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					Prepayment and Mortgage Rate
         Modeling

Recent Advances in Mathematical Finance
       Chicago, December 8, 2007

          Yevgeny Goncharov
              Department of Mathematics
               Florida State University
              goncharov @ math.fsu.edu



    partially supported by NSF grant DMS-0703849   1
                 Mortgage Securities

   ?             ?            ?               ?           ?         ?


             Pooled cash flow: interest + principal payments


prepayment

                     Rule for distribution of cash flow        default

 curtailment

                         A           B            C

                                  Investors                              2
                              A Mortgage
                                         • m – mortgage rate
 Borrower              Lender
                                         • P(t) – outstanding principal
                              $          • gt – Ft -intensity of prep.
          P0 at time “0”                          (“prepayment rate”)
                                         •     c – payment rate ($/time)
cash flow up to time    min( , T )
                                         •     – prepayment time
     c  mP(t )  P(t )                 •    Ft – “relavant” information


                     E  - t rθ dθ ( s) m  -rt rθeg-dθrθ g- t rds F 
                                                          s
                                                                  
                                                                  dθ 
       M t  E (t )E  c e  c  P(t )gIs  e s  P( ds F tθ G 
                                  sT                s


                                  
                       T 
           = P [   Pds                              t               dθ
                   
                         T
       Mt Mt =
                  t
                     
                       t t               {t  T }         )e
                                                                       
                                                                           t ]
                                                                           t
                                                                             

                                                                                 3
       Prepayment (Intensity) Specification

   Prepayment:           Empirical           or    Model-based
        g t :               g (Xt )         or        g ( t )
                                                                      refinancing
                                                                      incentive
                           market factors                  ( X t )
                                                                  measure of
                                                                  borrower’s
                                                                  reaction
            m0  mt , comparison of mortgage rates
      t : 
            Lt  P(t ), "market price" of profitability

Note: the incentive translates “market” to “resident” money-language
                                                                            4
                         Prepayment
                                  or
           Intensity g t : g ( t ) Modeling

Intensity function g (.)                  Refinancing incentive t
   (“Low frequency”)                            (“High frequency”)
Estimates the borrower‟s “response”    Estimates “usefulness” of the
(in probabilistic terms) to certain    refinancing from the borrower‟s
market situations.                     point of view.
                                       Based on the “market” information
                                       (interest, unemployment rates,
                                       home prices).


                                                                           5
                       Why to Model?
                            sensitive to time

Empirical   g 1 ( X t1 )   g 2 ( X t2 )   g 3 ( X t3 )   g 4 ( X t4 )

                X t1           X t2           X t3         X t4 time


Modeling    g (1 )
                t           g ( t2 )     g ( 3 )
                                               t          g ( t4 )

                           not sensitive to time

                                                                        6
Mortgage Modeling
 Calibration

       Numerical
     implementation



           Mortgage Model


                             gt
 rt , X t , etc...
                            t

                                  7
   Mortgage Model Classification
• Data, calibration                    Not mortgage-specific.
• Computational Method
                                       “Standard” problems
• Interest Rate Model,

• Prepayment intensity function
• Additional predictors                Statistics
  (house prices, media effect, etc)

• Refinancing incentive
                                      1. Mort-Rate-Based
    Mortgage Model = Ref. Inc.
                                      2. Option-Based
                                                           8
                                  Implied
                   Mortgage Rate Process
Let mt be mortgage rate at time t :
                          t T                                                   
                                                               s

M t = P(0)  E   P( s)  m  rs  e
                                                            -  rθ g  dθ
   new                                            t            t
                                                                           ds F t   0
                            t
                                                                                  
           t T                                      
                                      s
                                  -  rθ g  dθ
       Et   rs P ( s ; m ) e
                           t         t
                                                 ds 
 mt                                                 
             t
                                                        : LT (mt {g s }t st T )
            t T                                           t
                                   s
                               -  rθ g  dθ
        Et   P ( s ; m t ) e t               ds 
              t
                                                    

      The rate mt implied by the prepayment process gt:

                            mt  LT (mt {g s }t st T )
                                  t

                                                                                          9
  Mortgage-Rate-Based Approaches
                         m0      c(m0 )
Examples:  t  m0  mt , t , or         , i.e., g t  g ( m0 , mt )
                         m       c( mt )

1. The process mt  the 10-year Treasury yield+const.
                                                                                 s
                                                                  t [ rθ g ( m ,m
                                           t T                                            t       

                                    E [                         t       -                             )] dθ
                                                  rs P( s ; m ) e                                              ds ]
2. m  L (m {g s }t  s t T ) 
     t    T
          t
               t                       t
                                                                             s
                                                                t [ rθ g ( m ,m
                                            t T                                       t       

                                     E [                    t       -                             )] dθ
                                                   P( s ; m ) e                                            ds]
                                            t

   in general mt  mt !

3. Endogenous mortgage rate {m t }t  0
                                    :                                                Pliska/Goncharov

                                                                                     MOATS
                   mt  LT (mt {g (mt , ms )}t st T )
                         t
                                                                                                           10
                    A Simple Example
  Consider a simple market which is completely described
  by a Markovian time-homogenous process rt . Then

  mt  LT (mt {g (mt , ms )}t st T )
        t                                                         LT (m0 {g (m0 , ms )}0sT )
                                                                   0


  Thus mt  m( rt ), where

                    m(r )  L(r, m(r ) m())


                                                              s

                 E [
                        T
                            rs P ( s; m(r )) e 0 r g ( m ( r ),m ( r )) d ds r  r ]
                                                      -
                                                                                   0
i.e., m( r )       0
                                                          s

                 E [
                            T
                                P ( s; m(r )) e 0 r g ( m ( r ),m ( r )) d ds r  r ]
                                                  -

                        0                                                           0            11
12
                           MRB Mortgage Rate
0.18

                                                                    0      
                  , c(m ) / c(m )  1  0.3
                                0         t
0.17
          gt                                                                  2
0.16
                ,          otherwise

0.15
                                                                                    0.6
0.14



0.13                                                                               0
0.12



0.11



 0.1



0.09
               0.05,   2
   0.06      0.07   0.08     0.09   0.1       0.11   0.12   0.13    0.14   0.15        0.16




                                                                                  13
0.18
                  Option-Based Mortgage Rate
                                                                           2 to 
0.17



                          , Lt  (1  0.3) P(t )              0
0.16
                  gt  
0.15                    ,         otherwise
                                                                                 0.6
0.14



0.13                                                                            0
0.12



0.11



 0.1



0.09
              0.05,   2
    0.06   0.07       0.08   0.09    0.1    0.11   0.12   0.13   0.14   0.15        0.16




                                                                               14
                      Citigroup‟s MOATS
                                   (generalized)
   1. For t from Tmoats  1 to Tmoats  T : mt  LTmoats t (mt {g (mt , m s )}t  s Tmoats )
                                                  t



       0                                                          t                   Tmoats

                                                                      T
    2. For t from Tmoats  T  1 to 0 :           mt  LT (mt {g (mt , m s )}t  s t T )
                                                        t



       0                       t                                                      Tmoats
                       T                      T                       T
Citigroup: • T=360 (30 yr), Tmoats=720 (60yr)
             comlpexity: (361*360/2+360*360)*N*I=194,580*N*I
           • Interest only? One factor only?
           • Historical dependence dropped, “calibrated” later…
                                                                                             15
                                  0.1
MOATS mortgage rates




                                 0.09

                                 0.08

                                 0.07

                                 0.06

                                 0.05

                                 0.04
                 60
                       40        0.03                         0.1
                            20                         0.08
                                                0.06
                                 0.02    0.04
                                     0
                                  0.02
                                                                16
         MOATS convergence

                    0.08




                    0.07




                                 MOATS mortgage rates
                    0.06




                    0.05




                    0.04
0   30    60   90          120                            150       180       210    240

                                                        time to Tmoats (quarterly)         17
                                       MOATS convergence
                       0.084




                       0.079
MOATS mortgage rates




                       0.074




                       0.069

                                                                                        time: 45/ term:15

                                                                                        time: 30/ term:30
                       0.064
                                                                                        time:22.5/term:30

                                                                                        time: 15/term:30

                                                                                        time: 0 / term:30
                       0.059




                       0.054
                            0.03   0.035   0.04   0.045   0.05   0.055   0.06   0.065        0.07       0.075   0.08

                                                                                                interest rate          18
                         MOATS convergence (interest only)
                       0.084




                       0.079
MOATS mortgage rates




                       0.074




                       0.069

                                                                                        time: 45/term:15


                                                                                        time: 30/term:30


                                                                                        time:22.5/term:30
                       0.064
                                                                                        time: 15/term:30


                                                                                        time: 0/ term:30



                       0.059




                       0.054
                            0.03   0.035   0.04   0.045   0.05   0.055   0.06   0.065      0.07            0.075   0.08



                                                                                               interest rate              19
Endogenous Mort Rate Iteration
           T                                                                
                                         s
                                -  r g ( m ( r ), m ( r )) dθ
        E   rs P( s, m(r ))e      0
                                                                  ds r0  x 
m( r )                                                                     
             0

            T                                                             
                                  s
                             -  r g ( m ( r ), m ( r )) dθ
         E   P( s, m(r ))e     0
                                                                ds r0  x 
              0
                                                                                x r


                                
                mi 1 (r )  L x, mi (r ) mi ()         xr


• mi+1() requires estimation of L() for “every” r?
• The result of L()-estimation is used at x=r only,
  other values discarded?
• Curse of dimensionality with growth of r-dimension?
                                                                                        20
                          Mortgage Rate “Iterations”
                                                     m i (r 4 )
                                                                                          m k  m 0  k m

                              m (r 3 )
                                                    m (r 4 )          m4
                   m (r 2 )                   3                           m3
                                         m i (r )
                                                                      m2
    0    m (r1 )
m (r )                                                                     m1
                                                                      m0
    r0        r1         r2          r3             r4                          r0   r1        r2 r3
                                                                                                       r04 ri 4 r 4
         Fix r then solve for m(r):                                        Fix m then solve for r(m):
         mi 1 (r )  L(r, mi (r ) mi ())                               m  L(ri 1 , m mi ())  L(r , m)
                                  the same…                                  the same!
                                          Refinancing region controlled                                               21
             Computation with Level Sets
                                 •m k  m 0  k m iterations if
                                   No need for
                                   Vm £m(r )"transaction costs"
x2                               • The conditional expectation in L()
                                 m4
                  m( x1 , x2 )   is used on a hypersurface (level set),
                                     m3
                                 i.e., “waste of one dimension” only
                                 m2
                     m4         • Number of L()-estimations is
                                 m0
                                     m1
                                 independent of the dimension/number
            m3                         r0       r1     r2 r3 r 4 r 4 r 4 r
      m2                        of the underlying factors i    0




     m1
                                                       
                                    m  x m  L x, m m()            

                                       x1                              22
                   Conclusion
 Endogenous mortgage rate is defined
   far from or implied by 10yr Treasury yield
   accented nonlinear behavior
 MOATS
   transparent definition      efficient implementation
   convergence to MRB is shown
 A general „level set‟ method is proposed
   flexibility of implementation: [RQ]MC or PDE
   reduces/eliminates the burden of iterations
   complexity of the same order as the underlying problem
   efficient and simple for the computation of implied
     mortgage rate given any prepayment model            23