550.448 Financial Engineering and Structured Products Week of March 31, 014 Structured Securitization Liability-Side Cash Flow Analysis & Structured ransactions Assignment Reading (this week, March 31 st ) Read Chapter 9, [1] & 13 of R&R Allman: Chapter 8 Reading (next week, April 7 th ) R&R: Chapter 11 (PAC Structures) Allman: Chapter 9 1.1 1. Plan for his Week Review Mid-term Analysis of the Dynamic Behavior of Structured Securitization Analysis & Design in Use of Monte Carlo Analysis Loss Distribution Model Key Measures: ranche Risk & Performance Yield, Weighted Average Life, Reduction in Yield Design so-called nonlinear convergence problem Analysis of Seasoning and (relative-) value 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 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 YM (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 he published rating vs. the market bid/offer 1.3 1.4 1
Monte Carlo analysis will generate a realization of the reduction in yield associated with the risk distribution of the assets in the pool he 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 Calculate the CF s yield, average life and reduction in yield for each class Repeat for a large sampling of realizations to generate a distribution of each class reduction in yield his 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 1.5 he key engineering/analysis task is to close the loop Were the assumptions used to construct each class consistent with that class risk For engineering the task: If so, 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) 1.6 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 Lognormal Loss Distribution Select a Brownian motion model for the credit loss process 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 dt dz so the change in ln x between 0 and some future is normally distributed with mean (µ - σ /) and variance σ ln x( ) ln x(0) ~, Since ln x() is normal, x() is lognormally distributed 1.7 1.8
Lognormal Loss Distribution A lognormal variable can take any value from 0 to and has distribution: From our expression for ln x() we have that Ex ( ) x0 e var( x ) x e ( e 1) 0 Determination of CF Properties YM, Average Life, DIRR ake 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 YM (IRR) and average life Note the analytic implementation in PMB (Allman Chapter 8) Monthly yield, BE yield & Average Life (8.4, 8.5, 8.7) Average Life = Average time for Return of Principal key for comparisons Now we look at calculating the DIRR, or Reduction in Yield 1.9 1.10 DIRR At the maturity of the transaction, the expected reduction of yield must be calculated Let the following be pool performance variables I 0 = Initial balance of the security R = Annual rate of interest on the security L() = Brownian motion loss index value at maturity of the transaction CE = Credit enhancement as subordination = Years to maturity t 0 = 0 So the logarithm of credit loss at time t between t 0 and is ln L( t) ln L( t0) ~ ( t t0), t And ln L( t0) ( tt0) is the average logarithm of credit loss at any t 1.11 DIRR Assuming interest is paid until liquidation, the yield on the ABS, IRR(L) is the solution to i 1 I0 max LCE,0 I0 RI0 i 1 1 IRR( L) 1 IRR( L) By construction, P 0 = I 0 + CE holds at closing, where P 0 is the initial balance of the collateral pool So yield is only a function of L(), the loss index at maturity Neglecting values of L() greater than P 0 : IRR REIRRL ( ) L ( ) P 0 Only remaining unknown is the starting value of L(t 0 ) At least as large as the bid-ask spread on a similar pool of collateral 1.1 3
DIRR he Credit Scale he average DIRR is used as a confirmation/ratings measure Impact of the Loss Distribution on Deal Structure and Performance 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 his scale can be used throughout the transaction life span 1.13 1.14 Design 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 reasuries Design he Method of Non-linear convergence (-tranche deal) Chose an initial set of (Par) coupon for the A- & B- classes From BOE or experience with similar deals hrough a Monte Carlo (MC) process (,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 1.15 1.16 4
Design he Method of Non-linear convergence (-tranche deal) his is a highly simplified case where we only varied Par coupon Could address size, add mezzanine classes Other changes? 1.18 Analysis he result of the design is not stationary 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 Non-performing Collateral 1.19 Assume the losses build up according to the mean value of the logarithm of the index ln L( t) ln L( t0) ( tt0) he associated structured rating will therefore change according to the distribution of remaining losses from t to 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 1.5) o 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 log 0.10 1.5 log 0.001 7 hen, at pricing: So 0.5 0.6846 and 0.343 1.0 And we find that the reduction in yield, ΔIRR, is 5.89bps mapping to a rating of BBB he 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 1.1 5
Graphically over time he advantage of a lower coupon of 5% he advantage of a shorter maturity, 6-years 1. 1.3 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) ( tt0) Now assume that run-rate losses are 5% worse than expected on a logarithmic scale: the mean value is 5% higher ln L( t) ln L( t0) 1.5 ( tt0) he remaining calculations are identical to the Performing scenario & Non-Performing Pool Graphically 1.4 1.5 6
Static Rating and Over Enhancement If the loss adjustment factor were 5% (vs. 5%) Static Rating and Over Enhancement Wasted Capital in a performing deal 1.6 1.7 7