The relationship between the yield curve & mortgage current coupon.

• Author: Belbase, Eknath

In this article we discuss the Andrew Davidson & Co., Inc. approach to forecasting mortgage current coupon as a function of either Treasury or swap rates. Along the way, we compare several statistical approaches to the problem and discuss software implementation issues. In addition, we examine the relative performance of using a single Treasury rate, two Treasury rates or two swap rates to forecast mortgage current coupon rates.

1. INTRODUCTION

   A dynamic prepayment model is an integral component of an option-adjusted valuation and risk management framework. Such a model typically requires a forecast of future mortgage rates and possibly index rates such as prime and COFI. Term structure models are used to forecast rates of various maturities on the Libor/Swap yield curve or on the Treasury yield curve. It is, therefore, necessary to model mortgage rates as functions of these yield curve rates.

   In this article we discuss the Andrew Davidson & Co., Inc. approach to forecasting mortgage current coupon as a function of either Treasury or swap rates. Along the way, we compare several statistical approaches to the problem and discuss software implementation issues. In addition, we examine the relative performance of using a single Treasury rate, two Treasury rates or two swap rates to forecast mortgage current coupon rates.

2. DATA

   The current coupon is the semi-annual equivalent of the parity-price interpolated coupon, based upon the two bonds whose price bracket the parity price. The mortgage current coupon rates used were based on month-end closing prices. The types used are Fannie Mae 7, 15 and 30 year, Ginnie Mae I 15 and 30 year, and Freddie Mac 5, 7, 15 and 30 year. These rates can be viewed on Bloomberg, e.g. MTGE FNCL for the FNMA 30 rate.

   The Treasury rates used were month-end closing par bond-equivalent yields for the two and ten-year on-the-run Treasuries (GT2 and GT10 ) or the two and ten-year zero coupon Treasuries (STRP02Y and STRP10Y). The swap rates used were either the two and ten-year Libor rates (USSWAP2 and USSWAP10 ) or the two and ten-year zero coupon Libor rates, which were derived from the par swap rates using a bootstrapping method with linear interpolation.

   All rates used for this analysis covered November 1991 through August 2000, and there were no missing values.

3. PRELIMINARY ANALYSIS

   Exhibit 1 is a scatter plot of FNMA 30 year current coupon versus ten-year Treasuries and versus ten-year swap rates on the same scale. This plot suggests that ten-year swap rates may be a better predictor of current coupon than ten-year Treasury rates because the swap versus current coupon points show much less variation about a line through their "center."

   Because we also need to predict rates of shorter maturities, such as five and seven-year balloon rates, it is useful to examine the relationship between ten-year rates and a balloon rate. Exhibit 2 displays scatter plots of a five-year balloon rate versus the ten-year Treasury and swap rates. In contrast to Exhibit 1, neither Treasury nor swap rates gives a tight fit.

   Exhibit 3 shows FHG five-year balloons plotted against two-year Libor and Treasury. It appears to be a much better fit than that in Exhibit 2.

   Next, we give an example of a model that simply uses the ten-year Treasury plus a spread to predict the FNMA current coupon. The spread between current coupon and the ten-year Treasury rate in the first month of analysis, November 1991, was used;

   FNMA30_CC = 10YR_TREAS + .91

   Now, using rates from August 2000, we have:

   10YR_TREAS + .91 = 5.72 + .91 = 6.63

   The actual value of the FNMA 30 current coupon for August 2000 was 7.7; therefore the prediction using the ten-year Treasury and spread was 107 basis points off.

   Next, we attempt to predict the FHG 5 using the ten-year Treasury plus spread, utilizing the same methods as the previous example:

   10YR_TREAS + .50 = 5.72 + .50 = 6.22

   The actual value of the FHG 5 current coupon for August 2000 was 7.3. Again, the prediction using only the ten-year Treasury ended up 99 basis points off.

   Before deciding on a final set of yield curve maturities to use, it is helpful to look at correlations between rates of different maturities and some mortgage current coupon rates. Ideally, the rates chosen to model mortgage current coupon would come from liquid points on the yield curve. In addition, they should have high correlations with mortgage current coupon rates, but low correlations with each other. A high correlation with mortgage current coupon indicates that there is a strong linear relationship between the two rates; a low correlation with each other implies that the additional rate adds to the strength of the relationship. The final set chosen should maximize fit with as simple a set of rates as possible.

   The correlation matrix of two and ten-year Libor rates and one, two, five, and ten-year Treasury rates versus the thirty-year FNMA and five-year Freddie balloon mortgage rates is shown Exhibit 4.

   The correlation matrix tells us several things. First, using the two-year rate is better than using the one-year rate because the two-year rate has both higher correlations with mortgage current coupon rates and low correlation with the ten-year rate. Second, while the five-year Treasury has a stronger relationship with the thirty-year mortgage rate than does the ten-year Treasury, it has a weaker relationship with the five-year balloon rate than does the two-year Treasury. Because the five-year Treasury has a high correlation with both of these Treasury rates, it is...