate and ~r H E½dZ t dZ t 

rr denotes volatility. We assume E½dZ Pt dZ~rt  ¼

¼ 0. We define the initial loan balance of Q 0 , the loan to value ratio of L, and the maturity of the mortgage as T. The loan includes continuous mortgage payments of at for all t 2 ½0; T and the terminal payment (i.e. balloon payment, sometimes also called the bullet payment) BT at maturity. The outstanding loan balance (i.e. OLB) at time 0; Q 0 , is equal to sum of discounted expected value of the future mortgage payments and the terminal balance such that

Z Q0 ¼

T

e~r0 ðsÞs E0 ½as ds þ e~r0 ðTÞT E0 ½BT ;

ð4Þ

0

where ~r 0 ðsÞ is the term structure of risk adjusted yield. For simplicity, we assume a non-amortizing mortgage, also called interest-only mortgage, in which there is a balloon payment consisting of the entire principal amount of the mortgage at maturity. Therefore, the outstanding loan balance for each period equals to the initial loan payment, implying Q t ¼ Q 0 ¼ BT for all t 2 ½0; T. Continuous mortgage payments are determined with a constant proportion i of OLB and at ¼ iQ t ¼ iQ 0 for all t 2 ½0; T, where i is the mortgage rate representing the cost of using mortgage determined at time 0. If there is no prepayment, default, and any other risk, then the mortgage rate i equals to the risk-free interest rate. In comparison to conventional mortgage, participating mortgages offer a participative contract between the lender and the borrower. In return for reduced mortgage rate, PM promises the lender to a part of either the excess payoff from the periodic net operating income or the gain of the sale proceeds or both. In other words, the borrower compensates the declined mortgage rate in the mortgage contract by giving a share of the excess profit flow (i.e. ðP t  KÞþ ) or the appreciation of the property value at the mortgage maturity (i.e. ðHT  H0 Þþ ) to the lender. K and H0 denotes the fixed threshold for the profit flow and the initial value of the property respectively. These threshold points can change depending on the agreement between the borrower and the lender. The share of the excess profit flow is binding by the contract, so both the lender and the borrower agree on the amount of the profit summed. Therefore, continuous mortgage payments, at , and the remaining balance at maturity BT in participating mortgages now becomes

Y. Varli, Y. Yildirim / Journal of Banking & Finance 61 (2015) 81–88

at ¼ iQ t þ hP ðPt  KÞþ

ð5Þ

and

BT ¼ Q T þ hH ðHT  H0 Þþ ;

ð6Þ

where hP and hH are the participation rates for the excess profit flow and the property appreciation value respectively. We can now write the loan balance at 0 in PM case as

Z

T

e~r0 ðsÞs E0 ½iQ s ds þ hP

Q0 ¼

Z

0

T

e~r0 ðsÞs E0 ½ðP s  KÞþ ds

0

þ e~r0 ðTÞT Q T þ hH e~r0 ðTÞT E0 ½ðHT  H0 Þþ :

ð7Þ

For simplicity, assuming non-amortizing loans, the OLB becomes

Z

T

e~r0 ðsÞs iQ 0 ds þ hP

Q0 ¼ 0

Z

T

cðP0 ; K; s; ~r 0 ðsÞÞds

0

þ e~r0 ðTÞT Q 0 þ hH cðH0 ; H0 ; T; ~r 0 ðTÞÞ;

ð8Þ

where cð:Þ represents the pricing formula for European call option. The value of call option written on the profit cash flow, with strike K and at time 0 for the any maturity time s 2 ð0; T is

cðP 0 ; K; s; ~r 0 ðsÞÞ ¼ P0 edP s Nðd1 ð0; sÞÞ  KB0;s Nðd0 ð0; sÞÞ;

ð9Þ

where N denotes the standard normal cumulative distribution function. The values as an input for N in Eq. (9) are given by

dk ð0;sÞ ¼

  ðlnP0 Þ  ðlnKÞ  ðlnB0;s Þ  dP s þ k  12 v 20;s

v 0;s

k 2 f1; 0g; ð10Þ

where

v 20;s

v

2 0;s

satisfies

  r~r 2 1 2as 3 þ r2P s ¼ 2 s þ eas  e  2a 2a a a 2qP~r r~r rP

þ

a

½s  ðD0;s Þ:

ð11Þ

Furthermore, B0;s represents the price of zero coupon bond with maturity s and formulation of that bond in this context is



B0;s ¼ e



D0;s ðk~r 0 Þsk

ffi

r~r D0;s p 2 a

2

;

ð12Þ 2 ~r 2a2

r a s where D0;s ¼ 1ea and k ¼ ~ h þ ra~r q  . q is market price of risk and assumed to be 0.25 (see Hull et al., 2014). As a reminder, the value of call option written on the property value cðH0 ; H0 ; T; ~r 0 ðTÞÞ can also be calculated the same way considering the respective parameters. The description of the term structure ~r t ðsÞ (see Merton, 1973; Brigo and Mercurio, 2006) is illustrated by

~r 0 ðsÞ ¼ 

lnB0;s ¼ s

 2

rD ffiffi D0;s ðk  ~r 0 Þ þ sk þ 2~rp0;s a s

:

ð13Þ

83

lowers the mortgage rate. Other factors effecting the mortgage rate are initial loan-to-value (i.e. LTV) ratio L and Q 0 , i.e. Q 0 ¼ L  H0 . Table 1 shows the parameters used in calculating the base case mortgage rate. In Fig. 1, we plot the mortgage rate for different state variables. It is clear that mortgage rate decreases with increasing level of participation rates. While the mortgage rate in a conventional mortgage (i.e. hP ¼ hH ¼ 0) for LTV of 80% is calculated as 8.04%, the rate is reduced to 0.89% in a participation mortgage case with 10% participation for profit cash and property value (i.e. hP ¼ hH ¼ 10%). The desired level of mortgage rates can be reached by altering loan-to-value ratio. The base case scenario indicates mortgage rate can be reduced in the participating mortgage setup by changing the values of participation rates and loan-to-value ratios. However, the reduction depends on the sensitivity of loan condition with respect to parameters chosen. Furthermore, the profit threshold and initial value of the property determine the continuous excess profit and appreciation of the property respectively. Fig. 1 also indicates that the mortgage rate moves up when each of these threshold points increases in participating mortgages. 2.2. Types of participating mortgage Depending on the values of participation rates, various kinds of agreements can be arranged between the borrower and the lender. Three common types of participating mortgages are defined in Ebrahim and Hussain (2010) and Ebrahim et al. (2011). These are Shared Income, Shared Appreciation and Shared Equity Mortgages. In the first one, the lender participates in the mortgage by receiving only a proportion of the net operating income in exchange for allowing a coupon rate below the market interest rate. In shared appreciation mortgages, the lender participates the mortgage by receiving only a proportion of the sale proceeds in exchange for allowing a coupon rate below the market interest rate. For shared equity mortgages, the lender waives interest payments of the borrower by receiving a proportion of the net operating income and sale proceeds. Fig. 2 compares the conventional and three types of participation mortgages. In the conventional case (i.e. hP ¼ hH ¼ 0%), the mortgage rate is 8.04% in the base case. The mortgage rate becomes 7.81% in shared appreciation mortgages (i.e. hP ¼ 0%; hH ¼ 5%) and 4.69% in shared income mortgages (i.e. hP ¼ 5%; hH ¼ 0%). It goes down to 4.46% in shared equity mortgages (i.e. hP ¼ hH ¼ 5%). Additionally, the lender fully trades off interest payments in a shared equity mortgage contract whenever the participation brings the same return. When both participation rates are nearly 11.24%, the mortgage rate becomes zero in the base case scenario. 3. Prepayment and default options

2.1. Mortgage rate calculation The borrower decides what percent of the excess profit flow (i.e. hP ) and the appreciation (i.e. hH ) to share with the lender in order to lower the mortgage rate. At the time of mortgage origination, we have

1  e~r0 ðTÞT i ¼ RT e~r0 ðsÞs ds 0 " RT # hP 0 cðP0 ; K; s; ~r 0 ðsÞÞds þ hH cðH0 ; H0 ; T; ~r0 ðTÞÞ ;  RT Q 0 0 e~r0 ðsÞs ds

Kalotay et al. (2004), Deng et al. (2000), Ciochetti and Vandell (1999) and Riddiough and Thompson (1993) among others use option pricing models in conventional mortgage pricing. In this section, we introduce the early termination clauses of prepayment and default for the PM using the option pricing model. 3.1. Prepayment

ð14Þ

where the part in brackets on the right hand side reflects the reduction in mortgage rate in the case of a participating mortgage in comparison to conventional mortgage. An increase in participation rate

Commercial mortgages differ from residential loans in several ways. Commercial mortgages are backed by income producing properties, like office buildings, retail shops, multifamily apartments, and hotels. Moreover, the commercial mortgages can be either non-amortizing or partially amortizing with a balloon payment. Their prepayment provisions also differ from single-family

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Y. Varli, Y. Yildirim / Journal of Banking & Finance 61 (2015) 81–88

Table 1 Base case parameters for the calculation of mortgage rate. Parameter

Definition

Value

P0 H0 K ~r 0 dP

Initial value of profit flow Initial value of property Threshold for profit flow Initial value of risk adjusted short rate Dividend rate Volatility of profit flow Service flow rate Volatility of property value Volatility of risky short rate Speed of mean reversion Long run mean of risky short rate

10 100 11 7.5% 3% 10% 3% 10% 1% 5% 7.5%

Life time of the loan Loan to value ratio Share rate for excess profit flow Share rate for appreciation of property value

30 80% 10% 10%

rP dH

rH r~r a ~ h T L hP hH

Values of all necessary parameters to calculate mortgage rate in participating mortgages.

residential mortgages. Commercial mortgage contracts generally include a penalty to restrict the borrower’s ability to refinance the loan. The most commonly used clauses are yield maintenance that enable the lender attain the same yield as if the borrower continues making the promised payments; lockout periods, allowing the lender to charge a diminishing penalty on the outstanding loan balance; and defeasance where the borrower pledges to the lender U.S. Treasury securities whose cash flows equal to the mortgage payment. On the other hand, residential mortgages can only have prepayment penalties. Therefore, we concentrate on both the prepayment penalty and defeasance in prepayment risk.

Dierker et al. (2005) indicates that prepayment conditions for residential mortgages and commercial mortgages differ. For example, residential mortgages may prepay when interest rates fall, while commercial mortgages may prepay when interest rates rise. For this reason, prepayment penalty and defeasance analyses are generalized for residential mortgages and commercial mortgages respectively. 3.1.1. Prepayment penalty In the case of declining short term rates, the borrower may want to adjust the costly loan balance by refinancing, or cashing out the equity built up to buy another property. The lender induces a penalty to avoid the prepayment. Thus, the saving comes from the difference between borrower’s new and existing mortgage interest payments while the cost is due to prepayment. Let us define the mortgage rate as ið~r 0 Þ in the contract at the beginning and it is constant for each period until maturity. However, there is a refinancing possibility in each time t after the mortgage originated at time 0 due to declining short term rates. If the borrower chooses to refinance, s/he has to pay the existing OLB in the amount of Q t ðið~r 0 ÞÞ. The new OLB at t after refinancing is Q t ðið~r t ÞÞ which is less than the old balance Q t ðið~r0 ÞÞ. We define the prepayment penalty as a proportion p of old OLB Q t ðið~r 0 ÞÞ at the time of prepayment. Therefore, prepayment takes place if the savings is more than the prepayment penalty. The condition of prepayment at time t for participating mortgage is

Q t ðið~r0 ÞÞ  Q t ðið~rt ÞÞ ¼ Sav ing P Prepayment Penalty ¼ pQ t ðið~r 0 ÞÞ;

ð15Þ

where p is the constant penalty rate.

Fig. 1. Effects of some state variables on the mortgage rate. Dependence of the mortgage rate on various state-variables in the model. The level of the mortgage rate is shown in each sub-figures with changing participation rates ðhP ¼ hH ¼ 0; 0:05; 0:10Þ.

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Y. Varli, Y. Yildirim / Journal of Banking & Finance 61 (2015) 81–88



Pt ¼

rt

L  Ht  M t 1L

   elðTtÞ  1 :

ð19Þ

We conclude that defeasance in participating mortgage occurs if the benefit from defeasance exceeds the cost of the defeasance, that is ~

Pt P Mrt t  Mrt t :

ð20Þ

Plugging Eqs. (17)–(19) into Eq. (20), we write the condition of defeasance given by

Fig. 2. Effects of participating mortgage type on the mortgage rate. Each line refers to different types of participating mortgages. The line with hP ¼ hH ¼ 0 stands for the conventional mortgages, hP ¼ 0 indicates Shared Appreciation Mortgages, hH ¼ 0 means Shared Income Mortgages and hP ¼ hH – 0 shows Shared Equity Mortgages.



  L  Ht  M rt t  elðTtÞ  1 1L Z T Z ert ðsÞðstÞ iQ 0 ds þ hP P t

Z

þhH cðHt ; H0 ; T;r t ðTÞÞ 

r

dr t ¼ aðh  rt Þdt þ rr dZ t :

ð16Þ

The long-run mean of the interest rates and the volatility has ~ and rr < r~r . We assume Brownian declining trends such as h < h motions in risky and risk-free short term processes are not corre~

lated. Mrt t and Mtrt are now defined as:

Z

M rt t ¼

T

ert ðsÞðstÞ iQ 0 ds þ hP

t

Z

T

cðPt ; K; s; r t ðsÞÞds

cðPt ;K; s;r t ðsÞÞds þ ert ðTÞðTtÞ Q 0

e~rt ðsÞðstÞ iQ 0 ds

t

Z

T

þhP 3.1.2. Defeasance Another clause of prepayment generally used in commercial mortgages is called defeasance. It can be defined as an exchange of risky mortgage payments with Treasury securities providing the same amount of payments. We follow Dierker et al. (2005) to describe the defeasance condition in participating mortgages. Here, we make a critical assumption that there is a liquid tradable participating mortgage market, and the borrower substitutes the payments due in a commercial participating mortgage with a treasury security providing the same amount of payments for the remaining term of the mortgage. Namely, at time t the investor pays Mrt t for the Treasury security to repay a participating mort~ gage, which is only worth M rt t (i.e. Q t ðið~rt ÞÞ). We use the risky rates, ~r t , to discount the risky mortgage payments of the developer to repay the participation loan at time t. We assume risk-free rate, r t ; follows the Vasicek (1977) process such that

t T

T

cðPt ; K;s; ~rt ðsÞÞds þ e~rt ðTÞðTtÞ Q 0 þ hH cðHt ; H0 ;T; ~r t ðTÞÞ : ð21Þ

t

3.2. Default A borrower defaults on her mortgage obligation when she stops making the promised monthly payments. The factors influencing the mortgage defaults are generally loan-to-value ratio (LTV) and debt service coverage ratio (i.e. DSCR) (see Vandell et al., 1993; Vandell et al., 1993; Ambrose and Sanders, 2003). Default happens when LTV ratio is higher than one, causing negative equity. Another condition of default depends on the DSCR, the ratio of income available for debt servicing of the mortgage. Whenever DSCR is less than one, the borrower may default on the mortgage. We define the default condition as

Q t P Ht ;

ð22Þ

and

iQ 0 þ hP ðPt  KÞþ P Pt :

ð23Þ

The first one represents the condition where the OLB Q t ðið~r 0 ÞÞ is higher than the value of the property Ht at time t 6 T. The second condition indicates the situation in which the periodic debt payment is not satisfied adequately by the net operating income of the property.

t

þe

r t ðTÞðTtÞ

Q 0 þ hH cðHt ; H0 ; T; r t ðTÞÞ

ð17Þ

4. Simulation results

and ~

M trt ¼

Z

T

e~rt ðsÞðstÞ iQ 0 ds þ hP

t

Z

T

cðPt ; K; s; ~r t ðsÞÞds

t

þ e~rt ðTÞðTtÞ Q 0 þ hH cðHt ; H0 ; T; ~r t ðTÞÞ:

ð18Þ

The risk free rates are less than the risky rates, r t ðsÞ < ~r t ðsÞ for all ~

s 2 ½t; T, and Mtrt < Mrt t . This relation explains the cost of defeasance. The advantage of defeasance comes from the increased equity position, which is given by Ht  M rt t , or increased interest rates. An investor may use the additional cash from defeasance as a down h r i H M t payment for another project, and may invest up to t1L t in the new project. As a result, this investment contributes an extra h r i LHt M t t amount of in comparison to property value Ht at the time 1L of defeasance. As proposed in Dierker et al., 2005, investing in the new project brings a benefit equal to the constant k per unit of capital such that ðelðTtÞ  1Þ, where l denotes the project’s excess return above its required rate of return. Consequently, benefit of defeasance Pt by liquidation is given by

In order to value early termination options for participating mortgages, we employ American options using the Longstaff and Schwartz’s (2001)3 simulation algorithm based on the use of least squares estimation to approximate the conditional expected payoff to the option holder. At every point in time, the holder of an American option compares the payoff from immediate exercise with the expected payoff from continuation. The key point in LSM approach is that the conditional expectation can be estimated from crosssectional information in the simulation by using least squares. The ex post realized payoffs from continutation is regressed on functions of the values of the state variables. The fitted value from the regression, an estimate of the conditional expectation function, is compared with the payoff from immediate exercise. This process is maintained until the initial time (t ¼ 0) and gives the value of the option. With this algorithm in simulation, the average of these values gives the current value of the American option (see section 2.2 of Longstaff and Schwartz (2001) for details). 3 Longstaff and Schwartz (2001) refers to this techniques as the least squares Monte Carlo (LSM) approach.

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Y. Varli, Y. Yildirim / Journal of Banking & Finance 61 (2015) 81–88

The parameters used in the simulation are documented in Table 2. We assume the long term mean of the interest rate process is ~ h ¼ 7:5%. Therefore, we use initial values for short rates of 5%, 7.5%, and 10% to indicate upward sloping, flat, and downward sloping term structure cases respectively. Table 3 illustrates the default, defeasance and prepayment option values using the base parameters in Table 2 for three term structure cases. Note that the default option value decreases as the interest rate increases. This is due to the increasing value of the property. As the asset value rises with increasing interest rates, default becomes less profitable. Therefore, the default option value decreases. On the other hand, the value of defeasance increases with an increase in the initial value of the short rate, because the treasury securities become cheaper. As an example, taking as ~r 0 ¼ 0:05, it is more advantageous to defease the loan later since the term structure has upward slope in this case. Furthermore, as the initial value of short rate increases, the term structure slopes downward. Since the short rate has a diminishing trend, the saving due to prepayment increases. As a result, a rise in the value of prepayment option is to be explicit. Table 4 investigates the American option values of default, defeasance and prepayment with different values of loan to value ratio. A significant increase in the value of a default option is associated with the ascent of indebtedness. The higher the loan-to-value ratio, the higher the default probability a participating mortgage will have. Higher rates of loan-to-value ratio increase the share of investors in the new project. Thus, the defeasance becomes more advantageous and the value of defeasance increases significantly. The effect becomes stronger with an increased short rate. Similar to the default case, the option value of the prepayment penalty also increases as a result of higher indebtedness. Additionally, the change in option values with respect to different term structures preserves the impacts mentioned in Table 3. The impacts of participation rates (hP and hH ) on option values are illustrated in Table 5. As it is mentioned in Section 3.2, the conditions of default are LTV (Eq. (22)) and DSCR (Eq. (23)). An increase in the participation rates affects the LTV condition in two opposite channels. First, all else being equal, the OLB goes up with increasing participation rates. The other channel comes from the negative effects of the participation rates on the mortgage rate, so increasing participation rates will lower the OLB. Although these two opposite channels are neutralized in LTV condition, the participation rate hP has a positive effect on DSCR condition of the default. Therefore, the value of default increases as participation rate hP for the profit flow raises. Moreover, no significant impact of the participation rate hH for appreciation of the property on the option of default is found. Table 5 also indicates that there is no significant effect of hP on the value of defeasance option in a given simulation setup. However, the defeasance option suffers from increasing value of hH . The advantage of defeasance coming from property appreciation decreases with an increase in the participation rate for the property appreciation. Thus, the value of the defeasance option decreases. For the flat term structure of short rates, a 20% increase in hH results the decrease of defeasance option value from 24.2200 to 18.3393. In the case of prepayment penalty, both participation rates have positive impact on the value of the prepayment option. An explanation depends on the parameters. It can be stated that as shares given to the lender increase, willingness to prepay increases since the difference between the saving and penalty increases. The saving due to prepayment becomes more advantageous in the case of rising participation rates. Table 6 documents the implications of changes in credit spread. We examine the effect of using riskier securities, instead of

Table 2 Base case parameters for option valuation. Parameter

Definition

Value

P0 H0 K ~r 0 dP

Initial value of profit flow Initial value of property Threshold for profit flow Initial value of risk adjusted short rate Divident rate Volatility of profit flow Service flow rate Volatility of property value Volatility of risky short rate Volatility of risk-free rate Speed of mean reversion Long run mean of risky short rate

10 100 11 7.5% 3% 10% 3% 10% 1% 0.6% 5% 7.5%

Long run mean of risk-free rate Life time of the loan Loan to value ratio Share rate for excess profit flow Share rate for appreciation of property value Penalty rate Excess return of the new project

4.5% 30 80% 10% 10% 2% 1%

rP dH

rH r~r rr a ~ h h T L hP hH p

l

Values of all necessary parameters to calculate the values for each option in participating mortgages.

Table 3 Values of options under the base case scenario.

~r 0 ¼ 5% ~r 0 ¼ 7:5% ~r 0 ¼ 10%

Default

Defeasance

Prepayment

2.2758 1.3413 0.7371

19.5679 24.2200 28.3394

9.0129 10.0278 10.7527

Comparison of option values for each termination case with different term structures.

Table 4 Values of options with changing loan to value ratio. Default

Defeasance

L ¼ 70% L ¼ 80% L ¼ 90%

1.0463 2.2758 3.8594

10.7286 19.5679 50.4824

Prepayment 8.3041 9.0129 9.3763

L ¼ 70% L ¼ 80% L ¼ 90%

0.6502 1.3413 2.8001

13.3207 24.2200 52.8099

9.7385 10.0278 10.5317

~r 0 ¼ 7:5%

L ¼ 70% L ¼ 80% L ¼ 90%

0.2986 0.7371 1.7915

14.9219 28.3394 66.2441

9.9961 10.7527 11.2218

~r 0 ¼ 10%

~r 0 ¼ 5%

Effects of increasing level of loan to value ratio on early termination options under different term structures.

Table 5 Values of options with changing participation rates. Default

Defeasance

Prepayment

hP ¼ 10% hP ¼ 30% hP ¼ 10%

hH ¼ 10% hH ¼ 10% hH ¼ 30%

2.2758 2.4916 2.2911

19.5679 20.2079 15.5412

9.0129 9.9902 21.9292

~r 0 ¼ 5%

hP ¼ 10% hP ¼ 30% hP ¼ 10%

hH ¼ 10% hH ¼ 10% hH ¼ 30%

1.3413 2.3544 1.3360

24.2200 22.8974 18.3393

10.0278 12.1905 22.6184

~r 0 ¼ 7:5%

hP ¼ 10% hP ¼ 30% hP ¼ 10%

hH ¼ 10% hH ¼ 10% hH ¼ 30%

0.7371 1.0166 0.3676

28.3394 26.3165 20.2324

10.7527 13.8538 22.9561

~r 0 ¼ 10%

Effects of increasing level of participation rates on early termination options under different term structures.

87

Y. Varli, Y. Yildirim / Journal of Banking & Finance 61 (2015) 81–88 Table 6 Values of options with changing difference between long term mean values of short rates.

~ h  h ¼ 0:03 ~ h  h ¼ 0:04

Default

Defeasance

2.2758

19.5679

Prepayment 9.0129

2.2347

17.6182

9.1335

~ h  h ¼ 0:05

2.2819

16.4695

8.9803

~ h  h ¼ 0:03 ~ h  h ¼ 0:04

1.3413

24.2200

10.0278

1.3408

23.6635

10.0848

~ h  h ¼ 0:05

1.3334

22.7894

10.1022

~ h  h ¼ 0:03 ~ h  h ¼ 0:04

0.7371

28.3394

10.7527

0.7296

27.9108

10.6597

0.7292

27.7001

10.7394

~ h  h ¼ 0:05

~r 0 ¼ 5%

~r 0 ¼ 7:5%

~r 0 ¼ 10%

Table 8 Values of options with changing maturity time. Default

Defeasance

Prepayment

T ¼ 10 T ¼ 30 T ¼ 50

1.1388 2.2758 2.9604

0.8996 19.5679 75.6260

4.0798 9.0129 17.1474

~r 0 ¼ 5%

T ¼ 10 T ¼ 30 T ¼ 50

0.3506 1.3413 2.2906

1.2177 24.2200 75.7552

5.0799 10.0278 15.6156

~r 0 ¼ 7:5%

T ¼ 10 T ¼ 30 T ¼ 50

0.1924 0.7371 1.4286

1.5428 28.3394 76.0779

5.6167 10.7527 12.7413

~r 0 ¼ 10%

Effects of increasing maturity time of property on early termination options under different term structures.

Effects of increasing difference between long term mean values of short rates on early termination options under different term structures.

treasury securities in the defeasance option. For this reason, we ~  h by changing modify the difference of long term mean rates h the long term mean value of risk-free rate h. Increasing credit spread decreases the defeasance option value in each term structure. For example, for ~r 0 ¼ 0:05, the value of defeasance option equals to 19.5679 as credit spread is 0.03. The option value decreases to 16.4695 when the credit spread increases to 0.05. Furthermore, we use only risky rate and assume that the credibility of the mortgage borrower remains the same throughout the time on the default and prepayment penalty options. Thus, changing the credit spread does not have any impact on the option values of the default and prepayment penalty. Table 7 presents the results of different option values for different term structure cases with various volatilities. For all term structure cases, the default option value becomes more valuable with increasing prices and short rate volatilities. These results are intuitive, because increasing volatilities raise the difference between the OLB and the prices. For the defeasance option, increases in the volatilities of risky short rate and the property value enhance the benefit of defeasance more than the cost of it. Therefore, the defeasance option value moves up for all term structure cases as the volatilities of risky short rate and the property value increase. However, the defeasance option values decrease with increasing volatility of profit flow due to the increase in the cost of defeasance. Additionally, the volatilities have different effects on the prepayment option. While an increase in the volatility of the profit flow raises the value of the prepayment option, volatility of the property value has no significant effect on the prepayment option. This result can be interpreted by referring the initial values of the profit flow and the property value. Lastly, the difference between the saving and the penalty in the prepayment

option goes up by increasing risky short rates. Thus, a positive change in the risky short rate volatility raises the prepayment option value. Lastly, Table 8 analyzes the impact of different mortgage maturities on option values. Since the option values depend on the mortgage prices, as the mortgage maturity increases the option value increases. We confirm this positive relationship for all term structure cases. 5. Conclusion A participation mortgage is a type of loan in which portions of the excess payoffs are shared across borrower and lender for any kind of real estate. In order to get a mortgage rate below market interest rate, the borrower pays a proportion of net operating income and sale proceeds of the property. This kind of risk sharing strategy brings beneficial outcomes for both parties in a mortgage contract especially in emerging economies. Thus, participating mortgages surpass conventional mortgage instruments for a high return investment. The borrower owes a rate less than the market coupon rate and the lender obtains a higher yield than the conventional mortgage rate. The mortgage rate, which is one of the most important variables in mortgage financing, is constructed and then examined under base case parameters. The mortgage rate reduces to 0.89% from 8.04% in the base case, giving 10% participation to the lender in a given mortgage. It is also shown that reduction in the mortgage rate varies according the type of participating mortgage. Additionally, early termination options such as default, defeasance and prepayment penalty are configured harmoniously with the modeling of participating mortgages. We also conducted a sensitivity analysis for option values under changing parameters in order to reach practical results. The slope

Table 7 Values of options with changing volatilities of state variables. Default

Defeasance

Prepayment

rP rP rP rP

¼ 10% ¼ 20% ¼ 10% ¼ 10%

rH rH rH rH

¼ 10% ¼ 10% ¼ 20% ¼ 10%

r~r ¼ 1% r~r ¼ 1% r~r ¼ 1% r~r ¼ 2%

2.2758 2.7665 8.2312 11.4539

19.5679 19.2484 29.6491 112.4735

9.0129 9.9799 8.9371 32.5240

~r 0 ¼ 5%

rP rP rP rP

¼ 10% ¼ 20% ¼ 10% ¼ 10%

rH rH rH rH

¼ 10% ¼ 10% ¼ 20% ¼ 10%

r~r ¼ 1% r~r ¼ 1% r~r ¼ 1% r~r ¼ 2%

1.3413 1.6221 6.8893 6.2757

24.2200 23.9559 30.2614 86.1719

10.0278 10.5658 10.5147 32.4122

~r 0 ¼ 7:5%

rP rP rP rP

¼ 10% ¼ 20% ¼ 10% ¼ 10%

rH rH rH rH

¼ 10% ¼ 10% ¼ 20% ¼ 10%

r~r ¼ 1% r~r ¼ 1% r~r ¼ 1% r~r ¼ 2%

0.7371 1.0771 4.9768 4.5416

28.3394 27.2805 34.1399 70.7710

10.7527 11.4073 10.7900 29.0989

~r 0 ¼ 10%

Effects of increasing volatilities of profit flow, property value and short term risky rates on early termination options under different term structures.

88

Y. Varli, Y. Yildirim / Journal of Banking & Finance 61 (2015) 81–88

of short rate term structure is influential for all prepayment options. One of the most interesting results of this paper is that the parameters of loan to value ratio, maturity time of the mortgage and variance of short term rates positively affect the values of each early termination option. That is, increasing either indebtedness or life time of the mortgage or volatility of short rates are critical factors that raise the values of all options in any kind of participating mortgage. For the purposes of a more comprehensive analysis, changes in almost all parameters were used to investigate the sensitivities of option values.

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