MORTGAGE VALUES OF LENDERS AND BORROWERS WITH TOP-UP PAYMENTS
Rose Neng LAI
University of Macau
Seow Eng ONG
National University of Singapore
Tien Foo SING
National University of Singapore
First draft: September 6, 2002
This draft: April 23, 2003
Abstract
A little known but widely adopted provision in mortgage documents endows the lender the right to request for a top-up negative equity when the property value falls below the loan outstanding. Although this clause is exercised only very sparingly, it fundamentally affects mortgage values to both the borrower and lender in that now the lender has a put option to request for top-up payments. We analyze the effect of the top-up option by appealing to a contingent claim framework. Specifically, we model the top-up option as a synthetic option comprising a long put to request for a top-up, a short put that cancels out the first option in the event of a default and a binary put option that is the difference between the mortgage outstanding and the property value. We show that it is sub-optimal for the lender not to call for top-ups, and that the value of the mortgage may drop substantially given the lender’s right to request top-ups.
1. Introduction
Negative equity has become a severe problem in many countries in Asia. A White Paper released by Heizo Takenaka in November 2002 reported that losses from decline in property prices amounted to 1.16 quadrillion yen (US$9.5 billion) since 1990. The loss is twice the size of Japan’s gross domestic product. It was estimated that a third of all homeowners in Hong Kong face negative asset values as property prices fell almost half from their peak in 1997 (Tan, 2001). Negative equity in Singapore was estimated to amount to S$14 million (US$7.8 billion) (Ang, 2002).
Negative equity is typically a concern for banks in terms of default risk; mortgagors with negative equity are better off defaulting than continue servicing the mortgage. Another concern, however, arises from a provision that gives banks wide leeway in requesting for a top-up when they are uncomfortable with the security margin in relation to the outstanding loan (Siow, 2002). Although banks in Hong Kong and Singapore exercise this top-up clause very sparingly (Siow, 2002; Tan, 2001), it was reported that banks in Japan have exercised the top-up provision (Hau, 2002). The consequence is that mortgagors had to take up a second mortgage, doubling the debt service burden for some borrowers.
Interestingly, banks rarely take action when the value of the property falls below the mortgage outstanding as long as the mortgagor continues to service the loan. However, a lack of action actually exacerbates the negative equity problem when property prices continue to fall. When a default occurs subsequently, the losses for mortgagors and mortgagees could be higher (Ang, 2002). Should the bank exercise the top-up clause, then that very action could lead the borrower to default.
The mortgagor’s right to default has been well studied in the existing mortgage literature. However, the existence of a top-up provision that is exercisable by the bank cast the right to default in a completely different light. In the following, we will study the values of a mortgage contract to both the mortgagor and the lender given the provision of a top-up payment whenever there is negative equity. For illustrative purpose, we will present such values in an over-simplified case with which there are only two payments in the mortgage contract; and then extend the models to one with three payments. Multiple payment contracts can be easily extended, although the analytical solutions will become more tedious and complicated. In our analysis, the mortgagor holds put options to default whereas the lender holds put options to request top-up payment. Specifically, the series of mortgage payments imply that both parties are essentially holding compound puts. From our model, unless other non-financial reasons for not defaulting exist, it is obvious that it would always be suboptimal for lenders not to request for top-up payments. On the other hand, the value of the mortgage value to the mortgagors may drop substantially given the lender’s right to request top-up payment.
This paper is organized as follows. We will firstly review previous literature in the following section. We will then present the model frameworks for the mortgagors’ option to default and the top-up payment option of the lenders in Sections 3 and 4 respectively. Section 5 provides some numerical illustrations associated with comparative static studies on the models. Finally, Section 6 concludes.
2. Literature Review
The options approach recognizes the value of the right to prepay or default in a mortgage, following the seminal work on pricing contingent claims in capital markets by Black and Scholes (1973) and Merton (1973). The choice to prepay a mortgage can be treated as an embedded call option because it allows the borrower to buy all future obligations remaining under the mortgage at a price equal to the loan’s outstanding balance (Maris and Yang, 1996). The right to prepay, i.e., the prepayment option, has value in that it may have exercise value and time value (Kolb, 1995). It has exercise value if by prepaying the mortgage the mortgagor gains a premium from giving up the mortgage with a higher interest for one with a lower one. The prepayment option gives the mortgagor, over and above the exercise value, the opportunity to postpone the prepayment by at least one more period to see if the prepayment will be more optimal.
The opportunity to default can likewise be treated as a put option since it enables the borrower to sell his property to the mortgagee at a price equal to the loan’s outstanding balance. The modeling of defaults on non-recourse mortgages as put options held by the mortgagor was developed by Foster and Van Order (1984, 1985) and extended by Epperson, et al. (1985). Such treatment has become standard in the mortgage termination literature. Simply stated, the borrower will default when the value of the home falls below the value of the loan balance (Crawford and Rosenblatt, 1995; Ambose and Capone, 1998).
Chatterjee, et al. (1998) note that a two-state variable model (using short rate and property value) is the most efficient in predicting primary market mortgage values. Market mortgage rates affect mortgages in two ways. First, if the market mortgage rate (which is the sum of the risk-free rate plus a premium) increases, default is less likely since the mortgagor values the low-cost mortgage and will hesitate to surrender it. Second, if the market mortgage rate decreases, refinancing of the loan (prepayment) becomes more likely since the mortgagor may be able to obtain a cheaper new mortgage (Capozza, et al., 1998).
Capozza, et al. (1998) further show that a two-factor model captures the competing hazards of prepayment and default. Typically, house prices are deemed to follow a stochastic process; the standard lognormal Geometric Brownian Motion (Ito process). Short-term interest rates are also stochastic following an Ornstein-Uhlenbeck mean reverting process introduced by Cox et al. (1985). See Kau, et al. (1994), Capozza, et al. (1998), Hilliard, et al. (1998), Ambrose and Capone (1998), Rebonato (1998) and Wilmott (1998) for details.
Kau, Keenan, Muller and Epperson (1995) (thereafter KKME) developed a model to price fixed-rate mortgages incorporating amortization, prepayment as well as default. The KKME model is a significant contribution to research in mortgage valuation since it is a flexible and general model of mortgage contracts that lends itself easily to modification to investigate creative financial techniques (Kau, et al., 1995).
While acknowledging the value of the KKME model, Hilliard, Kau and Slawson (1998) present a simpler model that uses the bivariate binomial options pricing technique[1] developed by Nelson and Ramaswamy (1990) and Hilliard, Schwartz and Tucker (1995). Like KKME, the model aims to simultaneously value the prepayment and default options in a fixed rate mortgage. It was found that the results generated by the bivariate binomial model are close to those of KKME especially when the interest rate and house price volatilities are low.
To the best of our knowledge, no previous work has examined the effect of a top-up provision on mortgage values.
3. The Model Framework
When a mortgage contract is considered as a contingent claim (or a series of contingent claims), its value will depend on two sources of uncertainty, namely the interest rates (r), and the property prices (S). A stochastic interest rate process follows from Cox et al. (1985) as mean-reverting, that is
(1)
where l is the speed of reversion factor, is the steady-state mean interest rate, sr is the volatility of the interest rates, and zr is a Wiener process. The property price process follows the standard stochastic process of
(2)
where m is the risk-adjusted expected growth rate of the property price, d is the convenience yield represented in terms of housing service flow, sS is the instantaneous standard deviation of the return on the asset, and zS is the Wiener process. Note that h, , m, d, sr and sS are all assumed known to the market. The correlation between dzr and dzS is r.
Assume that the mortgage contract expires at time T, and the current time is t so that the life span of the contract is t = T – t. At the beginning of the mortgage contract when t = 0, the amortized contractual payment for each payment date will be set according to
(3)
where m is the additional interest above the risk-free rate and M0 is the initial mortgage principle when the contract is determined. A negative m will mean that the interest rate charged on the mortgage will be lower than the risk-free rate, which is often true when the intention is to attract borrowers. The mortgage principle amount outstanding will be[2]
(4)
and M(r,S,0) = C for a fully amortized mortgage. Otherwise, M(r,S,0) will be equal to the amount of final payment in case of a balloon payment.
For purpose of parsimony, next assume c = r + m as the coupon payment rate, or the mortgage borrowing rate. If discrete payments rather than continuous payments are assumed, then the amount of each payment in (3) will be replaced by
(5)
and the principle amount outstanding after the ith payment becomes
" i = 1 to n (6)
where n = T ´ k is the total number of payments in the contract, with k as the number of payments per year. In order to facilitate the formulation of default put options as well as keeping the models as close to reality as possible, we will hereafter utilize expressions (5) and (6) for discrete payments methods instead of continuous payment streams.
3.1. Mortgage Values for Two-payment Case
3.1.1. Mortgagor’s Default Option
Following the risk-neutrality argument, the value of the default put option, at any time t is governed by[3]
(7)
where h represents the housing service flow. As our emphasis is on default activities, we will assume that prepayment is not possible.[4] Thus, the stochastic term structure can be ignored; and the PDE in (7) becomes
(8)
With a ruthless default opportunity in a two-payment mortgage contract, the boundary conditions for (8) are
(9)
if S ® ¥
and the solution to (8) subject to the boundary conditions in (9) is the default option value at time t (with t = T – t period remaining until maturity) according to the Black-Scholes model, which is
(10)
where and N(×) is the cumulative standard normal distribution function. Note that the stochastic property price, S, is according to expression (2), which is therefore a function of time. Nevertheless, in order to maintain parsimony, we would denote S to be the property price right at any time t.
Notice that the nature of the default option is analogous to an American put in that it can optimally be exercised as soon as the boundary condition holds. Hence, the default option value derived above based on the Black-Scholes formula for a European put can only serve as the minimum value of the American counterpart. Fortunately, having the minimum value to a purchaser is equivalently having a maximum value to the lender in market equilibrium. Therefore, we can safely assume that the default option above is the equilibrium value that all market participants perceive and agree with. In fact, Kau et.al (1992) mention that default will occur only when payment is due because the borrower can enjoy the services of the house until such time.[5]
3.1.2. Mortgage Value to the Mortgagor
Consider the commencement of a mortgage contract right after the first payment, C, is made. Upon knowing the value of the default option, the value of the mortgage to the mortgagor (borrower) at any time t between the first and last payment dates will be equal to the value of the residential unit less the present value of the second payments if there is no default, plus the value of the default option, which is
(11)
Notice that if the default option is deep-in-the-money.
Thus, the mortgagor will decide whether to default or pay the second payment to complete the mortgage contract by the following option payoff function
(12)
3.1.3. Mortgage Value to the Lender
Following the value of the mortgage to the mortgagor, that of the lender will be
(13)