cmbs ppt.pdf

7/27/2019 cmbs ppt.pdf

1/23

Mortgage Backed Securities

Jesse Laeuchli, Paul Romine, Katherine Reid, HuiFang


2/23

Introduction to MBS Valuation


3/23

Mortgage Backed Securities

A security backed by

monthly mortgage

payments Is an important sector

of the bond market $468.9 billion

Need to price security

18%

8%

36%

17%

16%

Municipal Sec.Treasury Sec.Fed. Agency DebtMBSAsset Backed Sec.Corporate Debt

Figure 1. Percentages of each security in the

Debt Market in 2006

http://www.bondmarkets.com/story.asp?id=98


4/23

Basic MBS Cash Flow Homeowners make monthly

payments of principal and

interest at the mortgage rate.

The service retains a portion of

the interest components of each

monthly payments as the

Monthly servicing fee.

The pass-through rate is themortgage rate net of the

servicing rate.

Homeowners

Service

Investors

Mortgage Rate

Pass-through Rate


5/23

MBS ValuationKey Idea:The price of any security can be written inThe price of any security can be written in

terms of the net present value (NPV) of its discountedterms of the net present value (NPV) of its discounted

cash flows under the risk neutral probability measure.cash flows under the risk neutral probability measure.

Where

d(t) : Discounting Factor, available from market.

C(t): Payment at time t.

Q: Risk-Neutral Measure.

0 0

( ) ( ) ( )M M

Q Q

t t

P E PV t E d t C t= =

! " ! "= =# $ # $

% & % &' '


6/23

Interests

tax

property

Casualty

insurance

Principal

Prepayment

Monthly Payment

WhereWhere

SP(tSP(t): Scheduled Payments): Scheduled Payments

TPP(t): Total Principal PaymentsTPP(t): Total Principal Payments

INT(tINT(t): Interest Payments): Interest Payments

PRIN(tPRIN(t): Scheduled Principal Payment): Scheduled Principal Payment

PP(t): Principal PrepaymentsPP(t): Principal Prepayments

C(t) =C(t) =SP(tSP(t) +) +PP(tPP(t))

SP(tSP(t) =) =PRIN(tPRIN(t)) ++INT(tINT(t))

TPP(tTPP(t) =) =PP(tPP(t) +) +PRIN(tPRIN(t),),


7/23

How to Value MBS?

Observation: Cashflow C(t) is interest-rate

path dependent.

Questions:

1. What is the Cashflows of MBS?

2. How to predictthe Interest Rate ?

Interest rate models!3. How to predictthe prepayment Behavior?

Prepayment models!


8/23

Interest Rate Modeling


9/23

Pricing Along One Path

Zero-Curve Interest Rate Models

Prepayment Model

Realized Forward-Rate

Predicted Cashflow

SMM(t)

0( ) ( ) ( )

N

i tPV t d t C t

=

=!C(t)

d(t)


10/23

IR Model Hull-White ModelImplement Hull-White Model via Trees

( ) ( ( ) ( ) ( )) ( ) ( )dr t t t r t dt t dW t ! " #= $ +


11/23

IR ModelLibor Market ModelArbitrage-Free Market Models


12/23

Prepayment Model


13/23

Prepayment Terminology

Where

Q(t): Fraction of Mortgage that has not been prepaid.

SMM(t): Fraction of pool prepaid during month t.

1

( 1) ( )( )( 1)

( ) (1 ( ))t

n

Q t Q t SMM tQ t

Q t SMM n=

! !=!

= !"

Key Idea:The survival factorQ(t) links MBS cash

flow

with prepayments models.


14/23

Prepayment Model--Cox ModelPrepayment is governed by a hazard function h(t)

( )

0( ) ( )v t

h t h t e!

=

Where

h0(t): baseline hazard function.

v(t): is a vector of explanatory variable for prepayment.

Notice:

h(t) is calibrated by historical data.

v(t) is functional of interest rates.

SMM(t) = h(t) dt


15/23

Probability of Prepayment

Figure 5. Example Prepayment Probability

Coxs proportional hazard specification


16/23

Numerical Example and Results


17/23

Numerical Examples

Mortgage Description:

1 Million 30y mortgage with 8.5% mortgage rate

and 7.25% coupon rate.

Model Input:Model Input:

1.1. Interest Rate Models (Libor Market Model)Interest Rate Models (Libor Market Model)

1.1 Zero Curve1.1 Zero Curve

1.2 Forward Rate Covariance Structure1.2 Forward Rate Covariance Structure

1.3 Instantaneous volatility function.1.3 Instantaneous volatility function.2.2. Prepayment Models.Prepayment Models.

3.3. OASOAS


18/23

Scheduled PaymentsC(t) =PRIN(t)+INT(t)


19/23

Cashflow with PrepaymentC(t) =PRIN(t)+INT(t)+PP(t)


20/23

Monte Carlo Simulation Results The model

demonstrate good

converged rate.

Able to predict value

of security.

Figure 6. Histogram of MBS Net Present Value


21/23

Questions?


22/23

Cash Flow with Prepayments

'( ) ( ) ( 1)

'( ) ( ) ( 1)

'( ) ( ) ( 1)

'( ) ( '( ) '( )) ( )

'( ) ( ) ( )

SP t SP t Q t

PRIN t PRIN t Q t

INT t INT t Q t

PP t BAL t PRIN t SMM t

BAL t BAL t Q t

= ! "

= ! "

= ! "

= " !

= !

Let SP(t), PRIN(t), INT(t), BAL(t) be the corresponding

quantities with prepayments

Qestion: How to compute SMM(t)?

Answer: History repeats!


23/23

CashFlow without Prepayment

( )

( ) 1

( ) 1

12

( )1

(1 )( )

1

( )1

R t

R t

N

R t

N

WACG

GSP t

U

G UINT t

U

GUPRIN t

U

+

+

=

=

!

!

=

!

=

!

Where,WAC: Wighted Average Coupon Rate

U =1/(1+G): Monthly Discouting Factor.

R(t) = N t: Remaining Loan Terms.

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