Implied Future Home Price Growth from Subprime ABS Prices

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1 Implied Future Home Price Growth from Subprime ABS Prices Jason Vinar January 7, 2010

2 Implied Future Home Price Growth from Subprime ABS Prices Project Goals Learn about mortgage loans and mortgage backed securities. Build a borrower behavior model including prepayments, defaults, and loss recoveries. Build a mortgage security cash flow waterfall tool Calibrate the a HJM model and simulate interest rate term structures Derive the implied future home price growth Project Expectations Use of statistical techniques Monte Carlo Simulation will be a common tool Hard work and have fun

3 Quick Overview of Mortgage Securitization Several loans (mortgages) belong to a trust The borrower (home owner) remits their monthly payments to the servicer The servicer remits those payments plus loss mitigation collections on defaulted loans to the trust. In most cases the servicer is required to advance scheduled payments for delinquent loans until the loan is resolved or current. Several bonds belong to a securitization (deal) The trust distributes cash to each bond in the deal in the form of principal and interest payments as specified in the deal documents A typical subprime distribution structure pays accrued interest to each bond, but the principal is paid sequentially while the losses are taken in reverse order This cash flow structure leads to bonds within a deal having different ratings. For example, the most senior bonds can be rated AAA while the junior bonds are rated BBB or lower. Our goal is to model each of these cash flow components to price each bond in a deal.

4 Loan Cash Flows Each month the borrower is expected to pay their mortgage payment of which part is accrued interest and the rest is principal paydown. At any point in time the borrower has the ability to prepay the entire remaining balance. The two most common reasons for prepayment are refinancing to a new mortgage with better terms or moving (housing turnover). Likewise, a borrower has at any time the right to default on their obligation to repay the loan. In doing so the borrower will typically lose their home through process of foreclosure. Reasons for default include: job loss, divorce, death of borrower, and lost home equity (more frequent of late). These two options are of the American style: able to exercise at any point in time from now until maturity. Unlike equity options, however, the borrower is rarely able to optimally exercise these options due to a number of reasons including available credit, underwritting standards, cost of refinance, etc. We ll use multinomial logistic regression to estimate borrower behavior.

5 Estimating Borrower Behavior Borrower behavior is path dependent, e.g. if a borrower misses a payment this month they will likely miss next month s payment too or similarly if a borrower misses an opportunity to refinance they will be more likely to miss the next one To model the path dependent behavior a stylized state space (S) is used Current loan, i.e. has made all of their mortgage payments Delinquent loan, i.e. has missed one or more payments Loan in foreclosure, i.e. servicer has begun the process taking the home to sell and recover some or all of the balance Paid off loan, i.e. the loan is either paid in full or resolved to a loss The probability of each state is estimated given its current state and all the information leading up to the current state The multinomial logit model is used to compute the transition probability e X i,m βj P(S i,m = j) = J j=1 ex i,m βj where X i,m is a vector of covariates for loan i in month m

6 Prepayment and Default Curves CPR CDR Percent Jan 02 Jan 04 Jan 06 Jan 08 Jan 10 Time

7 Predictive Variables (X) Age of the loan Seasonality (time of the year) Refinance incentive (current rate relative to market rate) Burnout (missed refinance opportunities) Credit (FICO Score) Product type (Fixed, ARM, IO) Equity (difference between property value and remaining balance) Months delinquent State foreclosure timelines Property type Underwriting types (occupancy, documentation, purpose) Prepayment penalties

8 Converting Borrower Behavior to Cash Flow Most loan cash flows are implied by state transitions. For example: From C to D no payment is made From C to P the borrower makes a payment equal to the remaining balance The loss (or recovery) for a default needs a separate model. Hayre and Saraf present a detailed approach to model the loss although it is written simply as L = B + C S M where B is the loan balance, C is the servcing costs, S is the property sale price, and M is the mortgage insurance payment. In the loss equation we are only sure of the loan balance B. All other items need to be modeled in terms of other factors. The costs C accumulate for foreclosure expenses, P&I advances to the securitization trust, and other property related expenses. The sale price S is a factor of both home price growth and a discount for a forced sale (and possibly also a poor or neglected property condition) Mortgage insurance M is derived from the coverage amount, the claim amount, and reduced for some probability of claim denied

9 Using Simulation to Generate Loan Cash Flow Since loans behave in a path dependent way Monte Carlo simulation is used to generate the loan cash flow. The steps for the simulation are Given the current loan state calculate the transition probabilities using e X i,m βj P(S i,m = j) = J j=1 ex i,m βj Select a transition using a uniform random draw and P(S i,m = j) Compute loan cash flows implied by the transition Compute any losses (or recoveries) for a default Update loan characteristics such as age, seasonality, equity, refinance incentive, etc. Repeat until the loan resolves or the balance is 0

10 HJM Model The HJM model simulates the entire forward term structure. To do this the model takes the current forward rate curve as an input and evolves the entire curve rather than a single point. Borrowers will often look to longer term rates to drive their refinance decisions and therefore a model like HJM is needed. The Model df (t, T) = µ(t, T)dt + σ(t, T)dW(t) The differential df is with respect to time t not maturity T and W is a d-dimensional Brownian Motion. Calibration Historical Current Market Calibration Discretization and Simulation Exact simulation is difficult except for in special cases of σ. A grid of time steps and maturities is used. Discretization of initial forward curve is fixed to the maturity dimension of the grid.

11 Bond Cash Flow Calculator Inputs are: aggregate loan cash flows and the interest rate path to compute the variable rate bonds Outputs are: Principal and Interest Payments for each bond Bond Metrics: Duration Convexity Price (given spread) B M = NX i=1 C i (1 + Z i + s) i where B M is the modeled price for the bond, C i is the modeled cash flow for month i, Z i is the swap zero rate in month i. Note, the current term structure of swap zero rates are used to discount the projected bond cash flows.

12 Historical Bond Prices 100 RAMP05 EFC Bond Price A 2/AAA A 3/AAA 30 Jul 08 Oct 08 Jan 09 Apr 09 Jul 09 Oct 09 Jan 10 Apr 10 Time

13 Putting All the Pieces Together to Generate Loan Cash Flows Set the home price path, e.g. -1% for 2 years and +1% thereafter Generate an interest rate term structure path using the HJM simulation Simulate loan transitions to generate cash flows given the interest rate and home price paths Aggregate loan cash flows and put through bond cash flow calculator Repeat loan and bond simulation Average bond cash flows and discount using the swap zero curve Repeat interest rate simulation Repeat for each stylized home price path Challenges There are a lot of loops to keep track of and save output at the appropriate time The number of simulations could be quite large given the nested simulations

14 Finding the Implied Home Price Growth The recipe to find the implied HPG is as follows: Generate several stylized home price paths, e.g. no growth for 2 years followed by 1% annual growth or no growth for 5 years followed by 3% annual growth Price each of the bonds, with known market prices b, using each of the stylized home price paths. Store the modeled prices in a matrix M with each row representing bond and each column representing a home price path. Solve Mw = b for w where w is the market implied weight for each stylized home price path. This approach leads to further analysis considerations. Are there equivalent home price paths in terms of the modeled bond prices? Can a series of geographically concentrated pools 1 be used provide geographic specific home price views? We hope to address each of these questions and perhaps others that we identify through the course of the project. 1 Conversly, can a region specific path be used to derive model prices?

15 References Baxter, M.; Rennie, A. Financial Calculus. Cambridge University Press, Glasserman, P. Monte Carlo Methods in Financial Engineering. Springer, Greene, W.H. Econometric Analysis. Prentice Hall, Hayre, L.; Saraf, M.; Young, R.; Chen, J. Modeling of Mortgage Defaults. Journal of Fixed Income, Spring 2008, Vol. 17 Issue 4, Hayre, L.; Saraf, M. A loss Severity Model for Residential Mortgages. Journal of Fixed Income, Fall 2008, Vol. 18 Issue 2, Heath, D.; Jarrow, R.; Morton, A. Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation. Econometrica, 60 (1992),