FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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MEASURING PREPAYMENT
AND DEFAULT
PREPAYMENT MODELLING
LN.05&06: FRL 383
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Measuring Prepayment and Default
• FBB.3, also
• Overlap is partial
• F.11 (parts of)
• S.8 (parts of)
• Overview
• Prepay and default conventions
• Not a model of either, more like a
benchmark
• Construct MPT cash flows
• CFY, spreads (Z-spread and OAS)
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Modeling Prepayment
• FBB.4, also F.9
• Move away from “benchmarks”
• Consider empirical behavior; two broad
categories
• Empirical models
• Short shelf-life
• Logit model
• Rational prepayment models
• Cumbersome
• Components of prepayment
• Variables that contribute to prepayment
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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MPT: Prepayment
• Start with agency deals – no defaults
• Must project CFs (since prepayment unknown)
in addition to future interest rates to value MPT
• Conventions important
• Pricing at origination
• Relative value (to other MBS, also other FI)
• Hedging, risk management
• Ex-post performance
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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MPT: Prepayment Conventions
• Originally: no prepayment, then all at year 12
• FHA prepayment experience
• Prepayment depends on interest rate environment
• FHA loans assumable, conventional not
• Conditional prepayment rate (CPR)
• Constant fraction of remaining pool prepays each
month
• Conditional on rate environment, pool
characteristics, etc.
• Public Securities Association (PSA)
convention
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Conditional Prepayment Rate
• Let
• Bt be the unpaid pool balance at date t
• Pt be the unpaid pool balance at date t if no prepayments
• St be the scheduled repayment for period t (i.e., date t to t+1)
• CPRt is an annual rate
• Convert to “Single Month Mortality,” SMMt = periodic
rate at which pool prepays in period t
• SMMt = 1 – (1-CPRt)1/12
• Convention is to compound survival rates
• Prepayment in period t is (Bt-1-St)×SMMt
• So new pool balance is Bt = Bt-1 - St - (Bt-1-St)×SMMt
• Note that (1-SMM) is the monthly survival rate
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Survival Factor Qt aka Bond Factors
• The fraction of mortgages in pool that have not
prepaid at date t is given by the survival factor, Qt
• Qt = (1-SMM1)×(1-SMM2)×… ×(1-SMMt)
•
= Bt /Pt
• Needed to construct MPT cash flows
• See example, next
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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PSA Benchmark
• Think of as way to quote MPT value
• Not really a model, but
• Based on FHA prepay experience
• Monthly series of CPR
• Function of age of pool only
• Independent of underlying term, interest rate environment, etc.
• For “100 PSA:” CPRt = 6% × min(t/30,1), where t is measured in
months
• See diagram over
• So, t=1: CPR = 0.2% and for t≥30: CPR = 6%
• Increasing portion called the ramp
• For any other “n PSA” benchmark, e.g. n=200, multiply
asymptotic CPR factor (=6%) by n/100, e.g., 200/100 ×
6%=12%
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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PSA Annual Conditional Prepayment Rate
16%
100 PSA
14%
50 PSA
200 PSA
12%
10%
8%
6%
4%
2%
0%
0
20
40
60
80
100
120
Pool Age (Month)
140
160
180
200
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Agency Pass-Through Cash Flows
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Other Conventions
• PPC: Prospectus prepayment curve
• E.g., ALT-A or subprime different from conforming
• Why?
• So prospectus will often include language like “8-20% CPR over 12
months”
• What does this mean?
• HEP: home equity prepayment curves
• Ramp is only 10 months
• “20% HEP” means the asymptotic CPR is 20%
• See Exh 3.6
• MHP: manufactured housing prepayment curve
• 100% MHP is 3.6% CPR at t=0 and 6% CPR at t=24months
• See Exh 3.7
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Delinquency, Default & Loss
• Now switch to credit sensitive MBS
• Private label, CMBS
• Terminology is not standard
• OTS (MBA):
• 0 days late; Current
• 1-30 days late: Current (30 days delinquent)
• Usually the cut-off for MBA is 15 days late
• 30-60 days late: 30 days delinquent
• 60-90 days late: 60 days delinquent
• More than 90 days delinquent: 90+ days delinquent
• Default
• Legally, when borrower loses title
• Rule of thumb: 90+ = defaulted
• Roll rate that far delinquent to default is near 100%
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Delinquency, Default & Loss
• CDR: Conditional default rate
• Again, convert to MDR via MDRt = 1 – (1 – CDRt)1/12
• Default rate for month t = (Default loan balance in month
t)/(Beginning balance for month t less Scheduled principal payment
in month t)
• SDA: Standard default assumption
• “100 SDA:” 0 to 0.6% in 30 month, 0.6% in months 30-60, 0.6% to
0.03% from month 60 to month 120, 0.03% thereafter
• See diagram over
• “n SDA” is, as before, n/100 times this curve
• CDX: Cumulative default rate
• Total face value defaulted/total face value of pool
• COR: Charge off rate (annualized)
• Liquidation rate for month t = (Liquidated loan balance in month
t)/(Beginning balance in month t less Scheduled Principal payment
in month t)
• Convert to COR via usual compounding
FRL 383 Fall 2016
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Prepayment and Default Measurement & Modelling
PSA SDA CDR Benchmarks
1.40%
1.20%
50 SDA
100 SDA
1.00%
200 SDA
0.80%
0.60%
0.40%
0.20%
0.00%
0
20
40
60
80
100
120
Mortgage Age Months
140
160
180
200
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Delinquency, Default & Loss
• SDA base benchmark often includes a 150
PSA prepayment
• Example: See spreadsheet below
• Loss-severity rate = (Liquidation balance
month t less Liquidation proceeds month
t)/(Liquidation balance month t)
• Numerator is also loosely the “loss-given-default”
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Prepayment Models
• Two broad categories
• Rational prepayment models
• Assume pre-payable and no recourse
• View prepayment (and default) as a call option (put option), i.e.,
• call the mortgage at a strike equal to the outstanding balance plus
prepay penalty (put the asset at a strike equal to the outstanding
balance)
• Mt = Bt - Ct – Pt,
• where M is the mortgage value,
• B is the equivalent option-free (amortizing) bond value,
• C is the call option value,
• P is the put option value
• When are C and P ITM, OTM?
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Prepayment Models (cont.)
• Rational prepayment models
• Need interest rate and house price models
• Generate “states-of-world”
• Solve backward in time, along each path and time choose whether to
•
•
•
•
•
exercise immediately or wait, based on comparison of expected present
values
Exercising one option extinguishes the other
Also need external hazard (mortgage termination due to non-financial
reasons) and a refinance cost (prevents continual termination)
Note that embedded options are path dependent, since their value at any
time will depend not only on current rate and future rates, but also on how
you got to the current interest rate
Strengths: structural, independent of interest rate environment
Weaknesses: many parameters – difficult to estimate, cobbling on of
hazard
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Yields and Spreads
• A word about spreads…. FBB.10
• Yield = discount rate that sets projected cash
flows equal to price (or market value);
• Called cash flow yield.
• Note that must be quoted with prepay/default
assumptions, e.g., 100 PSA, 100 SDA, etc.
• Realized CFY will depend on realized PSA/SDA
• Convention: convert MEY to BEY
• Ease of comparison to Treasuries
• Reinvestment risk ignored!
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Yields and Spreads
• Yield spread to Treasuries
• Somewhat meaningless (why?)
• Must use comparable
• Duration, or
• WAL, weighted average life:
t principal received at time t
WAL
12 total principal
t 1
T
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Yields and Spreads: Simple & Static
• Simple aka nominal spread
• Difference between MBS CF yield and the
Treasury yield for maturity = average life of MBS
• Ignores fact that some of this spread is extra
return for taking on prepayment risk
• Static (or zero volatility) spread
• aka Z-spread
• Single number that must be added to the entire
Treasury spot rate curve (term structure) to set
expected discounted MBS cash flows equal to its
market price
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Yields and Spreads: OAS
• Option-adjusted spread (OAS)
• Adds interest rate volatility
• Write down an interest rate model,
simulate paths
• Compute cash flows, including effect of
options
• Need prepayment and default model
• These two steps comprise a valuation
model
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Yields and Spreads: OAS
• The single fixed spread that must added to
all paths in the interest rate model to match
model value and market price is called the
OAS
• If IR model is realistic, OAS reflects
cheapness/richness of traded price relative
to model
• OAS “corrects” (“removes”) added value of
options embedded in mortgages relative to
the base index (e.g., Treasury)
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Z-spread, OAS: Examples
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Prepayment Models (resumed)
• Empirical prepayment models
• E.g., Bear Stern’s agency prepay model (pool level)
• Identify critical factors, e.g. current market rate vs. WAC, house price
appreciation etc.
• Postulate a functional form based on observations
• Fit parameters
• Strengths: flexible, easy, accurate
• Weaknesses: short shelf life
• Bear Stern’s components
• Turnover
• Due-on-sale clause in almost all but government loans
• External hazard (life-cycle reasons for termination)
• CPR looks very much like the PSA prepay function
• Why is this reasonable
• Factors: mainly seasoning (PSA), add housing appreciation (LTV),
seasonality (higher spring/summer) as “perturbations”
FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
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Prepayment Models (cont.)
• Typical components (cont.)
• Cash-out Refinancing
• Principal components are HPI and WAC vs. current rate (see below)
• Rate/Term refinancing
• Critical variable “refinancing ratio” = WAC/Current Rate, found to be
better than difference
• HPI also important
• S-curve for prepayments (explain shape). Schematically:
CPR
HPA1
HPA2 ramp time
• D = WAC – CC-2, and let
• T = value of D at which prepay increases, and P = value of D at
which prepay reaches max, then
• I = min(D-T,P), and estimate
• CPR = α + β1A + β2I + … + ε
• Or, e.g., use atan function
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FRL 383 Fall 2016
Prepayment and Default Measurement & Modelling
Empirical Prepayment Modeling
• Alternative: Logit transformation
• SMM = 1/[1+ exp(-xβ)], so
• y = -ln[(1/SMM) – 1] = xβ
• Then run OLS/GLS on y
• Example
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