# «Daniel Domb An honors thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Science Undergraduate College Leonard ...»

the 10-year treasury had. However, the 1-year Treasury still moves in similar rhythm to the fixed rate. Furthermore, to back up my assertion I ran another simple regression between the 1-year Treasury as the independent variable and the fixed rate mortgage as the dependent variable. This regression is statistically significant with an f-statistic of almost 345 and an R-sq (adj.) of 67.8%. The 10-year Treasury has a much higher R-sq value, proving that it is a better explainer of the fixed mortgage rate. Looking at Exhibit 4 one can see the residuals from this regression are similar in pattern to the previous regression, but are much tighter together. This finding leads me to believe that there other variables that will help further explain the fixed rate mortgage. An explanation as to why the 10-year treasury does a better job explaining the fixed rate mortgage is the difference in the inflation premium built into the 10-year rate that is not as prevalent as the 1-year rate.

I have already explained 93.1% of the fixed rate mortgage through the 10-year Treasury; I only have 6.9% left to attempt to explain. To avoid multi-collinearity I subtracted each monthly fixed rate mortgage from the 10-year treasury rate to use as my dependent variable for future regressions. First, I plotted my new variable to test for any distinctive patterns in the variation between the two rates. Exhibit 5 displays the plotted variable. The data looks fairly volatile, with no real visible pattern. The data varies from a minimum of.83% in mid 1994 to 2.57% as the maximum during early 2001. The mean difference between the 10-year Treasury and the fixed rate mortgage is 1.78% and the median is 1.74%. According to my earlier explanation of what makes up a basic rate,

variable is the risk premium. However, I believe that there are several other variables that can be measured to explain the deviation between the fixed rate and the 10-year Treasury rate.

Before I run simple regressions with each of my independent variables, I need to check my data for seasonality. Even though the 10-year Treasury already describes the fixed rate mortgage fairly well, I thought that it was possible for there to be additional seasonality effects that, if removed, could increase the R-sq. First I created a dummy variable for each individual month of the year, and tested the dummy variables against the fixed rate. The results showed no months as statistically significant at a 95% confidence interval. Actually, none of the months were significant at a 90% confidence interval either. Next I tested for seasonality among quarters. Again, no conclusive evidence of seasonality effects was found. I repeated similar tests for each of my independent variables that I tested in the model, and I found that all seasonality effects had been removed from the variables before I collected them.

**Simple Regressions:**

First, I will run a simple regression between the deviation between the fixed rate and 10-year Treasury and the percentage of mortgages that are taken out with adjustable rate mortgages (ARM percentage). Before I ran the regression I examined a graphical plot of the percentage of adjustable rate mortgages over the 13 ½ year period. There are three large humps in the graph where the ARM percentage increases significantly

occurs in a time period where the fixed rate chart in Exhibit 1 shoots up for a brief period and drops off soon afterward. I will address this point later in my conclusions when I discuss borrowers’ choices and how they affect rates. Even over this relatively short time period the data ranges from a low of 8.00% to a high of 59.00%. Looking back from another 3 years before 1990, in 1987 the percentage of ARM’s hit 69.00%.

An important factor in considering the percentage of adjustable rate mortgages in a given month is the difference in the rate between the effective adjustable rate mortgage and the effective fixed rate mortgage. Exhibit 7 plots the difference in the monthly ARM and fixed rate against the percentage of ARM’s. Across the plot in Exhibit 7 there are similarities in patterns between the two variables. The similarities tell me that there is some relation to the spread between the adjustable rate and the fixed rate compared with the percentage of ARM’s, but the spread does not tell the entire story. Running a regression between the two variables there is an R-sq of 36.2, confirming that there is a relationship between them, but one is not a complete explanation of the other. Instead, there are other factors that contribute to the median borrower choosing an adjustable rate mortgage over a fixed rate mortgage.

Running a simple regression with the percentage of ARM’s as the independent variable and the difference between the fixed rate and the 10-year Treasury as the dependent variable will help clarify if there is any relation between these two variables.

After running the regression I found that there is a statistically significant relationship

regression is just above 40 and the R-sq is 20.2%. Exhibit 8 shows the residuals for this regression, and the residuals look fairly random, although there are large areas either above or below the 0 residual line.

The next variable I tested is the loan-to-value ratio. The loan-to-value ratio is the percentage of the purchase price or appraised value that is financed with debt. The median loan-to-value ratio ranges from 72.60% to 80.80%. Overall the range is fairly tight, however when one looks at a graph in Exhibit 9, for the first time, you don’t see the same pattern as the fixed rate mortgage. This observation makes me suspect that there are other factors that determine the loan-to-value ratio. To check this observation I ran a simple regression testing the statistical significance. The regression has an f-statistic of

27.58 and an R-sq of 14.5%. Although the R-sq is low, it is statistically significant. The residuals for this simple regression are in Exhibit 10. The residuals look fairly scattered, but similarly to the residuals for the percentage ARM regression, there is a distinctive pattern that is visible within the scatter plot. Both percentage ARM and loan-to-value have descriptive value for the fixed rate mortgage, but neither explains away this emerging pattern.

One would expect that the median price of homes sold over the past 13 ½ years has been obviously. As long as the United States is in an inflationary period, home prices are expected to increase year after year. However, the amount of the increase, or lack of increase, may have descriptive value in modeling a representation of the fixed mortgage

the data period. Instead, the first 6 years of the data set show the median home price fairly stagnant. Between 1990 and 1996 home prices fall some months and rise other months, with ultimately, prices ending in the same place in 1996 that they started in 1990.

In 1997 prices begin to move on the upswing through the rest of the data set eventually hitting a high of $253,900. The early nineties are explainable because we were in a very bad housing market, which is vastly in contrast to the late nineties when we were in one of the largest housing booms seen in decades. The simple regression with the difference between the fixed rate and the 10-year Treasury against the median price is highly statistically significant with an f-statistic of over 91 and an R-sq of 36%. The residuals are scattered, but again this mysterious pattern exists, meaning there is still an unknown variable out there that is determining the fixed rate mortgage, and I have yet to regress this variable. Overall, the median home price does play a role in determining the fixed rate mortgage, and will surely be apart of my final models.

One concern I have with this regression is that there is a chance that these results are coincidental. In the early nineties prices were much lower and interest rates were much higher, towards the late nineties and into the next millennium the opposite is true.

Although I agree that interest rates definitely have a strong impact on the amount of house individual borrowers can afford, I would feel more comfortable seeing several housing booms and recessions built into the model. I would hypothesize that there still will be a negative correlation between the median price and the interest rates, but I believe that the R-sq value may be slightly lower than this model suggests.

area for future research.

As I mentioned earlier, fixed rate mortgages are usually either 30 years or 15 years. The rates I have been using are effective rates, so it makes sense for me to run a regression that contains the average term to maturity against the effective fixed rate mortgage. The range of the data over the 13 ½ years is just over 4 years from 24.7 to

28.8 years. Exhibit 13 shows a line graph of the term to maturity data, which does not display any obvious patterns in the data spread. I expect the term to maturity to be statistically significant because the longer the term to maturity, the higher I expect the interest rate to be, because there is more of an inflation risk with longer terms to maturity.

Conversely, the shorter the term to maturity, the more similar the fixed rate should be to the adjustable rate mortgage. However when trying to compare the term to maturity graph with the graph in Exhibit 7 of the fixed rate minus the ARM rate I do not see any distinctive relationship. Modeling a simple regression in similar fashion as the other independent variables thus far, I determined that there is statistical significance in the median monthly term to maturity. The f-statistic for this simple regression is 30.19, and the R-sq is 15.7%. Exhibit 14 shows the residuals for this regression. Interestingly, the residuals around April of 1994 seem to dip further, with more points clustered in that area compared to some of the earlier residual plots. Looking back at the fixed rate from March to April of 1994, rates jumped an entire percentage point in that short time frame.

Later I may remove some data points that I believe are outliers. Either way, the term to maturity is statistically significant will be included in my final model.

variable, I began to believe that there was a variable out there that encompassed some sort of supply and demand feature of the money supply of commercial banks and lending institutions to issue residential mortgages. I concluded that a ratio described as the investment to loan ratio is a good representative of the supply of money to commercial banks. The investment to loan ratio is equaled to the total investments at all commercial banks divided by total loans and leases at commercial banks.6 The ratio should be a good indicator of when the real estate market is heating up. I expect this variable to hopefully describe the faint pattern in my residuals and make them more random. Unfortunately, this simple regression is barely statistically significant with an f-statistic of 8.95 and an R-sq of 5.2%. In addition, the residuals of the mildly correlated regression between the investment to loan ratio and the spread between the fixed rate and the 10 year Treasury, still displays signs of the pattern I am attempting to eliminate with each additional variable. Since the R-sq is so low in this regression, I may or may not use this variable in my final model. I will do one multi-linear regression with the investment to loan ratio built in and one regression without the variable, and choose the model with the higher adjusted R-sq.

I used the investment to loan ratio because I was looking for a variable that will give me a sense of the volume of the housing market, and will sense if the market is overheated, or very slow. Next I looked at a good volume indicator of the housing http://pages.stern.nyu.edu/~cliu/refin_MktStats_Fall2003.pdf

States Census website. The data is available in two forms, either as seasonally unadjusted monthly numbers or seasonally adjusted annualized data, but given on a monthly basis. I ran a regression between my dependent variable and the seasonally adjusted, annualized data, because my original data had no seasonality effects, and I did not want to enter this issue into my model. Before running the regression, I examined a line graph of the data as seen in Exhibit 17. Looking at the graph we see that at the start of the 1990’s new home sales were on the decline, followed quickly by a rise in new home sales. Although some months were negative over the data period, overall, new home sales increased year after year.

I figured that the new home sales in units would be a good indicator of the housing market because total volume in dollars would be dependent on the average price of a home, which I have isolated with a separate variable. As I suspected, the simple regression with the new home sales in terms of units sold annualized variable as the independent variable is statistically significant with an f-statistic of 37.58 and an R-sq of 18.8%. Exhibit 18 displays the residuals from this regression. The residuals are again random with a faint pattern in the background. The new home sales units appear to be a much better indicator of how heated the real estate market is compared with the regression of the investment to loan ratio. Two cautions I must remember as I use this variable is that the time period of my data is limited, and secondly, new home sales are driven partially by homebuilders desire to produce more homes.

new homes is the monthly housing starts. Housing starts are collect by the US government on a monthly basis, and are an important economic indicator of the real estate market. Housing starts appears to be a very similar variable to new housing sales.

However, they are actually very different. Comparing Exhibit 19, which is a graph of the housing starts over the 13 ½ year period, to Exhibit 17 you can see that the general direction of the line graphs are the same, but the month to month patterns are actually very different. Intuitively, housing starts measure builders’ perceptions of the market, or the supply side, and the number of new home sales shows consumers’ perceptions, or the demand side of the housing market. Together these two variables give a great indicator of where the housing market is and where it is going in the near future.