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Valuing participation mortgage loans using proﬁt caps and ﬂoors M. Shahid Ebrahim Mark B. Shackleton† Rafał M. Wojakowski‡ January 14, 2008 Abstract V ERY PRELIMINARY. P LEASE DO NOT CIRCULATE . We proceed with systematic investigation of different variants of par- ticipating mortgages as deﬁned by Ebrahim and Hussain (2007). They include several narrow forms observed in practice: Shared Income Mortgage (SIM), Shared Appreciation Mortgage (SAM) and Shared Equity Mortgage (SEM). To this end we use the closed-form formu- lae to value proﬁt caps and ﬂoors recently established in Shackleton and Wojakowski (2007). Nottingham University Business School, Nottingham, U.K. † Lancaster University Management School, Lancaster, U.K. ‡ Corresponding author. Accounting and Finance, Lancaster University Management School, Lancaster LA1 4YX, UK. Tel: +44(1524)593630; Fax: +44(1524)847321; Email: r.wojakowski@lancs.ac.uk 1 Contents 1 Introduction 3 2 The model 3 3 Participating mortgage loans 6 3.1 Shared Income Mortgage (SIM) . . . . . . . . . . . . . . . . . 6 3.1.1 Comparative statics . . . . . . . . . . . . . . . . . . . 8 3.2 Shared Appreciation Mortgage (SAM) . . . . . . . . . . . . . 12 3.2.1 Comparative statics . . . . . . . . . . . . . . . . . . . 12 3.3 Shared Equity Mortgage (SEM) . . . . . . . . . . . . . . . . . 13 2 1 Introduction It may take several years for an investment to reach cash ﬂow levels sufﬁ- cient to repay coupons of a traditional ﬁxed-rate mortgage or loan. While being exposed to increased default risk during this initial investment pha- se, lenders are usually excluded from sharing the upside potential of a project. Participation mortgages can help reconcile diverging interests of lenders and investors. Investors can obtain ﬁnancing which would be im- possible to obtain otherwise, given low initial level of cash ﬂows. Lenders get improved contracts with participation clauses on expected future cash ﬂows. Participation mortgages, also known as participation loans, are a fam- ily of commercial loans secured on property where the lender gets a pro- portion of cash ﬂows generated by the property. The borrower is rewarded for letting the lender participate by having the coupon rate set below the prevailing market interest rate and/or being granted a higher loan to value ratio. Depending on the contract, usually non-standardised and negoti- ated “over the counter,” the lender participates in either gross or net oper- ating income, cash ﬂows after senior debt service or proceeds from sale of the property. As pointed out by Alvayay, Harter and Smith (2005), there are not many papers on participation mortgages. Ebrahim (1996) establishes Pare- to-superiority of participating mortgages in a general equilibrium overlap- ping generations setting. Ebrahim and Hussain (2007) argue that partici- pating and convertible securities are characteristic at the developed stage of a system dominated by ﬁnancial markets. In case of mortgages, partic- ipating variants reduce agency costs leading to an increase in the value of property. 2 The model We assume that the commercial plant operating proﬁt cash ﬂow Pt real world process follows a geometric Brownian motion dPt = ( g δ) Pt dt + σPt dWt where g is the required growth rate for this class of risk and δ is a constant “cash yield” typical to this type of business activity, which is analogous to 3 the dividend rate in case of a stock. Assuming that agents are risk-neutral or, equivalently, that trading in underlying assets is somehow possible so that hedging is possible, the proﬁt process can be written as dPt = (r δ) Pt dt + σPt dZt (1) where Zt is the corresponding Brownian motion under the risk-neutral measure and r is the risk-free interest rate. Using option pricing tech- niques, the total current value of risky proﬁt ﬂow can be computed as a discounted risk-neutral expectation and is given by Z ∞ rt P0 A0 = e E [ Pt ] dt = 0 δ Throughout this paper we adopt the convention that E [ x ] is the expecta- tion taken at time t = 0 under the equivalent martingale measure Q i.e. E [ x ] = EQ [ x j F0 ] and Et [ x ] = EQ [ x j Ft ], where Ft is information avail- able at time t. Assuming that the current value of the real estate A0 is solely deter- mined by the type of business activity hosted, this quantity can then be interpreted as the current value of commercial property. Note that the property value A is then driven by the same dynamics as P i.e. (1). Fur- thermore, because P0 = δA0 , the current proﬁt cash ﬂow P0 can be repre- sented as a constant proportion δ of the property value. In a more general case the value of commercial property need not be perfectly correlated with proﬁt cash ﬂows or equal to the present value of thereof. Moreover, the commercial property usually represents value on it’s own, even without hosting a business. Let the real estate value H be driven by a process only imperfectly correlated to proﬁt cash ﬂow P dHt = (r δ H ) Ht dt + σ H Ht dZtH (2) where δ H can then be explained as “rental rate.” Such interpretation of the cash yield parameter δ H is typical in the context of housing ﬁnance, as e.g. in Capozza, Kazarian and Thomson (1998). A more general meaning of δ H involves “service ﬂow” from using the house over time as in Kau, Keenan, Muller and Epperson (1992). The σ H is then the volatility of real estate, possibly different from the dispersion parameter σ of the proﬁt cash ﬂow process P. In (2), ZtH represents the risk-neutral standard Brownian motion 4 process driving real estate prices. Correlation of house price returns with the stochastic component dZt driving proﬁt cash ﬂows is h i Et dZt dZtH = ρdt In what follows we will assume that the ﬁnancing of “tools” necessary to generate the proﬁt ﬂow is exogenously given, concentrating on ﬁnanc- ing the real estate property instead. It is, however, straightforward to ex- tend our analysis to the case where lender ﬁnances everything, buildings but also business equipment, for example. Consider a conventional ﬁxed-rate mortgage, known as a standard mort- gage loan. At time t = 0 the investor makes the initial deposit D0 against property valued at H0 .1 The lender ﬁnances the net loan amount Q0 and the time to maturity or tenure is T. The lender may wish to impose a max- imum permitted loan-to-value ratio L (typically L < 1) such that Q0 = H0 D0 LH0 (3) implying that the initial deposit, D0 , must be greater than D, where D = H0 1 L D0 The other agreed term of the loan is typically the coupon at . The value of the loan at time t = 0 must be equal to the discounted expected value of future cash-ﬂows Z T rt rT Q0 = e E [ at ] dt + e E [QT ] (4) 0 This is a very general expression. It must be satisﬁed by intermediate cash ﬂows and by the terminal cash ﬂow. The borrower is required to pay con- tinuous interest coupon ﬂow at and reimburse the remaining principal Q T at maturity. Note that the inﬁnitesimal coupon amount paid during time dt is at dt, while the coupon ﬂow (amount per unit of time) is at . To begin with, consider non amortizing loans. The outstanding balance remains constant, equal to the initial loan amount Q0 , so that Qt = Q0 for all t, implying Q T = Q0 . In this case the agreed coupon structure is 1 We assume that the market price of the property and its “estimated value”—normally provided by an expert hired by the lender—do not differ. 5 typically constant in time and can be fully characterized by a single para- meter i. It is known as cost of funds and is equal to the lender’s required, continuous coupon rate. at = iQt = iQ0 If the loan is riskless—in particular there is no default or prepayment risk—cash ﬂows bear no risk and, as expected, simplifying expression (4) gives i = r. 3 Participating mortgage loans Different variations of participating mortgages are deﬁned by Ebrahim and Hussain (2007). They include several narrow forms observed in prac- tice: 1. Shared Income Mortgage (SIM) 2. Shared Appreciation Mortgage (SAM) 3. Shared Equity Mortgage (SEM). 3.1 Shared Income Mortgage (SIM) In the case of participating mortgage the required coupon rate i is re- duced so that i < r to accommodate lender’s participation. Instead, the lender is compensated proportionally to the excess intermediate proﬁt ﬂow ( Pt K )+ where K is the ﬁxed proﬁt threshold above which partici- pation is payable Z T Z T h i rt rt + rT Q0 = e E [iQt ] dt + θ P e E ( Pt K) dt + e E [QT ] (5) 0 0 where θ P is the excess proﬁt ﬂow participation rate. To be more precise, this type of mortgage is often termed Shared Income Mortgage (SIM), be- cause the ﬁnancier subsidizes the interest component in return for a share in the income from operations. 6 If the lender is also compensated proportionally to proceeds A T from re-sale of business assets at maturity, then Z T Z T h i Q0 = e rt E [iQt ] dt + θ P e rt E ( Pt K )+ dt (6) 0 0 h i +e rT E [QT ] + θ A e rT E ( AT X )+ where X is the ﬁxed threshold above which participation is payable at maturity and θ A is the proceeds from sale participation fraction. The last component can be valued using the Black-Scholes formula for European call option on assets A with strike X. If the loan is non-amortizing Q0 = Qt = Q T and (6) gives the following condition iQ0 rT rT Q0 = 1 e + θ P C ( P0 , K, T ) + e Q0 + θ A c ( A0 , X, T ) (7) r where C ( , , ) and c ( , , ) represent the proﬁt cap and the call option func- tions, respectively. To begin with, function c ( A0 , X, T ) represents an European call option value on assets A of the company, with strike price X and maturity T. It can be computed using the Black-Scholes (1973) and Merton (1973) for- mula X,A X,A c ( A0 , X, T ) = A0 e δT Φ d1 0 Xe rT Φ d0 0 (8) where Φ is n cumulative normal probability distribution function and the o X,A0 X,A0 coefﬁcients d0 , d1 are given by 1 ln A0 ln X + r δ+ β 2 σ2 T d X,A0 β = p (9) σ T for β = f0, 1g. Moreover, equation (7) also contains C ( P0 , K, T ) which is the proﬁt cap written on the proﬁt ﬂow P, with threshold ﬂow K and maturity T. The value of the proﬁt cap can be computed using explicit closed form formu- 7 lae established in Shackleton and Wojakowski (2007) P0 K,P C ( P0 , K, T ) = 1 P0 >K e δT Φ d1 0 (10) δ K K,P 1 P0 >K e rT Φ d0 0 r b K,P + A (b, a) P0 1 P0 >K Φ db 0 a A ( a, b) P0 1 P0 >K Φ dK,P0 a where the four coefﬁcients dK,P0 for β 2 fb, 0, 1, ag can be computed using β (9) with K, P0 in place of X, A0 while parameters a, b and the function A are given by s 1 r δ r δ 1 2 2r a, b = + 2 2 σ2 σ2 2 σ K1 a b b 1 A ( a, b) = j a bj r δ 3.1.1 Comparative statics Since formulae for all components are available in closed form, much pro- gress is possible in determining dependence of relevant contract variables on input parameters. Having described the setup of the problem, this re- mains one of the main objectives of the paper. Consider a simpliﬁed version of (6) where participation is limited to intermediate proﬁt cash ﬂows. This is done by setting2 θ A = 0 iQ0 rT rT Q0 = 1 e + θ P C ( P0 , K, T ) + e Q0 r This condition will provide possible combinations fi, θ P g between reduced interest and participation rate. Furthermore, the lender may ﬁne-tune the loan by offering higher or lower maximum loan-to-value ratio L. Assum- ing that the constraint (3) is fully saturated, so that Q0 = LH0 , where H0 is 2 Alvayay, Harter and Smith (2005) analyse a similar case, which is a particular form of (6), such that θ P = θ A . Their discrete-time formulation involves ﬁnite sums in place of integrals and requires numerical simulations. 8 0.1 0.08 0.06 i 0.04 0.02 0 0 20 40 60 80 100 T Figure 1: Dependence of the contractual interest rate i on maturity T of the mortgage loan. Parameters are P0 = 250000, K = 1000000, r = 0.1, δ = 0.05, σ = 0.05, H0 = 100000, L = 0.9. Participation ratios are: θ P 2 f1, 0.5, 0.25, 0.15, 0.1, 0.05g. the current market value of the property, gives θP i=r 1 rT ) C ( P0 , K, T ) (11) LH0 (1 e The interpretation of this result is straightforward. The contractual inter- est rate on the mortgage can be reduced provided positive participation rate θ P > 0. The higher the participation, the greater the reduction. Re- duction is moderated by higher loan-to-value ratio L. For sufﬁciently low L and sufﬁciently high participation θ P it is possible to achieve i = 0. At this point all cash ﬂows to the ﬁnancier become uncertain, contingent on proﬁt ﬂow and there are no ﬁxed interest payments. Fixed income ﬂow is substituted for risky stream. When i becomes negative i < 0, the lender periodically pays the entrepreneur to extract portion θ P of random excess proﬁt ﬂow ( Pt K )+ . It is relatively easy to evaluate the impact of increased time to maturity 9 10000 100 P 1 15 20 25 30 35 40 T Figure 2: Dependence of the participation ratio θ P on maturity T. Pa- rameters are P0 = 250000, K = 1000000, r = 0.1, δ = 0.05, σ = 0.05, H0 = 100000, L = 0.9. Contractual interest rates are: i 2 f0.095, 0.05, 0g. The dashed line represents the case i = 0. on the contractual interest rate i by computing the partial derivative ∂i rθ 1 ∂C re rT = P rT rT C (12) ∂T LH0 1 e ∂T 1 e The term in brackets contains the theta hedge ratio of the cap. It is equal to the expected, discounted terminal caplet payoff on the (unknown) ﬂow PT at time T struck at K, which corresponds to evaluating at time T the integrand in the second integral appearing in (5) or (6). Evaluating the risk-neutral expectation produces the Black-Scholes formula for European call and is positive ∂C h i = e rT E ( PT K )+ = c ( P0 , K, T ) ∂T K,P K,P = P0 e δT Φ d1 0 Ke rT Φ d0 0 > 0 The theta is positive in the sense that as the horizon T gets longer by ∆t, the cap C becomes more valuable by c ( P0 , K, T ) ∆t. The impact of maturity 10 T on i can thus be obtained by establishing the sign of the term in brackets in (12). For inﬁnite maturity loans C ( P0 , K, ∞) does not depend on T, implying ∂C ∂i lim =0 and therefore lim =0 T !∞ ∂T T !∞ ∂T as expected i.e. for T ! ∞, the contractual rate i will gradually converge to some i∞ given by θP i∞ = r 1 C ( P0 , K, ∞) LH0 See Figure 1. In this stylized example the continuum of caplets on proﬁt ﬂow is initially deeply out-of-the money (P0 K). However, it is expected that participation will kick in in the future. Under assumption of exponen- tially growing cash ﬂow at the rate r—which is 10% in our example—it will reach the threshold K = 1, 000, 000 at tK 13.9 years. Again under the same assumptions, the excess cash ﬂow should be comparable to the property price at about t H 14.8 years. It is clear from Figure 1 that, in- dependently of the rate of participation θ P , the lender is able to grant a decent discount only if the maturity T of the loan exceeds both tK and t H . Once Pt K, at this future point in time it will be further expected that the level of available cash ﬂow to share will continue to grow exponentially. The higher the participation level θ P and the longer the maturity T, the higher the discount below r = 10% the lender is willing to offer. If the maturity T of the loan is too short, the excess proﬁt cash ﬂow is null, the discount is quasi null and so i r. Alternatively, the participation rate θ P can be expressed in terms of the contractual interest rate i LH0 1 e rT i θP = 1 (13) C ( P0 , K, T ) r Clearly, higher loan-to-value ratios L necessarily imply higher participa- tion rates θ P . Similarly, the lower the contractual interest rate i < r, the higher the participation θ P . Furthermore, impact of variations in time to maturity T or threshold proﬁt ﬂow level K can be assessed by employing appropriate sensitivities of the cap C with respect to underlying variables, which are readily available in closed form. 11 Figure 2 illustrates the dependence of the participation ratio θ P on ma- turity T. It conﬁrms intuition that, the longer the maturity of the loan, the lower the participation required to grant the same discount. Similarly, the higher the discount granted the higher the required participation. The dashed line represents the highest discount (no interest charged i = 0) and thus participation here must dominate other offers, for the time to matu- rity of mortgage loans kept constant. For loans of maturity T = 40 years it is sufﬁcient to participate at the level θ P 0.51 in order to generate full discount (i = 0). On the other hand logarithmic scale is used for θ P in or- der to illustrate the fact that for shorter maturity loans even participation parameter substantially higher than one θ P 1 may prove insufﬁcient to compensate for given discount r i. 3.2 Shared Appreciation Mortgage (SAM) In this case the lender subsidizes the interest component in return for a share in the appreciation when the property is sold. Under assumption that the property can only be sold at maturity of the loan T we have Z T h i Q0 = e rt E [iQt ] dt + e rT E [QT ] + θ H E e rT ( HT H0 )+ (14) 0 where θ H is the share of real estate appreciation parameter and i is the below-market loan rate. Note that the last component can be valued using the Black-Scholes formula for European call option on real estate H issued at the money. Structurally this component, ( HT H0 )+ , resembles a similar component, ( A T X )+ , present in (6), which is the analogous condition for a SIM. However, both these components are conceptually different. The former represents the net appreciation of the real estate (decrease or increase in value of e.g. a warehouse), while the latter is an option payoff on business assets (excess above X after selling off e.g. machines). 3.2.1 Comparative statics Since all quantities can be expressed in closed form, sensitivity analysis of a SAM is rather simple. In particular, since there is no sharing of interme- diate option-like components, there is no need using the cap formula (10). 12 For non amortizing loans (14) becomes i Q0 = Q0 1 e rT + Q0 e rT + θ H c ( H0 , H0 , T ) (15) r where represents an European call option on real estate. It can be computed as h i δH T H0 ,H0 rT H0 ,H0 c ( H0 , H0 , T ) = H0 e Φ d1 e Φ d0 where r δ H + β 1 σ2 T 2 H H d β 0 ,H0 = p β = 0, 1 σH T It is straightforward to solve (15) for i or θ H and proceed with sensitivity analysis. When constraint (3) is fully saturated (Q0 = LH0 ), we obtain θ H c ( H0 , H0 , T ) i=r 1 (16) LH0 (1 e rT ) where H0 is the current market value of the property. This formula is anal- ogous to (11) and similar conclusions apply. Furthermore, Q0 1 e rT i θH = 1 (17) c ( H0 , H0 , T ) r The main difference is that (16) and (17) do not require using cap formula (10). Their analysis is thus simpler but similar conclusions will apply. 3.3 Shared Equity Mortgage (SEM) Here, the ﬁnancier is the co-owner. He will grant Q0 at the onset. He may wish to waive coupon payments by setting interest to zero i = 0, but he will usually require participation in the proﬁt ﬂow and in the appreciation of the property Z T h i Q0 = θ P e rt E ( Pt K )+ dt + e rT E [ Q T ] 0 h i h i +θ A e rT E ( A T X )+ + θ H e rT E ( HT H0 )+ Note that two terminal options are present here: one linked to the “pro- duction” component and the second to the “real estate” component of the business activity. 13 References A LVAYAY, J. R., C. H ARTER , AND W. S. S MITH (2005): “A Theoretic Analy- sis of Extent of Lender Participation in a Participating Mortgage,” Re- view of Quantitative Finance and Accounting, 25(4), 383–411. B LACK , F., AND M. S CHOLES (1973): “The Pricing of Options and Corpo- rate Liabilities,” Journal of Political Economy, 81(3), 637–654. C APOZZA , D. R., D. K AZARIAN , AND T. A. T HOMSON (1998): “The Con- ditional Probability of Mortgage Default,” Real Estate Economics, 26(3), 359–389. E BRAHIM , M. S. (1996): “On the Design and Pareto-Optimality of Partici- pating Mortgages,” Real Estate Economics, 24(3), 407–419. E BRAHIM , M. S., AND S. H USSAIN (2007): “Financial Development and Asset Valuation: The Special Case of Real Estate,” Nottingham Univer- sity working paper. K AU , J. B., D. C. K EENAN , W. J. M ULLER , AND J. F. E PPERSON (1992): “A Generalized Valuation Model for Fixed-Rate Residential Mortgages,” Journal of Money, Credit and Banking, 24(3), 279–299. M ERTON , R. C. (1973): “The theory of rational option pricing,” Bell Journal of Economics, 4(1), 141–183. S HACKLETON , M. B., AND R. W OJAKOWSKI (2007): “Finite Maturity Caps and Floors on Continuous Flows,” Journal of Economic Dynamics and Control, 31(12), 3843–3859. 14

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