A Fast Track to Structured Finance Modeling, Monitoring, and Valuation: Jump Start VBA By William Preinitz Copyright 2009 by William Preinitz

A Fast Track to Structured Finance Modeling, Monitoring, and Valuation: Jump Start VBA By William Preinitz Copyright 2009 by William Preinitz APPENDIX A Mortgage Math OVERVIEW I have included this section as a quick introduction to (or, for those of you who are already familiar with the material, a quick review of) the cash flow characteristics of loans and how to calculate them. In this book we are dealing with a collection of loans to small business. Even though these are loans to commercial entities or business establishments, they are no different in their cash flow characteristics than ordinary residential mortgages. Each loan at its inception has a starting balance. This balance is to be retired through the application of a monthly payment. The monthly payment will be the minimum amount to pay both the current month s interest charges on the balance of the mortgage and a portion of the principal. There are a fixed number of payments (e.g., 30 years 12 payments per month 360), and a beginning payment amount that is set at the time of the issuance of the loan. If the coupon rate of the mortgage is fixed that is, if the interest rate is not subject to change over the life of the loan, this combined monthly interest/portion of principal payment is fixed for the life of the loan. If the coupon rate is floating, the coupon may change periodically. The periodicity of these changes and the amount by which the payment is allowed to change are specified in the terms of the loan agreement. A typical floating rate loan is described as a spread to an index; an example might be 200 basis points over Prime. In that case, the initial rate of the loan, unless otherwise specified, will be the rate of the Prime Rate on the day the loan was issued, and the contractual spread, in this case set at 2.00%. Thus, if on the day of issuance of the loan, the Prime Rate was 6.00% the interest rate of the loan (or its coupon rate) would be 6% 2% 8%. The payment of the loan can be reset at specific intervals based on the issuance date of the loan. At these times, the spread is applied to the index and a new coupon rate for the loan is determined. It is not, however, the only calculation that needs to be accomplished. The new coupon rate is compared to a series of limits that may govern the manner in which this adjustment is made. There are, with all the loans in this portfolio, a set of caps and floors. These are minimum and maximum limits that the coupon rate is subject to at the time of adjustment. These levels take two broad forms: limits on what can happen at the reset date, and limits that apply to the lifetime of the loan. Once the new coupon rate is applied, the payment is recalculated to ensure that the loan s balance can be repaid in the number of payments remaining until maturity. 671

672 APPENDIX A: MORTGAGE MATH After we have determined what the new regular monthly payments are, we should also be aware of other loan payment behaviors. These are loan prepayment and loan default. In the case of prepayment, the Obligor makes a payment larger than the minimum monthly amount required by the current conditions of the contract. This extra money is always applied to the outstanding balance of the loan s principal. Often, this extra payment is the entire outstanding balance of the loan. Once the loan is paid in full via prepayment, the contract of indebtedness between the lender and the Obligor is no longer in force the principal has been fully retired and the scheduled future interest payments are obviated. The second type of payment behavior with which to concern ourselves is nonpayment. In this case, the Obligor defaults on the loan and stops making monthly payments altogether. The lender then moves to seize the property and sell it, with the goal of receiving enough money through the sale to repay the loan. It may take an appreciable period of time for this process to occur, especially if the Obligor contests the proceedings. If there is a subsequent realization of cash from a sale, it is called a recovery of principal. The lender is usually allowed to recover an amount equal to the cost of repayment of the loan, all interest due from the time of the default to the time of the recovery, and any costs involved in the repossession, repairs, and selling. Anything in excess of those amounts are returned to the Obligor. In this section, we start with how to calculate the monthly payment on a loan, both fixed and floating, and then look at how coupon income is calculated and principal is retired. We then will learn how to calculate prepayments and anticipate their effects on the cash flows. Lastly, we will look at the issue of defaults and recoveries. LOAN AMORTIZATION TERMS Before we start learning how to crunch mortgage numbers, we should review the language of mortgage math. Original Term. The number of months from the date mortgage issuance to the final payment date. For the purposes of these discussions, we will assume that all the mortgages we are dealing with are monthly payment mortgages. That is, the borrower will make one payment per month until all principal is retired, the loan prepays, or the loan defaults. For example, the original term of a 20-year mortgage is 240 months. Remaining Term. The number of months remaining until the end of the loan. For example, the original term of the mortgage is 300 months. A total of 120 monthly payments have been made to date. The remaining term of the loan is therefore 300 120 180 months. Seasoning or Age. The age of a mortgage is the difference between the original term and the remaining term. To continue our example, the age or seasoning of the mortgage above is the original term less the remaining term, or 300 months minus the remaining term of 180 120. Original Balance. The initial amount lent by the lender to the Obligor, the amount to be repaid over the life of the loan through the monthly payments.

Loan Amortization Terms 673 Remaining Balance. The current, also referred to as outstanding, balance of the loan principal. Appraisal Amount. The value of the property and chattel that secures the mortgage. This lender makes an estimation of the value of the property and its contents, then uses this as a backstop, a final source of liquidity (through sale of the asset) in case the Obligor defaults on the mortgage. Loan-to-Value (LTV) Ratio. This is the ratio between the remaining balance of the mortgage and the appraisal value of the property and chattel securing the loan. Thus, if the appraisal value of the Obligor s property and chattel is $1,000,000 and the remaining balance of the loan is $700,000, the LTV ratio is $700,000/$1,000,000 70%. The LTV ratio will change as both the appraisal value changes and the mortgage balance of the mortgage decreases through payment activity. Equity. This is the difference between the remaining balance of the mortgage and the current appraisal value. In the previous example, the equity is $1,000,000 $700,000 $300,000. Equity can be expressed either as a dollar amount or a percentage. When expressed as a percentage it is the ratio between the equity amount and the appraisal amount, $300,000/$1,000,000 30%. Payment Amount. The level of the current monthly payment. Scheduled Amortization. The amount of the payment that is applied to retire the remaining balance of the loan. Coupon Income. The interest expense component of the monthly payment. Prepaid Principal. The payment of remaining balance that exceeds the minimum monthly amount required by the loan terms. It is any extra money received, usually with the monthly payment, once all previously scheduled principal payment and current month s coupon income payment are met. Defaulted Principal. Defaulted principal is the amount of remaining balance when the Obligor stops making the monthly payment. Principal Recovery. A principal recovery is the amount of money realized through the disposition of the property and chattel, less any expenses incurred by the lender during the recovery period, but not greater than the remaining balance plus any capitalized interest added to the balance during the default period of the mortgage at the time of the default. Coupon Reset Frequency. Applicable to floating rate notes, this is the interval, in months, between the recalculation of the mortgage s coupon level based on the spread of the mortgage and the current level of the index. Payment Reset Frequency. Again for floating rate notes, the interval, in months between the times that the monthly payment is recalculated to reflect the current coupon level of the mortgage. Applicable to floating rate notes, this is the number of times each year that the loan s payment amount may be adjusted. Periodic Floor and Periodic Cap. Respectively, the lowest point and the highest point to which the mortgage coupon rate can be reset. Lifetime Floor and Lifetime Cap. Respectively, the lowest and highest coupon rate levels allowable at any time in the life of the mortgage.

674 APPENDIX A: MORTGAGE MATH COMPONENTS OF THE CASH FLOWS OF A LOAN Having tied down the basic language we are about to use, let s get to the task of understanding how to calculate mortgage cash flows! Calculating the Monthly Payment One of the first things we have to do is calculate the monthly payment of the mortgage. In that the payments of fixed-rate mortgages are much easier to explain and calculate, we will start with them. Fixed Rate Mortgage Payment Calculation To calculate the level monthly payment for a fixed-rate mortgage one merely sums the following series where: N number of payments r periodic interest rate (9.00% annual (9% 12)monthly rate) n current period, from 1 to N Sum (1 (1 r) 1st ) (1 (1 r) 2nd ) (1 (1 r) 3rd ) (1 (1 r) Nth ) Payment original balance sum Thus, the payment calculation for a fixed-rate 9% mortgage with a 24-month original term and a $100,000 balance is $4,568.47. See Exhibit A.1 for the calculation of the payment. If we look at cell C9 we can see that 1/(1 r) 0.992556. The sum of this series for the 24 months of the loan is shown in cell C10 21.889146. The payment is therefore $100,000/21.889146 $4,568.47. If we apply this payment monthly to the outstanding loan balance, we see that the mortgage amortizes to zero in month 24, as scheduled. We also see that the total coupon payments are $9,643.38, the principal payments are $100,000.00, and the total cash flows are $109,643.38, the combination of the two. Floating-rate Mortgage Calculation To calculate the payment of a floating-rate mortgage, we will need to know more information than we did for a fixed-rate mortgage. With a floating-rate mortgage, we will need to know the following: 1. The current index level 2. The spread to the index 3. The remaining term of the mortgage 4. The coupon reset schedule 5. The payment reset schedule (if it is different than the coupon reset schedule) 6. The periodic reset floor and cap 7. The lifetime reset floor and cap

Components of the Cash Flows of a Loan 675 EXHIBIT A.1 original term Amortization table of a fixed-rate mortgage with a 9% coupon and 24-month There is an excellent example of a series of calculations that show the step-by-step payment reset process in Chapter 14, so no need to replicate that work here. It is important to point out that the process is broadly similar to that of a fixed-rate loan. There are two obvious differences. The first is that with a fixed-rate loan the coupon is fixed and never changes, while with a floating rate note we must calculate what the new coupon is at each reset period. To do this: 1. Take the current index level and add the mortgage spread to it. This is our provisional new coupon. We don t know if this is the coupon level we will use yet, because it is subject to two tests. 2. Compare the provisional coupon to the reset floor or cap limits. If the change in the coupon limit is greater than the cap it is limited in its increase by the cap amount. If the decrease is greater than the floor, it is limited in its decline by the floor amount. For example, assume the cap increase is 2.00% and the previous coupon level is 7.00%. The index is at 9.00% and the spread is 3.00%, so the provisional coupon would then be 9% 3% 12%. The most that the coupon can increase, however, is the previous level plus the reset cap of 7% 3% 10%, which is lower than 12%. The new provisional coupon level is therefore limited to 10%.

676 APPENDIX A: MORTGAGE MATH $5.000 Cash Flow Components (short tenor mortgage) $4.500 $4.000 Monthly Amount $3.500 $3.000 $2.500 $2.000 $1.500 $1.000 $500 $- 1 2 3 4 5 6 7 8 9 101112131415161718192021222324 Months Principal Coupon EXHIBIT A.2 Principal and coupon components of the payment over 24 months of the loan 3. Compare the provisional coupon level to the lifetime floor and cap constraints. The lifetime floor and cap on this mortgage are 5.5% and 22.0% respectively. Since the provisional coupon level is now at 10% and fits well within the 5.5% to 22.0% Range, we are fine to leave it the way it is. 4. Now, using exactly the same formulas we used for the fixed rate mortgage, we calculate the new payment level, based on 10% as the new coupon rate to be applied for the remaining term of the mortgage. Exhibit A.3 illustrates the above example in table form. The initial coupon level at the issuance of the mortgage was 7.00%. Instead of being a fixed-rate mortgage, as we saw in the earlier example, where the coupon did not change over the life of the loan, the current mortgage is a floating-rate note. The first reset period is at month 13, at which point the loan resets to a 10% coupon for months 13 to 24. Looking at Exhibit A.3, we then see the new cash flows. If we apply the earlier formulas for calculating the payment, we will use the following values to determine our new reset payment level: N R 12 remaining term 10% 12 periodic interest rate Periodic Factor 1 (1 r) 0 991736 Sum of the Periodic Factors 11 374508 Remaining Balance $51 744 21 Monthly Payment $51 744 21 11 374508 $4 549 14

Components of the Cash Flows of a Loan 677 EXHIBIT A.3 Floating rate mortgage with an initial coupon of 7% that reset to a coupon of 10% in the second year We can see that this monthly payment, extended out, retires the last of the outstanding principal balance just when it should, in the 24 th month of the mortgage. Just remember that when you are calculating the new payment levels for each of the payment reset dates, you simply follow the steps above until you determine a new coupon level and then treat the loan as if it were a fixed-payment loan. In the case of multiple payment reset dates prior to the remaining term, you just need to treat each as though it is independent of the others. If you have a 120-month loan with annual reset dates, you simply assume the current coupon is to be the coupon in effect for the rest of the life of the mortgage. As each subsequent reset date occurs, just follow the same procedures to readjust the monthly payment. Cash Flow Components of a Long-Tenor Loan The graph of the components of scheduled principal amortization and coupon income for the monthly payments of the example 24-month term loan in Exhibit A.2 is typical of those of a shorttenor loan. The preponderance of the monthly payment will be used to retire the outstanding principal. The story is quite different for long-term mortgages. If instead of an original term of 24 months we have a mortgage with an original term of 360 months, you will see that coupon income is the predominate component of the monthly payment until well into the 25 th year of the mortgage. See Exhibit A.4.

678 APPENDIX A: MORTGAGE MATH $900 Cash Flow Components (long tenor mortgage) $800 $700 Monthly Amount $600 $500 $400 $300 $200 $100 $- 1 25 49 73 97 121 145 169 193 217 241 265 289 313 337 Months Principal Coupon EXHIBIT A.4 Relative components of coupon income and scheduled amortization for a long-term mortgage, in this case 360 months, fixed-rate, level payment, and 9% coupon. The amount of the coupon income is the area below the curve, the amount of the principal paid is above it to the $800 line Coupon Income Coupon income is one of the easiest of the cash flow components to compute. In any given period simply multiply the prorated coupon rate by the current remaining principal balance of the period. For the purposes of simplification, the above examples simply prorated the coupon to the monthly level by dividing it by 12. Depending on the convention of the mortgage, there may be several ways of determining the prorated coupon based on the number of days in the period. The model allows the use of three different types of day count. The types are 30/360, which is to say 30 days per month in a 360-day year, or 12 payments. The second is Actual/Actual which is the number of days in each month, (corrected for Leap Years) divided by the actual days of that year, also Leap Year-corrected. The final method is Actual/360. In this method the actual number of days in the month are divided by 360 days for the year. On individual months the differences can be small, but can add up quickly if you are not aware of this possibility. See Exhibit A.5. Scheduled Amortization A discussion of the calculation of the scheduled amortization portion of a mortgage loan payment is also given in Chapter 14.

Components of the Cash Flows of a Loan 679 EXHIBIT A.5 Calculation of 9% coupon on $1,000,000,000 for the month of February 2008, (a Leap Year) February 2008 Interest Balance Outstanding Coupon Rate $ 1,000,000,000 9.00% Methodology Factor Interest 30/360 0.0833333 $ 7,499,997.00 Actual/360 0.0805600 $ 7,250,400.00 Actual/Actual 0.0794500 $ 7,150,500.00 The formula for the percentage of outstanding principal retired by scheduled payment activity in any given period is: N total number of remaining term period n period of the principal retirement n 1 period immediately prior to the retirement period r periodic coupon rate Principal amortization of period n: Amort Factor [(1 r) Nth (1 r) nth ] [(1 r) Nth (1 r) nth 1] This is a particularly useful formula because it makes the calculation of the period principal a function of the remaining principal balance outstanding. Thus, all you ever need to know is the remaining balance of the mortgage at the beginning of the period, the coupon rate, the current period, and the original period of the sequence. A tabular display of this calculation can be seen in Exhibit A.6. This is the table of principal retirement factors for the original 9%, 24-month, $100,000 mortgage we looked at in the payment calculation section. In columns M, N, and O we can see the calculation of the factors for the principal retirement for each period. Column M contains the monthly factors of the hopefully now familiar series of (1 (1 r) 1st ) (1 (1 r) 2nd ) (1 (1 r) 3rd ) (1 (1 r) Nth )

680 APPENDIX A: MORTGAGE MATH EXHIBIT A.6 Calculation of the scheduled principal amortization factors, (some columns hidden for clarity) for each of the periods. In cell N9 we take the maximum value of the sequence that will be the term: (1 (1 r) Nth ) We now have everything we need to populate the Principal Retired column N. Once these monthly factors are in hand, we need only multiple them by the previous period remaining balance of the mortgage to arrive at the scheduled principal amortization for the month. We can then take these numbers from column O and place them into the mortgage amortization table column H. Prepayments of Principal If the world were a simple place, we could stop right here! If everyone sent in each payment month after month, we could drop the subject of mortgage math at this point and go home. Fortunately for us, the world is much more complicated. I say fortunately, because if it were simple, we would not be needed as bankers, risk analysts, etc. (and therefore not get paid the big bucks). Prepayments can occur for both financial and non-financial reasons. A business may prepay an existing mortgage because it has sold its current location and is expanding. It may prepay because it wants to take advantage of a relatively better

Components of the Cash Flows of a Loan 681 financing rate. Regardless of the reason, prepayments in a mortgage portfolio occur and can at times significantly change the expected cash flows. There are two basic prepayment methodologies discussed in this book and our model. The first is called the Constant Percentage Rate, or CPR method. The second uses CPR factors as its base and constructs a ramped table over time. This is the Public Securities Administration (PSA) method and the method that is used in most of the structuring activity outlined in Chapter 18. Let s look at each in turn. Constant Prepayment Rate (CPR) Method The CPR is an annual rate. We are producing monthly cash flows for the model. What we need to do is to first restate the annual rate of the CPR prepayment speed into a one-month prepayment speed that we can use here in our spreadsheet, and later in our VBA calculations. To reduce the annual rate to a monthly one-month rate we can use the following formula: One Month Rate 1 (1 CPR) 1 12 Thus, an 8% annual CPR represents a one-month percentage prepayment attrition of: One Month Rate 1 (1 08) 083333 One Month Rate 1 ( 92) 083333 One Month Rate 006924 or 0 6924% This means that 0.6924% of the outstanding principal balance of the mortgage will prepay in a given month. Keep in mind that prepayments occur only after the retirement of the scheduled amortization of the mortgage, and are assessed against the post-scheduled amortization balance, not the beginning balance of the period. Once we have determined the one-month factor, we are practically done! All that is left to do is to multiple this factor against the post-scheduled amortization outstanding balance then we will have our prepayments for the period. Public Securities Administration (PSA) Method As the loan securitization industry grew, it became clear that the CPR methodology could be improved upon. What was needed was a way to account for the fact that the probability of a loan prepaying grows as the loan ages. The more time passes, the greater chance there was that some set of circumstances would emerge that would favor a prepayment event. As a result, the PSA method was invented. The PSA method uses a series of CPR factors to construct a graduated increase of prepayment activity in the first 30 months of the life of the mortgage, and then levels out to a plateau rate that stays constant for the remaining life of the loan (no matter how long that is). The pattern starts with a base rate of 0.2% CPR for the first month. It then increases by 0.2% CPR each month thereafter, until it reaches the 30 th month, when it reaches the plateau rate of 6% CPR for the remaining life of the mortgage. This pattern is called 100% PSA. If the analyst wishes to apply a more robust prepayment assumption, a multiple of the curve can be used; if a less robust speed assumption is made, a fraction of the curve can be used. An example of using multiples or fractions

682 APPENDIX A: MORTGAGE MATH EXHIBIT A.7 Table of PSA curve multiples expressed in monthly CPR speeds Month 1 6 12 18 24 30 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 PSA Rate (Monthly Speeds in CPR%) 50% 100% 150% 200% 300% 0.10% 0.20% 0.30% 0.40% 0.60% 0.60% 1.20% 1.80% 2.40% 3.60% 1.20% 2.40% 3.60% 4.80% 7.20% 1.80% 3.60% 5.40% 7.20% 10.80% 2.40% 4.80% 7.20% 9.60% 14.40% of the PSA base curve is shown in Exhibit A.7. The CPR speeds for selected months of a 20-year mortgage at various multiples of the base PSA curve are shown in Exhibits A.7 and A.8. Exhibit A.8 shows the monthly speeds, expressed in CPR, of the various multiples of the base PSA curve. PSA speeds are often quoted in multiple units of the basic curve. To use a PSA-based prepayment speed, we first need to convert the PSA speeds to CPR speeds and then convert the CPR speeds to the one-month prepayment percentages. The Effects of Prepayments on a Pool of Mortgages When a prepayment occurs, the entire remaining principal balance of the loan is immediately retired. Unfortunately, that means that there will be no more coupon income from that loan. This

Components of the Cash Flows of a Loan 683 20% PSA Speed Curve Multiples 18% CPR Spead (Annual Rate) 16% 14% 12% 10% 8% 6% 4% 50% PSA 100% PSA 150% PSA 200% PSA 300% PSA 2% 0% 1 12 24 36 60 84 108 132 156 180 204 228 Months From Issuance EXHIBIT A.8 Graph of PSA curve multiples expressed in monthly CPR speeds can be an especially nasty little problem if the mortgages that start to prepay rapidly are those that have the highest margin to the funding costs of your deal. What this means is the following: If you are borrowing funds at a rate of 5% and the average coupon on the mortgages is say, 8%, you have a 3% margin. If, however, the top coupon mortgages all prepay in a group because of a change in interest rates and the yield drops to 6.5%, then your margin has collapsed to 1.5% and the aggregate amount of cash flows will be greatly reduced. They may be so reduced that you may have trouble making up the cash flows missing from the defaulted mortgages in the pool and the deal might collapse. Quantifing Prepayment Effects We will now expand our sample mortgage from 24 periods to 180 periods. The effect of prepayments is to accelerate the receipt of principal payments and dilute or lessen the receipt of coupon payments. Exhibit A.9 shows the effects of various CPR prepayment speeds on the receipt of scheduled amortization, coupon income, and total cash flows. Here we can clearly see the reduction in coupon income from a pool of mortgages subject to higher and higher prepayment rates. Exhibit A.10 displays this material in a graphic format. Principal Defaults The discussion of prepayment activity is a natural lead in to the next subject on the list of mortgage cash flow analysis, defaults of principal. The effects of defaults on total cash flows are much more severe than that of prepayments. See Exhibit A.11. In Exhibit A.12 we can see the devastating effects that defaults can inflict on the cash flows of a mortgage portfolio. With a 25% CPR prepayment rate, we retained

684 APPENDIX A: MORTGAGE MATH EXHIBIT A.9 The decrease in total cash flows from an increase in prepayments Effects of Prepayment Activity on Cash Flow Components CPR Rate Coupon Scheduled Prepaid Total % of 0% Payments Principal Principal Cash Flows Cash Flows 0% $ 82,567.99 $ 100,000.00 $ - $ 182,567.99 100.00% 1% $ 78,195.70 $ 91,347.99 $ 8,652.01 $ 178,195.70 97.61% 2% $ 74,115.15 $ 83,517.52 $ 16,482.48 $ 174,115.15 95.37% 5% $ 63,419.35 $ 64,206.70 $ 35,793.30 $ 163,419.35 89.51% 10% $ 49,727.73 $ 42,410.67 $ 57,589.33 $ 149,727.73 82.01% 25% $ 26,925.77 $ 15,319.02 $ 84,680.98 $ 126,925.77 69.52% 69.52% of the total 0% prepayment rate. Here, with a 25% CPR default rate, we retained 22.59% of the original cash flows. The difference is even more striking when we put Exhibits A.12 and A.13 together. In Exhibit A.13, we see the difference most clearly between the effect of a prepayment and a default. If the default rate and prepayment rate are both 10%, we will lose an incremental 31.99% of our original cash flows. Almost 1/3 of the original cash flow-generating power of the collateral is gone. The total amount of foregone payments is $58.4 thousand dollars on an original amount of $182.6 thousand. The fortunate thing about defaults is that it is rare that, as we are assuming in the tables above, there are no recoveries whatsoever. If we have a recovery rate of even 25%, we can realize a definite improvement in these results. $200,000 Effect of Prepayments on Coupon and Scheduled Amortization Amounts Total Cash Flows $180,000 $160,000 $140,000 $120,000 $100,000 $80,000 $60,000 $40,000 $20,000 Prepaid Principal Coupon Income Scheduled Principal $- EXHIBIT A.10 0% 1% 2% 5% 10% 25% CPR Rate of Prepayment A graph showing the loss of total cash flows as prepayment rates increase

Components of the Cash Flows of a Loan 685 Total Cash Flows $200,000 $180,000 $160,000 $140,000 $120,000 $100,000 $80,000 $60,000 $40,000 $20,000 $- EXHIBIT A.11 Effect of Defaults on Coupon and Scheduled Amortization Amounts 0% 1% 2% 5% 10% 25% CPR Rate of Defaults A graph showing the loss of total cash flows as default rates increase Coupon Income Scheduled Principal EXHIBIT A.12 A table showing the loss of total cash flows as default rates increase Default CPR Rate Effects of Default Activity on Cash Flow Components Defaulted Principal Coupon Payments Scheduled Principal Total Cash Flows % of 0% Cash Flows 0% $ - $ 82,567.99 $ 100,000.00 $ 182,567.99 100.00% 1% $ 6,301.51 $ 78,130.24 $ 91,271.51 $ 169,401.75 92.79% 2% $ 12,224.59 $ 73,990.48 $ 83,377.03 $ 157,367.51 86.20% 5% $ 27,920.53 $ 63,148.85 $ 63,932.83 $ 127,081.68 69.61% 10% $ 48,224.02 $ 49,293.03 $ 42,039.94 $ 91,332.97 50.03% 25% $ 80,509.05 $ 26,287.94 $ 14,956.14 $ 41,244.08 22.59% EXHIBIT A.13 A table showing the comparative loss of total cash flows between identical default and prepayment rates Difference in Effects of Default vs Prepayment Activity on Cash Flow Components CPR Rate Prepayment Scenarios Default Scenarios Loss of Cash Flows Prepayment % Original Default % Original Difference % Original 0% $ 182,567.99 $ 182,567.99 $ - 100.00% 100.00% 0.00% 1% $ 178,195.70 $ 169,401.75 $ 8,793.95 97.61% 92.79% 4.82% 2% $ 174,115.15 $ 157,367.51 $ 16,747.64 95.37% 86.20% 9.17% 5% $ 163,419.35 $ 127,081.68 $ 36,337.67 89.51% 69.61% 19.90% 10% $ 149,727.73 $ 91,332.97 $ 58,394.76 82.01% 50.03% 31.99% 25% $ 126,925.77 $ 41,244.08 $ 85,681.69 69.52% 22.59% 46.93%

686 APPENDIX A: MORTGAGE MATH Recoveries of Principal The last elements of mortgage portfolio cash flows that we will address are recoveries. When a mortgage default occurs, the lender has the right to move to secure the pledged interest of the borrower. Most commonly this interest is in the form of real property, a home, a building, equipment, materials, and other chattel. Unless the Obligor voluntarily relinquishes use and holding of the asset, the lender must resort to some type of legal proceedings to recover principal. The first move is to establish the lender s right to gain legal control over the pledged property. In the case of residential and commercial mortgages, this usually means a foreclosure. Once the property is under the lender s control, it can attempt to liquidate the property. Upon liquidation the lender can apply the realized proceeds to the outstanding loan balance, plus any allowable expenses. Depending upon the laws of the state in which the legal action takes place, these expenses can vary. They generally, but may not necessarily, include: 1. Legal expenses. Foreclosure and legal notice service, title policy expenses, eviction expenses. 2. Capitalized interest. This is interest that is charged to the loan between the time of the default and the time of the liquidation of the assets. The interest is added to the loan balance outstanding at the time of the loan default. 3. Servicing expenses. Any expenses involved in monitoring and servicing the loan or the property, e.g., hiring a premises protection service for a commercial property. 4. Rehabilitation expenses. The expenses are typically repairs and renovations required to make the property conform with building codes, and to prepare the property for sale. 5. Selling expenses. These expenses are typically realty fees of various sorts, and sales commissions paid to real estate brokers. In aggregate these expenses can be significant. Recoveries are dependent upon the amount of net proceeds that can be realized relative to the outstanding balance of the mortgage at the time of the default, plus any or all of these expenses. If the lender is in a position in which there is sufficient Obligor equity in the property and these expenses are reasonable, recoveries can be substantial. At this point I would direct the reader to the illustrations in Exhibits 3.4 and 3.5 in Chapter 3, Securitizing a Loan Portfolio, for a graphic depiction of the relationship between loan-to-value ratios, severity of loss upon liquidation, and the recovery amount. A set of example cases in Exhibits A.14 and A.15 will illustrate these points. Exhibit A.14 is representative of normal economic times, while Exhibit A.15 is representative of a recessionary period. Recoveries of whatever level are an additional source of cash flows for a deal. The one other aspect that needs to be considered when dealing with the subject of recoveries of defaulted principal is the timing of their receipt. How quickly the property or chattel can be disposed of is of critical importance. The longer the delay between seizure of the property and the liquidation activity, the

Components of the Cash Flows of a Loan 687 EXHIBIT A.14 Various recovery scenarios based on differing loan-to-value ratios during normal economic times Calculation of Recovery Amounts & Percents # Item Calc Case 1 Case 2 Case 3 1 Original Appraisal Value 125,000 125,000 125,000 2 Outstanding Mortgage Balance 80,000 60,000 40,000 3 Current Owners Equity 1 2 45,000 65,000 85,000 4 Loan-to-Value Ratio 2/1 64.00% 48.00% 32.00% 5 Market Value Decline 20% 1*20% 25,000 25,000 25,000 6 Net Owners Equity 3 5 20,000 40,000 60,000 7 Liquidation Expenses 15,000 15,000 15,000 8 Net Proceeds Upon Liquidation 6 7 5,000 25,000 45,000 9 Recovery Amount Min(2,8) 5,000 25,000 40,000 10 Recovery Percentage 9/2 6.25% 41.67% 100.00% greater the possibility of damage to the property or other events that might trigger a value decline. In addition, the longer the period the more capitalized interest accrues. The model allows the user to input assumptions about the Recovery Lag Period. A reasonable recovery lag time assumption for residential and commercial real estate is a minimum of 12 months, and more likely 18 months. In recessionary environments, 24 to 30 months is not an unreasonable assumption. EXHIBIT A.15 Various recovery scenarios based on differing loan-to-value ratios during moderately stressed economic times Calculation of Recovery Amounts & Percents # Item Calc Case 4 Case 5 Case 6 1 2 3 Original Appraisal Value Outstanding Mortgage Balance Current Owners Equity 1 2 125,000 80,000 45,000 125,000 60,000 65,000 125,000 40,000 85,000 4 Loan-to-Value Ratio 2/1 64.00% 48.00% 32.00% 5 Market Value Decline 30% 1*30% 37,500 37,500 37,500 6 Net Owners Equity 3 5 7,500 27,000 47,500 7 8 Liquidation Expenses Net Proceeds Upon Liquidation 6 7 25,000 17,500 25,000 2,500 25,000 22,500 9 Recovery Amount Min(2,8) 0 2,500 22,500 10 Recovery Percentage 9/2 0.00% 4.17% 56.25%

688 APPENDIX A: MORTGAGE MATH NEXT STEPS This concludes our discussion of mortgage math. All of these exercises concerned the behavior of the loan collateral. Appendix B will address the measurement of the performance of the debt side of the deal. In Appendix B we will learn the specifics of calculating a series of measurements, such as average life, final maturity, duration, internal rate of return, and several other statistics. We will do this to be able to compare the performance of the debt under various scenarios. These terms, in addition to what we have learned here, will allow us to better describe the performance of the deal in more succinct and precise ways.