550.448 Financial Engineering and Structured Products

Where we are 

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Module 6 – Structured Securitization:

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Previously: Static Analysis and Credit/CE; Asset- & Liability- Side Cash Flows Now: Dynamic Behavior and Analysis Next: Advanced Liability Structures – PAC & TAC  March 23rd

Liability-Side Cash Flow Analysis & Analysis of Dynamic Behavior in Structured Transactions

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Spring Break: March 16th – 20th (no class on 13th) Midterm 2: April 8th (Wednesday)

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Assignment 

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What’s Ahead

Reading (Dynamic Behavior & Analysis)

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 Read Chapter 9, [12] & 13 of R&R  Allman: Chapter 8

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Reading (Advanced Liability Structures)

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 R&R: Chapter 11 (PAC/TAC Structures)  Allman: Chapter 9 

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Allman Chapter 6/7 Project

2 Weeks hence: Spring Break (March 16 - 20) Midterm 2: Wednesday, April 8th Last Day of Class: Wednesday, April 29th Final: Tuesday May 12th at 2:00pm – 5:00pm  In the classroom: Ames 234

 Choice: Trigger Variance, Reserve Variance. …  Proposals Due: April 7, 2014 

Allman Chapter 8/10 Project – MC analysis

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Schedule 

Plan

Date

Module

1/26/2015 (Mon) 1/28/2015 (Wed) 1/30/2015 (Fri) 2/2/2015 (Mon) 2/4/2015 (Wed) 2/6/2015 (Fri) 2/9/2015 Mon) 2/11/2015 (Wed) 2/13/2015 (Fri) 2/16/2015 (Mon) 2/18/2015 (Wed) 2/20/2015 (Fri) 2/23/2015 (Mon) 2/25/2015 (Wed) 2/27/2015 (Fri) 3/2/2015 (Mon) 3/4/2015 (Wed) 3/6/2015 (Fri) 3/9/2015 (Mon) 3/11/2015 (Wed) 3/13/2015 (Fri) 3/16/2015 (Mon) 3/18/2015 (Wed) 3/20/2015 (Fri)

Intro & Overview Mtg/MBS (M1) Intro & Overview Mtg/MBS (M1) Intro & Overview Mtg/MBS (M1) Intro & Overview Mtg/MBS (M1) Legal, Accounting, & the SPE (M2) Legal, Accounting, & the SPE (M2) Static Valuation & Credit (M3) Static Valuation & Credit (M3) Asset Side Cash Flows (M4) Asset Side Cash Flows (M4) Midterm 1 Review Section: HW Problem Review Section: Midterm Review Midterm 1 Exam Midterm 1 Exam Return Liability-Side Cash Flows (M5) Liability-Side Cash Flows (M5) Liability-Side Cash Flows (M5A) Dynamic Behavior (M6) Dynamic Behavior (M6) Section - No Class Spring Break Spring Break Spring Break

Due

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Comments R&R(1-3), MBS, Allman(1-2), Preinitz(3)

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R&R(1-3), MBS, Allman(1-2) R&R(4-5), Allman(3-4) Assignment 1

Analysis of the Dynamic Behavior of Structured Securitization

R&R(6)

All HW Returned (NLT)

Analysis & Design (Engineering) in Structured Finance Use of Monte Carlo Analysis Loss Distribution Model Key Measures: Tranche Risk & Performance  Yield, Weighted Average Life, Reduction in Yield

Assignment 2

R&R(7), Allman(6-7)

 Design – so-called nonlinear convergence problem  Analysis of Seasoning and (relative-) value

R&R(9,12-13), Allman(8)

 The published rating vs. the market bid/offer 1.5

Analysis of Dynamic Behavior in Structured Finance 

With a CF model we have a basis to do dynamic analysis – scenario analysis based on the credit/prepayment uncertainty embedded in a structured transaction  We will use the spreadsheet based model to exemplify the approach for credit uncertain transactions

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Structured Securities are assessed and compared based on the promise to pay and the fulfillment of that promise  Specifically, the definitive measure is the difference between the promised yield (nominal coupon) and the YTM (IRR) of the actual set of CFs over the life of the transaction  Any reduction of yield (for a tranche/class) is the central focus of the credit analysis and of the associated Monte Carlo simulation  Monte Carlo analysis will generate a realization of the reduction in 1.7 yield associated with the risk distribution of the assets in the pool

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Analysis of Dynamic Behavior in Structured Finance 

The general idea is as follows:  Select a single realization from the scenario population of credit loss alternatives  Generate the associated CF realization for each class/tranche  Calculate the CF’s yield, average life and reduction in yield (if any) for each class  Repeat for a large sampling of realizations to generate a distribution of each class’ reduction in yield  This will provide an indicator of the class risk in each case – in the easiest form, we use average reduction in yield  Also apparent is the weighted average life

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Analysis of Dynamic Behavior in Structured Finance 

Analysis of Dynamic Behavior in Structured Finance

The key engineering/analysis task is to close the loop

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 Were the assumptions used to construct each class consistent with that class’ risk?  For the engineering task:

Credit Loss Distribution Model  Analytic Loss Distribution models are only an approximation to reality, but they provide insight and go a long way to quantifying resulting transaction performance  Lognormal distribution provides a good starting place for analysis (though they don’t give quite the iconic S-curve convergence)  We now look at some attributes/application of a lognormal loss model and how that translates into credit performance analysis

 If so (consistent assumptions), the task is complete;  If not, update the design and rerun the analysis

 For the analysis task:  Is the deal performing better or worse than initially believed, and, does the market price reflect that

 We will come back to address closing the loop a little later 

Considerations for Analyzing Dynamic Behavior  Credit Loss Distribution Model  Determining tranche yield, average life & reduction in yield (DIRR) 1.9

Lognormal Loss Distribution 

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Lognormal Loss Distribution

Select a Brownian motion model for the credit loss process

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A lognormal variable can take any value from 0 to ∞ and has distribution:

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From our expression for ln x(T) we have that E ( xT )  x0 e T

 For an index x(t) , geometric Brownian motion is defined by dx   dt   dz , where  is the drift and  is the volatility x and dz is a Wiener process  For example, let x(t) be a credit loss index at t

 Using Ito’s lemma with y = ln x we find 1   dy      2  dt   dz 

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so the change in ln x between 0 and some future T is normally distributed with mean (µ - σ2/2)T and variance σ2T ln x(T )  ln x(0) ~      2 2  T ,  T   Since ln x(T) is normal, x(T) is lognormally distributed

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var( xT )  x0 2 e 2 T (e T  1)

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Analysis of Dynamic Behavior in Structured Finance 

Analysis of Dynamic Behavior in Structured Finance – DIRR

Determination of CF Properties – YTM, Average Life, DIRR     

At the maturity of the transaction, the expected reduction of yield must be calculated

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Take a random sample from the distribution Suppose for example, it is cumulative loss at maturity Apply the sample to our CF model Analyze the CF to determine YTM (IRR) and average life Note the analytic implementation in PMB (Allman Chapter 8)

 Let the following be pool performance variables      

 Monthly yield, BE yield & Average Life (8.4, 8.5, 8.7)  Average Life = Average time for Return of Principal – key for comparisons

I0 = Initial balance of the security R = Annual rate of interest on the security L(T) = Brownian motion loss index value at maturity of the transaction CE = Credit enhancement as subordination T = Years to maturity t0 = 0

 So the logarithm of credit loss at time t between t0 and T is ln L(t )  ln L(t0 ) ~      2 2  (t  t0 ),  t  ln L (t0 )      2 2  (t  t0 ) is the average logarithm of credit loss at  And any t

 Now we look at calculating the DIRR, or Reduction in Yield

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Analysis of Dynamic Behavior in Structured Finance – DIRR 

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Analysis of Dynamic Behavior in Structured Finance – DIRR

Assuming interest is paid until liquidation, the yield on the ABS, IRR(L) , is the solution to i T   I 0  max  L  CE , 0  1 I 0  R   I 0  max  L  CE , 0       T i 1 1  IRR ( L )  1  IRR( L)

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The Structured Finance Credit Scale  The average DIRR is used as a confirmation/ratings measure

 By construction, P0 = I0 + CE holds at closing, where P0 is the initial balance of the collateral pool  So yield is only a function of L(T) , the loss index at maturity  Neglecting values of L(T) greater than P0 : IRR  R  E  IRR  L(T )  L(T )  P0 

 Only remaining unknown is the starting value of L(t0)  At least as large as the bid-ask spread on a similar pool of collateral

 This scale can be used throughout the transaction life span 1.15

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Analysis of Dynamic Behavior in Structured Finance 

Analysis of Dynamic Behavior in Structured Finance – Design

Impact of the Loss Distribution on Deal Structure and Performance

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 In design and analysis  Design – choosing deal parameters consistent with market expectations for risk/reward  Analysis – as a deal seasons and expectations for performance are replaced by observations of reality, a deal will perform or not and we can establish value based on that

With the Excel-based CF model, we now systematically address the design of the securities & their convergence in yield space – a “nonlinear convergence”  In its simplest form this is a problem of assuming a coupon level for each security issued from a deal – their Par yield – and then verifying consistency of delivering that yield performance, given the associated risk  Bonds are priced for each risk level based on a spread to US Treasuries

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Analysis of Dynamic Behavior in Structured Finance – Design 

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Analysis of Dynamic Behavior in Structured Finance – Design

The Method of Non-linear convergence (2-tranche deal)

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The Method of Non-linear convergence (2-tranche deal)

 Chose an initial set of (Par) coupon for the A- & B- classes  From BOTE or experience with similar deals

 Through a Monte Carlo (MC) process (2,500 scenarios) on the risk factors – credit/prepays/etc.  A simple situation for PMB is cumulative credit loss

 From the MC output establish the distribution for DIRR and weighted average lives for each, the A- and B- , class  Compare with the market pricing expectation – yield, WAL, credit  If the comparison is unfavorable, adjust the Par coupons and repeat the MC process  If the result validates a “fixed point”, then done 1.19

 This is a highly simplified case where we only varied Par coupon  Could address size, add “mezzanine classes”  Other changes?

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Analysis of Dynamic Behavior in Structured Finance – Analysis 

Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning

The result of the design is not stationary

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 No matter how well-conceived the transaction, as the deal seasons and expectations are replaced by measurement the deal may get better or worse  Agencies typically perform periodic analysis on credit ratings and update their finding – on watch or up-/down- grade  Investors and PF managers do this too  Monthly as reports are issued by servicers and the trustee  Whenever they are considering a buy/sell so they have a measure of value to compare with market bids/offers  We now look at some scenarios of a deal as time progresses and seasoning occurs  Performing Collateral 1.22  Non-performing Collateral

 Assume the losses build up according to the mean value of the logarithm of the index ln L(t )  ln L(t0 )      2 2  (t  t0 )  The associated structured rating will therefore change according to the distribution of remaining losses from t to T  Suppose security coupon is 8%; it has already been structured at closing to a BB level of payment certainty – loss coverage between 1.0 and 1.5 (assume the midpoint 1.25)

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Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning 

Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning

Performing Pool

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 Suppose security coupon is 8%; it has already been structured at closing to a BB level of payment certainty – loss coverage between 1.0 and 1.5 (assume the midpoint 1.25)  To find the parameters μ and σ, assume the bid-ask spread is 10bps, the coefficient of variation (  ) is 0.5, subordination is 10%, and the initial bullet maturity is 7 years  Expected loss by 7 years: CE  10%  Coverage  Loss  Loss  .1/1.25  8%  Then, at pricing: log  0.10 1.25   log 0.001     2 2  7

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0.5    2  So solving   8  5.008  0

Performing Pool

Performing Pool  And we find that the reduction in yield, ΔIRR, is 25.89bps mapping to a rating of BBB  Where the reduction in yields is (for the PAR rate of interest, R ) IRR  R  E  IRR  L(T )  L(T )  P0 

 And the IRR is the solution to

T   I 0  max  L  CE , 0  1 I 0  R   I 0  max  L  CE , 0       T i 1 1  IRR ( L )  1  IRR( L) i

 Where I0 is the PAR value of the security

for the smallest root > 0 gives

  0.6846 and   0.3423

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Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning 

Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning

Performing Pool

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 The remaining entries in the table below are computed assuming the loss index’s actual path follows its expected throughout  Note that ratings transition show an improvement toward maturity

Performing Pool  Graphically over time

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Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning 

Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning

Performing Pool

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 The advantage of a lower coupon of 5%

Non-Performing Pool  Again, we assume the losses build up according to the mean value of the logarithm of the index ln L(t )  ln L(t0 )      2 2  (t  t0 )  Now assume that run-rate losses are 25% worse than expected on a logarithmic scale: mean value is 25% higher (loss adjustment factor) ln L(t )  ln L(t0 )  1.25      2 2  (t  t0 )  The remaining calculations are identical to the Performing scenario &

 The advantage of a shorter maturity, 6-years

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Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning 

Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning

Non-Performing Pool

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 Graphically

Static Rating and Over Enhancement  If the loss adjustment factor were 5% (vs. 25%)

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Analysis of Dynamic Behavior in Structured Finance – Analysis of Seasoning 

Static Rating and Over Enhancement  Wasted Capital in a performing deal

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