Abstract:
A method for modeling a current expected prepayment spread for a mortgage-backed security (MBS) as a function of a current refinance activity is disclosed. According to various embodiments, the method comprises the steps of computing a historical MBS prepayment spread value and determining a historical refinance activity value for each of a plurality of historical time value. The historical MBS prepayment spread value and the corresponding historical refinance activity value define a data pair. The method further comprises the step of processing the data pairs corresponding to the plurality of historical time values to determine terms of a multiple-term mathematical formula that is used to calculate the current expected prepayment spread, wherein the terms of the formula include as a variable the current refinance activity.

Description:
BACKGROUND OF THE INVENTION 
   The present invention relates generally to the valuation of mortgage-backed securities. 
   Mortgage-backed securities (MBSs) are popular among investors seeking a fixed-income investment having a yield generally greater than that of U.S. Treasuries or corporate bonds. A MBS is an ownership position in a pool of mortgage loans. Each mortgage loan underlying a MBS may originate in the primary mortgage market with the issuance of a mortgage loan by a loan originator to a mortgagor, typically a homebuyer. The originator may then resell the mortgage loan on the secondary mortgage market for securitization by a government-sponsored enterprise (GSE) such as, for example, Ginnie Mae, Fannie Mae, or Freddie Mac. Securitization refers to the process of MBS creation whereby mortgage loans purchased by, for example, a GSE are grouped into pools based upon loan characteristics, such as, for example, interest rate and time to maturity. Each pool is then divided into ownership shares (i.e., the MBSs) for sale to MBS investors, with each investor receiving a pro rata share of the principal and interest cash flows as the underlying mortgage loans are repaid. MBSs created by a GSE are generally referred to as agency MBSs. 
   One feature differentiating MBSs from other types of fixed income securities is the risk of prepayment associated with the underlying mortgage loans. In particular, because a mortgagor is typically allowed to prepay their mortgage loan in whole or in part without penalty, a MBS investor must assume the risk that MBS cash flows will be received sooner than anticipated. Prepayment risk depends upon a variety of factors, the most significant being the market interest rate. Generally, the value of a fixed-income investment increases when the market interest rate decreases. The same is true of a MBS, but only to a limited extent. In particular, if the market interest rate decreases below the interest rates of mortgage loans underlying a MBS, some mortgagors may find it financially advantageous to refinance their mortgage loans at the lower market rate. The prepayment of mortgage loans resulting from such refinancing activity forces a MBS investor to re-invest the prepaid amounts at the lower market interest rate, thus decreasing the return on investment. To compensate for this prepayment risk, MBSs typically provide higher returns compared to other fixed-income securities. 
   MBS valuation methodologies typically model the prepayment privilege as an embedded option written by the MBS investor. Thus, the price of an agency MBS may be stated as the price of an agency bond price less the value of the prepayment option. For MBS investors utilizing these valuation methodologies, the ability to accurately quantify and forecast changes in the prepayment option value is vital for realizing an optimal investment strategy. One known technique for valuing the prepayment option is to decompose a MBS yield curve into its constituent yield components so that the yield attributable to prepayment risk may be analyzed separately from that attributable to non-prepayment risk factors (e.g., the credit quality of the MBS issuer and market liquidity). To perform this analysis, a valuation benchmark curve positively correlated to the MBS yield curve, such as, for example, the swap yield curve or the spread between the swap yield curve and the U.S. Treasury yield curve, may be used as a surrogate for the MBS yield attributable to non-prepayment risk factors. The basis point difference between the MBS yield curve and the benchmark curve (the “prepayment spread”) provides an estimated value of the prepayment option. A widening of the prepayment spread indicates an increasing option value and decreasing MBS price, whereas a tightening of the prepayment spread indicates a decreasing option value and increasing MBS price. 
   For MBS valuation methodologies currently in use, understanding and predicting changes in the prepayment spread is often based on a qualitative analysis of past prepayment spread changes. Such analysis, however, does not facilitate an understanding of prepayment spread dynamics in terms of one or more current macroeconomic factors and may thus result in a less-than-optimal MBS investment strategy. 
   SUMMARY 
   In one general respect, embodiments of the present invention are directed to a method for modeling a current expected prepayment spread for a MBS as a function of a current refinance activity. According to various embodiments, the method comprises the steps of computing a historical MBS prepayment spread value and determining a historical refinance activity value for each of a plurality of historical time values. The historical MBS prepayment spread value and the corresponding historical refinance activity value define a data pair. The method further comprises the step of processing the data pairs to determine terms of a multiple-term mathematical formula that is used to calculate the current expected prepayment spread. The terms of the formula may include as a variable the current refinance activity. 
   In another general respect, embodiments of the present invention are directed to a method for determining a current value of a MBS. According to various embodiments, the method comprises the steps of computing a first current prepayment spread value for the MBS, computing a second current prepayment spread value for the MBS, and determining the current value of the MBS based upon a comparison of the first current prepayment spread to the second current prepayment spread. 
   In yet another general respect, embodiments of the present invention are directed to a computer-readable medium containing computer-executable instructions for computing a historical MBS prepayment spread value and determining a historical refinance activity value for each of a plurality of historical time values. The historical MBS prepayment spread value and the corresponding historical refinance activity value define a data pair. The computer-readable medium may further contain computer-executable instructions for processing the data pairs to determine terms of a multiple-term mathematical formula that is used to calculate the current expected prepayment spread. The terms of the formula may include as a variable the current refinance activity. 
   In yet another general respect, embodiments of the present invention are directed to a system for modeling a current expected prepayment spread for a MBS as a function of a current refinance activity. According to various embodiments, the system may include a derivation module for computing a historical MBS prepayment spread value and determining a historical refinance activity value for each of a plurality of historical time values. The historical MBS prepayment spread value and the corresponding historical refinance activity value define a data pair. An additional function performed by the derivation module may include processing the data pairs to determine terms of a multiple-term mathematical formula that is used to calculate the current expected prepayment spread. The terms of the formula include as a variable the current refinance activity. The system may further include a database in communication with the derivation module for storing the mathematical formula, and an application module for retrieving the mathematical formula from the database and for computing the current expected prepayment spread based upon the mathematical formula and a current refinance activity value. 

   
     DESCRIPTION OF THE FIGURES 
     Various embodiments of the present invention will be described by way of example in conjunction with the following figures, wherein: 
       FIG. 1A  shows a data set containing data for computing prepayment spread values; 
       FIG. 1B  shows a MBS-to-treasury spread curve and a MBS-to-swap spread curve obtained by plotting MBS-to-treasury spread values and swap-to-treasury spread values of  FIG. 1A , respectively, as a function of date; 
       FIG. 2  shows a MBS-to-swap spread curve and an refinance activity curve obtained by plotting MBS-to-swap spread values and refinance activity values of  FIG. 1A , respectively, as a function of date; 
       FIG. 3  shows an application of a linear regression analysis for developing a one-factor model that may be used to generate an expected MBS-to-swap spread value based on a corresponding refinance activity value; 
       FIG. 4A  shows a data set containing an expected MBS-to-swap spread value generated using the one-factor model defined in  FIG. 3  for each refinance activity value; 
       FIG. 4B  shows a comparison of the MBS-to-swap spread curve of  FIG. 2  to a MBS-to-swap spread curve defined by the one-factor model of  FIG. 3 ; and 
       FIG. 5  shows a computer system for deriving and applying a one-factor model. 
   

   DETAILED DESCRIPTION 
   The present invention, according to various embodiments, is directed to methods and related systems for computing an expected prepayment spread value for an MBS using a one-factor model having as input a current refinance activity value. Such methods and systems are useful for, among other things, identifying pricing discrepancies between fair MBS price and market MBS price, thus uncovering arbitrage investment opportunities. 
     FIG. 1A  shows a data set  5  containing historical data for computing prepayment spread values associated with a MBS over a selected period of time. According to various embodiments, the MBS may be an agency MBS such as, for example, a 30-year Ginnie Mae (GNMA) MBS. Data set  5  entries are ordered by date  10  and include a MBS current yield value  15 , a MBS weighted-average-life (WAL) value  20 , a five-year swap value  25 , a ten-year swap value  30 , a WAL-adjusted swap value  35 , a five-year treasury yield value  40 , a ten-year treasury yield value  45 , a WAL-adjusted treasury yield value  50 , a WAL-adjusted swap to WAL-adjusted treasury (“swap-to-treasury”) spread value  55 , a MBS to WAL-adjusted treasury (“MBS-to-treasury”) spread value  60 , a MBS to WAL-adjusted swap (“MBS-to-swap”) spread value  65  (i.e., the prepayment spread), and a refinancing activity value  70 . For purposes of clarity, only a representative portion of the entries of data set  5  is shown in  FIG. 1A . It can be appreciated that other data sets similar to data set  5  can be constructed for other types of MBSs. 
   The MBS current yield value  15  is computed as the current coupon rate of the MBS divided by the market price of the MBS on the corresponding date  10  and may be obtained from any of a variety of financial reporting services and/or publications such as, for example, the below, the MBS WAL value  20  may be used for defining a point on a valuation benchmark curve having a maturity equal to the MBS WAL value  20 . 
   According to various embodiments, the spread between the swap yield curve and the U.S. Treasury yield curve may be used as a valuation benchmark for representing the amount of MBS yield attributable to non-prepayment risk factors. The swap yield curve reflects the rates received by fixed-rate payors in return for paying LIBOR for a pre-determined time period, typically three or six months. Data for swap yield curves and U.S Treasury yield curves of various maturities is available from a variety of sources such as, for example, the Federal Reserve Board. Because the maturities for the reported swap and U.S. Treasury yield curves are typically different than the MBS WAL value  20 , swap and U.S. Treasury yield values having maturities equal to the MBS WAL value  20  may be computed by linear interpolation. For example, as seen in  FIG. 1A , the MBS WAL value  20  on Oct. 2, 1998 was 6.34 years. The reported five-year and ten-year swap values  25 ,  30  on that date were 4.92 and 5.18, respectively. Accordingly, the WAL-adjusted swap value  35  corresponding to the MBS WAL value  20  of 6.34 years may be computed as: 
             WAL   ⁢     -     ⁢   adjustment   ⁢           ⁢   Swap     =       4.92   ×       (     5.18   -   4.92     )     5     ×     (     6.34   -   5     )       =   4.99           
Similarly, the WAL-adjusted Treasury yield value  50  corresponding to the MBS WAL value  20  of 6.34 years may be computed using the reported five-year and ten-year Treasury yield values  40 ,  45  as follows:
 
   
     
       
         
           
             WAL 
             ⁢ 
             
               - 
             
             ⁢ 
             adjusted 
             ⁢ 
             
                 
             
             ⁢ 
             Treasury 
             ⁢ 
             
                 
             
             ⁢ 
             Yield 
           
           = 
           
             
               4.24 
               × 
               
                 
                   ( 
                   
                     4.47 
                     - 
                     4.24 
                   
                   ) 
                 
                 5 
               
               × 
               
                 ( 
                 
                   6.34 
                   - 
                   5 
                 
                 ) 
               
             
             = 
             4.30 
           
         
       
     
   
   The swap-to-treasury spread value  55 , i.e., the basis-point differential between the WAL-adjusted swap value  35  and the WAL-adjusted Treasury yield value  50 , may then be computed as follows:
 
Swap-to-Treasury Spread=(4.99−4.30)×100=69
 
   The MBS-to-treasury spread value  60 , i.e., the basis-point differential between the MBS yield value  15  and the WAL-adjusted Treasury yield value  50 , may then be computed as:
 
MBS-to-Treasury Spread=(5.95−4.30)×100=165
 
   The MBS-to-swap spread value  65 , i.e., the basis-point differential between the MBS yield value  15  and the WAL-adjusted swap value  35 , may then be computed as:
 
MBS-to-Swap Spread=(5.95−4.99)×100=96
 
   The refinancing activity value  70  corresponds to the most recent value of the refinance index for corresponding date  10 . The refinance index value is updated weekly by the Mortgage Banker Association (MBA) based on mortgage banker survey data and represents the level of refinancing activity by residential and commercial mortgagors. Increases and decreases in the refinance index are thus indicative of increases and decreases in refinancing activity, respectively. As discussed below in connection with  FIG. 2 , a high degree of correlation may be shown to exist between the refinancing activity value  70  and the MBS-to-swap spread value  65 . 
     FIG. 1B  shows a MBS-to-treasury spread curve  75  and a MBS-to-swap spread curve  80  obtained by plotting each MBS-to-treasury spread value  60  and swap-to-treasury spread value  55  of data set  5 , respectively, as a function of their corresponding date  10 . Changes in the value of the prepayment option over time is indicated by changes in the spread between the curves  75 ,  80  (i.e., changes in the MBS-to-swap spread value  65 ). First and second regions  85 ,  90  highlighting portions of curves  75 ,  80  are instructive in this regard. In particular, as seen within the first region  85 , the MBS-to-treasury spread curve  75  correlates in an overall manner with the MBS-to-swap spread curve  80 , and the MBS-to-swap spread values  65  within the first region  85  are generally smaller than those outside of its bounds. Accordingly, the first region  85  marks a tightening of the spread between the MBS-to-treasury spread curve  75  and the MBS-to-swap spread curve  80  and, thus, higher MBS prices. Within the second region  90 , by contrast, the MBS-to-treasury spread curve  75  is characterized by a series of sharp divergences relative to the MBS-to-swap spread curve  80 , giving rise to MBS-to-swap spread values  65  that are large compared to those of the first region  85 . Accordingly, the second region  90  marks a widening of the spread between the MBS-to-treasury spread curve  75  and the MBS-to-swap spread curve  80  and, thus, lower MBS prices. 
     FIG. 2  shows a MBS-to-swap spread curve  95  and a refinance activity curve  100  obtained by plotting each MBS-to-swap spread value  65  and refinance activity value  70  of data set  5 , respectively, as a function of their corresponding date  10 . Scales for each of the curves  95 ,  100  are shown on the left and right vertical axis of  FIG. 2 , respectively. As seen in  FIG. 2 , the general features of the MBS-to-swap spread curve  95  correspond closely to those of the refinance activity curve  100 , thus demonstrating the usefulness of the refinance index for quantifying MBS-to-swap spread values  65  and for forecasting changes therein. 
     FIG. 3  shows an application of a linear regression analysis for developing a one-factor model that may be used for generating an expected MBS-to-swap spread value based on a corresponding refinance activity value  70 , according to various embodiments. The use of a linear regression of the form y=α+βx+ε, wherein α and β are determined using least-squares fitting techniques and ε represents the error, is well known in the art and is thus not discussed at length herein. In  FIG. 3 , a scatter-plot  105  constructed by plotting each MBS-to-swap spread value  65  of data set  5  versus its corresponding refinance activity value  70  is shown. The application of a linear regression analysis to the data points of the scatter plot  105  defines a one-factor linear model for generating an expected MBS-to-swap spread value based upon a corresponding refinance activity value  70  as follows:
 MBS-to-Swap Spread (Regression)=0.0121×Refinance Activity+62.386 
As shown in  FIG. 3 , value of the coefficient of determination, R 2 , is 0.8613, indicating that a large degree of the variation in the MBS-to-swap spread values  65  is accounted for by the refinance activity values  70 . A plot of the regression-based MBS-to-swap spread output generated by the one-factor model as a function of refinance activity is shown in  FIG. 3  as trendline  110 .
 
     FIG. 4A  shows a data set  115  containing an expected MBS-to-swap spread value  120  generated using the one-factor model defined in  FIG. 3  for each refinance activity value  70 .  FIG. 4B  shows a comparison of the MBS-to-swap spread curve  95  of  FIG. 2  to a regression-based MBS-to-swap spread curve  125  constructed by plotting each expected MBS-to-swap spread value  120  of data set  115  versus its corresponding date  10 . As seen in  FIG. 4B , the expected MBS-to-swap spread values  120  computed using the one-factor model correspond closely to the MBS-to-swap spread values  65  of data table  5 . The one-factor model may thus be used to accurately model prepayment spread dynamics in terms of a macroeconomic factor, i.e., the level of refinance activity as measured by the refinance index. 
   According to various embodiments, the above-described one-factor model may be used to determine the “fair” prepayment spread value for a MBS as a function of refinance activity. Disparities between fair MBS price and market MBS price may be identified by comparing model-generated prepayment spread values to prepayment spreads values observed in the market. Such pricing disparities may arise, for example, from market frictions or from an imbalance in MBS supply and demand. Knowledge of pricing disparities enables a MBS investor to profit using an arbitrage-based investment strategy. For example, when the prepayment spread value generated by the one-factor model is larger than that observed in the market (i.e., the fair MBS price is less than the market price), a MBS investor may take a short position in MBSs in anticipation that the market price will decrease to reflect the price dictated by the model-generated prepayment spread. Conversely, where the prepayment spread value generated by the one-factor model is smaller than that observed in the market (i.e., the fair MBS price is more than the market price), a MBS investor may take a long position in MBSs in anticipation that the market price will eventually increase to reflect the price dictated by the smaller model-generated prepayment spread. 
     FIG. 5  is a diagram of a computer system  130  for deriving and applying the above-described one-factor model, according to various embodiments. The computer system  130  may include a computing device  135 , which may be implemented as one or more networked computers, such as personal computers, servers, etc. The computer system  130  may include a derivation module  140  and an application module  145 . The modules  140 ,  145  may be implemented as software code to be executed by a processor (not shown) of the computing device  135  using any suitable computer language such as, for example, Java, C, C++, Virtual Basic or Perl using, for example, conventional or object-oriented techniques. The software code for each module  140 ,  145  may be stored as a series of instructions or commands on a computer-readable medium, such as a random access memory (RAM), a read-only memory (ROM), a magnetic medium such as a hard drive or a floppy disk, or an optical medium, such as a CD-ROM or DVD-ROM. 
   The derivation module  140  may be configured to derive a one-factor model that may subsequently be used for computing an expected MBS-to-swap spread value  120  based on an input refinance activity value  70 . As discussed above in connection with  FIG. 3 , the process of deriving a one-factor model for a particular MBS may include application of a linear regression algorithm to a plurality of historical MBS-to-swap spread values  65  and corresponding historical refinance activity values  70 . Historical data necessary for the derivation of a one-factor model may be stored in a database  150  that is in communication with the derivation module  140 . The historical data may include, for example, a plurality of dates  10  and corresponding MBS current yield values  15 , MBS WAL values  20 , five and ten-year swap values  25 ,  30 , five and ten-year treasury yield values  40 ,  45 , and refinance activity values  70 . Although not necessary, historical data may also include WAL-adjusted swap values  35 , WAL-adjusted treasury yield values  50 , swap-to-treasury spread values  55 , MBS-to-treasury spread values  60 , and MBS-to-swap spread values  65 . Alternatively, these values  35 ,  50 ,  55 ,  60 ,  65 , may be computed as needed by derivation module  140  based on stored values  15 ,  20 ,  25 ,  30 ,  40 ,  45  in accordance with the calculations discussed above in connection with  FIG. 1A . Because the database  150  may be configured to store historical data for any number of MBSs, the derivation module  140  may store a derived model for each MBS in the database  150  for future use. 
   According to various embodiments, a time range for specifying the historical data to be used for deriving a one-factor model for a particular MBS may be manually input into the derivation module  140  by a user of the computer system  130 . For example, one user may wish to derive a one-factor model for a particular MBS using historical data from the most recent three months, whereas another user may wish to derive a one-factor model for the same MBS using historical data from the most recent year. Furthermore, historical data stored in the database  150  may be augmented with new historical data as it becomes available. Thus, according various embodiments, a user of the computer system  130  may be provided with an option to periodically update (i.e., “re-derive”) one or more stored one-factor models to reflect the most recent historical data. According to other embodiments, the one-factor model derivation module  140  may be configured to automatically update stored one-factor models when new historical data is added to the database  150 . 
   The application module  145  may be configured to compute an expected MBS-to-swap spread value  120  based on an input refinance activity value  70  using a one-factor model previously derived and stored in the database  150 . According to various embodiments, a user of the computer system  130  may provide as input into the application module  145  a name of a particular MBS for which an expected swap spread value  120  is needed. Based upon the MBS name received as input from the user, the application module  145  may retrieve the appropriate one-factor model from the database  150  and compute an expected swap spread value  120  corresponding to the input refinance activity value  70 . According to various embodiments, the refinance activity value  70  input may be automatically input from an external data source. According to other embodiments, a user may manually input a refinance activity value  70  into the application module  145 . 
   While several embodiments of the invention have been described, it should be apparent, however, that various modifications, alterations and adaptations to those embodiments may occur to persons skilled in the art with the attainment of some or all of the advantages of the present invention. In embodiments of the computer system  130  of  FIG. 5 , for example, the derivation module  140  may be configured to communicate one-factor models directly to the application module  145 . According to such embodiments, the derivation module  140  may derive the needed one-factor model “on-the-fly,” thus eliminating the need for storing one-factor models in the database  150 .