Valuing participation mortgage loans using profit caps and floors

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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

      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:

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

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
    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

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

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

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)

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.

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-

    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.

    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
          iQ0              rT                                       rT
   Q0 =       1        e          + θ P C ( P0 , K, T ) + e              Q0 + θ A c ( A0 , X, T )   (7)
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-
                                          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

                           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-

lae established in Shackleton and Wojakowski (2007)
                                    P0                    K,P
                C ( P0 , K, T ) =        1 P0 >K e δT Φ d1 0                            (10)
                                      K                    K,P
                                           1 P0 >K e rT Φ d0 0
                                                  b            K,P
                                    + A (b, a) P0 1 P0 >K Φ db 0
                                        A ( a, b) P0 1 P0 >K     Φ dK,P0

where the four coefficients dK,P0 for β 2 fb, 0, 1, ag can be computed using
(9) with K, P0 in place of X, A0 while parameters a, b and the function A
are given by
                           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
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.






                0      20          40                   60               80   100

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

                    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




                  15       20          25         30                    35   40

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 )
                                 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

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
                        i∞ = r 1         C ( P0 , K, ∞)
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.

     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)

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).

For non amortizing loans (14) becomes
            Q0 = Q0        1 e rT + Q0 e rT + θ H c ( H0 , H0 , T )     (15)
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
                          r  δ H + β 1 σ2 T
                                          2    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 ]
                            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.

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