Bargaining, Mortgage Financing and Housing Prices*
Xun Bian College of Business & Economics Longwood University
[email protected] Zhenguo Lin Tibor and Sheila Hollo School of Real Estate College of Business Florida International University
[email protected] Yingchun Liu Department of Finance, Insurance, Real Estate and Law College of Business University of North Texas
[email protected]
Abstract Bargaining and mortgage financing have been extensively studied in the literature. However, they have only been studied separately. This paper is the first to embed financing into a bargaining model, and our model yields several new insights. First, we show that financing creates new ground for trading. In contrast to conventional wisdom, our model shows a buyer does not have to value a property more than its seller for a mutually beneficial trade to exist, and transaction prices are not bounded by the buyer’s and the seller’s valuations. It can be above the buyer’s valuation and below the seller’s reservation price. Second, our analysis shows that when financing is omitted in a bargaining model, the total gain from trade is incorrectly defined, and price is thus miscalculated. Third, our model can be used to analyze many commonly used financing arrangements in real estate transactions such as assumable loan, seller financing and seller-paid closing costs. Closed-form solutions of equilibrium prices are derived for various financing arrangements.
March 11, 2017 ________________
* This paper received the best paper award in the category of “Real Estate Brokerage/Agency” sponsored by the National Association of Realtors (NAR®) at 2015 American Real Estate Society annual conference.
Bargaining, Mortgage Financing and Housing Prices
[Abstract] Bargaining and mortgage financing have been extensively studied in the literature. However, they have only been studied separately. This paper is the first to embed financing into a bargaining model, and our model yields several new insights. First, we show that financing creates new ground for trading. In contrast to conventional wisdom, our model shows a buyer does not have to value a property more than its seller for a mutually beneficial trade to exist, and transaction prices are not bounded by the buyer’s and the seller’s valuations. It can be above the buyer’s valuation and below the seller’s reservation price. Second, our analysis shows that when financing is omitted in a bargaining model, the total gain from trade is incorrectly defined, and price is thus miscalculated. Third, our model can be used to analyze many commonly used financing arrangements in real estate transactions such as assumable loan, seller financing and seller-paid closing costs. Closed-form solutions of equilibrium prices are derived for various financing arrangements.
1.
Introduction Real Estate transactions typically possess two important characteristics. First,
prices are often established through bargaining between buyers and sellers. According to Nash (1950), bargaining is a situation where individuals (“players”) have the possibility of concluding a mutually beneficial agreement. At the same time, there is a conflict of interest about which agreement to conclude, and no agreement can be imposed on any individual without the individual’s approval. In the context of real estate transactions, the buyer naturally wants to pay the lowest price. The seller, however, desires to sell at the highest price. In addition to its bargaining element, acquisitions of real properties are often financed through mortgages. In many cases, affordability dictates that it is necessary for the buyer to have debt financing. Furthermore, with a mortgage, a buyer is able to take advantage of tax benefits, increase his financial leverage, and achieve diversification. 1 Given mortgage financing is often complementary to real estate 1
Throughout the paper, we use “she” when referring to the seller and “he” when referring to the buyer.
transactions, it is reasonable to assume that when contemplating a real estate transaction, a buyer not only considers his valuation of the property, but also weigh his payoff from different financing options. For instance, high cost of financing reduces buyers’ payoff and, in turn, shrinks or even eliminates the pie to be divided between buyers and sellers. Therefore, when studying real estate price bargaining, it is essential to consider financing as a part of the transaction process. Both bargaining and mortgage financing have been extensively studied in the literature. However, they have only been studied separately. Standard bargaining models assume gain from trade results from the difference between buyer’s and seller’s valuations. If a buyer values a property more than its seller, the surplus can be divided. Mortgage financing has been largely ignored in previous game theoretical models of real estate bargaining, and this omission leads to several limitations. First, the total gain from trade is incorrectly defined. If mortgage financing is used, the size of the pie is affected by it. Total gain from trade diminishes when financing is costly and grows when better financing options are available. Second, when omitting financing, we fail to acknowledge that a seller can often influence buyers’ financing options and alter bargaining outcomes by offering assumable loans, seller financing and seller-paid closing costs. In other words, buyers and sellers do not just split a pie with a predetermined size. They can collaborate to make the pie bigger. Our study is the first to incorporate mortgage financing into a bargaining model. We contribute to literature in several ways. First, we show that financing can create new ground for trading. In contrast to conventional wisdom, our model shows a buyer does not have to value a property more than its seller for a mutually beneficial trade to exist.
2
Furthermore, with mortgage financing, we show that transaction price can be above the buyer’s valuation and below the seller’s reservation price. Without being bounded by the buyer’s valuation and the seller’s reservation price when mortgage financing is used, transaction prices may not accurately reflect intrinsic values of properties. As a result, commonly used financial measures for credit risk, such as the loan-to-value ratio (i.e., the ratio of loan amount to transaction price), are noisier than common perception. Furthermore, we show that financing arrangements affect total gains from trade and, in turn, impact transaction prices. By embedding mortgage financing into a bargaining model, we achieve a more accurate and complete understanding of property price formation. Specifically, we derive a closed-form solution of the transaction price when bargaining and mortgage financing are considered simultaneously. Second, by adding mortgage financing, we enrich the classic bargaining framework. Assumable loans, seller financing and seller-paid closing costs, which are frequently used in real estate transactions, can now be analyzed within a bargaining framework. We derive closed-form solutions of equilibrium prices for each of those financing arrangements. As a result, we gain new insights on how real estate price bargaining are influenced by various financing options. As we will show, in a bargaining model with financing, the total gain from trade is endogenously defined and may be improved by assumable loans and seller-financing. As a result, both the buyer and the seller are better off. This paper is organized as follows. Section 2 reviews related literature. Section 3 builds a model of real estate price bargaining with mortgage financing and uses our developed framework to analyze mortgage interest deduction and a variety of financing
3
arrangements, such as assumable loans, seller financing and seller-paid closing costs. Section 4 provides concluding remarks.
2.
Literature Review The strategic interactions between sellers and buyers in the real estate market
have traditionally been modelled as a bargaining process using game theory. Since the Nash bargaining solution does not allow for the players’ strategic behavior per se, and “obscures [the details of price negotiations] in its black box” (Albrecht et al., 2007), the proper model for the real estate transaction is the non-cooperative bargaining model of Rubinstein (1982). 2 Unfortunately, the only paper that studies in some detail the noncooperative bargaining problem in the context of the real estate market is Arnold (1999). Arnold (1999) constructs a search-bargaining model to study the strategic role of asking price. In his model, asking price not only influences the rate at which offers arrive, but also serves as an initial offer made by the seller in the bargaining game. As the asking price decreases, the seller benefits from more offers arrived. However, a lower asking price may generate a loss, because high-valuation buyers normally will not pay a price greater than the asking price. 3 An optimal asking price balances these two effects. Arnold (1999) suggests that although search is external to the bargaining process, it can influence 2
Since Rubinstein (1982), the bargaining literature has been extended to incorporate features such as, outside options (Binmore, 1985), the risk of breakdown (Binmore, Rubinstein and Wolinsky, 1986), incomplete information (Myerson and Satterthwaite, 1983, Samuelson, 1984, Abreu and Gul, 2000, Fuchs and Skrzypacz, 2010), and multilateral bargaining (Baliga and Serrano, 1995, Krishna and Serrano, 1996). Furthermore, the Rubinstein bargaining game has been examined within different market structures, such as decentralized, two-sided markets (Rubinstein and Wolinsky, 1990) and network markets (CorominasBosch, 2004).
3
However, under certain not-so-normal circumstances, it is well recognized that potential buyers can get into a “bidding war” in which they bid a price above the listing price; however, this happens rarely and only when the market is exceptionally “hot” or the listing price of a property is dramatically underpriced. Based on the data from the National Association of Realtors, Green and Vandell (1998) find that such a situation occurs in about five percent of transactions.
4
bargaining outcomes through asking price. In this paper, we adopt non-cooperative Rubinstein bargaining structure and examine the connection between bargaining and another integral element of real estate transactions, mortgage financing. We contribute to the literature by showing that other than search, mortgage financing also plays a critical role in determining real estate price through bargaining. Mortgages are widely used when purchasing real estate assets, and prices resulted from the transactions are affected by how mortgage financing is constructed. Numerous empirical studies have examined the capitalization of financing premiums into transaction prices. Although the extent to which such capitalization occurs is debatable, almost all studies have documented the impact of financing arrangements on prices. 4 Zerbst and Brueggeman (1977), Brueggeman and Zerbst (1979) and Colwell, Guntermann and Sirmans (1979) compare residential property transactions financed through Federal Housing Administration (FHA) loans or Veteran Affair (VA) loans, under which discount points are paid by sellers, to transactions financed through conventional mortgages. The studies find that sellers shift the cost of discount points to FHA buyers and VA buyers by adjusting prices upward. Sirmans, Smith and Sirmans (1983) and Ferreira and Sirmans (1986) examine house prices resulted from transactions in which assumable mortgages are used. These studies show that houses with assumable loans tend to sell at higher prices than homes financed with conventional mortgage loans with higher interest rates. Rosen (1982) studies creative financing arrangements, such as teaser rates, assumable mortgages, and seller financing at below market rates. He finds that the advantages of creative financing are capitalized in housing prices leading to a “creative financing 4
For example, Bible and Crunkleton (1983), Clauretie (1984) and Rosen (1982) documents full capitalization of financing premium. On the other hand, less-than-complete capitalization is found in Ferreira and Sirmans (1986), Sirmans, Smith and Sirmans (1983) and Smith, Sirmans, and Sirmans (1984).
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premium”. Overall, ample evidence suggests that mortgage financing can affect real estate prices. However, no effort has been devoted to studying such an effect within a bargaining framework. We attempt to fill this gap in this paper.
3.
The Model We consider the real estate market in which a seller (S) has a property for sale,
potential buyers (B) arrive to inspect the property and to determine the value of this property to them. We denote the gain from the trade between the seller and a buyer as Π . When Π > 0 , there is a gain from trade. Thus, the seller and the buyer engage in a bargaining game described in Rubinstein (1982) to divide the gain. The seller first makes an offer by proposing a split ( x, Π − x) . If the buyer accepted the proposal, the game ends and a deal is reached. The payoffs received by the seller and the buyer are x and Π − x respectively. If the proposal is rejected, the buyer makes a counter-offer by proposing a new split ( x' , Π − x' ) . If the seller accepts it, the game ends, and the payoff to the seller and the buyer are now equal to θ SBgn x' and θ BBgn (Π − x') , where θ SBgn < 1 and θ BBgn < 1 represent the seller’s and the buyer’s bargaining power, respectively. There are many factors affecting θ SBgn and θ Bgn such as seller’s and buyer’s search cost and their anxiety B of waiting. Alternatively, if the seller rejects the offer, she makes a counter offer. Thereafter, the seller and the buyer take turns to make proposals. We follow the technique developed by Shaked and Sutton (1984) to determine the perfect equilibrium of this bargaining game. We define M as the supremum of the payoffs which the seller can obtain in any perfect equilibrium of the game. Suppose the buyer rejects the seller’s first offer, the seller’s payoff becomes θ SBgn M . Now consider a 6
proposal made by the buyer after rejection. Any split proposed by the buyer which gives the seller more than θ SBgn M will be accepted by the seller, so there is no perfect equilibrium in which the seller receives more than θ SBgn M . It follows that the buyer will get at least Π − θ SBgn M in any perfect equilibrium of the subgame beginning from that point. In fact, Π − θ SBgn M is the infimum of the payoff received by the buyer in this subgame. Now consider a proposal made by the seller in the first offer. Any split proposed
(
by the seller which gives the buyer anything less than θ BBgn Π − θ SBgn M
)
will not be
(
)
accepted by the buyer. Hence the seller will obtain at most Π − θ BBgn Π − θ SBgn M . In fact, as before, this is the supremum of what the seller will receive in the first offer. But the game of the seller’s second offer is identical to the game of the seller’s first offer, apart from shrinkage of all payoffs due to θ BBgn and θ SBgn . Hence it follows that the supremum of the seller’s payoff here must equal M. Therefore, M should satisfy,
(
M = Π − θ BBgn Π − θ SBgn M
Solving for M, we have M =
)
(1).
1 − θ BBgn Π . Hence, the split of the gain between the 1 − θ SBgnθ BBgn
seller and the buyers is as follows, a) The seller’s gain from trade is: 1 − θ BBgn ΠS = M = Π 1 − θ SBgnθ BBgn
(2).
b) The buyer’s gain, which is simply the difference between the total gain and the seller’s gain, is,
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ΠB = Π − ΠS =
θ BBgn (1 − θ SBgn ) 1 − θ SBgnθ BBgn
Π
(3).
Equations (2) and (3) indicate that the seller’s and the buyer’s gain from trade depend on two factors: 1) the size of total gain from trade, Π ; 2) the seller’s and the buyer’s bargaining power ( θ SBgn , θ BBgn ). In particular, both the seller and the buyer receive a greater payoff if the total gain from trade is greater. In addition, seller’s (buyer’s) gain decreases with buyer’s (seller’s) bargaining power and increases with her (his) own bargaining power. We next examine how the total gain from the trade, Π , is determined when mortgage financing is used. A defining feature of real estate transaction process is the sequential but random arrival of offering prices that characterizes the mutual search between the seller and potential buyers. During the search process, the seller receives offers over time from a stream of buyers whose offering prices and timing of arrival are stochastic in nature. Suppose the buyers’ stochastic arrival is assumed to follow the Poisson process at rate λ . This assumption is consistent with the findings of Bond et al. (2007), in which UK data is used to investigate a number of assumptions about the distribution of times to sale, such as the normal, chi-square, gamma and Weibull distributions and find that the exponential distribution explains the data better than the others. 5 Following Read (1988), and Lin and Vandell (2007), we assume buyer’s valuation of the property is uniformly distributed over [V , V ] . 6
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Arnold (1999), Glower, Haruin and Hendershott (1998) and Miceli (1989) have also assumed that potential buyers’ arrival follows the Poisson process at rate λ . 6 Regarding the distribution of buyers’ valuations, Arnold (1999) assumes that the bid distribution is over [V , V ] with density function f (V ) , where V and V are respectively the upper and lower bounds. Read (1988), and Lin and Vandell (2007) adopts a more specific assumption about buyers’ valuations. They
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Denote t i as the waiting time between the arrival of the (i − 1) th and the ith N
buyers, then the random arrival time of the Nth buyer satisfies TN = ∑ ti . At time TN , i =1
the seller has received N offer prices. We assume that the total holding cost is g (TN ) = hTN as in Haurin (1988), where h is the holding cost per unit of time. The first question to the seller is when to stop the search. That is, how many offers should the seller wait for before considering the deal? Intuitively, the seller should wait for as many offers as possible. However, beyond certain point, the marginal benefit of waiting for another offer may no longer justify the cost of doing so. In addition, some earlier offers may find other appropriate properties and are no longer available, and thus exit the bidding. We denote π as the percentage of bidders who are still interested in the property, thus ( 1 − π ) (where 0 < π < 1) is the percentage of exiting offers. Other things being equal, a higher π implies fewer exiting offers and a smaller supply of similar properties, hence a tighter housing market. With our setup of the search process, we are able to show that
Theorem 1: The optimal N * and the expected optimal waiting time on market (TOM) is given by = N*
1 λπ (V − V ) − 1 h π
(4),
and
assume the bid density function f (V ) is uniformly distributed. For technical simplicity, we adopt the same assumption. However, our essential results would hold under a wide variety of more complex distribution function assumptions.
9
N*
E[TN * ] E= = [ ∑ ti ] i =1
N*
λ (5).
1 λπ (V − V ) = − 1 h λπ
The optimal N * and the optimal TOM have the following properties (a)
∂E[TN * ] ∂N * ∂N * < 0 ; (b) > 0 ; and (c) < 0. ∂λ ∂h ∂h
Proof: See Appendix I.
Theorem 1 suggests that a seller with a higher holding cost chooses to wait for fewer buyers, hence facing less marketing period risk. In addition, Theorem 1 also indicates that, other things being equal, a seller chooses to wait for more buyers when a market is strong with a higher λ . From Equation (A5), we can rewrite the highest valuation of the property among available offers under the seller’s optimal waiting time on market TN as follows, *
E[V max ] =
πN *V + V πN * + 1
(6).
Suppose that if a purchase agreement is reached between the seller and the buyer with the highest valuation of the property at the optimal waiting time on market TN * , the buyer chooses to finance his purchase through a mortgage. The buyer incurs a transaction cost c in order to search and apply for a loan. This mortgage-related cost includes search cost incurred to find a lender as well as fees paid at settlement for mortgage origination and loan underwriting. Similar to Sirmans, Smith, Sirmans (1983), we assume that a buyer makes an equity investment E and finances his property purchase through a one10
period mortgage by borrowing P − E , where P is the transaction price. 7 At the end of the period, the buyer repays principal plus interest, (1 + i )( P − E ) , where i is the mortgage interest rate. If we denote buyers’ discount factor as δ B (δ B < 1) , the buyer’s gain from the transaction is P B = E[V max ] − c − E − δ B (1 + i )(P − E ) .
(7).
Generally speaking, the mortgage rate i is related to the amount borrowed (P-E), and the buyer first needs to choose an optimal equity investment ( E * ) to maximize his gain from trade, i.e., E * = arg max (E[V max ] − c − E − δ B (1 + i )(P − E )) . Equation (7) reveals E
that the buyer’s gain is equal to his private value of the property, V
max
, less three costs: 1)
transaction cost, c; 2) equity investment, E * , and 3) the present value of the repayment of principal plus interest at the end of period. Suppose that the seller’s value of the second-best use of the property is R. 8 For a given transaction price P, the seller’s gain from trade is PS = P − R
(8).
7
The one-period analysis has been widely adopted in the literature, such as Sirmans, Smith, and Sirmans (1983). Although it is simple, however, the model does capture the core function of debt financing: paying interest in exchange for delayed repayment. As for factors such as collateral and borrower characteristics (e.g. income, FICO, assets, and so forth), these are captured by the mortgage rate i through underwriting and impact the transaction price indirectly. 8
R can be also be interpreted as the seller’s reservation price. Hereafter, we use “the seller’s reservation price” and “the seller’s value of second-best use” interchangeably. The reservation price is the lowest price the seller is willing to sell the property. The determination of reservation price can certainly be impacted by many factors such as the seller’s tax status, opportunity costs (e.g. availability of other investments), and financial constraints. For example, a greater urgency of cash may prompt the seller to lower her reservation value. Modeling the determination of the seller’s reservation value is a complicated issue and is beyond the scope of our analysis. Instead, we follow existing bargaining literature by assuming a reservation value exists, and the gain from trade is given by the difference between the transaction price and the reservation price. Previous studies that adopted this assumption include Schelling (1956), Chatterjee (1982), Rubinstein and Wolinsky (1985), Fudenberg, Levine and Tirole (1987), and Arnold (1999).
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In other words, the seller’s gain is simply the difference between the transaction price, P, and the value of the seller’s second-best use of the property. We are able to derive a closed form solution of the transaction price, P, when bargaining and mortgage financing are simultaneously considered.
Theorem 2: the closed-form solution of the transaction price when bargaining and mortgage financing are considered simultaneously is given by R+ P=
1 − θ BBgn πN *V + V − R − c − (1 − δ B (1 + i )) E * ] [ 1 − θ SBgnθ BBgn πN * + 1 1 − θ BBgn [1 − δ B (1 + i)] 1− 1 − θ SBgnθ BBgn
(9).
Proof: See Appendix II.
Four conclusions can be immediately drawn from Equation (9). 9 First, consistent with standard bargaining theory, we show that P is strictly increasing in the bargaining power of the seller (i.e., ∂P ∂θ SBgn > 0 ) and strictly decreasing in the bargaining power of the buyer (i.e., ∂P ∂θ BBgn < 0 ). Second, the transaction price increases with the number of available buyers (i.e., ∂P ∂ (πN * ) > 0 ). It is worth noting that as we have shown in Theorem 1, a larger π N * must be accomplished through a longer time-on-market. Therefore, our model predicts a positive relationship between TOM and transaction price. A longer TOM is also associated with a greater seller’s gain from trade, because a higher 9
In the remainder of the paper, when referring to the buyer we mean the buyer who has the highest valuation of the property and is expected to outbid the others. In our model, we assume buyers are homogeneous. As a result, all buyers face the same mortgage rate, and the buyer with the highest valuation wins the bid. In reality, some buyers may qualify for better rates because of their excellent FICO score while others do not. In this case, the buyer with the highest valuation may not always be able to outbid others. Our model can be easily modified to allow for heterogeneity among buyers.
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transaction price translates to more seller’s gain. The intuition is straightforward. Costly waiting must be justified by the benefit of search, which takes the form of a higher transaction price. Our prediction is consistent with empirical evidence documenting a positive relationship between TOM and price. 10 Third, the transaction price is strictly increasing with the value of seller’s second-best use of the property (i.e., ∂P ∂R > 0 ). Finally, the transaction price decreases with the buyer’s transaction cost (i.e., ∂P ∂c > 0 ), because transaction cost shrinks the buyer’s gain from trade and, therefore, negatively affects price. The effect of mortgage interest rate i on the transaction price is more complex.
(
)
Note that ( P − E * ) − δ B (1 + i ) P − E * is the present value of the mortgage, and it should be greater than the transaction cost c for a buyer to take a mortgage. In other words, with mortgage financing, buyer’s discount rate δ B (δ B < 1) satisfies,
δB
