Mortgage Contracts and Underwater Default
MMORTGAGE CONTRACTS AND SELECTIVE DEFAULT
YERKIN KITAPBAYEV AND SCOTT ROBERTSON
Abstract.
We analyze recently proposed mortgage contracts which aim to eliminate selectiveborrower default when the loan balance exceeds the house price (the “underwater” effect). We showthat contracts which automatically reduce the outstanding balance in the event of local house pricedecline remove the default incentive, but may induce prepayment in low price states. However, lowstate prepayments vanish if borrower utility from home ownership, or outside options such as rentalcosts, are too high. We also show that capital gain sharing features, through prepayment penaltiesin high house price states, are ineffective, as they virtually eliminate prepayment in such states.For typical foreclosure costs, we find that contracts with automatic balance adjustments becomepreferable to the traditional fixed rate mortgage at contract rate spreads of approximately 50 − Introduction
It is by now incontrovertible that the housing crisis of 2007-2009 was exacerbated by the “under-water” effect, where homeowners owed more on their house than it was worth on the market. Thenegative effects of being underwater are well known, having been documented at the government([14]), academic ([4]) and public ([32]) levels.Underwater mortgages powered a vicious cycle within many United States metropolitan areas,most prominently in the Southwest. Borrowers, having purchased homes initially worth far morethan their incomes could support, but recently having lost a large portion of their value, were stuckin houses which they could neither afford nor sell. In response, they engaged in large scale selectivedefaults on their loans (c.f. [5]). This led banks to incur significant losses, either directly throughthe foreclosure process, or indirectly through the resultant fire sales, in which the repossessed homewas sold at a depressed value (c.f. [21, 8, 4]). The fire sales further depressed home prices andappraisal values, putting more homeowners under water, repeating the cycle.In short, underwater mortgages posed, and continue to pose, significant risks for the homeowner,the lending institution, and the broader health of the economy. Furthermore, there is an inherentasymmetry in that traditional mortgage contracts have built-in protections against interest ratemovements (e.g. adjustable rate mortgages, refinancing with no penalties), but there are no such
Date : May 8, 2020. a r X i v : . [ q -f i n . P R ] M a y YERKIN KITAPBAYEV AND SCOTT ROBERTSON protections for house price decline. Indeed, default associated to house price decline has tradi-tionally been considered a “moral” issue ([22]), to be worked out in the (lengthy, expensive) legalsystem.To mitigate risks associated to underwater mortgages, as well as to avoid the legal system, anumber of alternative mortgage contracts have been proposed. At heart, each contract aims toinsulate the borrower in the event of area wide house price decline, by suitably adjusting eitherthe outstanding balance or monthly payment of a traditional fixed rate mortgage (FRM). Fromthe bank’s perspective, the idea is that if one accounts for foreclosure costs and other negativeexternalities associated to underwater default, then, despite the lower payments (compared to theFRM), the contracts are competitive or even preferable.The purpose of our paper is to analyze three such proposals, and to see which is “best” for boththe borrower and lending institution, while also identifying any unforeseen risks. We consider the“adjustable balance mortgage” of [1]; the “continuous workout mortgage” of [43]; and the ”sharedresponsibility mortgage” of [31, 32]. We choose these contracts because they span a wide rangeof possible adjustments, such as lowering payments immediately, lowering payments if house pricesfall sufficiently far, and including a prepayment penalty, and comment that our method of analysisis not limited to just these contracts.All contracts start with a certain level payment, and then adjust payments according to themovements of a (local) house price index H , which acts as a proxy for the home value. An indexis used, as opposed to home appraisals, for two reasons. First, appraisals are cumbersome andexpensive. Second, an index removes moral hazard, as the borrower cannot profit from intentionallylowering his home value. Lastly, local house price indices exist. Indeed, both the “S&P CoreLogicCase-Shiller Home Price Indices” ([9]) and the Federal Housing Finance Agency House Price IndexReports ([17]) track national and local house price movements, with Case-Shiller having indices fortwenty U.S. metropolitan areas.While detailed formulas (in continuous time) are provided for each contract in Section 2, webriefly describe the payments at a given time t prior to the loan maturity at T . To fix notation, set B Ft as the outstanding balance and c F the level payment for a traditional FRM with mortgage rate m , unit purchase price, and initial loan-to-value of B (e.g. B = 0 . (cid:2) B Ft , H t (cid:3) . Here, we have normalized H = 1, and remark that fora 20% down payment, house prices would have to fall by at least 20% before any payments areadjusted. The monthly payment is then derived to be c F × min (cid:2) , H t /B Ft (cid:3) (c.f. (2.4)), so that italso never exceeds the originally scheduled payment.Alternatively, the continuous workout mortgage (CWM) and shared responsibility mortgage(SRM) take as primitive the monthly payment, and unlike the ABM, begin adjusting payments There are also numerous papers which design optimal mortgage contracts based upon principal-agent and/orequilibrium considerations. For example, see [36, 37, 7, 16]. These are discussed at the end of the introduction.
ONTRACT DESIGN 3 upon any decline in H . The CWM monthly payment is c × min [ H t ,
1] for a certain payment cap c ,which along with the outstanding balance, are determined using risk neutral pricing theory. TheSRM sets the monthly payment to c F × min [ H t , B Ft × min [ H t ,
1] (c.f. (2.7)). Additionally, the SRMhas a profit sharing feature, designed to make the loan more valuable to the bank . Should theborrower prepay at t , he must pay the penalty α × max [ H t − , α × α = 0 .
05. The idea is to insulate the bank againstthe possibility of the borrower refinancing into another SRM when house prices are at historicminimums, because if so-refinanced, any future prepayment will incur a large penalty.Unfortunately, there are two immediate problems with the CWM. First, the outstanding balanceis not publicly observable. Rather, it is the expected value of a discounted future cash flow stream,computed under a risk-neutral measure (c.f. Remark 2.1). Unless there is a liquid market for thisstream (which, to the best of our knowledge, there is not), a model for the house price and stochasticdiscount factor must be used to compute B Ct . While this might be fine for the bank, it is bad for theborrower, who must know the balance to make a prepayment decision. Questions abound regardingimplementation, such as which model is used, and who chooses the model parameters. Second, asshown Section 3.3, under very general conditions the ABM and SRM remove the default incentive,but the CWM might not. For these two reasons, we pay primary attention to the ABM and SRM.Our analysis is performed using American options pricing methodology, where we assume bothlocally and globally, the bank takes a worst-case approach to valuation. Locally worst-case meansthat given a termination time (either default or prepayment), the bank assumes it will receive thelower of the two possible payments. Globally worst-case means the bank values the mortgage byconsidering the worst possible termination time, which is modeled as the optimal stopping time.While this avoids explicitly identifying the borrower’s rational for default or prepayment, it doesimplicitly assume a level of financial sophistication upon the borrower, as now discussed.Application of options pricing theory to value mortgage backed securities is well known in theliterature: see [27, 28, 26, 46, 12], as well as [25, 10, 11, 23]. However, it was quickly recognizedthat borrowers do not always act in a financially optimal manner (c.f. [30] for a more recentexposition), and there are individual reasons (“turnover”) for contract termination such as loss ofjob, injury, divorce, death, etc.. This led to the more commonly used reduced form models formortgage valuation: see [15, 40, 41, 29] and the many extensions therein.Despite its pitfalls, we believe the options pricing approach is necessary to differentiate thecontracts. Simply put, as the contracts’ stated objective is to reduce selective default, we mustassume the borrower is sophisticated enough to selectively default. Otherwise, we are left witheither the ad-hoc task of defining default intensities for each contract, or if we use a commondefault intensity, we may not capture contract-specific features . Unlike the ABM and CWM, the SRM is nearing commercial availability: see [34]. Our analysis can incorporate reduced form prepayment models, where for example, turnover is governed by aconditionally independent Poisson process which in turn adjusts the discounting rate.
YERKIN KITAPBAYEV AND SCOTT ROBERTSON
Using the American options approach, we identify the contract values along with optimal stoppingboundaries. We assume the house price index H follows a geometric Brownian motion with volatility σ and “dividend” rate δ , which measures either the utility the homeowner obtains by living in thehouse, or an outside option, such as renting the house. The interest rate is either constant, orfollows a Cox-Ingersoll-Ross (CIR) process which shocks partially correlated to the shocks driving H (see [29, 10, 23] for similar models). The numerical methods we used for computations shouldbe applicable also for wide class of diffusion models of house price and interest rate.While the associated free boundary problems are easy to solve numerically, the lack of explicitsolutions makes a qualitative comparison between the contracts difficult. To enable comparison, thebulk of our analysis takes place under two simplifying assumptions: constant interest rates and aninfinite maturity. Constant interest rates allow us to focus on the relationship between house pricesand default (house prices are the primary driver of default), while the perpetual horizon enables“explicit” solutions to the free boundary problems. As we later show in the stochastic interest ratecase, default boundaries are insensitive to the interest rate, and hence assuming a constant ratedoes not alter the main message. As the typical mortgage contract length is 30 years, the perpetualassumption is mild, given that selective default or prepayment decisions occur near the beginningof the term.In this setting, our main conclusions are(1) The SRM capital gain sharing feature is ineffective. At low percentages (e.g. α = 1% − α . The 5% penalty suggested in [32] is too high to have the intended effect.(2) The ABM is competitive with the FRM at relatively low mortgage rate spreads, while theSRM requires a larger spread. For example, at 35% foreclosure costs (c.f. [21, 8, 4]), the ABMrequires a spread over the FRM of 50 basis points to have the same value to the bank. TheSRM requires a spread of 150 basis points. These spreads rapidly decrease with δ .(3) If δ is sufficiently low, both the ABM and SRM endogenously cause prepayment in low houseprice states. Here, the borrower compares her mortgage rate m to the utility dividend rate δ .Should δ < m , for cash flow purposes, she has an incentive to prepay. However, for δ > m thisregion vanishes: low-state prepayment will not occur if the borrower is sufficiently happy livingin the house (or, e.g., if rental costs are too high).We conclude the ABM is generally superior the SRM, because of the ineffective SRM profitsharing feature, and because the ABM waits for larger house price declines before adjusting pay-ments. Furthermore, the SRM profit sharing feature essentially locks the borrower into her loanif home prices rise. Especially after large house price gains, the (negative) prepayment penaltywill dominate the (positive) capital-loss protection, and we envision the borrower will be frustratedwith the contract. We also conclude the ABM could be effectively marketed as a product whichoffers protection against house price decline for a reasonable mortgage spread. Furthermore, if theborrower sufficiently enjoys living in her house, it will not be sold in low house price states. ONTRACT DESIGN 5
Briefly, we comment on the literature which obtains optimal mortgage contracts based uponprincipal-agent and/or equilibrium considerations, as well as [18], which analyzes the SRM ina general equilibrium framework. The papers [36, 37, 7, 16] (among others) as well as the recent[38, 6, 20] identify optimal mortgages from a contract design viewpoint. Here, the borrower, lender,as well as the depositors to the bank, are rational agents who derive utility through consumptionand housing, and who have stochastic investment opportunities and income processes.Though not a uniform conclusion, the above papers indicate the superiority of an option ARM(for two counter-examples, see [24] which considers the “ratchet” mortgage, which is an ARM withonly negative rate resets; and [38] which, very interestingly, produces a contract similar to theABM, indexing the balance to a house price process.). This is an ARM with the added feature thatthe borrower, should he encounter a negative income shock, is allowed to defer principal payments.The mortgage would then negatively amortize, up until a point (written into the contract) atwhich the borrower defaults. Option ARMs have been issued in the market, with varying degreesof effectiveness depending upon the financial sophistication of the borrower (c.f. [19, 39] for anoverview, as well as [3] for a broader empirical discussion on mortgage contracts with complexfeatures). Especially for less sophisticated borrowers, option ARMs did not perform as intended,because given the option to lower their monthly payment, the borrowers took it, even if it putthem at greater risk. By contrast, the contracts we consider have automatic payment adjustments,removing the borrower’s discretion.Lastly, we highlight [18] which shows that indexing (especially at a national level) may leadto macroeconomic instability. Underlying this analysis is the assumption that the SRM has beenadopted on a large enough scale to cause feedback effects. For example, this could happen if FNMAdecides to back SRM mortgages. However, currently these mortgages are either in the theoreticalstages (ABM, CWM), or are being advertised on a very small scale (SRM, c.f. [34]). As such, weoffer a “first implementation” analysis, where the bank is considering offering these products on asmall scale (to sophisticated borrowers) and wishes either to know what might happen in the worstcase, or more generally, how to most effectively market the product.This paper is organized as follows. Section 2 provides details of continuous time versions of thethree contracts. Section 3 formulates corresponding optimal stopping problems, and shows thatthe CWM may not remove the default incentive. Section 4 analyzes the FRM, ABM and SRMcontracts in the infinite horizon and constant interest rate setting. Section 5 extends to finitehorizon case and allows for stochastic interest rates. Appendix A contains the proofs.2.
The Mortgages
Each mortgage involves a loan of B at time 0 with maturity T . We normalize the purchaseprice to 1 so that B is the initial loan to value (LTV). We do not assume B = 1: typical initialLTVs are 0 . . H = { H t } t ≤ T , with H = 1 scaled to the purchase price. YERKIN KITAPBAYEV AND SCOTT ROBERTSON
Traditional.
The baseline contract is a continuous time, fully amortized, level payment FRMwith mortgage rate m . The outstanding balance B F solves the ordinary differential equation (ODE)˙ B Ft = mB Ft − c F , B FT = 0where c F is the (to-be-determined) payment rate. This admits explicit solution(2.1) B Ft = B (cid:0) − e − m ( T − t ) (cid:1) − e − mT ; c F = mB − e − mT for t ≤ T . The interest portion of the payment rate is mB Ft , while the principal portion is c F − mB Ft .Next, we remark that (2.1) implies(2.2) c F = mB Ft − e − m ( T − t ) for t ≤ T , which is useful equality, identifying c F as the level payment for a loan of B Ft at rate m with maturity T − t .2.2. Adjustable Balance.
The ABM was proposed in [1], and here we present a continuous timeversion. As with the FRM, m is the mortgage rate. With B F as in (2.1), the remaining balance B A of the ABM is(2.3) B At := min (cid:2) B Ft , H t (cid:3) for t ≤ T . To compute the payment rate c A , assume at t the homeowner has borrowed B At in afixed rate, level payment, loan with maturity T − t and contract rate m . From (2.2) and (2.3) wededuce(2.4) c At = mB At − e − m ( T − t ) = c F × min (cid:20) , H t B Ft (cid:21) for t ≤ T . By design, the ABM is never underwater, with B At = B Ft , c At = c F when H t ≥ B Ft . Forexample, with a 20% down payment, house prices would have to drop by at least 20% before theABM and FRM begin to differ.2.3. Continuous Workout.
The CWM was proposed in [43], and has been subsequently analyzedin [44, 45]. Unlike the FRM and ABM, in the CWM there is no mortgage rate m . Rather, theCWM starts with a (to-be-determined) upper bound c C for the payment rate. Given the housevalue H t (which starts from H = 1 at t = 0) the payment rate is c Ct := c C × min [1 , H t ]for t ≤ T . To obtain the outstanding balance, the CWM borrows from risk-neutral pricing theory,positing the existence of a pricing measure Q , and a stochastic model for both H and the moneymarket process r . The outstanding balance is then defined via the “no arbitrage” pricing formula,assuming no future prepayments or defaults. Thus,(2.5) B Ct := E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv c Cu du (cid:21) = c C × E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv min [1 , H u ] du (cid:21) ONTRACT DESIGN 7 for t ≤ T , where E t represents taking a conditional expectation at time t . As B C = B is fixed, wemay solve for c C , obtaining the outstanding balance and payment rate(2.6) B Ct = B × E t (cid:104)(cid:82) Tt min [1 , H u ] e − (cid:82) ut r v dv du (cid:105) E (cid:104)(cid:82) T min [1 , H u ] e − (cid:82) u r v dv du (cid:105) ; c Ct = B × min [1 , H t ] E (cid:104)(cid:82) T min [1 , H u ] e − (cid:82) u r v dv du (cid:105) . Remark . Absent a liquid market for a security which promises the cash flow stream min [1 , H · ], B C is not publicly observable, but rather dependent upon a model. This raises many questionsregarding implementation. Who decides to choose the model, model parameters, and parametervalues? Is a precise description of the modelling assumptions written into the contract?2.4. Shared Responsibility.
The last contract is the SRM, proposed in [31] as well as [32, 33].It is similar to the CWM in that the payment rate declines linearly in the house price when thelatter is below 1. However, unlike for the CWM, the remaining balance of the SRM is publiclyobservable. Second, and this is a key departure from the FRM and ABM as well, upon prepaymentthe borrower must split a portion of the capital gains with the lending institution.Like the FRM and ABM, the SRM takes the mortgage rate m as given. With the FRM levelpayment c F of (2.1), the SRM payment rate is c St := c F × min [1 , H t ]for t ≤ T . To define the remaining balance, we appeal to (2.2) and (2.1), setting for t ≤ T (2.7) B St := c St × − e − m ( T − t ) m = B Ft × min [1 , H t ] . Let us compare B S , c S with the ABM-counterparts from (2.3) and (2.4), respectively. While theABM lowers payments once the house price index falls below the originally scheduled remainingbalance, the SRM adjusts payments downward for any decline in H .To compensate the bank for the reduced payment rate, the SRM requires the borrower, uponprepayment, to share a portion of the capital gain with the lender. Specifically, for α ∈ (0 , t < T the bank receives (recall H = 1) B St + α × ( H t − + = B Ft × min [1 , H t ] + α × ( H t − + . Above, the first component is the remaining balance, and the second a fraction of the capital gain.In [32], α = 5% is recommended, and as an essential feature of the SRM contract, throughout, weassume α is strictly positive.The SRM just outlined is a continuous time version of the contract described in [33], whereboth the outstanding balance and payment rate are adjusted. However, in [31] (see also [18]) thecontract adjusts the payment rate but not the outstanding balance. This is achieved by takingthe payment reduction c F − c St from the interest portion mB Ft , while leaving the principal portion c F − mB Ft alone . To be consistent with [33], we will assume reductions in both the payment rateand balance. For large house price declines, a reduction in principal may take place as well.
YERKIN KITAPBAYEV AND SCOTT ROBERTSON
Perpetual Contracts.
To conclude, we present perpetual analogs of the above mortgages,which are more amenable to analysis. As reinforced in Section 3.4, due to the (typically) lengthycontract maturity (e.g. T = 30 years), perpetual mortgage valuations are close to their finitematurity counterparts at the beginning of the term, where the effects of selective default andprepayment are most pronounced.Each of the balances and payment rates are easily derived from their finite maturity analogs bytaking T = ∞ (2.8) Contract Balance Payment RateFRM B m × B ABM min [ B , H t ] m × min [ B , H t ]SRM B × min [1 , H t ] m × min [1 , H t ]We omit the CWM (c.f. Remark 2.1 and Theorem 3.1 below).3. Optimal Stopping Problems and Default Incentives
In this section, we formulate optimal stopping problems associated to the contract values, andwell as the respective default and prepayment option values. We also show that the ABM and SRMeliminate the borrower’s incentive to default, but the CWM contract might not.3.1.
Model and Assumptions.
Results are valid assuming the house price index follows a (constant-dividend) geometric Brownian motion, but with essentially arbitrary money market process. Thereis a filtered probability space (Ω , F , F , Q ). The money market rate r is a non-negative adaptedprocess with (cid:82) T r u du < ∞ almost surely, and the house price index has dynamics(3.1) dH t H t = ( r t − δ ) dt + σdW t , H = 1where W is a Brownian motion and δ, σ > τ be a termination time. If the borrower prepays(e.g. home sale, refinancing) the bank receives B τ (the remaining balance for whichever contractis being used). If the borrower defaults, the bank receives H τ , the house price . Therefore, takinga worst-case perspective, the bank assumes it will receive min [ B τ , H τ ] with prepayment when B τ < H τ and default when H τ ≤ B τ . In Section 4.6 we account for foreclosure costs , which are significant. Indeed, foreclosure can take up to 3 years([13]), with total costs (due to maintenance, marketing and discounted “fire sale” pricing) approaching 35 −
40% ofthe home value: see [2, 21, 8, 4].
ONTRACT DESIGN 9
Second, we assume the bank has access to a liquid market which trades in H and r , and uses Q as a risk-neutral pricing measure. Furthermore, the bank assumes the borrower will do what is“globally” worst, and values the mortgage by minimizing the expected discounted payoff over alltermination (stopping) times.Importantly, we do not examine the borrower’s rationale for prepaying (i.e. refinancing versusselling) or defaulting. However, in the absence of frictions (e.g. foreclosure costs, prepaymentpenalties, refinancing costs, moving costs), there is a direct connection between the bank applyinga worst case analysis, and assuming the borrower is a financial optimizer. By contrast, whenincorporating frictions such as foreclosure costs for the bank (c.f. Section 4.6), this connection isnot as strong, as these costs do not factor into a homeowner’s decision to default.3.2. Mortgage, Prepayment, and Default Option Values.
First consider either the FRM,ABM or CWM, and with i ∈ { F, A, C } , denote by c i be the cash flow rate, and B i the outstandingbalance. In light of the above discussion, at t ≤ T the bank assigns the value(3.2) V it = essinf τ ∈ [ t,T ] E t (cid:20)(cid:90) τt e − (cid:82) ut r v dv c iu du + e − (cid:82) τt r v dv min (cid:2) H τ , B iτ (cid:3)(cid:21) where the essential infimum is taken over all stopping times τ with values in [ t, T ]. Note that beingan essential infimum, the mortgage value is an F t -measurable random variable. However, when wespecify a (Markov) model for r , the value will be a deterministic function of the current time, houseprice, and interest rate: i.e. V it = V i ( t, H t , r t ), and there will be an optimal termination policy, soone can replace “infimum” with “minimum” (see (3.7) below).To understand the above formula, note that for a given termination time, the expectation issimply the arbitrage-free price for a cash flow of c i until τ , followed by a lump-sum payment ofmin (cid:2) H τ , B iτ (cid:3) at τ . Then, the mortgage value is found by applying the worst-case analysis over allsuch stopping times. The SRM value is similarly obtained, but we must account for the capitalgain sharing feature, setting(3.3) V St = essinf τ ∈ [ t,T ] E t (cid:20)(cid:90) τt e − (cid:82) ut r v dv c Su du + e − (cid:82) τt r v dv min (cid:2) H τ , B Sτ + α ( H τ − + τ P P it := V NoP P,it − V it for i ∈ { F, A, C, S } .3.3. Preventing Ruthless Defaults. We now show the ABM and SRM ensure the default optionhas no value, but this is not necessarily the case for the CWM. Intuitively, this is because the ABMand SRM by construction ensure the mortgage is never underwater, whereas the CWM defines thebalance via risk neutral pricing, and there is no guarantee that the contract is never underwater. Theorem 3.1. For each t ≤ T , D At = D St = 0 almost surely. By contrast, D Ct = 0 almost surelyif and only if δ (cid:16) − e − δ ( T − t ) (cid:17) ≤ B E (cid:20)(cid:90) T e − (cid:82) u r v dv min [1 , H u ] du (cid:21) . (3.5)Let us consider the case r t ≡ r > 0. At t = 0, using the identity min [ a, b ] = b − ( b − a ) + for anynumbers a and b , the condition (3.5) simplifies to(3.6) (cid:90) T C (1 , u ; 1 , r, δ, σ ) du ≤ − B δ (cid:16) − e − δT (cid:17) where C ( H , T ; K, r, δ, σ ) is the Black-Scholes call option price for stock price H , maturity T ,strike K , interest rate r , dividend rate δ , and volatility σ . Using the explicit formula for (cid:90) T C ( H , u ; K, r, δ, σ ) du obtained in [42, Section 3], Figure 1 compares the left-hand side and right-hand side of (3.6) at t = 0 as a function of δ . We see that only for δ above a certain threshold, the default optionis worthless. This is intuitive, as the higher the utility from living in the house, the more thehomeowner will be reluctant to default, and above a certain level, there is no default incentive. ONTRACT DESIGN 11 Figure 1. This figure displays left-hand side (thin dash) and right-hand side (solid)of the equation (3.6) with respect to δ . The shaded region is where the default optionhas value. The critical threshold is δ = 5 . T = 30, r = 2 . B = 0 . 8, and σ = 25%. Immediate Prepayment for the FRM, ABM. We next present a useful result which states that(essentially) irrespective of the model, it is never optimal to prepay either the FRM or ABM whenthe current interest rate r t exceeds the mortgage contract rate m . Specifically, when r t > m and B Ft < H t immediate prepayment is always dominated by waiting until either r t = m or maturity. Proposition 3.2. For the FRM and ABM, it is never optimal to prepay on (cid:8) r t > m, H t > B Ft (cid:9) . Free Boundary Problems and Verification. We close this section by identifying freeboundary problems for the mortgage contract value, as well as both the default and prepaymentoption values. It is standard procedure in the option pricing literature to reduce the early exerciseproblems to free boundary PDE problems. The latter can be tackled using numerical methods suchas finite difference, which, in our setup which involves at most two spatial variables, is numericallyefficient. We exclude the CWM contract, as it may not rule out default, and requires a model tocompute the outstanding balance. Throughout, we either take r > r is a CIRprocess driven by a Brownian motion B constantly correlated with the Brownian motion W driving H (c.f. (5.2) below). Hence, we are in a “Markovian” setting with the pair of processes ( r t , H t ).To simplify notation, we write conditional expectations as E r,ht [ · ], rather than E t [ ·| r t = r, H t = h ].In this setting, each of the contract values takes the functional form(3.7) V ( t, r, h ) = inf τ ∈ [ t,T ] E r,ht (cid:20)(cid:90) τt e − (cid:82) ut r v dv c ( u, H u ) du + e − (cid:82) τt r v dv f ( τ, H τ ) (cid:21) for certain functions c ( t, h ) and f ( t, h ) of time and the house price. The respective c and f aregiven in (5.1) below, but, for example, the SRM has c ( t, h ) = c F × min [1 , h ] and f ( t, h ) = B Ft × min [1 , h ] + α ( h − + .Standard arguments show that the optimal timing problem (3.7) can be reduced to the followingfree boundary PDE problem(3.8) min [ V t + L V − rV + c, f − V ] ( t, r, h ) = 0; t ∈ (0 , T ) , r, h > where L is the second order operator associated to ( r, H ). The exercise region is { V = f } , whilethe continuation region is { V < f } . These regions must be determined, along with the solution V to the PDE. As usual, continuous and smooth pasting conditions at optimal stopping boundariesare imposed to obtain C solutions amenable to Itˆo’s formula and hence verification (c.f. [35] forextension of Itˆo’s formula to C functions).In the perpetual case, the value function is given as(3.9) V ( r, h ) = inf τ ≥ E r,h (cid:20)(cid:90) τ e − (cid:82) u r v dt c ( H u ) du + e − (cid:82) τ r v dv f ( H τ ) (cid:21) with associated free boundary problem(3.10) min [ L V − rV + c, f − V ] ( r, h ) = 0; r, h > r is constant, V becomes function of ( t, h ) and h , respectively, with r thought of as a parameter.4. Perpetual mortgage and constant interest rate As discussed in the introduction, we first take the interest rate r as constant, and use theperpetual contracts of (2.8). We assume r < m , both in view of Proposition 3.2, and becausenegative mortgage spreads are unrealistic in practice. Crucially, the perpetual case allows for“explicit” solutions to the free boundary problem (3.10), which will go a long way in providingboth intuition and comparative statics. Indeed, in (3.10), the homogeneous ODE L V − rV = 0 hasgeneral solution(4.1) V ( h ) = Ah p + Bh p for free constants A and B where p = − r − δ − σ / σ + 1 σ (cid:113) ( r − δ − σ / + 2 rσ > p = − r − δ − σ / σ − σ (cid:113) ( r − δ − σ / + 2 rσ < . (4.2)4.1. FRM. Let us first consider the perpetual FRM with c ( h ) = mB and f ( h ) = min [ B , h ] in(3.9). In the continuation region, the general solution to L V − rV + c = 0 is V F ( h ) = Ah p + Bh p + mB r for to-be-determined constants A, B . Next, we will identify two boundaries b < B < b suchthat default occurs for h ≤ b , prepayment occurs for h ≥ b , and continuation occurs within. Thesolution is obtained by finding ( A, B, b, b ) such that V F satisfies the continuous and smooth pastingconditions at b and b . Furthermore, to ensure verification, we must also show A, B < V F ( h ) ≤ min [ h, B ]. Lengthy but straightforward calculations prove the resulting system of four ONTRACT DESIGN 13 equations with four unknowns has a unique solution (see the Appendix for details). We summarizethe solution to the FRM problem in the following Proposition. Proposition 4.1. The value function V F is increasing, C , concave with the following actionregions h ≤ b ∈ ( b, b ) ≥ b Action Default Continue Prepay V F ( h ) h Ah p + Bh p + mB r B where b and b are optimal thresholds such that < b < B < b . ABM. Next we consider the perpetual ABM where c ( h ) = m × min [ B , h ] and f ( h ) =min [ B , h ] in (3.9). Here, in the continuation region L V − rV + c = 0 has solution V A ( h ) = Ah p + Bh p + mB r , h > B ˜ Ah p + ˜ Bh p + mhδ , h < B where A, B, ˜ A, ˜ B are to be obtained from boundary conditions. We recall that default is explicitlyruled out for the ABM, while in the prepayment region V A ( h ) = min [ B , h ]. The next Propositioncharacterizes the value function, showing the (surprising) existence of a prepayment region in lowhousing states, at least when the utility from occupying the house is sufficiently low. Proposition 4.2. The value function V A is increasing, C and concave. When m ≤ δ , V A hasaction regions h ≤ B ∈ ( B , b ) ≥ b Action Continue Continue Prepay V A ( h ) Ah p + Bh p + mhδ ˜ Ah p + ˜ Bh p + mB r B where b is the optimal prepayment boundary. When m > δ , V A has action regions h ≤ b ∈ ( b, B ] ∈ [ B , b ) ≥ b Action Prepay Continue Continue Prepay V A ( h ) h Ah p + Bh p + mhδ ˜ Ah p + ˜ Bh p + mB r B where b and b are the optimal prepayment thresholds. Here we give some intuition for why there is a “lower” prepayment region when δ < m , which atfirst may be surprising, but actually has a clear explanation. Indeed, when h < B , if the borrowerprepays and sells the house he receives h − min [ B , h ] = 0. Conversely, by continuing, on the net,he instantaneously pays ( m − δ ) hdt where we take into account the utility flow δh dt . Thus, he hasan incentive to prepay. Of course, by prepaying the borrower is giving up the opportunity to prepayin the future, but when the current home price h is very low, the future prepayment is of lesservalue. This is why prepayment occurs only when house price falls below some optimal threshold b < B . When m ≤ δ , the instantaneous net payment flow ( m − δ ) hdt is non-positive so there is noprepayment region below B . More formally, prepaying yields min [ h, B ] = h which is sub-optimal,as continuing forever yields the lower value E h (cid:20)(cid:90) ∞ e − ru m × min [ B , H u ] du (cid:21) < m (cid:90) ∞ e − ru E h [ H u ] du = mδ h ≤ h. To determine uniquely the six unknowns ( b, b, A, B, ˜ A, ˜ B ), we impose both continuous and smoothpasting conditions at b, b . We again refer to the Appendix for details.At first glance, the ABM low prepayment region and FRM default region appear similar. How-ever, there is an important difference. For the FRM, the borrower is defaulting, which inducessignificant foreclosure costs to the bank. For the ABM the borrower is not defaulting, rather she isrefinancing, or selling the home. Her desire to prepay is based primarily on cash flow considerations.That the low prepayment region disappears when δ ≥ m provides a key insight into the valueof the ABM (and, as we will see, the SRM as well). Having removed the default incentive, thehomeowner will remain in the mortgage provided his utility is high enough in comparison to theinterest he pays. Especially when this utility is high (e.g he likes the neighborhood or house; rentsare expensive) the borrower will not prepay at low values, and the bank will not receive the housevalue in the depressed state.4.3. SRM. We lastly consider the perpetual SRM, where c ( h ) = mB × min [1 , h ] and f ( h ) = B × min [1 , h ] + α ( h − + . For this contract, the solution to the corresponding ODE takes theform V S ( h ) = Ah p + Bh p + mB r , h > H = 1˜ Ah p + ˜ Bh p + mB hδ , h < H = 1in the continuation region while in the prepayment region V S ( h ) = B × min [1 , h ]+ α ( h − + . As wewill show in the Proposition below, α drastically increases the complexity of solution. Additionally,we need to define(4.3) m ∗ := − − p p r > m to m ∗ . Proposition 4.3. The value function V S is increasing, C , concave. Furthermore,(1) If m ≤ δ , there is a unique α ∗ such that for α < α ∗ , V S has action regions h < ∈ [1 , b ) ∈ [ b, b ∗ ] > b ∗ Action Continue Continue Prepay Continue V S ( h ) Ah p + Bh p + mB hδ ˜ Ah p + ˜ Bh p + mB r B + α ( h − 1) ˇ Bh p + mB r while for α ≥ α ∗ , V S has action regions h < > Action Continue Continue V S ( h ) Ah p + Bh p + mB hδ ˇ Bh p + mB r ONTRACT DESIGN 15 (2) If δ < m ≤ m ∗ , there is a unique α ∗ such that for α < α ∗ , V S has action regions (4.4) h ≤ b ∈ ( b, ∈ [1 , b ) ∈ [ b, b ∗ ] > b ∗ Action Prepay Continue Continue Prepay Continue V S ( h ) B h Ah p + Bh p + mB hδ ˜ Ah p + ˜ Bh p + mB r B + α ( h − 1) ˇ Bh p + mB r while for α ≥ α ∗ , V S has action regions (4.5) h ≤ b ∈ ( b, > Action Prepay Continue Continue V S ( h ) B h Ah p + Bh p + mB hδ ˜ Bh p + mB r (3) If m ∗ < m and α < B , V S has action regions h < b ∈ [ b, ∈ [1 , b ] ∈ [ b, b ∗ ] > b ∗ Action Prepay Continue Continue Prepay Continue V S ( h ) B h Ah p + Bh p + mB hδ ˜ Ah p + ˜ Bh p + mB r B + α ( h − 1) ˇ Bh p + mB r The capital gain sharing feature creates a continuation region for large house price index values,which is absent for both the FRM and ABM. This region appears because if h >> 1, prepaymentincurs too large a penalty. Furthermore, if α is large enough, then no matter how close h is to H = 1, the penalty is too severe, and the borrower will never prepay. Through numerical examplesbelow we will further see that in both cases when m ≤ δ and δ < m ≤ m ∗ , the threshold α ∗ is notvery high (e.g. 2% − h below H = 1 can be explained in the similar manner as for the ABM contract. Remark . The case m > m ∗ and α ≥ B is left untreated. Economically it is of no interestbecause typical values for α are around 5% and typical values for B are 80% − Option Values. In order to compute the default option value (FRM) and prepayment optionvalue (FRM, ABM, SRM), we need to identify the respective mortgage value functions eliminatingthe possibility of default and prepayment. Here, we present results for all mortgages simultaneously.We note that as these contracts are not the actual contracts, but rather “artificial” contracts used toisolate the value of default and prepayment, we will not use the terms “default” and “prepayment”when describing the actions. Rather we will use “stop” and “continue”. Proposition 4.5. (a) FRM: V NoDef,F ( h ) = B for h > : i.e., immediate prepayment is optimal. The value function V NoP P,F has action table (4.6) h ≤ b > b Action Stop Continue V NoP P,F ( h ) h Bh p + mB r where b is the optimal stopping threshold.(b) ABM: V NoDef,A ( h ) = V A ( h ) for h > as the ABM contract prevents selective default. Thevalue function V NoP P,A has action table(i) m ≤ δ h ≤ B > B Action Continue Continue V NoP P,A ( h ) Ah p + mhδ ˜ Bh p + mB r (ii) δ < m ≤ m ∗ h ≤ b ∈ ( b, B ) ≥ B Action Stop Continue Continue V NoP P,A ( h ) h Ah p + Bh p + mhδ ˜ Bh p + mB r (iii) m ∗ < m (4.7) h ≤ b > b Action Stop Continue V NoP P,A ( h ) h Bh p + mB r where b > B .(c) SRM: V NoDef,S ( h ) = V S ( h ) for h > . The value function V NoP P,S has action table(i) mB ≤ δ (4.8) h ≤ > Action Continue Continue V NoP P,S ( h ) Ah p + mB hδ ˜ Bh p + mB r (ii) δ < mB ≤ m ∗ h ≤ b ∈ ( b, ≥ Action Stop Continue Continue V NoP P,S ( h ) h Ah p + Bh p + mB hδ ˜ Bh p + mB r (iii) m ∗ < mB h ≤ b > b Action Stop Continue V NoP P,S ( h ) h Bh p + mB r Numerical Analysis. We now perform an extensive numerical comparison of the three con-tracts. To run the numerical analysis, we use late February 2019 rates r = 2 . m = 4 . 56% (Mortgage Banker’s Association 30 jumbo rate for B = 80%). We take σ = 25%, in accordance with the local house price index volatility during high uncertainty states asin [18, Section 3]. Lastly, we set α = 5%, the value suggested in [31, 32]. However, for comparison’ssake we will sometimes consider α = 1% and 2% as well. We consider two cases for δ : low when δ = 4% < m ; and high when δ = 7% > m . ONTRACT DESIGN 17 As summarized in the introduction, our main findings are(1) The SRM value is insensitive to the capital gain sharing proportion α because, even for small α (e.g. 1 − h > B . As such,while the prepayment option values for the FRM and ABM are similar, the prepayment optionfor the SRM is of little to no value, and may actually decrease with the home value.(2) Relative to the ABM, the SRM has a lower value, even ignoring the capital gain sharing feature.This is because the SRM lowers payments once H falls below 1, rather than once H falls below B (as the ABM does), which typically is 0 . . δ , the ABM becomes more valuable than the FRM atrelatively low foreclosure costs. Specifically, foreclosure costs which equate the ABM and FRMcontracts range from 20% (low utility) to 60% (high utility). For the SRM, the equivalentforeclosure costs are much higher (40% to 80%).(4) For fixed foreclosure costs, the endogenous spread (i.e. the spread over the FRM mortgage ratewhich equates the two contract values) of the ABM is very low, compared with the SRM, butboth increase substantially with the utility. For example, with low utilities and 15% foreclosurecosts, the ABM spread is 7 . α , we conclude the ABM is most effective at preventing ruthless defaults, beingpalatable to the borrower, and not introducing unexpected prepayment behaviors.Figure 2 plots the contracts’ value as a function of the house price index. Recall that we havenormalized the index so that at initiation it has value 1. Thus, Figure 2 indicates how the contracts’value will change with movements of the index. For example, if after one year, the house priceindex falls to 0 . δ = 7% case, the FRM will have value 0 . . 440 and the SRM( α = 5%) 0 . δ , with the effect much morepronounced for the ABM and SRM mortgages. Intuitively this is clear: once the default incentiveis removed, the higher utility the borrower obtains from owning the house, the less likely she will beto prepay. As such the payment rate comprises the bulk of the mortgage value, and this paymentrate is lowered for the ABM and SRM both by construction and since H decreases with δ .The ABM and SRM remove the default incentive. To gain an understanding of how valuableprepayments are, in Figure 3, we plot the relative prepayment option values for the ABM and SRM, Figure 2. This figure plots the mortgage value functions for FRM (thin dash),ABM (thick dash), and SRM ( α = 5%) (solid) with respect to the house price h for δ = 4% (left) and δ = 7% (right). Figure 3. This figure plots the relative prepayment option values for FRM (thindash), ABM (thick dash), SRM ( α = 1%) (dot-dash), and SRM ( α = 5%) (solid) asa function of the house price h for δ = 4% (left) and δ = 7% (right).along with the FRM. Here, we see a striking difference between the FRM, ABM contracts and theSRM contract. For the first two, the option value significantly increases for large home values, asone would expect. However, for the SRM, depending on the sharing proportion α , the prepaymentoption may actually decrease with the home value. This is entirely due to the capitalization sharingfeature, which penalizes prepayment for high home prices. Indeed, in the low utility δ = 4% case, forthe small sharing proportion α = 1%, the SRM prepayment option increases with the home value,though much more gradually than either the ABM or FRM. However, for α = 5% the penalty istoo large and the prepayment option decreases. Lastly, note that for all the contracts, the relativeprepayment option values substantially decrease as δ increases: as the homeowner derives moreutility from living in the house, he has less incentive to prepay.Figure 3 suggests one study the sensitivity of the SRM contract value with respect to α . Figure4 shows the map α → V S (1). Here, (note the y axis scaling) we see α has a minimal effect on thecontract value. This is because if α exceeds the threshold α ∗ ( m ) from Proposition 4.3 (6 . 55% forlow δ , and 1 . 58% for high δ ), the borrower never prepays the SRM when h > 1, and hence thecontract value is constant in α . The insensitivity of V S (1) to α is even more striking if the borrowerreceives a high utility from living in the house. ONTRACT DESIGN 19 Figure 4. This figure plots the map α → V S (1; α ) for δ = 4% (left) and δ = 7% (right).4.6. Foreclosure Costs. To gain a clearer picture of the respective contracts’ effectiveness, wenow account for foreclosure costs. Indeed, should the borrower default at τ , the bank may receivefar less than the home price H τ , due to both direct and indirect foreclosure costs, which may be30 − 40% of the home value ([8, 4]). Therefore, it is imperative to account for foreclosure, andin fact, this can only present the ABM and SRM contracts in a better light, since they explicitlyremove the default incentive.We assume that upon default of the FRM at τ , there is a fractional loss φ incurred by the bank,so that rather than receiving H τ , the bank receives (1 − φ ) H τ . The borrower, of course, does notcare about φ , so φ will not affect the default time. Thus, as the optimal stopping time, for agiven starting house price level of h , is τ ( h ) = inf (cid:8) t ≥ | H t ≤ b or H t ≥ b (cid:9) with default at b andprepayment at b , the FRM has foreclosure-adjusted value V Fφ ( h ) = E h (cid:34)(cid:90) τ ( h )0 e − ru mB du + e − rτ ( h ) (cid:0) (1 − φ ) b τ ( h ) ≤ τ ( h ) + B τ ( h ) ≥ τ ( h ) (cid:1)(cid:35) ;= V F ( h ) − φb E h (cid:104) e − rτ ( h ) τ ( h ) ≤ τ ( h ) (cid:105) . where we have written τ ( h ) and τ ( h ) as the first hitting times to b and b , respectively, given H = h .The function u ( h ) := E h (cid:2) e − rτ ( h ) τ ( h ) ≤ τ ( h ) (cid:3) clearly satisfies the boundary conditions u ( b ) = 1 and u ( b ) = 0. In region ( b, b ) it satisfies the ODE L u − ru = 0. As such, it admits the explicit solution u ( h ) = (cid:18) bh (cid:19) − p (cid:32) b p − p − h p − p b p − p − b p − p (cid:33) so that(4.9) V Fφ ( h ) = V F ( h ) − φ × b − p h − p (cid:32) b p − p − h p − p b p − p − b p − p (cid:33) for h ∈ ( b, b ). Using (4.9) we may easily identify the foreclose proportions φ A = φ A ( h ) and φ S = φ S ( h ) which equate the adjusted FRM value V Fφ ( h ) with the respective ABM and SRMvalues V A ( h ) and V S ( h ). This will tell us how large foreclosure costs need to be, in order for theproposed contracts to have the same value as the FRM. Figure 5 plots the map h → φ A ( h ) and h → φ S ( h ) for house prices below the initial H = 1. It isvery interesting that, except at extremely low prices, the equivalent foreclosure rates are insensitiveto the house price. Thus, once foreclosure costs are estimated, the contracts may be adoptedwith relatively little risk (in terms of losing value compared with the FRM) due to house pricemovements. Second, equivalent foreclosure costs are relatively low, especially in the low utilitycase. For example, when δ = 4%, the ABM becomes more valuable than the FRM once foreclosurecosts approach 20% of the home value. The SRM fares worse, with equivalent costs in the 40%range, but none-the-less both contracts outperform at the observed foreclosure costs of [8, 4]. Forlarge δ foreclosure costs must be very high before the ABM and SRM contracts are competitive.Here, the significant utility the homeowner receives by living in the house, induces the homeownerto neither prepay (nor default). As such, the lower payment rate dominates, reducing the ABMand SRM contract values.A second way of comparing the contracts’ performance accounting for foreclosure costs is toidentify endogenous mortgage rates . Here, for a given foreclosure percentage cost φ and FRMcontract rate m , the idea is to find rates m A , m S at which all three contracts have the same value.More precisely, if we think of the contracts’ value as functions of both the house price and mortgagerate, then we use (4.9) to seek m A , m S such that V Fφ (1 , m ) = V A (1 , m A ) = V S (1 , m S ) . Figure 6 then plots the maps φ → , × ( m A − m ) and φ → , × ( m S − m ) at h = 1, toexpress the results in basis points. For example, in the case of δ = 4%, if foreclosure costs are 15%of the home value, then the ABM contract need only offer a spread of 7 . δ = 7%the spreads significantly increase, with the ABM needing to offer 98 basis points and the SRM 178basis points.Especially for the ABM, Figure 6 enables a powerful message, even in the high utility case. Fora “low” spread of 1% over the traditional FRM rate, the ABM will ensure the depositor is neverunderwater. As the borrower utility from remaining in the house is large relative to her rate, shewill not prepay the mortgage at low price values. Thus, the borrower gets default protection, andthe bank gets a fairly valued mortgage, with little to no possibility of receiving the house in lowhouse price states. 5. Finite Horizon Case Our last section considers the finite maturity version of the contracts. We are interested in seeinghow the perpetual conclusions hold up when there is a finite horizon, with a particular interest inthe prepayment and default regions. ONTRACT DESIGN 21 Figure 5. This figure plots equivalent foreclosure costs (in %) for ABM (thickdash) and SRM ( α = 5%) (solid) as a function of the house price h for δ = 4% (left)and δ = 7% (right). Figure 6. This figure plots endogenous mortgage rate spreads (in basis points) at h = 1, as a function of the foreclosure cost for ABM (thick dash) and SRM ( α = 5%)(solid) for δ = 4% (left) and δ = 7% (right).5.1. Constant interest rate. To identify the action regions, we numerically solve (3.8), whichfor constant interest rate r > (cid:20) V t + 12 σ h V hh + ( r − δ ) hV h − rV + c, f − V (cid:21) ( t, h ) = 0 , t ∈ (0 , T ) , h > V ( T, h ) = f ( T, h ) , h > c ( t, h ) and f ( t, h ) determined by the three contracts(5.1) Contract c ( t, h ) f ( t, h )FRM c F min (cid:2) B Ft , h (cid:3) ABM c F × min [1 , h/B t ] min (cid:2) B Ft , h (cid:3) SRM c F × min [1 , h ] B Ft × min [1 , h ] + α ( h − + t This figure plots FRM prepayment and default boundaries (thick line),and ABM prepayment boundaries (thin line) as a function of time t for δ = 4%(left) and δ = 7% (right). The dot-dashed line represents the remaining balance.; Figure 8. This figure plots SRM prepayment boundaries as a function of time t for α = 5% (left) and α = 2% (right). Also the utility rate δ = 4%. Prepaymentregions are below the lowest curve and within the wedge. Everywhere else is thecontinuation region.To numerically compute the boundaries and value functions, we use a backward finite differenceexplicit scheme . Parameter values are the same as in Section 4.5 and T = 30 years. Figures 7 Our code is available upon request. ONTRACT DESIGN 23 and 8 broadly verify our intuition. Indeed, Figure 7 shows the ABM continuation region containsthe FRM region in that the FRM borrower will default or prepay first. In fact, this can be shownanalytically as both contracts have the payoff functions upon termination, but the value of FRMhas a higher payment rate). Furthermore, for δ high enough there will be no prepayment in lowhouse price states at t = 0. Interestingly, for the ABM, low house state prepayment arises furtherinto the life of the loan in the high utility case. This was of course absent in the perpetual case,where a simple comparison of m versus δ determined if low-state prepayment should ever occur.Heuristically, this can be seen by noting for H t = h < B Ft that immediate prepayment yields h ,while waiting until maturity yields E ht (cid:20)(cid:90) Tt e − r ( u − t ) c F min (cid:20) , H u B Fu (cid:21) du (cid:21) = h × (cid:90) Tt e − r ( u − t ) c F E (cid:34) min (cid:34) h , e ( r − δ − σ / u − t )+ σ √ u − tZ (1 − e − mT ) B (1 − e − m ( T − u ) ) (cid:35)(cid:35) du for Z ∼ N (0 , ∞ as h ↓ 0. Therefore, for finite horizon, immediate prepayment is notdominated by waiting forever. This of course, does not prove immediately that prepaying is optimalamong all policies, but does point to show that the residual horizon is finite, immediate prepaymentis a more viable option, and in fact, Figure 7 shows if the remaining horizon is small enough, thereis a low-house state prepayment region.For the SRM, the interesting part of Figure 8 is for large h . Here, there is a prepayment “wedge”.We have already seen that for h > α ( h − 1) will eliminateearly prepayment. Figure 8 shows this phenomena in the time dimension: namely that for any h > T − t below which the borrower will not prepay because thesharing feature dominates the gain from early prepayment. In fact, this is very easy to see: for h > t < T yields B Ft + α ( h − E ht (cid:20)(cid:90) Tt e − r ( u − t ) c F × min [1 , H u ] du (cid:21) ≤ c F × min (cid:34) − e − r ( T − t ) r , h − e − δ ( T − t ) δ (cid:35) . As such all ( t, h ) such that B Ft + α ( h − 1) exceeds min (cid:2) (1 − e − r ( T − t ) ) /r, h (1 − e − δ ( T − t ) ) /δ (cid:3) mustlie in the continuation region. This set induces a wedge-shaped prepayment region, and it easy tosee for any h > t, h ) will lie in the continuation region for t close enough to T .5.2. Stochastic interest rate. We lastly allow for stochastic interest rates. Namely, recall (3.1),and assume r follows a CIR process so that(5.2) dr t = κ ( θ − r t ) dt + ξ √ r t dB t ; d (cid:104) W, B (cid:105) t = ρdt where | ρ | < κ, θ, ξ > κθ ≥ ξ / 2, to ensure r stays strictly positive Thefree boundary problem (3.8) specifies tomin (cid:20) V t + 12 ξ rV rr + ξσρ √ rhV rh + 12 σ h V hh + κ ( θ − r ) V r + ( r − δ ) hV h − rV + c, f − V (cid:21) = 0 for t ≤ T and r, h > 0, with terminal condition v ( T, r, h ) = f ( T, h ) and ( c, f ) given in (5.1). Wesolve the PDE backwards using an explicit scheme. Our main interest lies in discovering how theaction regions vary jointly with the interest rate and house price. As we will see, at a very broadlevel, the conclusions obtained in the perpetual, constant interest rate case transfer over. Theparameter values for the rate m and house price volatility σ are the same as in Section 4.5, and T = 30 years. The interest rate parameters: long term mean θ = 2 . κ = 0 . 25, the volatility is ξ = 0 . 10, and correlation coefficient is ρ = − . δ = 4%, the upper panels of Figures 9 (FRM) and 10 (ABM) showthe default, continuation and prepayment regions at times t = 0 and 10 years. The prepaymentregions for both contracts are consistent with Proposition 3.2: prepayment only occurs for largehouse price values and when r t < m .As for default (or lower prepayment for the ABM), the regions are rather insensitive to the interestvalue r , lending validity to conclusions drawn in the perpetual case of Section 4. Furthermore, astime evolves both the prepayment and default regions decrease (along with the balance). Lastly,though hard to tell, the upper prepayment region for the ABM is above that for the FRM, and thelower prepayment region for the ABM is lower than the default region for the FRM. As such, theABM borrower waits longer to make a decision, and as noted above, this can be shown rigorously.The lower panels of Figures 9 and 10 give the analogous plots in the high utility δ = 7%case. Here, the FRM regions are essentially unchanged (a close inspection will see the prepaymentboundaries are shifted upwards as δ → 7% as one would expect). By contrast, the increase in δ hasa drastic effect on the ABM. The upper prepayment region sees a modest increase, but the lowerprepayment region is virtually eliminated. This is consistent with the infinite horizon, constant r case (see Proposition 4.2) as well as the finite horizon, constant interest rate case (c.f. the rightplot in Figure 7).Figure 11 shows the SRM regions in the low utility δ = 4% case for respective prepaymentpenalties α = 5% and α = 2%. Here, in both instances, the continuation region is significantlywider, compared with FRM and ABM (note the house price scaling), with the upper prepaymentregion essentially disappearing by 10 years (in fact, as shown for the constant r case the upperprepayment region is rather a wedge, bounded from above by a second continuation region). Thisis entirely due to the profit sharing penalty. Appendix A. Proofs Proof of Theorem 3.1. First consider the ABM. From (3.2) and (3.4) it suffices to show B A ≤ H .But, this is obvious from (2.3). Next, consider the SRM. From (3.3) and (3.4) it suffices to show B S + α ( H − + ≤ H . From (2.7) we know for t ≤ T that H t − (cid:0) B St + α ( H t − + (cid:1) = H t − B Ft min [1 , H t ] − α ( H t − + . On H t > − α )( H t − 1) + (1 − B Ft ) ≥ B Ft ≤ B ≤ 1. On H t ≤ H t (1 − B Ft ) ≥ ONTRACT DESIGN 25 Figure 9. This figure plots stopping regions for FRM when δ = 4% (upper panels)and δ = 7% (lower panels) for t = 0 (left panels) and t = 10 (right panels) years.The y axis represents the house index and x axis the interest rate. The prepaymentregion is darkly shaded and the default region is lightly shaded.We now turn to the CWM. As a preliminary result, we first show almost surely on [0 , T ] × Ω(A.1) D Ct = esssup τ ∈ [ t,T ] E t (cid:104) e − (cid:82) τt r v dv (cid:0) B Cτ − H τ (cid:1) + (cid:105) . Indeed, let τ be any fixed (bounded) stopping time and recall the formula for B C in (2.5). By theoptional sampling theorem B Cτ = E τ (cid:104)(cid:82) Tτ e − (cid:82) uτ r v dv c Cu du (cid:105) . Therefore, by the tower property E t (cid:20)(cid:90) τt e − (cid:82) ut r v dv c Cu du + e − (cid:82) τt r v dv B Cτ (cid:21) = E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv c Cu du (cid:21) = B Ct , and this holds for all τ . Thus, (3.4) implies V (bwr) ,NoDef,Ct = H t − B Ct . Continuing, as min [ a, b ] = b − ( b − a ) + E t (cid:20)(cid:90) τt e − (cid:82) ut r v dv c Cu du + e − (cid:82) τt r v dv min (cid:2) H τ , B Cτ (cid:3)(cid:21) = B Ct − E t (cid:104) e − (cid:82) τt r v dv (cid:0) B Cτ − H τ (cid:1) + (cid:105) As this holds for all τ it is clear from (3.2) that V (bwr) ,Ct = H t − essinf τ ∈ [ t,T ] (cid:16) B Ct − E t (cid:104) e − (cid:82) τt r v dv (cid:0) B Cτ − H τ (cid:1) + (cid:105)(cid:17) ;= V (bwr) ,NoDef,Ct + esssup τ ∈ [ t,T ] E t (cid:104) e − (cid:82) τt r v dv (cid:0) B Cτ − H τ (cid:1) + (cid:105) , Figure 10. This figure plots stopping regions for ABM when δ = 4% (upper panels)and δ = 7% (lower panels) for t = 0 (left panels) and t = 10 (right panels) years.The y axis represents the house index and x axis the interest rate. The prepaymentregions are darkly shaded.which gives the result. With this identity, we first show (3.5) implies D (bwr) ,Ct = 0 almost surely.Define the martingale M · := E (cid:0)(cid:82) · σdW v (cid:1) · and set (cid:102) M = E (cid:104)(cid:82) T min [1 , H u ] e − (cid:82) u r v dv du (cid:105) . Let t ≤ s ≤ T . From (2.6) we see B Cs = B (cid:102) M E s (cid:20)(cid:90) Ts e − (cid:82) us r v dv min [1 , H u ] du (cid:21) ;= B (cid:102) M H s M s E s (cid:20)(cid:90) Ts min (cid:104) e − (cid:82) u r v dv + δt , e − δ ( u − s ) M u (cid:105) du (cid:21) ; ≤ B (cid:0) − e − δ ( T − s ) (cid:1) δ (cid:102) M H s ≤ B (cid:0) − e − δ ( T − t ) (cid:1) δ (cid:102) M H s , so that (3.5) implies B C ≤ H on [ t, T ] and hence D (bwr) ,Ct = 0 almost surely (c.f. (A.1)). We nowprove the opposite direction, assuming (3.5) is violated. As min [ a, b ] = b − ( b − a ) + we deduce B Ct H t = B (cid:102) M H t (cid:90) Tt E t (cid:104) e − (cid:82) ut r v dv H u (cid:105) du − B (cid:102) M H t E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv ( H u − + du (cid:21) ;= B (cid:0) − e − δ ( T − t ) (cid:1) δ (cid:102) M − B (cid:102) M H t E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv ( H u − + du (cid:21) . ONTRACT DESIGN 27 Figure 11. This figure plots stopping regions for SRM when α = 5% (upper panels)and α = 2% (lower panels) for t = 0 (left panels) and t = 10 (right panels) years inthe low utility case δ = 4%. The y axis represents the house index and x axis theinterest rate. The prepayment regions are darkly shaded.As (3.5) is violated, we may write B (cid:0) − e − δ ( T − t ) (cid:1) = (1 + ε ) δ (cid:102) M , for some ε > 0. Therefore, B Ct H t = 1 + ε − B (cid:102) M H t E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv ( H u − + du (cid:21) . We next claim(A.2) essinf (cid:18) H t E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv ( H u − + du (cid:21)(cid:19) = 0 . Given this, there is a set A t ∈ F t with positive probability such that on A t H t E t (cid:20)(cid:90) Tt e − (cid:82) ut r v dv ( H u − + du (cid:21) ≤ ε (cid:102) M B , so that B Ct ≥ (1 + ε/ H t on A t . Now, define the stopping time τ := t A t + T A ct . As B CT = 0 wesee D (bwr) ,Ct ≥ ε/ A t and the result follows since A ˆ t has strictly positive probability. It remainsto show (A.2), but this is clear by considering the map h (cid:55)→ E t (cid:34)(cid:90) Tt (cid:18) e − δ ( u − t ) M u M t − h e − (cid:82) ut r v dv (cid:19) + du (cid:35) , as h ↓ (cid:3) Proof of Proposition 3.2. For the SRM, by definition, prepayment never occurs when H t ≤ B t (should the borrower terminate at this time, it is labeled a default). Next, for each contract,on the set { r t > m, H t > B t } prepaying immediately yields B t . Consider the stopping time τ =inf [ u ≥ t | r t ≤ m ] ∧ T . For the FRM, straightforward calculations (integration by parts along with(2.1)) show E t (cid:20)(cid:90) τt c F e − (cid:82) ut r v dv du + e − (cid:82) τt r v dv min [ B τ , H τ ] (cid:21) = B t + B − e − mT E t (cid:20) − (cid:90) τt ( r u − m ) e − (cid:82) ut r v dv du + e − m ( T − t ) (cid:16) − e − (cid:82) τt ( r v − m ) dv (cid:17)(cid:21) . For ω fixed, the map w → − (cid:90) wt ( r u − m ) e − (cid:82) ut r v dv du + e − m ( T − t ) (cid:16) − e − (cid:82) wt ( r v − m ) dv (cid:17) has derivative − ( r w − m ) e − (cid:82) wt r v dv (cid:16) − e − m ( T − w ) (cid:17) . Thus, for the given τ we know almost surely that E t (cid:20) − (cid:90) τt ( r u − m ) e − (cid:82) ut r v dv du + e − m ( T − t ) (cid:16) − e − (cid:82) τt ( r v − m ) dv (cid:17)(cid:21) ≤ , with strict inequality on { r t > m } , giving the result for the FRM. The ABM case follows immedi-ately since c A ≤ c F . (cid:3) Proof of Proposition 4.1. Throughout we use m > r , (4.2) as well as the identity(A.3) 1 − p − p × p − p = δr . The value matching and smooth pasting conditions identify A, B in terms of b, b (and show A, B < V F ), and, writing b = B z , b = B y for 0 < z < < y , lead to thefollowing system of equations1 r z − p − p − p m z − p = (cid:18) r − m (cid:19) y − p ;1 r z − p − − p − p m z − p = (cid:18) r − m (cid:19) y − p . (A.4)For 0 < z < 1, the left-hand side of the first equation above is strictly positive. However, for theleft hand side of the second equation to be positive we need z < ( − p / (1 − p ))( m/r ). For m < m ∗ from (4.3) this requires 0 < z < z := ( − p / (1 − p ))( m/r ) < 1. For m ≥ m ∗ there is no restriction.With this proviso, from the first equality in (A.4) we obtain(A.5) y = z (cid:32) r − p − p z m r − m (cid:33) − /p . ONTRACT DESIGN 29 For now, let us ignore if y > 1. Plugging y into the second equality in (A.4) and simplifying wewish to solve 1 r − − p − p m z = (cid:18) r − m (cid:19) z (cid:32) r − m r − p − p z m (cid:33) − p /p , This will have a solution if and only if1 = f ( z ) := (cid:18) k (cid:18) − p − p z (cid:19)(cid:19) p × (cid:18) − k (cid:18) − p − p z − (cid:19)(cid:19) − p ; k = m r − m > . Direct calculation shows f (0) > f is strictly decreasing. Now, if m ≥ m ∗ then f (1) < (cid:18) − k − p (cid:19) (cid:18) − p p k − p (cid:19) − p /p < . But, for any c > (cid:96) → (1 − (cid:96) )(1 + (cid:96)/c ) c is strictly decreasing. At (cid:96) = k/ ( − p ) and c = − p /p we thus see the above inequality holds. Therefore, when m ≥ m ∗ there is a unique z ∈ (0 , 1) such that with y as in (A.5) the system of equations in (A.4) has a solution. If m < m ∗ then f ( z ) = 0 and there is a unique z ∈ (0 , z ) such that with y as in (A.5) the system of equationsin (A.4) has a solution ˆ z . We also claim ˆ z ≤ m/δ . Indeed, this is obvious if either a) m ≥ δ or b) m < δ ≤ m ∗ , where b) follows sinceˆ z < z = − p − p mr = mm ∗ ≤ mδ . In the remaining cases m < m ∗ < δ or m ∗ ≤ m < δ calculation shows (since m/δ < m ∗ ≤ m < δ and m/δ < z if m < m ∗ < δ ) f (cid:16) mδ (cid:17) − /p = (cid:32) − (cid:32) − p δ r − m (cid:33)(cid:33) (cid:32) − p p (cid:32) − p δ r − m (cid:33)(cid:33) − p /p < , where the inequality follows since (1 − (cid:96) )(1 + (cid:96)/c ) c is decreasing in (cid:96) for 0 < (cid:96) < c > 0. Thus,in all cases, ˆ z ≤ m/δ .The next thing to show is that for this z , the y from (A.5) exceeds 1. To see this, note that forany solution ( y, z ) to (A.4) with 0 < z < y > z . Next, with b = B z , b = B y ,as A, B < V F is C , concave, and strictly concave in ( b, b ). Thus, since V F ( h ) = h on (0 , b ), bystrict concavity on ( b, b ) we know V F ( h ) < h for h > b . But, this implies V ( b ) = B < b and hence y > V F is the value function. As it is C . piece-wise C and H ofunbounded variation, Itˆo’s formula applies ([35]) and thus e − rt V F ( H t ) = V F ( h ) − mB (cid:90) t e − ru b 0. Thisshows the quantity on the second line above is non-negative for all stopping times τ , and, if wedefine τ to be the first time H hits b (prepayment) or b (default) then the quantity on the secondline is 0. This gives the result. (cid:3) Proof of Proposition 4.2. The calculations are similar in spirit to those in Proposition 4.1, andhence an outline will be given. First, for m ≤ δ the value matching and smooth pasting conditionsdirectly give b = B (cid:32) p r − p − δp r − p m (cid:33) − /p , along with explicit formulas for ˜ A, ˜ B, A which show ˜ A, ˜ B, A < b > B because p /m > ( p − /δ when m ≤ δ . By strict concavity on ( B , b ) and ˙ V ( b ) = 0, V is strictly increasing on ( B , b ) so V ( h ) ≤ B on [ B , ∞ ). Again by concavity, ˙ V (0) = m/δ ≤ V ( h ) ≤ h on (0 , B ).Therefore, V ( h ) ≤ min [ B , h ]. From here, the verification argument is identical to that in theproof of Proposition 4.1 (noting that | ˙ V ( h ) | ≤ K h ≤ b abm ).When m > δ we have six equations (value matching and smooth pasting at b, B , b ) andsix unknowns ( b, A, B, ˜ A, ˜ B, b ). However, A, B, ˜ A, ˜ B can all be expressed in terms of b, b with A, B, ˜ A, ˜ B < 0, and, writing b = B z for0 < z < b = B y for y > 1, we obtain the system ofequations − (1 − p ) (cid:18) δ − m (cid:19) z − p − p (cid:18) r − m (cid:19) y − p = − p r − − p δ ; − ( p − (cid:18) δ − m (cid:19) z − p + p (cid:18) r − m (cid:19) y − p = p r − p − δ . Solving the first equation for z , calculation shows 0 < z < y > m < m ∗ , while if m ≥ m ∗ there exists y > < z < < y < y . Plugging z = z ( y ) fromthe first equation into the second leads us to solve an equation of the form g ( y ) = 0 on y > m < m ∗ ) or 1 < y < y ( m ≥ m ∗ ). In each case, one can show g (1) < 0; ˙ g ( y ) > y > m < m ∗ ) or 1 < y < y ( m ≥ m ∗ ); and lim y ↑∞ g ( y ) = ∞ ( m < m ∗ ) or g ( y ) > m ≥ m ∗ ).Thus, there is a unique ˆ y in the allowable range such that g (ˆ y ) = 0. This gives the V . As V isconcave with V ( h ) = h for small h and V ( h ) = B for large h ; and b < B < b , it is easy to show V ( h ) ≤ min [ h, B ]. From here, the verification argument is the same as in Proposition 4.1 (again,noting that | ˙ V ( h ) | ≤ K h ≤ b ). ONTRACT DESIGN 31 (cid:3) Proof of Proposition 4.3. While similar in spirit to Propositions 4.1, 4.2 the calculations are verylong and involved for the SRM. As such, only an outline is given for the δ < m ≤ m ∗ case .Let V be defined via (4.5). Direct calculation shows V is C for b = (cid:32) δ − m δ − − p − p r (cid:33) p − , along with explicitly given A, B, ˜ B < 0. This shows V is also non-decreasing and concave. Thislatter fact implies V ( h ) ≤ B h for 0 < h < 1. Since E h (cid:104)(cid:82) ∞ e − ru σ H u ˙ V ( H u ) du (cid:105) < ∞ , for any α > 0, verification will follow provided V ( h ) = ˜ Bh p + mB r ≤ B + α ( h − h > . A lengthy calculation proves the existence of a unique α ∗ such that the above holds for α ≥ α ∗ ;and that α ∗ is the unique 0 of α → − p − p (cid:16) p ˜ B (cid:17) − p α − p − p − α − mB (cid:18) r − m (cid:19) , in (0 , − p mB (1 /r − /m ). Therefore, V = V S for α ≥ α ∗ .Next, we turn to α < α ∗ . Define V via (4.4). Direct calculation shows V is C on ( b ∗ , ∞ ) for b ∗ = − p − p (cid:18) mB α (cid:18) r − m (cid:19) + 1 (cid:19) , and that ˇ B < b ∗ . Note that α < α ∗ < − p mB (1 /r − /m ) implies b ∗ > 1. Next, value matching and smooth pasting give A, B, ˜ A, ˜ B < b, b , and leave usto solve the system of equations( b ) − p − p p − (cid:18) mB α (cid:18) r − m (cid:19) + 1 (cid:19) ( b ) − p + mB α (cid:18) δ − m (cid:19) ( b ) − p = mB α (cid:18) δ − p p − r (cid:19) ;( b ) − p − − p − p (cid:18) mB α (cid:18) r − m (cid:19) + 1 (cid:19) ( b ) − p + mB α (cid:18) δ − m (cid:19) ( b ) − p = mB α (cid:18) δ − − p − p r (cid:19) , in the region 0 < b < < b < b ∗ . Solving the second equation for b we find b = δ − m δ − − p − p r + (cid:16) − p − p (cid:16) r − m + αmB (cid:17) − αmB b (cid:17) ( b ) − p p − . Calculation shows 0 < b < < b < b ∗ . Plugging this into the first equation above, b is obtainedby solving an equation of the form g ( y ) = 0 for 1 < y < b ∗ . Analysis shows g (1) > g ( y ) < b ∗ depends on α . A long calculation shows that for α = − p mB (1 /r − /m ), g ( b ∗ ( α )) > 0, but there exists a unique ˆ α ∈ (0 , − p mB (1 /r − /m )) such that g ( b ∗ ( ˆ α )) = 0 and g ( b ∗ ( α )) < < α < ˆ α . Thus, for such α there is a unique b ∈ (1 , b ∗ ( α )) solving our system. Please contact the authors to receive a complete copy of all the proofs in the perpetual, constant interest ratecase. Lastly, a long calculation shows that ˆ α = α ∗ as previously computed. The rest of the verificationargument follows along the lines of the previous propositions. (cid:3) Proof of Proposition 4.5. We first treat the FRM. When there is no default V NoDef,F = inf τ E h (cid:20)(cid:90) τ mB e − ru du + e − rτ B (cid:21) . As m > r the function t → (cid:82) t mB e − ru du + e − rt B is strictly increasing. Thus, τ ≡ B . Next, removing the prepayment option V NoP P,F = inf τ ≥ E h (cid:20)(cid:90) τ mB e − ru du + e − rτ H τ (cid:21) . Direct calculations show V defined via (4.6) is C for b = − p − p mr B ; B = 1 p b − p . This also shows V is strictly increasing, concave, and hence V ( h ) ≤ h . Next, note that for h ≤ b we have mB − δh ≥ mB (cid:18) − − p − p δr (cid:19) ≥ , where the last inequality follows from (A.3). By Itˆo’s formula we may deduce e − rτ V ( H τ ) = V ( h ) − (cid:90) τ mB e − ru du + (cid:90) τ ( mB − δH u )1 H u ≤ b e − ru du + (cid:90) τ σH u ˙ V ( H u ) e − ru dW u . It is easy to show E h (cid:104)(cid:82) ∞ σ H u ˙ V ( H u ) e − ru du (cid:105) < ∞ , so the local martingale term vanishes inexpectation. Therefore, V = V NoP P,F .For the ABM we already know V NoDef,A = V A from Theorem 3.1. As for V NoP P,A , for thesake of brevity we will outline the case m ∗ < m for m ∗ from (4.3) (the other cases involve similaranalysis). From (A.3) we see that m > δ here as well. Define V via (4.7). Direct calculation shows V is C with b ( − p / (1 − p )) mB /r = mB /m ∗ and ˜ B = (1 /p )( b ) − p < 0. Note that b > B because m > m ∗ . This means V is concave, increasing with V ( h ) ≤ h (because V ( h ) = h before b ). Clearly, E h (cid:104)(cid:82) ∞ e − ru σ H u ˙ V ( H u ) du (cid:105) < ∞ and from Itˆowe know e − rτ V ( H τ ) = V ( h ) − (cid:90) τ m min [ B , H u ] e − ru du + (cid:90) τ ( m min [ B , H u ] − δH u ) 1 H u ≤ b e − ru du + (cid:90) τ σH u ˙ V ( H u ) dW u The martingale term vanishes in expectation and( m min [ B , h ] − δh ) 1 h ≤ b = ( m − δ ) h h ≤ B + ( mB − δh )1 B ONTRACT DESIGN 33 For the SRM, Theorem 3.1 ensures V NoDef,S = V S . As for V NoP P,S , in the interest of brevitywe will outline the case mB ≤ δ . Define V via (4.8). V is C with A = − mB p − p (cid:18) − p δ − − p r (cid:19) < 0; ˜ B = − mB p − p (cid:18) p r − p − δ (cid:19) < . This also implies V is concave, increasing with 0 ≤ ˙ V ( h ) ≤ ˙ V (0) = mB /δ ≤ mB ≤ δ .Therefore, V (0) = 0 implies V ( h ) ≤ h . From here, verification easily follows. (cid:3) References [1] B. W. Ambrose and R. 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