Financial Engineering and Structured Products

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1 Financial Engineering and Structured Products Week of March 10, 2014 Structured Finance Analysis of Liability-Side Cash Flows Assignment Reading Allman: Chapter 6-7 Read Chapter 7 of R&R Assignment See Website (2 problems) Problem 1: handed-in, Wednesday, February 26 Problem 2: due today, Monday, March Assignment Next Week Spring Break In 2 Weeks Midterm Exam In 3 Weeks Allman: Chapter 8 (tentative) Chapter 9 of R&R (tentative) Plan for This Week Continue from last week Static Valuation Model (BOTE) Elements of Credit Analysis for Structured Finance Asset Side Cash Flows Allman Chapter 3-4 Material R&R Prepays, Defaults and Recoveries

2 Plan for This Week Liability Side Cash Flows Liability Side CF protocol and hierarchy The Cash Flow Waterfall Types of Liabilities & PMB Fees Interest Principal & PMB Reserve Account Triggers Swaps Recoveries Model Builder First some terminology Gross Loss was defined as an asset that is defaulted and assumed not to pay (total value lost) Recovery is the cash that results if/when the defaulted asset is repossessed and sold Net Loss is Gross Loss minus Recovery Net Loss = Gross Loss Amount Recovery Amount Two more terms Recovery Rate = Recovery Amount / Gross Loss Amount Loss Severity = Net Loss / Gross Loss Amount Finally, since a recovery can take from a few weeks to even years, we define the Recovery Lag as the length of time it takes from default date to the recovery of actual cash Recoveries Model Builder An Example (A $1580 default) Recovery Amount ($850) minus Cost of Liquidation ($200) gives a Net Recovery Amount of $650 Recovery Rate = 650/1580 = 41.13% Loss Severity = 930/1580 = 58.86% = 1 Recovery Rate Recoveries Model Builder Data has been collected on Liquidation Proceeds & Costs and Average Months to Liquidation (by vintage year) See the data in MB5-1 Raw Data From (raw) Loss Data, can compute Recovery Rate Need to do historical Analysis of recoveries and lags to model cash flows, but usually available data permits only simplified approaches (not like delinquencies and default data). 1.8 Since data is only for 2 years, take worst case to be conservative. Recovery Rate = 40.34% Liquidation Lag = 5 months Now we have rationale to add back losses (as recovered) with appropriate rate and timing (lag) 1.9 2

3 What s Next? Model Builder Now we have all we need to establish the cash flow generated each period available to pay liabilities Total Cash to Fund Liabilities = Principal (Voluntary & Scheduled) + Actual Interest + Principal Recoveries Complete model to this point is MD5.2 Next we move on to the liability side But first, lets step back for analytic formality distinct from cash flow & spreadsheet empiricism Dollar-Space vs. Account-Space 1.10 Analysis of Asset Side Cash Flows In PBM we focus on dollar space characterization of a pool for prepays and defaults Alternatively, account or loan space allows delinquency/default treatment at the individual loan-level Moving back and forth is facilitated through the idea of average loan size (or individual loan size) In a loan level approach, the credit state of each loan is tracked; thus, an additional column needs to be included to record this state Otherwise, the approaches are closely related 1.11 Analysis of Asset Side Cash Flows Loss Curves & Loss Distributions Loss Distribution adds the standard deviation of losses to the expected loss Analysis of Asset Side Cash Flows Forecasting the Expected Loss We will see later that the credit loss curve is best modeled using the logistic curve for the cumulative distribution function (CDF) a F() t CDF ct ( t0 ) 1be From which we can relate the probability density function (PDF) as ct ( t0 ) () () () Ft () abc e df t f t dt f t t ct ( t 2 0 ) 1be And in the spreadsheet approximation, the marginal or monthly losses in pool percentage terms follows from f(t) = F(t) F(t-1)

4 Analysis of Asset Side Cash Flows Forecasting the Expected Loss As noted, there are equivalences between account-space treatment (at the loan level) and the summary representative lines (so called, dollar-space) It can be convenient to consider account-space treatment so the credit state is expressed on a loan-by-loan basis We can forecast defaults as a % of initial pool balance, F(t), and Map to account-space, L(t), % of loans L(t) = F(t) x N(0) By multiplying F(t) by N(0), the number of loans at t = 0 Consider initial pool of 2000 loans where we wish to simulate an expected loss of 10% of the account base 200 loans Over the life of the pool, 200 loans will default according to the CDF on the last slide That is, L(T) = 200 The loss curve L will not reach 200 until t becomes arbitrarily large We deal with this by normalization in account space 1.14 Analysis of Asset Side Cash Flows Forecasting the Expected Loss Convert F(t) to L(t) and then L(t) to l(t); l(t) is the non-normalized account space loss at t : l(t) = L(t) L(t-1) (during month t) Then if D r is the cumulative account space default rate, then by definition E(L) = N(0) x D r = L(T) And we may normalize l(t) in account space, call it n D (t), as follows ltn () (0) Dr ltn () (0) Dr nd () t T LT ( ) L(0) () lt t1 Then summing up to T gives T T N(0) Dr N(0) Dr nd () t l() t L( T) L(0) E( L) t 1 LT ( ) L(0) t1 LT ( ) L(0) Next we move on to the liability side cash flow 1.15 Liabilities in a structured Transaction include any cost that is to be paid from the cash generated by the assets Foremost are costs which enable the transaction Servicer, trust, rating agency fees, etc. Also paramount are fees that insure the integrity of the transaction e.g., CE & IR risk Swap fees/payments, insurance, etc. Finally, payment of principal & interest due to the parties (note holders) that funded the transaction 1.17 The liability structure is a set of rules for allocating and distributing cash collected from the assets The typical payment promise to note holders is for timely interest and ultimate principal (when?) At the heart of the liability structure is Allocation: Setting aside (identifying) the amounts due Distribution: Paying the amounts identified Pass-Through Structures (cf. MBS pass-through) a participation structure: symmetric allocation and distribution on either side of the balance sheet Single Tranche (usually)

5 Pay-Through Structures, OTOH, exhibit a hierarchy of credit: payment speed symmetry with top-down payment priority and asymmetrical allocation of principal that favors some investors over others Multiple Tranche Payment Logic of pay-through structures and the hierarchy of claims for repayment Sequential serial repayment Pro Rata partitioned ratably or in proportion to the invested amount Due Paid patterns of pay-through principal Sequential: First fully to class A, then to class B, Pro Rata: Allocated pro rata on initial participation %, but Class A receives first, then as available to Class B, Para Passu Pro Rata: Allocated on participation percent and distributed on a similar pro rata basis Usually for interest payment rules in a single credit class of notes Pari Passu parallel repayment where subclasses have varying maturities with associated coupon The Raw Deal A pseudo-model for Liabilities To introduce an algebra of allocation and distribution Simple Pay-Through structure w/a-b Tranche Structure Only source of CE is XS and Subordination No reserve, trigger, surety, etc. Glossary: Bond Structure usually refers to principal allocation method, but can refer to entire set of CF modalities in the waterfall Waterfall (WF) Rules that specify the way cash is distributed to claim-holders; each item in the WF is a level (6 levels in the below) 1.21 The Raw Deal A pseudo-model for Liabilities Glossary (continued): Pari passu Separate distributions made at the same level Sequential Principal allocation is done so that all principal of the senior class is retired before any principal is allocated to the junior class Pro rata Total principal is allocated proportionally to all classes according to the initial principal balance of each class We follow with nomenclature: t integer collection periods ra Class A interest rate (annualized) s f Servicing fee rate (annualized) s Servicing fee shortfall rate rate (annualized) r

6 The Raw Deal A pseudo-model for Liabilities The Raw Deal A pseudo-model for Liabilities Nomenclature (continued): Nomenclature (continued): St ( ) Servicing fee due at time t PP( t) Prepaid receivable balances at time t B ( t) Class A principal Balance at the end of time t A I ( t) Class A interest due w/r to time t A P ( t) Class A principal due at time t A F( t) Remaining funds at time t and level i in WF i P ( t) Total Principal Due at time t D Ct ( ) Cummulative Principal Due at time t R( t) Residual payment due to SPE stockholders at time t R ( t) Recoveries on defaulted receivables at time t C P ( t) Regular Principal Collections at time t It ( ) Interest Collections at time t Vt ( ) Pool balance at end of time t V(0) B (0) B (0) BA(0) Advance rate V (0) Dt ( ) Defaulted receivable balances at time t R A B P ( t) Class A Principal Paid at time t AP P ( t) Class A Principal Shortfall at time t AS The Cash Flow Water Fall The priority of payments is described in the term sheet Waterfall because the distribution of available cash starts at the top and trickles down in an ordered way Such movement transfers well to the 2-D spreadsheet Time and order is organized by row and column, respectively Fundamental Four-Item Structure: what-you-have and what-you-need, starting with available funds Amount Due Amount Paid Shortfall (Unpaid) The Cash Flow Water Fall The Raw Deal comprises six levels Servicing Fee(s) Class A Interest Class B Interest Class A Principal Class B Principal Residual Payment Remaining

7 The Cash Flow Water Fall Required Data before entering any time step Available Funds : F0 ( t) P( t) RC ( t) I( t) Pt ( ) PPt ( ) PR ( t) Total Principal Due: PD() t D() t PR() t PP() t Starting Class A Principal Balance: BA( t1) Cumulative Principal Due: Ct ( ) P( t) Ct ( 1) Next We Look at the WF Levels 1 6 D The Cash Flow Water Fall (Level 1-6) The Cash Flow Water Fall (Level 1-6) The Cash Flow Water Fall (Level 1-6)

8 The Cash Flow Water Fall (Level 1-6) The Cash Flow Water Fall (Level 1-6) Reserve Account and the Liquidity/Credit Facility Reserve fund is a mechanism for placing, holding, and releasing cash in the transaction Can be prefunded and/or be funded from XS capture and growth Control release if a fixed target percentage or amount is used Liquidity Facility is cash available to facilitate payments When need for timing: precision is high and low default likelihood Credit Enhancement is capital that facilitates payment when certainty of return is quantifiable but not high The distinction is driven by position that reimbursements occupy within the WF before or after Principal Payment Types of Liabilities Three Basic Types (can be whole dollar or, more usually, as a percent of assets or debt balance) Fees Interest Principal PMB on Liability Side Liability Input on Inputs Page Sr/Sub Structure Structure Inputs on Input Page fees, reserve, etc. Generating Liability Cash Flows The raw deal as per R&R (groan) We ll come back to this later but first

9 Types of Liabilities Fees Support Operation of Transaction Servicer of the loans Trustee for taking cash and distributing it property on behalf of the note holders Rating agencies for analyses and ratings To insure the integrity of credit and other risks (IR Swaps, CDS, credit lines, etc.) more on this when we address advanced liability structures PMB on Liability Side Fees Fee per cent One for each debt tranche (tranche balance) One for the structure against assets (asset balance) Waterfall/Cash Flow Structure: due/paid/unpaid/cash remaining Types of Liabilities Interest Floating Fixed Type mismatch to Assets PMB on Liability Side Interest Interest Type Fixed/Floating/Custom Floating Index (Curve) and Margin Custom Schedule (e.g., step-up) Waterfall/Cash Flow 1.37 PMB on Liability Side Interest Waterfall/Cash Flow Set Up Cash Flow Sections on CF Sheet Senior & Subordinate Structure: rate/due/paid/unpaid/cash remaining for each period Due Amount Depends on Period Principal Balance See Principal in the Waterfall Shortly (Next Section) Senior paid First Subordinate paid Next What about unpaid? Usually capitalized or carried over as due the next period For PMB, it is just carried as unpaid

10 Types of Liabilities Principal Repayment Sequential Senior first, then Subordinate Pro Rata According to % of original face Changing Principal Allocation by schedule, event, parameter PMB on Liability Side Principal Advance Rate Tranche per cent of Transaction original face Principal Allocation Type Sequential/Pro Rata Waterfall/Cash Flow 1.39 PMB on Liability Side Principal Waterfall/Cash Flow Add Loan Balances to facilitate Interest Calculations Return to Cash Flow Sections on CF Sheet Senior & Subordinate Structure: due/paid/unpaid/cash remaining for each period Principal Due Sequential scheduled + prepaid + defaulted until tranche fulfilled or none left Pro Rata scheduled + prepaid + defaulted as pro rata share due tranche Basic Liability Waterfall is Complete Not operational additional structures yet to be added Reserve Account, Swaps, Triggers, etc Summarizing the Results So Far Everything we need to model CF is in place Need to consider valuation and relative performance Key measures are yield (actual vs. promise) and average life Still, a few things are needed Three types of Reserve Account Trigger(s) Swaps Loss protection Single most important rationale for credit enhancement Motivates these advanced liability structures

11 Reserve Account A mechanism for placing, holding and releasing cash in a transaction Can provide liquidity protection and/or credit enhancement depending on location in the waterfall In the BOTE analysis funding from XS was distinguished from a closing-funded reserve for credit measurement In modeling, funding at the beginning of the transaction just changes the initial state (to which operational XS may be added) for CF analysis Can be funded to a minimum of a fixed percent of PMB on Liability Side Reserve Account Reserve Active: Y/N (for each liability) Reserve Account % of initial collateral principal PMB Assumes the Reserve funded at closing How would PMB be modified if reserve were not fully funded at closing? If the reserve account goes below the %, then XS is tapped Reserve is a fixed amount in PMB amortizing is not modeled How would you add an amortizing reserve, instead? Assumption: Only senior tranche has access to reserve Waterfall/Cash Flow liabilities or a fixed dollar amount PMB on Liability Side Reserve Account Waterfall/Cash Flow Cash Flow Sections on CF Sheet Period Reserve Account Structure: minimum/beginning/withdrawals/reimbursements/ ending balance/cash remaining for each period Reimbursements are made to the account if it is below minimums and the covered tranches are not paid off A bug in PMB? Senior Interest/Principal CF Structure: augment to show unpaid covered by reserve and remaining still unpaid This amount now constitutes withdrawals XS Released after any required application to Senior Principal Senior Principal Due when trigger breached another PMB bug? 1.45 Triggers The simplest, most efficient, and most cost effective method of mitigating loss is by altering the structure of the transaction when problems arise In many deals, as investors see collateral starting to sour, they prefer to see their principal returned OTOH, early return of principal results in a lower yield and a shorter average life Conflict between payment speed and reward makes defining and setting a trigger tricky PMB defines some simple trigger structures

12 Triggers A classic trigger is based on cumulative default rate If a static pool of assets have an expected default rate of 3%, a transaction could set a trigger at 5% such that if breached there is a rapid amortization of senior principal Alternatives trigger definitions include Negative Excess Spread: If XS becomes negative Delinquency: If delinquencies exceed a specified level Rolling Average: Rather than a period test, a moving average is compared to a trigger breach level Qualitative: Could include trust missing a payment, failing to send reports, failing to meet any of a set of criteria 1.47 Triggers Trigger breach can force a variety of changes and consequences In the case of a breach not being very severe, it could mean trapping extra cash for a period and increasing minimum reserve to keep it in the deal Alternatively, for more severe cases, a deal could go into full rapid amortization with all cash being redirected to the senior tranches Provisions for trigger breaches to cure can also be specified Only quantitative triggers depending on CF state are usually modeled 1.48 PMB on Liability Side Triggers PMB models four classes of triggers Capture All XS used in stress test scenarios; all XS is used to pay down senior debt Capture All XS: Y/N No is OFF Waterfall/CF Boolean Indicator shows state of Capture All XS: T/F XS is used to accelerate the pay down of senior debt if T uses remaining cash at the end of the waterfall and applies to senior principal 1.49 PMB on Liability Side Triggers PMB models four classes of triggers Post Default Trigger Month implements the deferral of rapid amortization; allows a period for the continued release of cash Number of Months for Deferral Zero is OFF How about applying this idea to other triggers a wait period? To see if delinquencies cure Waterfall/CF Boolean Indicator shows state of Capture All XS: T/F All XS is used to pay down senior debt if T subordinate tranche is cut off

13 PMB on Liability Side Triggers PMB models four classes of triggers Default Trigger Per Cent Tracks Default and trips if Defaults Exceed a predefined level Default Trigger % Level to Trip Trigger SDA Curve? Waterfall/CF Cumulative Default Percentage is Tracked for each Period Boolean Indicator shows if the Default Trigger is Tripped: T/F All XS is used to pay down senior debt if T subordinate tranche is cut off PMB on Liability Side Triggers PMB models four classes of triggers Custom Event of Default Allows the modeler to customize a default state on a period-by-period bases and input it directly None Entered for each period of CF Waterfall/CF Event of Default state is entered for each period: T/F All XS is used to pay down senior debt if T subordinate tranche is cut off Swaps A mechanism for hedging risk by swapping a party s exposure to a counterparty Interest Rate Fixed-to-Floating Basis Rate-to-Rate Credit Default F/X PMB implements fixed-for-floating IR swap Swaps usually entail a cost in the deal for the risk benefit received paid prior to fees PMB on Liability Side IR Swap Structure Parameters Swap Active: Y/N Swap Rate In: Vector Swap Rate Out: Vector Waterfall/Cash Flow Cash Flow Sections on CF Sheet Swap Cash Flows Structure: notional/rate in/cf in/rate out/cf out/swap Net (in out)/cf available = CF from Assets + Net

14 A Note About Ratings We have everything we need to start to perform primitive CF-based ratings analysis We can run PMB to see if the deal performs on any given scenario Alternatively, we can look at a variety of scenarios and judge as to whether the deal performs as expected, or and more important if & when it doesn t 1.55 A Note About Ratings S&P and (to some extent) Fitch perform ratings analysis using stress tests Here, a variety of scenarios are run to judge deal performance under adverse condition Alternatively, one could generate scenarios under a modeled variation of likely future states and in doing so generate an ensemble of outcomes with statistical properties this is the Moody s approach This is what we will look at next: R&R Chapter