# ChicagoGoncharov2007

Document Sample

```					Prepayment and Mortgage Rate
Modeling

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
(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,
• 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

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 46 posted: 6/25/2011 language: English pages: 23