Forwarded by: "SIEC" Forwarded to: siec Date forwarded: Fri, 22 May 1998 11:37:44 MET Date sent: Wed, 20 May 1998 18:04:29 +0100 (BST) From: Gaia Garino To: SIEC Subject: Re: Question about poster session CAUSES OF MORTGAGE REPAYMENT FAILURES Gaia Garino* and Peter Simmons** ABSTRACT The paper uses a three period model to explain the optimal behaviour of risk averse consumers borrowing for house purchase with competitive risk neutral lenders providing the necessary mortgage funds, under income and house price uncertainty. Assuming CARA preferences, rational expectations and normally distributed individual incomes and house prices we test our theory using non linear least squares on UK data 1969-95. We find that the jump in mortgage arrears and house repossessions observed in the late 80's and early 90's is an equilibrium to which agents adjust upon realisation of adverse real shocks in an increasingly liberalised mortgage market. Keywords: Property Repossessions; Mortgage Arrears; Housing Finance JEL Classification: D1,D8,R2 * Dept. of Economics and Finance, Brunel University, Uxbridge UB8 3PH, email: Gaia.Garino@brunel.ac.uk ** Dept. of Economics, University of York, Heslington, York YO1 5DD, email: ps1@york.ac.uk CAUSES OF MORTGAGE REPAYMENT FAILURES G.Garino & P.Simmons Brunel University & University of York Introduction Various studies have examined the causes of the sharp increase in the rate of default on personal housing debt in the UK in the late 80's and early 90's. A direct telephone survey by Coles (1992) revealed that income or health shocks of debtors and personal relationship breakdown were important factors. Earlier econometric work by Brook and Mahmoud (1990) and Breedon and Joyce (1992) has pointed to chronic unemployment, high nominal interest rates and negative housing equity. Muellbauer and Cameron (1996) have focused on the role of court orders in the rate of repossession. The first aim of our paper is to present a theoretical framework of the mortgage process which allows a role for most of the above factors in the determination of default. Our second aim is to use this theoretical framework to estimate the rate of mortgage arrears on the available UK data. Our theoretical framework consists of a three period model where risk averse borrowers who derive utility from consumption and housing services buy a house with the aid of a mortgage loan in the first period; and then face a second period decision between: (i) repaying their mortgage and keeping their house into the third period; (ii) repaying their mortgage and selling their house in the second period; (iii) extending their mortgage to the third period; (iv) defaulting on their mortgage in the second period, with the lender seizing their income and house before canceling the debt. The optimal choice between these options depends on the individual borrower's expectations of future incomes and house prices and on the utility of housing, given present realised incomes and house prices and given the interest rates set by competitive risk neutral lenders. In our model borrowers start off with common preferences, incomes and expectations; but in the second and third periods they differ in their income and house price realisations, which then dictates the nature of their optimal behaviour. This fits in with the empirical observation that most mortgage defaults are triggered by individual bad shocks. Consequently we see the late 80's and early 90's rise in arrears and repossessions as a phenomenon resulting from optimal individual behaviour in the face of adverse exogenous shocks. For the purposes of an empirical test we adopt CARA utility and a normal distribution of incomes and house prices. We also measure house price risk by the regional variation in house prices; and income risk by the regional variation in the rate of unemployment; and assume that borrowers and lenders have rational expectations. Solving the relevant borrower optimisation problems we obtain an estimable equation for the borrower's arrears decision, where the degree of risk aversion, the utility of housing and the parameters associated with the dynamic adjustment terms of the equation are estimated by non linear least squares. We summarise the approach of earlier studies in section 1; we introduce our framework in section 2 and our results in section 3. Section 4 concludes. 1. Earlier Studies Brook and Mahmoud (1990) interpolate the available arrears data to a quarterly basis and then apply cointegration methods to the arrears logit, using as long run exogenous variables unwithdrawn housing equity (the rationale being that loans will be renegotiated up to this level), real personal disposable income, long term unemployment and the debt service ratio, which is measured by the ratio of the average nominal mortgage interest repayment to nominal income, divided by the real value of the owner occupied housing stock. The sample period is 1969.1-1987.4. In the error correction representation of the default rate, changes in inflation and real interest rates are significant in explaining changes in arrears, after these have been corrected for long run influences. In the long run all the exogenous variables except real income are significant; and in general the estimated equations are a satisfactory representation of the data. However, the interpolation process casts doubts on the interpretation of the dynamics. Breedon and Joyce (1992) build on the Brook and Mahmoud study with a three equation model for arrears, repossessions and house prices. Arrears and repossessions affect house prices through their impact on housing demand and supply and on expected capital gains; while house prices affect arrears and repossessions through their impact on housing equity, although the theory governing these interactions is not fully worked out. The arrears and repossessions equations are estimated 1970-1991 using bi-annual data, interpolated from annual data prior to 1982. The approach is also a cointegration one, with long run variables in the arrears equation being the debt service ratio, the unemployment rate, unwithdrawn equity, real personal disposable income and the loan to income ratio for first time buyers. In the repossessions equation the long run variables are the arrears ratio, unwithdrawn equity and the nominal mortgage rate. The equations again seem to represent the data and to have satisfactory diagnostics, but there are some weaknesses in the analysis: the authors do not model the range of decisions open to borrowers and expectations of future house prices and incomes do not affect arrears. They also use a limited range of restriction testing: for example weak exogeneity of variables in the long run relation (and crucially in the relation between arrears and default) is not checked; the coefficient of arrears in the repossessions equation is restricted to unity. Our empirical work builds on the above studies by avoiding the issues of data interpolation. Working with annual data also removes the problems of seasonal cointegration and of assigning events to particular quarters when the events themselves take several months. 2. The Model 2.1 Assumptions We consider risk averse borrowers and risk neutral lenders living three periods t=1,2,3. In each period borrowers derive utility from a composite consumption good c , whose price is normalised to unity, and from a homogeneous housing stock h , priced ã . Lifetime utility is additive, strictly concave and stationary over time: SU(c ,h ). To simplify, we assume that each borrower buys one unit of h at t=1 and that there is no physical depreciation: so h is either equal to 1, if the borrower owns the house in the period, or to 0, if the borrower sells her house or has her house repossessed by the lender. The first period house price ã is certain and common across borrowers; while, for any individual borrower i, future house prices ã and ã are uncertain at t=1. Borrowers have exogenous incomes y in each period: as for house prices, we assume that y is certain and common across borrowers; while for individual borrower i y and y are uncertain at t=1. Future incomes and house prices have a continuous joint distribution which is common knowledge to borrowers and lenders. Both borrowers and lenders can also observe house price and income realisations at the start of the relevant period, prior to choice. At t=1 all borrowers take out a mortgage for an amount ã -d, where d is an optimally chosen housing deposit paid out of initial income y . The repayment of the mortgage plus interest, m = (1+r )m , is due at t=2. r is the mortgage interest rate, set by competitive risk neutral lenders to make zero expected profits, with expectations being taken over future random variables and borrowers' decisions. There is also a safe one period bond s , with a nominal interest rate of r which is constant through time: on this bond, the consumer can borrow or lend up to the values of her expected future income and house price . Since in the second and third periods individual borrowers differ in their income and house price realisations, different individuals will select different repayment options. 2.2 Optimal Behaviour Borrowers choose optimally the mortgage deposit d and saving/borrowing s to maximise lifetime expected utility, given the interest rates set by lenders. In the last period, t=3, terminal utility of the borrower depends on realised third period variables and on past optimal choice at t=2, for which there are four possibilities: the borrower has repaid the mortgage at t=2 and kept the house into t=3; the borrower has repaid the mortgage at t=2 and sold the house at t=2; the borrower has defaulted on her mortgage at t=2; the borrower has not repaid the mortgage at t=2 and gone into arrears. Let us examine each of these possibilities, beginning with arrears. 2.2.1 Second Period Choice: Arrears A borrower i going into arrears at t=2 consumes y -s and owes the lender a repayment of m = (1+r )m at t=3. For individuals i such that y + ã < m , the individual has no other choice but to default on her mortgage at t=3. This occurs with probability F(m ) where F(.) is the cdf of y + ã . In this case, we assume that the lender seizes both the borrower's income and house, y + ã , but leaves any other remaining asset - in our framework, the one period bond proceeds (1+r)s - to the borrower . Conversely, if y + ã > m , which occurs with probability 1-F(m ), then the borrower consumes y + ã + (1+r)s and repays m to the lender at t=3 . Hence the representative borrower's second period expected utility in case of arrears is: V = max U(y -s ,1) + F(m )E U((1+r)s ,0) + + (1-F(m ))E U((1+r)s + y + ã -m ,0) (1.1) where E (E ) denote i's expectations formed at t=2 over variables at t=3, conditional on the event of third period default (no default). The arrears rate r is set ex ante by the risk neutral lender to satisfy: (1+r)m = F(m )E (y + ã ) + (1-F(m ))m (1.2) Note that expression (1.1) can accomodate various situations: de facto arrears by the borrower, an agreed loan extension between borrower and lender; or a case of remortgage to finance further consumption. 2.2.2 Second Period Choice: Default No mortgage repayment at t=2 may also signify voluntary second period default by the borrower (the phenomenon of "handing the keys over to the lender"): in this case on our assumptions the lender takes y + ã and the borrower's expected utility is simply: V = max U(-s ,0) + E U((1+r)s + y ,0) (2) Note that choice of this option can only be rational if both realised second period variables and expectations of third period variables are very low: we will come back to this point in the empirical section. 2.2.3 Second Period Choice: Repay and Sell In this case the borrower repays the mortgage but also sells the house at t=2, due to comparatively low expected third period house prices and utility: V = max U(y + ã - m - s ,0) + E U((1+r)s + y ,0) (3) 2.2.4 Second Period Choice: Repay and Keep This option gives the borrower utility of housing in the second period and the chance of a risky third period capital gain. If third period house prices are high and/or enough can be borrowed against expected future incomes and house prices, it is likely to be the preferred option: V = max U(y - m - s ,1) + E U((1+r)s + y + ã ,0) (4) 2.2.5 First Period Choice Back to the first period, the borrower chooses the housing deposit d to maximise lifetime expected utility: max U(y -d) + E max V ,V ,V ,V (5.1) while the risk neutral lender sets r ex ante to ensure zero expected profits: (1+r)m = E (y + ã ) + E m (5.2) where m satisfies (1.2). It should be noted at this point that a three period model has the minimum number of dates required to represent the mortgage contract: if the periods were two, it would be impossible to distinguish arrears from default. 2.3 Empirical Implementation In our model the interaction between the utility of housing and the effects of alternative options on uncertain financial wealth makes each borrower's optimal behaviour depend on the values of her present realised circumstances, her expectations of future incomes and house prices, and her degree of risk aversion. So for the purposes of an empirical test we need to assume specific forms for the utility function and the distribution of individual incomes and individual house prices. While the latter are common, borrowers may still differ by the realisations of their circumstances. The distribution of house prices and incomes determines the proportion of borrowers who select each of the second period options; so although there are no individual data available, we can predict the probability that a typical borrower will choose a particular option. Each step of the empirical implementation of our model is explained below . 2.3.1 Expected Incomes and House Prices At t=2 each individual i has a common distribution of income and house prices at t=3: we denote the means of these distributions as y and ã , which are the observed per capita values averaged over all individuals in the sample. We then assume rational expectations for each i, which implies that all individuals forecast the mean of the distribution . Thus: E (y ) = y (6.1) E (ã ) = ã (6.2) From expression (1) in the theory it is also clear that borrowers default at t=3 whenever their realised value of y + ã is below some critical level, given by the mortgage repayment m . We assume that house prices and incomes are normally distributed each period with means E (y ) = y , E (ã ) = ã , variances å , å and correlation r : hence their sum is also normally distributed with mean E (y + ã ) = y + ã and variance å . Third period default occurs with probability F(m ) = è((m -y -ã )/å ) where è(.) is the standard normal cdf. 2.3.2 The Utility Function As an approximation for utility we take a negative exponential form, partly for reasons of tractability and partly because this has a clear risk aversion parameter. So at t=1,2,3: U(c ,h ) = 1-n(h )exp(-bc ) (7) where b is a strictly positive constant and j=A,D,K,S indicates the available options of arrears, default, repay and keep and repay and sell. n(.) is decreasing and convex: since h = 0 for all j we set n(0) = 1. Substituting the period utility functions U(.) in (1)-(4) with the above form (7) and solving the resulting optimisation problems then gives us an analytic solution for each value function V , which we can use to obtain the condition for any option j to be preferred to any other option i#j among A,D,K,S. For reasons of space, the full routine calculations are reported in the appendix. The final forms of the value functions are: V = 2 - exp -b(1+r)y /(2+r) n(1) J (1+r) (2+r)/(1+r) (8.1) V = 2 - Z (1+r) (2+r)/(1+r) (8.2) V = 2 - exp -b(1+r)(y -m )/(2+r) n(1) J (1+r) (2+r)/(1+r) (8.3) V = 2 - exp -b(1+r)(y +ã -m )/(2+r) J (1+r) (2+r)/(1+r) (8.4) where: J = è(m )+exp -b(y +ã -m )+b å /2 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (8.5) J = J = exp -by +b å /2 (8.6) J = exp -b(y +ã )+b å /2 (8.7) The optimal behaviour of each borrower is then determined by max V ,V ,V ,V . We estimate the aggregate joint probability that v > v (i=D,K,S), generalising the model to t=2,3...T periods (=25 years in the existing time series). 2.3.3 The Estimable Equation Any individual i for whom the three conditions: V > V ; V > V ; V > V (9) hold simultaneously will go into arrears on her mortgage. Substituting the value functions (8) in each of the above inequalities and collecting terms gives us that any individual i will go into arrears iff: y > (logJ -logJ )/b(1+r) (9.1) and 0 < m + (logJ -logJ )/b(1+r) (9.2) and ã < m - logn(1)/b + (logJ -logJ )/b(1+r) (9.3) With y and ã normally distributed, the above conditions define the probability of going into arrears: è m -logn(1)/b+(logJ -logJ )/b(1+r) 1-è (logJ -logJ )/b(1+r) (10) where (9.2) is not used since it only includes deterministic variables at t=2. Then denoting: z = 1-è (logJ -logJ )/b(1+r) (10.1) z = è m -logn(1)/b+(logJ -logJ )/b(1+r) (10.2) z = z z (10.3) and calling ARR the available time series on the proportion of mortgages in arrears (which can be identified with the probability of arrears under any given form of the distribution of y and ã ) we have a model for the longrun level of arrears ARR*: ARR* = z (11) Just looking at the data, and even using the average new mortgage rather than the average outstanding mortgage to measure debt, the mean new mortgage over the sample is of the order of œ25,000 whilst the mean real lowest observed regional house price is of the order of œ40,000. This suggests that empirically we should almost never observe voluntary default, unless a loan extension were for some reason denied; and so z might be expected to be very close to 1. There is evidence that lenders only seek a court order for repossession after borrowers have built up some arrears: for example, Breedon and Joyce obtained a unit coefficient on arrears in the longrun relation for repossessions; and the survey undertaken by the Council of Mortgage Lenders in 1992 found that on average arrears amounted to six months before lenders initiate repossession. In addition, as mentioned, the 1991 agreement between lenders and the government stipulates that repossession will not commence if the mortgage interest repayments alone are being met. This is quite plausible: prior to arrears, the lender has no information that the borrower is or is likely to become in trouble. But arrears are a signal that the future repayment prospects of the borrower are questionable: in our theoretical approach this could lead the lender to separate borrowers into two groups defined according to their arrears position; to revise its probability of next period default for the bad risks who are in arrears; and hence to also revise upwards the default premium included in the interest rate for a further loan extension. If the borrowers cannot afford or do not wish to pay this increased risk premium then they default. As against this argument Breedon and Joyce report that nearly 50% of repossessions in the 90's have been instigated by borrowers; interpreting this as preempting repossession by lenders. But again this might be seen as a downward revision of (pessimistic) expectations by the borrowers themselves. From this we conclude that there is scope to include default within the set of possible optimal decisions by borrowers in the face of adverse shocks and lenders interest rates; and to estimate a model representing the probability that arrears dominate all other options. 3. Results We start by checking the dynamic properties of the arrears series ARR: although we have a small sample (the Council of Mortgage Lenders publishes the only available UK data on mortgage arrears and house repossessions starting in 1969), the ADF tests make it difficult to reject the hypothesis that ARR is I(1). Indeed, ADF tests on the first differences of ARR do consistently reject unit roots. We also try modelling ARR with a structural break in 1990 to account for the changed basis of the data collection, as well as with impulse dummies for the "critical" years 1973 and 1974, but again find it difficult to reject the unit root hypothesis. There are at least two reasons to expect a structural shift around 1990: on the one hand, in 1990 the Council of Mortgage Lenders was formed out of the earlier Building Societies Association; the main effect being that in the data the mortgage activity by commercial banks was added to that of building societies. On the other hand, in 1991 there was an informal agreement between the government and lenders to try to avoid repossession of mortgages on which the interest payments were being met. We measure the structural change in two alternative ways: first, as mentioned, with a step dummy which is 0 before but 1 after 1990; and secondly we try using Muellbauer and Cameron's time effects dummy resulting from their cohort analysis (op.cit., 1996). We find that these two approaches give very similar results but the graphs of fitted and actual values appear a bit better with the step dummy. Table 1: Unit Root Tests ARR,DARR 1971-94 Critical values: 5%=-1.958 1%=-2.682 t-adf lag ARR -0.960 2 ARR -1.170 1 ARR 0.168 0 DARR -3.276** 2 DARR -3.550** 1 DARR -2.144** 0 Since ARR is I(1) and the reasons above suggest there may be a structural break in 1990, we take a dynamic ecm corresponding to the longrun model in the form : z = àDUM + z z (12.1) DARR = á DARR + á (ARR - z ) + á Dz + e (12.2) where D is the first difference operator; DUM is the step dummy for 1990; and where all incomes, house prices, interest rates and mortgage debt terms included in z are measured in real terms. We assume that e is iid and normal with zero mean. The results of estimating (12) for 1971-94 are reproduced below: Table 2 : Equation (12), Non Linear Least Squares, 1971-94 Coefficient Standard Error b 76.920 (0.027505) à 0.015 (0.000007) v 0.004 (0.000042) á 0.544 (0.000879) á -0.552 (0.000877) á 0.451 (0.000547) R = 0.905 LM [AR(1)] 1.713 RESET 0.543 White Hetero 0.591 The following graphs indicate actual and fitted values for Table 2: Having estimated (12), we can then evaluate the equation by treating the estimated z as data; exploring its dynamic properties and then estimating a linear ecm between ARR and z. z has been estimated by nonlinear least squares; on the assumption that the disturbance in (12) is normal, a linear approximation to z is asymptotically normal with a variance that includes estimation error. So testing dynamic properties of z should provide some useful information. The unit root tests strongly suggest that z is I(1): Table 3: Unit Root Tests z,Dz, 1971-94 Critical values: 5%=-1.958 1%=-2.682 t-adf lag z 0.878 2 z 0.381 1 z -0.263 0 Dz -2.879** 2 Dz -4.227** 1 Dz -6.200** 0 We can then consider the possibility of cointegration between ARR and z: applying ADF tests to the residuals of a regression of the levels of ARR on the levels of z rejects the hypothesis of unit roots in the residuals; similarly the cointegrating DW has a value of 2.0, which confirms stationarity of the residuals. Estimating an ecm equivalent of (12) between ARR and z gives the results of Table 4: Table 4: ECM, DARR 1971-94 Dependent variable: DARR Coefficient Standard Error Constant 0.0005 0.0003 DARR_1 0.451 0.133 Dz 0.472 0.091 ARR -z -0.650 0.163 R = 0.683 F(3,20) = 14.370 å = 0.001 DW = 1.830 AR1-2 F(2,18) = 0.552 [0.585] ARCH1 F(1,18) = 0.008 [0.928] Normality Chi = 7.500 [0.023]* Hetero F(6,13)= 0.340 [0.904] RESET F(1,19) = 0.119 [0.734] The coefficients estimated for the dynamic effects are very similar to those of the nonlinear least squares estimation; and the diagnostic tests indicate no mispecification, as did the equivalent tests based on the nonlinear least squares regression. We conclude that the theoretical approach is data consistent. The following graph indicates actual and fitted values for Table 4: The results show that since the estimated z is very close to 1 at all data points, default is very much a second best option to arrears. This is consistent with our earlier arguments. On the other hand, the longrun elasticity of arrears with respect to z is only about 0.5, whereas a strict interpretation of the theory suggests that it should be unity. The ecm correction term of around -0.5 suggests quite rapid adjustment to the longrun equilibrium. 4. Conclusions In this paper we have proposed a stylised three period model to explain the optimal behaviour of risk averse consumers borrowing for house purchase with competitive risk neutral lenders providing the necessary mortgage funds under income and house price uncertainty. Assuming CARA preferences, rational expectations and a normal distribution for income and house prices we have empirically implemented our theory, using non linear least squares on the limited available UK data (1969-95). Our results indicate that mortgage arrears are an optimal decision by borrowers, identifiable with a loan extension under risk of future default, and explained by borrowers risk aversion, their expectations - shared with lenders - of future incomes and house prices, and by the lenders interest rates set to include feasible future default premia. So from our theory and our empirical analysis the jump in arrears of the late 80's and early 90's is substantially an equilibrium, to which agents have adjusted upon realisation of adverse real shocks in the context of an increasingly liberalised mortgage market. We conclude that in the face of rational behaviour and full information it is hard to believe that the distribution of real shocks has shifted so radically during the late 80's and early 90's to justify the increased number of mortgage arrears and house repossessions: it is much more likely that the financial liberalisation and the end of mortgage rationing have contributed to overheat the UK mortgage market, with the subsequent observed depression in the housing market. This conclusion is similar to that reached by Muellbauer and Cameron (1996), although for slightly different reasons (they give a major role to institutional factors). Whether our results depend on a "socially optimal" degree of risk aversion, one could argue that liberalisation has served to let lenders and households attitudes to risk find expression in market behaviour. Or one could also argue that non cooperative outcomes between multiple lenders trying to capture the marginal borrower may have partly prevented the renegotiation of debt that could have been viable in the longer term. Appendix The optimisation problems (1)-(4) of the text under assumptions (6)-(7) of rational expectations, normal distribution of incomes and house prices and negative exponential utility are solved below. (i) Arrears The borrower's problem is: V = max 1-n(1)exp[-b(y -s )] + F(m )[1-exp(-b(1+r)s )] + + i [1-exp(-b(1+r)s +y +ã -m )]dF(y +ã ) = 2 - max n(1)exp[-b(y -s )] + exp(-b(1+r)s ) è(m ) + + exp(bm )i exp[-b(y +ã )]dè(y +ã ) (1) As y and ã are normally distributed with means E (y )=y , E (ã ) = ã and variances å , å we have: i exp[-b(y +ã )]dè= i exp[-b(y +ã )]dè ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (2) (see Johnson, Kotz and Balakrishnan (1995), p.241) where: i exp[-b(y +ã )]dè = exp -b(y +ã )+b å /2 (3) Thus combining (2) and (3) we obtain expression (8.5) of the text: è(m ) + exp(-bm )i exp[-b(y +ã )]dè = = è(m )+exp -b(y +ã -m )+b å /2 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = J and substituting back into (1) we get: V = 2 - max n(1)exp[-b(y -s )] + exp(-b(1+r)s )J (4) The first order condition is: n(1)exp[-b(y -s )] - (1+r)J exp[-b(1+r)s ] = 0 (5) yielding saving/borrowing: s = [logJ + log(1+r) - logn(1) + by ]/b(2+r) (6) Substituting back in the optimisation problem (4) then gives us the value function (8.1) of the text. (ii) Default The borrower's problem is: V = max 1-n(1)exp[-b(y -s )] + i [1-exp(-b(y -(1+r)s )]dF(y ) = = 2 - max n(1)exp[-b(y -s )] + exp[-b(1+r)s ]i exp(-by )dè(y ) (7) where: i exp(-by )dè = exp -by +b å /2 = J as in (8.6) of the text. The first order condition is: exp(-bs ) - (1+r)J exp[-b(1+r)s ] = 0 (8) yielding borrowing: s = [logJ + log(1+r)]/b(2+r) (9) substituting back into (7) gives us the value function (8.2) of the text. (iii) Repay and Keep The borrower's problem is: V = max 1-n(1)exp[-b(y -m -s )] + + i [1-exp(-b(1+r)s +y +ã )]dF(y +ã ) = = 2 - max n(1)exp[-b(y -m -s )] + + exp[-b(1+r)s ]i exp[-b(y +ã )]dè(y +ã ) (10) where: i exp[-b(y +ã )]dè = exp -b(y +ã )+b å /2 = J as in (8.7) of the text. The first order condition is: n(1)exp[-b(y -m -s )] - (1+r)J exp[-b(1+r)s ] = 0 (11) yielding saving/borrowing: s = [logJ + log(1+r) - logn(1) + b(y -m )]/b(2+r) (12) Substituting back in (10) gives us the value function (8.3) of the text. (iv) Repay and Sell The borrower's problem is: V = max 1-n(1)exp[-b(y +ã -m -s )] + + i [1-exp(-b(y -(1+r)s )]dF(y ) = = 2 - max n(1)exp[-b(y +ã -m -s )] + + exp[-b(1+r)s ]i exp(-by )]dè(y ) (13) where: i exp(-by )dè = exp -by +b å /2 = J = J as in (8.6) of the text. The first order condition is: exp[-b(y +ã -m -s )] - (1+r)J exp[-b(1+r)s ] = 0 (14) yielding saving/borrowing: s = [logJ + log(1+r) - logn(1) + b(y +ã -m )]/b(2+r) (15) Substituting back in (13) gives us the value function (8.4) of the text. Data Appendix (1) Base variables (UK 1969-95): avnmort = total value of average new building society mortgage nloans = number of building society mortgages outstanding nhouse = stock of existing houses avhprice = average national house price mortrate = average annual mortgage interest rate on building society mortgages reghpric = regional house prices totdin = total personal disposable income gdpdefl = gross domestic product deflator arr = proportion of mortgages 6-12 months in arrear nposs = number of properties taken into possession tax = average income tax rate trate = annual treasury bill rate suppben = supplementary benefit allowance for married couples uemale = long term male unemployment (2) Sources: Economic Trends Annual Supplement: totdin gdpdefl tax trate suppben uemale Building Societies Gazette/Housing Finance: avnmort nloans nhouse avhprice reghpric mortrate arr nposs References [1] F.J.Breedon & M.A.S.Joyce (1992), "House Prices, Arrears and Possessions", Bank of England Quarterly Bulletin, May, 173-179. [2] M.Brook & P.Mahmoud (1990), "Mortgage Arrears, Inflation and Interest Rates", mimeo, Bank of England. [3] A.Coles (1992), "Causes and Characteristics of Arrears and Possessions", Housing Finance, n.13, February. [4] J.Muellbauer & G.Cameron (1996), "Economic Fundamentals vs. Policy Shifts in the UK's Mortgage Possessions Crisis", mimeo, Nuffield College, Oxford. [5] J.Doling & J.Ford (1991), "The Changing of Home Ownership: Building Societies and Household Investment Strategies", Policy and Politics, vol.19, n.2, 109-118. [6] N.Johnson, S.Kotz & N.Balakrishnan (1995), Continuous Univariate Distributions, Wiley.