Mortgage Backed Securities.
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Mortgage Backed Securities. Mortgage Backed Securities. Document Transcript

  • THE SUBPRIME MORTGAGE CRISIS QBS A STEP BY STEP EXPLANATION QBS: Asset Pricing Series Abdulla AlOthman Abdulla Alothman 1
  • Abstract No deep understanding of the subprime crisis is possible without an understanding of how Mortgage Backed Securities (MBS) are priced and accounted for. This paper aims at developing just such an understanding. Part1:In order to keep technicalities to a minimum and to obviate the need for introducing heavy mathematical machinery, a simple - almost cartoon like - setting is used. Nevertheless, it turns out, that such a setting, is sufficient to provide the reader with the in depth understanding of the key issues. Part2: Shows in detail how a simple three period mortgage backed security, with no prepayment options, can be valued. Part3: Extends the analysis of part 2 to include - most mortgages at least in the US fall into this category - prepayment options. It is shown that such an instrument is equivalent to a portfolio consisting of a long position in a simple MBS and a short position in a call option. It is then shown how such an option may be valued , what is meant by model risk and finally, how the option's premium can be incorporated - industry standard practice - into the asset's IRR. An algorithm - in pseudo code- for doing this is also provided. Part4: Analyzes the performance of two traders, one of whom invests in an MBS with no prepayment clause, the other in an MBS with a prepayment option. It is shown how the industry practice of incorporating the option premiums into the IRR and standard accounting practices, which subsequently, allow for the accounting of this inflated IRR as income -rather than treat it as the insurance premium which it is - creates strong incentives for a traders to adopt cavalier strategies at shareholder expense. Part5: Extends the analysis of Part3 to allow for the possibility of the borrower defaulting. It is shown that such a security is equivalent to a portfolio consisting of a long position in a default free MBS (with or without a prepayment option) and a short position in a put option, the value of which, depending on the likelihood - represented by an exogenous parameter - of default. The greater this likelihood, the more valuable the option will be. A model for the underlying physical asset is presented Using this model, we then show how the default option may be valued and, finally, as in part 3 above, we show how to incorporate this into the asset's IRR using a numeric algorithm. Part6: Extends the analysis of part4 to the case where one of the assets is no longer default free. The conclusions are similar to those in part 4, the effects however, are magnified by the inclusion of the default option into the asset's IRR. Abdulla Alothman 2
  • 1.The Data 1. We consider a three time period mortgage: t = 0, 1 2 2. The lender's cost of funding is 5.05% per annum 3. Interest for t=0 and t=1 rates are listed below Interest Rates Today t=0 Time t=1 Time t=1 Scenario 1 Scenario 2 Probability 0.5 (Probability 0.5) 1 Period 5% 4% 7% 2 Periods 5.5% 5% 8% Table 1.1 Abdulla Alothman 3
  • 2.The Pricing of a Simple Default Free Mortgage Backed Security with No Prepayment Option (MBS1) Consider a client wishing to borrow 100 for 2 periods in order to purchase a house, i.e. a two period mortgage. Assuming the above interest rates hold. The rate on this mortgage and associated amortization table are calculated as follows: Step1: We first need to calculate the payment amount. The client will make two payments - each including principle + interest - at times t = 1 and t = 2 respectively. So, we need to solve for x (prepayment amount) in: x x 100   (1.1) (1.05) (1.055)2 This says that the sum of the payments adjusted for the time value of money (first payment at the year 1 interest rate, second payment at the year 2 interest rate ) must equal the money advanced. Solving gives: 100 x  54.0297 (1.2) 1.850833 This represents the amount the borrower needs to repay each period. Step2: Calculate the IRR We solve for r in: 54.0297 54.0297 100   1  r / 100 (1  r / 100)2 (1.3) To get: r = 5.3269 Abdulla Alothman 4
  • Step3: Deriving the amortization table Starting Payment Interest Cost of Net Ending Balance Income Funding Income* Balance 0 -100 1 -100 54.03 5.33 5.05 0.28 -51.30 2 -51.30 54.03 2.73 2.59 0.14 0 Table 2.1: Interest Income - Cost of Funding Table 2.1, is what the client sees. Abdulla Alothman 5
  • 3.The Pricing of a Default Free Mortgage with an Option to Prepay (MBS2) Step 1: Analysis At time t = 1, the borrower can choose to refinance. Whether or not he chooses to do so will depend on which of the above scenario's prevails: Scenario1 4% 5% 1/2 t=0 5% 5.5% 1/2 Scenario2 7% 8% Figure 3.1: Interest Rate Scenarios At time 1, after making his payment, the mortgage holder has a one period note outstanding with a face value of 51.374 (see table 2.1 above) and a market value of: Scenario1 54.0297/1.04 = 51.9516 Rates Fall 1/2 100 1/2 Scenario2 54.0297/1.07= 50.4950 Rates Rise Figure 3.2: Time 1 Scenarios Abdulla Alothman 6
  • The gain from refinancing is: Scenario1 (51.952 -51.297)= 0.655 1/2 Rates Fall V 1/2 Scenario2 (50.495 - 51.297) = -0.802 Rates Rise Figure 1.3: Gain from refinancing Clearly a rational borrower will only refinance if the gain is positive, so his actual time 1 refinancing payoffs will look like: Scenario1 (51.952 -51.297)= 0.655 Rates Fall 1/2 V 1/2 Scenario2 0 Rates Rise Figure3.4: Gain from rational refinancing Figure 3.4, is the payoff of a one period call option on a bond with one period left to maturity and a strike price of 51.297. Abdulla Alothman 7
  • Step 2: Pricing the Option 1. If we look at the payoffs in figure 3.4 above, it seems clear that such an asset cannot have a value greater than its maximum payoff of 0.655 nor a value less than its minimum payment of 0. So the time t=0 price will perforce lie in the range (0,0.655) 2. Noting that the average payoff is 0.3275 - if we assume that investors on average require to be compensated for bearing risk (i.e. are risk averse) the above range can further be restricted to (0,0.3275). 3. Where exactly in this range the price should lie, will depend on how risk averse the market actually is. More risk averse and the value will be closer to 0, less risk averse and the value will be closer to 0.3275. At this point, there are two ways to proceed:  Try and estimate the level of the risk adjustment process directly  Build a model for the underlying asset process( most canned software use one of standard models e.g. Black and Scholes, Hull and White, CIR etc) , and calibrate its parameters using market data (in fact, though this is far from obvious, it turns out that this approach is equivalent to assuming a specific (parameter dependent) form for the risk adjustment process) To keep the analysis as simple as possible, I will assume an adjustment for risk of (0.62148,1.2833). This means that the average investor values a dollar less in an upstate (when the economy is booming for example) and more in a down state (during a recession). These risk adjustments can be factored in to the probability assumptions of (1/2, 1/2) to give the following pricing model : Abdulla Alothman 8
  • 0.32625 X1=Max(Aup-K,0)=0.655 V=(0.32625X1+0.67 375X2)/1.05 0.67375 X2=Max(Adown-K,0)=0 Table 3.5 Option Model Using the above model we can value the refinancing option as follows: 1. Multiply each payoff by its risk adjusted probability 2. Add these, to get the risk adjusted expected payoff 3. Divide by the one period interest rate (1.05) to adjust for the time value of money The value of the refinancing option is: V  (0.32625  0.655  0.67375  0) / 1.05  0.2035 (2.1) Digression - A Note on Model Risk Suppose we had picked a different risk adjustment factor, One consistent with a higher degree of average risk aversion. For concreteness suppose we had chosen to adjust for risk using (0.6, 1.304) instead*. This would have resulted in a model with scenario probabilities of - see appendix - (0.315,0.685) and an option value of: V  (0.315 * 0.815  0.685 * 0) / 1.05  0.2445 (2.2) So, the option value we obtain, depends on which model we decide to use. So long as the risk adjustment process implied by the model differs (and this will almost always be the case) from the underlying true process, the price implied by the model Abdulla Alothman 9
  • will differ from the true price needed to replicate the option. This is is known as model risk. *Choosing a different risk in this case (recall we already specified the future movement of rates, and the spot rates - and in so doing implicitly pined down the market price of risk -would result in an arbitrage opportunity). With the above choice - one way to insure we do not introduce arbitrage, is to change the two year spot rate to 5.516, which would result in the payment being 53.86, the IRR 5.1048, the strike 51.2444 and the option value 0.1633. End of Digression Step 3: Incorporating the Option Price into the Asset's IRR In practice, this option is not paid for at time 0, rather, it is build into the IRR of the bond. The following algorithm - in pseudo code - shows how to do this: The Algorithm{ Set Quit = NO Set V = 0.2035 ( The option value from (2.1)) Do While (Quit = NO) { a)Solve for mortgage payment amount PMT: PMT PMT V   (1.05) (1.055)2 0.2035  PMT (1.8508333)) PMT  0.10995 b) Calculate the IRR 54.0297  PMT 54.0297  PMT 54.0297  0.10995 54.4462  0.10995 100     1  r / 100 (1  r / 100)2 1  r / 100 (1  r / 100)2 r  5.47097 c) Use the this IRR to calculate the strike price: 54.13965 K  51.3313 (1  5.47097 / 100) d) Use the option model in (2.1) to value the option for this new value of K Abdulla Alothman 10
  • 54.13965 Vnext  (0.32625 * (  51.3313)  0.67373 * 0) / 1.05  0.22559 1.04 e) (Test if we are ready to quit) if (Vnext V  0.0000003 ) V  Vnext Else (If we are, then stop) Quit = YES Return Vnext } PrintResults Iteration Strike Option Value Period Payment Mortgage Equivalent IRR 1 51.29715 0.203355 0.1098725 5.326890 2 51.33130 0.22557 0.1218958 5.470874 3 51.335033 0.2280049 0.123190438 5.48662886 4 51.335434348 0.228266912 0.123331967 5.488325249 5 51.33547873 0.22829555 0.1233474 5.488510698 6 51.33548351 0.228298692 0.1233491 5.488530969 7 51.33544841 0.228299018 0.1233493 5.488533144 8 51.335484 0.228299061 0.1233493 5.4885334 Table 3.1: Option Premium Payment Comment: PMT PMT 54.03  0.123 54.03  0.123 100     (1  IRR / 100) (1  IRR / 100) (1  IRR / 100) (1  IRR / 100)1 (2.3) 2 IRR  5.49 }End of Algorithm Abdulla Alothman 11
  • Step 4: Deriving the amortization table t Beginning Payment Interest Cost of Net Ending Balance Income Funding Income** Balance Income 0 1 -100 54.15 5.49* 5.05 0.44 51.34 2a1 -51.34 54.16 2.82 2.59 0.23 0 2b2 -51.34 53.39 2.05 2.59 -0.54 0 Table 3.2: Amortization 1 No prepayment scenario 2. Mortgage is prepaid monies received are reinvested in the market Beginning Payment Interest Cost of Net Ending Balance Income Funding Income* Balance 0 -100 1 -100 54.03 5.33 5.05 0.28 -51.30 2 -51.30 54.03 2.73 2.59 0.14 0 Table 2.1 Reproduced for ease of comparison Abdulla Alothman 12
  • 4. A Story of Two Traders: Consider two traders, A and B, each with 1000 million of his institutions money to invest. Assume each is paid 20% of net annual income as an end of year bonus. Assume further that each can invest in only one of the above securities. Trader A chooses to invest in the no prepayment MBS (MBS1), Trader B chooses to invest in the MBS with the prepayment option (MBS2). Based on industry standard accounting practices, the profit that will accrue to their respective institutions is as follows: t Trader A Trader A Trader B Trader B Trader B Trader B Income Bonus (Scenario 1) Bonus Scenario 2 Bonus Income Income 0 1 2,800,000 560,000 4,400,000 880,000 8,800,000 880,000 2 1,400,000 280,000 2,230,000 446,000 -5,400,000 0 Table 4.1 Trader A will receive a bonus of 560,000 in year 1, and, assuming he does not get fired for "poor performance", a 280,000 bonus in the following year. Based on this performance, the markets view of him -especially if rates remain high, a random outcome- will probably be that of a "mediocre performer". Trader B will receive a bonus of 880,000 in year 1 and, assuming rates stay high - i.e. no refinancing takes place - a bonus of 446,000 in the following year. Moreover, after year1's results his market reputation will be that of a "star" performer If, after collecting his year 1 bonus rates drop, he can simply - something only too easy for a "star" trader to do - "jump ship" Leaving his institution and ultimately the shareholders to foot a loss of 5,400,000 to be realized at the end of year 2!! The problem in the above example, is that all of the IRR - this is standard accounting practice - including the part representing the option premium - is being booked as interest income!! This tantamount to an insurance company, booking all the premiums it receives on policies it writes, as profit!! From the perspective of standard Abdulla Alothman 13
  • accounting practices however, trader B, is simply long a risky bond, and the rules governing the book keeping of such an instrument are clear. Analysis Pausing for a moment and comparing (1.3) with (2.3), which for the readers convenience have been reproduced in modified form below: 54.03 54.03 100   (2.4) 1  r / 100 (1  r / 100)2 54.03  0.123 54.03  0.123 100   (2.5) (1  r / 100) (1  r / 100)2 We see that the extra yield accruing to trader B is a result of the option premium payment. The present value of this is - see table 3.1 - 0.2283 per 100 dollars invested i.e. 2.283 million. A natural question to ask at this point is, what does that premium really represent? To understand what is really going on, let is consider the following, time t = 0, portfolio: 1. An investment of 272.533 million in two period zero coupon bonds (B(0,2)) 2. A one year loan of 242.575 million (at 5%) 3. A loan of 2.283 million, the present value of the option premiums - currently being accounted for as income, to be repaid in 2 instalments of 1.233 million each. The value of this portfolio is: 1 V0  272.533   242.575-2.283  0 (1.055)2 Abdulla Alothman 14
  • It's payoff, excluding the loan repayment amounts, in millions, is: 0.5 X1**=7.347235 V=0 0.5 X2**=0 X1*  (272.5334379 * 1 / 1.04)  242.5753787*(1.05) * X 2  (272.53344379 * 1 / 1.07)  242.5753787*(1.05) Now consider a third trader C, who invests 1000 million in MBS2 and in addition invests in the above portfolio: t Beginning Payment Interest Cost of Loan Portfolio Net Ending Balance /Principle Income Funding Payment Income** Balance Income 0 1 -100 54.15 5.49* -5.05 -0.12 0.32 51.34 1 2a -51.34 54.16 2.82 -2.59 -0.12 0 0.11 0 2b2 -51.34 51.34 2.05 -2.59 -0.12 0.7 0.11 0 Table 4.2 Beginning Payment Interest Cost of Net Ending Balance Income Funding Income* Balance 0 -100 1 -100 54.03 5.33 5.05 0.28 -51.30 2 -51.30 54.03 2.73 2.59 0.14 0 The payoff to Trader C, is almost identical to trader A's. Which shows that the extra payoff to trader B was not the result of superior performance. But rather, a Abdulla Alothman 15
  • direct result of being able to book option premiums - needed to create the necessary replicating portfolios to protect institutions from market risks associated with prepayment - as income. To protect against such behaviour, a simple change in accounting rules, is all that is needed. To insure that such new rules are not violated, accountants need to be able to recognize cases - such as in the above case - when they apply. This in general - especially with complex structures - will not be possible without at least some advanced training in the theory of asset pricing. Abdulla Alothman 16
  • 5. The Pricing of a Mortgage with Prepayment Option and Risk Of Default. (MBS3) Continuing with the framework above. Suppose we allow for the possibility of default in period 2 on MBS2. That is, the possibility the borrower will not make the final payment. Step1: Analysis The payoff structure is represented below: Borrower Prepays 54.15* 53.38 51.34 1.04 100 54.15* 1 54.16  max(54.16  Adu , 0) 51.34 Adu  54.16   *First Payment 1 54.16  max(54.16  Add , 0) Add  54.16   Fig 5.1 MBS3 payoff diagram What the above shows is that: MBS3 = MBS2 - Put Option(K,2,) Here  is a proxy for the cost of default ( credit rating, social stigma etc). It is high (the option less valuable) for prime mortgage holders and lower (the option more valuable) for subprime borrowers. Abdulla Alothman 17
  • Step2: Building the Property Model Let's assume (this will be our choice of model) that market adjusts for risk and time value of money on real estate assets according to:   {0 , 1, 2 } where : 0  {1} 1  {1.3333, 0.571428} 2  {1.79442, 0.7690077, 0.91268, 0.15541} Given our interest model in part1 above, and a spot property price of 100, this implies the following real estate pricing model: 0.7 Auu=132 100 q=0.7 r=4% 1-q=0.3 Aud=108 100 r=5% q=0.8545 1-q=0.3 Adu=80.5 r=7% 1-q=0.1455 Add=42 Fig 5.2. Real Estate Asset Model Step2: Estimating the Default Cost Proxies Let us assume, for simplicity, that these are exogenously given: Prime Borrower  15 Subprime Borrower  5 Abdulla Alothman 18
  • Step 3: Valuing the Default Option 0.7 Puu=0 0 P(15)=0 q=0.7 r=4% P(5)=0.4724379 1-q=0.3 Pud=0 r=5% Pd(5)=1.769 q=0.8545 1-q=0.3 Pdu=0 r=7% Pd=q Pdu+1-q Pdu/1+r 1-q=0.1455 Pdd(15)=0 The other nodes are calculated similarly Pdd(5)=12.16 Fig 5.3 Valuing the Default Option Step 3: Incorporating the Option Price into the Asset's IRR a)Solve for option payment amount PMT: PMT PMT V   (1.05) (1.055)2 0.4724379  PMT (1.850833)) PMT  0.25525685 b) Calculate the IRR 54.0297  0.10995  0.255256 * 54.0297  0.10995  0.255256 100   1  r / 100 (1  r / 100)2 r  5.805291188 *The sum of the payment on MBS1+montly Call Option Premium + Monthly Subprime Put Option Premium c) The same algorithm as in part 3 above, then yields: 1. K=54.40 (Strike) 2. PMT = 54.40 3. P(5)=0.481739 4. MBS3 IRR = 5.81188 Abdulla Alothman 19
  • .Step 4: Deriving the Amortization Table t Beginning Payment Interest Cost of Net Ending Balance Income Funding Income** Balance Income 0 1 -100 54.40 5.81* 5.05 0.76 51.41 2a1 -51.41 54.40 2.99 2.69 0.30 0 2b2 -51.41 53.27 2.06 2.60 -0.54 0 3 2c -51.41 42.00 -9.41 2.60 -12.01 0 Table 5.1 1) No exercise 2) Borrower refinances, monies reinvested at lower rate of 4% 3) Borrower defaults Beginning Payment Interest Cost of Net Ending Balance Income Funding Income* Balance 0 -100 1 -100 54.03 5.33 5.05 0.28 -51.30 2 -51.30 54.03 2.73 2.59 0.14 0 Abdulla Alothman 20
  • 6. The Story of Two Traders Revisited: t Trader Trader TraderB Trader TraderB Trader TraderB TraderB A A Income Bonus Income B Income Bonus Income Bonus Case1 Case2 Bonus Case3 0 1 2,800,00 560,000 7,600,00 1,520,00 7,600,000 1,520,0 7,600,00 1,520,00 0 0 0 00 0 0 1,400,00 280,000 3,000,00 600,000 (5,400,000) 0 120,100, Sub 2 0 0 000!!! Prime Crisis Analysis: The analysis here is -mutatis mutandis - exactly the same as in Part 4. The only differences being: 1. Trader B has even a bigger incentive to invest in the high yielding asset 2. The Shareholders are left with a larger bill to foot!! Their time 2 payoff distribution is: Payoff Probability 2,400.000 25.63% -5,400,000 70% -120,100,000 4.37% Abdulla Alothman 21