NETWORKING Is the Home Equity Conversion Program in the United - - PowerPoint PPT Presentation

Is the Home Equity Conversion Program in the United States Sustainable?

Hua Chen, Temple University (with Samuel H. Cox and Shaun Wang)

NETWORKING

• Mortality improvement

– life expectancy at birth was 60.95 in 1933, 77.87 in 2004

• “House Rich & Cash Poor” dilemma

– Aging population: 34m elderly now; 71m elderly by 2030

• Reduced monthly incomes
• Rising health-care costs
• Decreasing pension plan benefits.
• Difficult to maintain financial independence and living standards.

– 12.5 million elderly have no mortgage debt, and the median value of these unmortgaged properties is $127,959 (American Housing Survey)

• Reverse Mortgage

– Enable elderly homeowners to convert their home equity into cash income without selling their home

Motivation

Figure 1: Comparison between Forward Mortgage and Reverse Mortgage

• Deferred repayment until

– a borrower sells the property, moves out, or dies – fails to pay property tax/homeowner insurance – fails to maintain the condition of the home)

Reverse Mortgage

• Borrower requirements

– 62 years of age or older – Own the property outright or have a small amount of balance – Occupy the property as the principal residence – Not be delinquent on any federal debt – Participate in a consumer session given by an approved HECM counselor

• Payment options

– Lump sum – Line of credit – Monthly cash advance (tenure/term)

• Initial principle limit (IPL) or principle limit factor (PLF)

– Age of the youngest borrower – Current interest rate – Adjusted property value (maximum claim amount)

HECM Program

Non-Recourse Provision

• Repayment under the non-recourse provision
• The borrower is holding a debt position and an exchange option

– exchange the property value for the loan outstanding balance

• The non-recourse provision (NRP) is equivalent to writing the borrower a

series of European exchange options with different times of maturity

   ≥ − < − =

t t t t t t

L H L L H H , , Repayment

) , max(

t t t

H L L − + − =

[ ]

∑

− − = − +

− =

1

] , max[ NRP

x t t t Q rt t x x t

H L E e q p

ω

t

H

t

L

• HECM loans are FHA insured

– protect lenders from suffering losses if nonrepayment occurs – guarantee borrowers receiving monthly payments if the lender defaults

• Mortgage insurance premiums (MIP) as of Oct. 4, 2010

– 2% of the property value at closing – 0.5% of the loan balance annually

• Under the equivalent principle

NRP = MIP

( )

∑

− − = −

+ =

1 1

005 . 02 . MIP

x t t rt x t

L e p H

ω

Mortgage Insurance Premiums

Insurance Risks in HECM

• Mortality risk

– HECM model: period life table

• Fail to capture the dynamics of mortality rates over time
• Fail to capture the mortality improvement jumps and adverse mortality jumps

– Our paper

• Generalized Lee-Carter model with asymmetric jump effects
• Dynamic life table
• Mobility risk

– Health-related (move into long-term health-care facilities or nursing homes) – Non-health-related (marriage, divorce, death of the spouse, disasters, etc) – Practice: 30% of mortality rate (Jacobs, 1988; Deutsche Bank, 2007)

• Interest rate risk

– HECM loans are opt for adjustable interest rates – Practice: A fixed interest rate with a risk adjustment (150 bps)

• House price depreciation risk

– Geometric Brownian motion (Cunningham and Hendershott, 1984; Kau, Keenan and Muller, 1993) – Autocorrelations (Case and Shiller, 1989; the Institute of Actuaries, 2005b; Li, 2007) – Time series analysis!

Insurance Risks in HECM

Mortality Modeling: The Lee-Carter Model

• .

The normalization conditions: A two-stage procedure for

– Apply the singular value decomposition (SVD) method to , – Re-estimate by iteration, s.t. where is the actual total number of deaths at time t, is the population in age group x at time t.

and 1 = =

∑ ∑

t t x x

k b

x t x

a m − ) ln(

, t

k

( )

∑

+ =

x t x x t x t

k b a Pop D ) exp(

, t

D

t x

Pop ,

t x t x x t x

e k b a m

, , )

ln( + + =

and

x t

b k

, 1

1 ln( )

T x x t t

a m T

=

⇒ = ∑

Mortality Modeling: The Lee-Carter Model

• Dynamic of from 1900 to 2006

t

k

Mortality Modeling: The Lee-Carter Model

• How to model ?

– Lee and Carter (1992): random walk with drift – Modeling mortality with jumps

• Biffis (2005)
• Cox, Lin and Wang (2005)
• Bauer and Kramer (2007)
• Chen and Cox (2009): transitory vs. permanent mortality jumps
• Cox, Lin and Pedersen (2008): combine two types of jumps, complicated model
• Chen, Cox and Wang (this paper): asymmetric jumps, normal distribution
• Brockett, Deng and MacMinn (2010): asymmetric jumps, double exponential

t

k

1 1

(1918)

t t t

k k Z Dummy µ σ

+ +

= + + +

Mortality Modeling: The Jump Process

• A model with permanent jump effects

– Jump severity – Jump frequency – Random variation – Compensation term

• A model with transitory jump effects

) , ( ~ s m N Y

) ( ) (

1 } 1 { 1 1 1

1

+ = + + +

+

+ + Λ − + =

t N t t t t

N Y Z k k

t

Ι σ µ

pm N Y E

N

= = Λ

=

)] ( [

} 1 {

I

1, with probability 0, with probability 1 p N p  =  − 

) 1 , ( ~ N Z

     + = + + =

+ = + + + + +

+

) ( ~ ~ ~

1 } 1 { 1 1 1 1 1

1

t N t t t t t t

N Y k k Z k k

t

I σ µ

Mortality Modeling: The Jump Process

• A model with asymmetric jump effects
• Compensation

Table 1: Parameter Estimates via CMLE

     + = + + Λ − + =

+ = + > + + + + = + < + + +

+ + + +

) ( ) ( ~ ) ( ) ( ) ( ~ ~

1 } 1 { 1 } { 1 1 1 1 } 1 { 1 } { 1 1 1

1 1 1 1

t N t Y t t t t N t Y t t t t

N Y Y k k N Y Y Z k k

t t t t

I I I I σ µ

{ 0} { 1}

[ ( ) ( )] [1 ( )] ( )

Y N

E Y Y N pm m s ps m s φ

< =

Λ = = − Φ − IΙ

mortality deterioration and transitory effect mortality improvement and permanent effect Y Y > ⇒ < ⇒

• Data: Nationwide House Price Index (HPI) from 1975 to 2009

Figure 2: HPI Log Returns (Y) Figure 3: The First Difference of HPI Log Returns (DY)

Model the HPI

Model the HPI

• ARMA(2,0) + GARCH(1,1)

, where

Figure 4: ACF of the Standardized Innovations Figure 5: ACF of the Squared Standardized Innovations t t t t

DY DY DY ε φ φ + + =

− − 2 2 1 1

2 1 1 2 1 1 2 − − +

+ =

t t t

d ε β σ α σ

) , ( ~ |

2 1 t t t

N σ ε

−

Φ

• Exponential Tilting
• Esscher Transform (Esscher, 1932)

– Justified by maximizing the expected power utility of an economic agent

• Conditional Esscher Transform (Buhlmann, Delbaen, Embrechts and

Shiryaev, 1996)

– Justified within the dynamic framework of utility maximization problems

[ ] [ ]

) exp( | ) exp( ) ( ) (

*

Y E x X Y E x f x f

X X

λ λ = =

[ ]

1 1 1 *

| ) exp( ) exp( ) | ( ) | (

− − −

Φ Φ = Φ

t t t t t X t X

X E x x f x f

t t

λ λ

[ ] [ ] [ ]

) exp( ) exp( ) ( ) exp( | ) exp( ) ( ) (

*

X E x x f X E x X X E x f x f

X X X

λ λ λ λ = = =

Conditional Esscher Transform

How to Transform?

• ARMA(2,0) + GARCH(1,1)

– , where –

• Under the physical measure P

– , where – , where

• Under the risk adjusted measure Q

– Choose s.t. (Buhlmann et al., 1996) –

t t t t

DY DY DY ε φ φ + + =

− − 2 2 1 1

2 2 1 1 − − +

=

t t t

DY DY φ φ µ

) , ( ~ |

2 1 t t t t

N DY σ µ

−

Φ ) , ˆ ( ~ |

2 1 t t t t

N Y σ µ

−

Φ

1

ˆ

−

+ =

t t t

Y µ µ

2 1 1 2 1 1 2 − − +

+ =

t t t

d ε β σ α σ

) , ( ~ |

2 1 t t t

N σ ε

−

Φ

      − Φ −

2 2 1

, 2 1 ~ |

t t t t

r N Y σ σ

) exp( ] | ); [exp(

1

r Y E

t q t t Qt

= Φ − λ

q t

λ

• 6-month delay from home exit until the actual sale of the property
• Transaction cost: 6%
• Risk-free interest rate: 10 year U.S. Treasury rate (3.42%)
• Interest rate charged on the loan: one year CMT rate (0.42%) plus a lender’s

margin (1.5%) and an additional MIP (0.05%)

• Rental yield: 2% per annum.
• Initial house value: $300,000
• Assume the property is located in Philadelphia, Zip code 19104.

Numerical Results: Assumptions

Table 2: Value of NRP and MIP at Different Ages

• NRP decreases with the age at origination
• MIP decreases, but MIPs are far more than NRPs.
• The HECM program is sustainable ?!

Numerical Results: MIP vs. NRP

• What if the housing price dropped by 5%, 10%, 15% or 20%?

Figure 6: MIP and NRP at Different Scenarios Figure 7: Ratios of the MPI to NRP

Numerical Results: Declining Housing Market

60 65 70 75 80 85 90 2 3 4 5 6 7 8 9 10 Ages at Closing Ratio of MIP to NRP H0=300,000 H0=285,000 H0=270,000 H0=255,000 H0=240,000

• Basic Risk?

– Fluctuations of individual property value may deviate from the HPI dynamics

• Mobility Risk?

– Health related: move to long-term care facility – Non-health related

• Interest rate risk?

– Lower interest rate – Higher lender’s margin

• Refinancing risk

– Housing price depreciation damped the incentive of refinancing risk

Discussions

New Changes on HECM

• Reduction in PLF (based on expected mortgage interest rate 5.5%)
• Increase in premiums

– Annual MIP increased from 0.5% to 1.25%

• Temporary loan limit increase

– American Recovery and Reinvestment Act (ARRA) of 2009 – Increase from $417,000 to $625,500, until the end of 2011

• Introduction of HECM saver (Oct 4, 2010)

– Upfront MIP 0.01% compared to 2% – Principal limit 10-18% lower

Additional Work: Normal Jumps

• Recall: the model with asymmetric jump effects

1 1 1 1

1 1 1 { 0} { 1} 1 1 1 { 0} { 1}

mortality deterioration and transitory effect mortality improvement and permanent ( ) Compensation term effect [

t t t t

t t t t Y N t t t Y N

k k Z Y k Y k Y E Y Y µ σ

+ + + +

+ + + < = + + + > =

 = + − Λ + +   = +   > ⇒ ⇒ Λ = < 1 1 1 1   

{ 0} { 1}

Disadvantage: Assume is normal distributed does not reflect different frequencies and severities ] .

Y N

Y

< =

⇒ 1 1

• Y ~ Double Exponential Distribution

{ 0} { 0}

( ) and 1 : proportion of positive and negative jumps and :inverse of the average severity of positive and negative jumps

u d

y y Y u y d y u d

f y p e q e p q p

η η

η η η η

− > <

= + = − 1 1

Additional Work: Double Exponential Jumps

• Model comparison via AIC and BIC

Additional Work: Model Comparison

Questions? Comments?