Data Analysis of choice between Variable and Fixed Interest Rates
The aim of this paper is to explore the determinants of mortgage borrowers’ choice between fixed rate and variable statistics problems with answers rate mortgage loans. The data consists of demographics, income and a variety of savings variables for a cross section statistics homework help sample of Danish borrowers. The paper is organized as follows: the first section discusses the variables selected for statistics project help inclusion in the model, the second section provides descriptive statistics and model estimates, and the concluding section briefly discusses results.
Model of choice of fixed or variable mortgage rate
Because the dependent variable is a binary variable coded as 1 if the individual chooses a variable rate mortgage and 0 otherwise, statistics homework solution the appropriate model is a logistic regression, which will restrict the predicted values from the regression to the [0,1] interval appropriate for probabilities. Perhaps the most important determinant of the choice of fixed versus variable interest rate for a prospective mortgage borrower is the level current fixed mortgage rate relative to the long term average of mortgage rates. Given the relatively long term of mortgages, a borrower would expect the mortgage interest rate to move toward its longterm average over the term of the mortgage. However, the fixed mortgage rate at the date of the origination of the mortgage is not included in the sample.
Income and wealth are a second set of variables that will be important to a borrower. The selection of a variable mortgage rate offers a return if the rate decreases over time at the cost of exposure to risk in the form of higher statistics problems with answers mortgage payments if the mortgage rate increases over time. Home buyers with higher incomes and with more savings can be expected to be more willing to bear this risk. The data includes a measure statistics homework help of household income which was converted to 100,000s of Danish krone to scale estimated coefficients appropriately. The data also statistics homework solution include measures of savings held as cash, stocks and bonds. Cash savings tend to be small and are not associated with the interest rate type choice. Bond and stock savings statistics project help were included in the statistics questions with answers model below after conversion to millions of Danish krone. In general, increased income or wealth should be associated with a greater likelihood of choosing a variable rate mortgage because the borrower’s ability to “selfinsure” against default on the mortgage in the event of an increase in the interest rate increases as income and more assets. However, bond rates may be correlated with mortgage interest rates. If bond rates are correlated with mortgage interest rates, then an increase in mortgage rates tends to occur with an increase in bond rates, and an increase in bond rates lowers bond prices, which implies that the borrower’s bond portfolio has lower value when the mortgage rate increases. Instead, large bond holdings statistics questions with answers may imply that a fixed rate mortgage statistics homework help provides some “hedge” against the risk of a generally higher interest rates that lower the value of the bond portfolio. For this reason, we should expect that bond wealth may be negatively statistics homework solution associated with the adoption of a variable rate mortgage.
Finally, the data include a number of individual characteristics. It is anticipated that the probability of choosing a variable rate decreases with age, since older borrowers have less time to amortize both benefits and costs of the variable rates and are lower for female borrowers, who may be more risk averse than males. Additionally, it is anticipated that more educated borrowers will in general be more sophisticated consumers who will be more willing to choose variable rate mortgages as part of some broad portfolio strategy. The data include a categorical variable for highest level of education completed with categories high school or less, some college, college and university and this variable is included make my statistics homework in the model below as a set of indicator variables, with high school or less the omitted category. Finally, urban consumers may both be more sophisticated financial managers and have better access to financial advice. The data contains a categorical variable on the location of the mortgage holder that includes Copenhagen, city of population 50,000500,000, town and village, and this variable is included in the model as a set of indicator variables, with Copenhagen the omitted category.
Descriptive statistics and results
The data contain 1,327 observations of different mortgage holders who have nonmissing values for all of the variables in the model. The sample was nearly evenly split between people who chose fixed rate mortgages (663, 49.96%) and people who chose variable rate mortgages (664, 50.04%). Of the 1,327 individuals, 375 (28.3%) are female and 952 (71.7%) are male. In terms of education, the sample includes 351 persons (26.5%) with high school or less, 386 persons (29.1%) with some college, 304 persons (22.9%) who completed college and 286 persons (21.6%) who completed university. In terms of locations of the mortgages, 288 persons (21.7%) lived in Copenhagen, 326 persons (24.6%) lived in large cities, 328 (24.7%) lived in towns and 385 (29%) lived in villages. Thus these demographic variables show a good mix of different types of people with adequate sample sizes.
Descriptive statistics for continuous variables are provided for the full sample of 1,327 persons in Table 1. The average age usa of a borrower was 52 years, with a range from 25 years to 75 years. Average income across borrowers statistics homework help was 639.2 thousand Danish krone, with a minimum of zero and a maximum of 4.8 million Danish krone. Average house value was 2.253 million Danish krone, average bond savings were 45,000 Danish krone, and average stock savings were 92,000 Danish krone. Note that more than 50% of the people in the sample have neither any bond savings nor any share savings.
Table 1 Descriptive statistics for continuous variables

Mean 
Std. dev. 
Min 
Median 
Max 
Age 
52.042 
16.190 
25 
54 
75 
Income 100,000s 
6.392 
4.794 
0.0 
6.000 
48.000 
House value millions 
2.253 
2.740 
1 
1.8 
50 
Bond savings millions 
0.045 
0.275 
0 
0 
5 
Stock savings millions 
0.092 
0.421 
0 
0 
5 
As is often true for distributions of wealth and income, the distribution of the income and savings variables in the sample is highly positively skewed. Figure 1 and Figure 2 provide box plots of assets, including house value, share savings and bond savings, and income, respectively, because these variables are scaled differently.
Figure 1 Box plots for asset variables
For house values and income, an outlier is do my statistics assignment defined by the box plot as any value greater than the 75^{th} percentile of the distribution plus 1.5 times the inter quartile range (IQR, i.e., the 75^{th} percentile minus the 25^{th} percentile. For stock savings and bond savings, 70% and 85% of the sample who reported do not have these australia types of savings, respectively. For the estimations below, a conservative approach of using the box plot definition of outliers conditional on positive share and bond savings was adopted. There were a total of 157 outliers in the sample.
The strategy adopted below is statistics homework help to estimate the logit model for type of rate chosen twice, once with outliers included and once Canada without outliers, in order to determine the effects of outliers on estimates, if they have effects.
Table 1 presents estimates for the logit regression model that includes outliers. For the test of the null hypothesis that the model does no better than the sample proportion at predicting the choice of rate type, the Wald chisquare statistics homework help variable has 12 degrees of freedom, a value of 313.75 and an associated p value of less than 0.0001, which implies we can overwhelmingly reject the null and conclude that the model outperforms the sample mean as a predictor. Further, using a cutoff value statistics assignment help of predicted probability greater than 0.5 for predicting a person chooses variable rate mortgage, the model predicts 71.7% of the sample correctly.
Table 2 Logit regression model for rate type

Coefficient 
Robust Std. Err. 
z 
P>z 
95% LCL 
95% UCL 
Female 
1.362 
0.175 
7.790 
0.000 
1.705 
1.020 
Age 
0.039 
0.004 
9.190 
0.000 
0.047 
0.030 
Education 






Some college 
0.529 
0.176 
3.010 
0.003 
0.185 
0.874 
College 
0.781 
0.179 
4.360 
0.000 
0.430 
1.131 
University 
0.953 
0.190 
5.020 
0.000 
0.581 
1.325 
Income, 100,000s 
0.033 
0.015 
2.180 
0.029 
0.003 
0.063 
Urban 






City 
0.795 
0.201 
3.940 
0.000 
1.189 
0.400 
Town 
0.723 
0.205 
3.530 
0.000 
1.125 
0.321 
Village 
1.191 
0.203 
5.870 
0.000 
1.589 
0.793 
House value, millions 
0.127 
0.081 
1.560 
0.118 
0.032 
0.286 
Bonds, millions 
0.855 
0.252 
3.390 
0.001 
1.349 
0.361 
Shares, millions 
0.399 
0.239 
1.670 
0.095 
0.069 
0.867 
Constant 
2.067 
0.340 
6.070 
0.000 
1.400 
2.734 
The signs statistics assignment help of individual coefficient estimates on income and asset variables are consistent with our expectations, as income and statistics homework help share savings are statistically significantly positively associated with choosing a variable rate and bond savings are statistically significantly negatively statistics homework help associated with choosing a variable rate. The education variables indicate that persons with some college, complete college or complete university have statistically significantly larger probabilities of choosing a variable rate mortgage than people who completed high pay for statistics homework school or less, although the 95% confidence intervals for the estimates imply that the probability of choosing variable rates may uae not differ among the three more educated groups. Also, females are statistics homework solution significantly less likely to choose a variable rate mortgage than males, and the probability of choosing a variable rate mortgage declines with age. Finally, people living in cities, towns or villages are significantly less likely to choose a statistics assignment help variable rate mortgage than people living in Copenhagen the omitted category.
Figure 2 Box plots for income
The logit model predicts the probability of choosing a variable rate mortgage by substituting the value of the regression statistics homework help function into the cumulative distribution function for the logistic distribution. Because the cumulative distribution for the logistic distribution is not a linear function, unlike the linear regression model, the effect of a one unit change in statistics assignment help an independent variable, or the effect of changing the value of a dummy variable, will depends on the values of the other covariates. One method for assessing statistics assignment solution the effect of variables on the probability of choosing a variable rate mortgage is to imagine that every person in the sample receives a one unit increase in the value of an independent variable and then compute the average change in probability of choosing a variable rate mortgage. The average change in the probability of selecting a variable rate mortgage is referred to as the statistics assignment help average marginal effect of a change in a dependent variable, and is presented in Table 3 below for the estimates in Table 2.
Table 3 Average marginal effects on probability of choosing variable rate

dy/dx 
Delta Method
Std.Err. 
z 
P>z 
95% LCL 
95% UCL 
Female 
0.262 
0.032 
8.120 
0.000 
0.326 
0.199 
Age 
0.007 
0.001 
10.410 
0.000 
0.008 
0.006 
Education 






Some college 
0.099 
0.033 
3.010 
0.003 
0.034 
0.163 
College 
0.146 
0.033 
4.410 
0.000 
0.081 
0.211 
University 
0.178 
0.035 
5.030 
0.000 
0.109 
0.247 
Income, 100,000s DKK 
0.006 
0.003 
2.190 
0.028 
0.001 
0.012 
Urban 






City 
0.152 
0.038 
4.020 
0.000 
0.226 
0.078 
Town 
0.138 
0.039 
3.580 
0.000 
0.213 
0.062 
Village 
0.229 
0.040 
5.690 
0.000 
0.308 
0.150 
House value, millions DKK 
0.023 
0.015 
1.570 
0.115 
0.006 
0.052 
Bonds, millions DKK 
0.156 
0.046 
3.410 
0.001 
0.245 
0.066 
Shares, millions DKK 
0.073 
0.043 
1.670 
0.095 
0.013 
0.158 
Table 3 indicates that the largest changes in the average probability of choosing a variable rate mortgage for females, who have a probability that is 0.262 lower than males on average, and people who live in villages, who have a probability that is 0.229 lower than people who live in Copenhagen. Education, completion of college and completion of university and city statistics assignment help town and village all change probabilities help with statistics homework by amounts in the 0.10 to 0.20 range relative to the omitted categories, less than high school education and Copenhagen, respectively. Averaged over the sample, the effect of a one million DKK increase in share savings would be an increase of 0.073 in the probability of choosing a variable rate mortgage, whereas a 1 million DKK increase statistics help tutors in bond savings would lower the probability of choosing a variable rate mortgage by about twice as much, 0.156. An additional 100,000 DKK of income raises the probability of a variable rate mortgage by 0.006. Clearly the effects of income and wealth, although statistically statistics assignment help significant, are not large, especially given that Table 1 indicates that a 1 million increase in share or bond savings is more than a two standard deviation increase for either of these variables.
Table 4 presents the same model as Table 3, with the outliers removed from the sample. The Wald chis square statistic for the full model again has 12 degrees of freedom and a value of 281.54 associated with a p value less than 0.0001, which again implies that the null hypothesis that help with statistics homework the model predicts no better than the sample mean can be rejected. Without the outliers and with the same statistics assignment help cutoff rule used earlier, the model predicts 72.3% of the sample correctly.
Table 4 Logit regression for rate type, savings and income outliers omitted

Coefficient 
Robust Std. Err. 
z 
P>z 
95% LCL 
95% UCL 
Female 
1.375 
0.181 
7.580 
0.000 
1.731 
1.019 
Age 
0.036 
0.004 
8.050 
0.000 
0.045 
0.027 
Education 






Some college 
0.524 
0.183 
2.870 
0.004 
0.166 
0.882 
College 
0.743 
0.184 
4.030 
0.000 
0.381 
1.104 
University 
0.946 
0.196 
4.840 
0.000 
0.562 
1.329 
Income, 100,000s 
0.084 
0.026 
3.250 
0.001 
0.033 
0.135 
Urban 






City 
0.790 
0.213 
3.720 
0.000 
1.207 
0.374 
Town 
0.802 
0.218 
3.690 
0.000 
1.228 
0.376 
Village 
1.180 
0.215 
5.480 
0.000 
1.602 
0.758 
House value, millions 
0.195 
0.074 
2.650 
0.008 
0.051 
0.340 
Bonds, millions 
3.651 
1.326 
2.750 
0.006 
6.249 
1.052 
Shares, millions 
1.062 
1.161 
0.910 
0.361 
1.214 
3.338 
Constant 
1.562 
0.345 
4.530 
0.000 
0.886 
2.238 
Although the omission of the outliers statistics assignment solution does not substantially affect the ability of the model to predict, it has quite a large effect on the estimated coefficients. First, with the outliers omitted, the value of statistics assignment help the house changes from being statistically insignificant at the 0.05 level of significance to being significantly positively correlated with the choice of a variable rate mortgage. Second, statistics project help although statistics assignment for money the estimated coefficient on share savings is nearly 50% larger than before, it is no longer statistically significant. Point estimates of the help with statistics assignment effects of female, age and education are similar for the two sets of estimates and each remains statistically significant at least the 0.05 level of significance. However, the positive association between income and variable rate and the negative association statistics assignment solution between bond savings and variable rate are more than doubled in magnitude and more than quadrupled in magnitude, respectively.
The estimates in Table 4 are reported as average marginal effects in Table 5. The largest changes as the result of omitting outliers is help with statistics assignment that the effect of bond wealth in this table occur for bond wealth, which indicates that the average change in probability from an increase of 1 million DKK in bind savings goes from 0.156 when outliers are included to 0.658 when outliers help with statistics homework are omitted. The average effect of income on statistics project help the probability of choosing a variable rate more than doubles but remains fairly small.
Table 5 Average marginal effects on probability of variable rate, income and savings outliers omitted

dy/dx 
Delta Method
Std. Err. 
z 
P>z 
95% LCL 
95% UCL 
Female 
0.262 
0.033 
7.960 
0.000 
0.326 
0.197 
Age 
0.006 
0.001 
8.920 
0.000 
0.008 
0.005 
Education 






Some college 
0.097 
0.034 
2.880 
0.004 
0.031 
0.163 
College 
0.137 
0.034 
4.070 
0.000 
0.071 
0.204 
University 
0.175 
0.036 
4.840 
0.000 
0.104 
0.245 
Income, 100,000s 
0.015 
0.005 
3.300 
0.001 
0.006 
0.024 
Urban 






City 
0.149 
0.040 
3.780 
0.000 
0.227 
0.072 
Town 
0.152 
0.041 
3.730 
0.000 
0.231 
0.072 
Village 
0.224 
0.042 
5.320 
0.000 
0.307 
0.141 
House value, millions 
0.035 
0.013 
2.680 
0.007 
0.010 
0.061 
Bonds, millions 
0.658 
0.236 
2.790 
0.005 
1.122 
0.195 
Shares, millions 
0.192 
0.209 
0.920 
0.360 
0.218 
0.601 
Conclusion
The logit regression model confirms the belief that the choice of variable rate versus fixed rate mortgage revolves primarily statistics homework for money around risk to a portfolio of wealth that help with statistics assignment includes the house. In particular, as income and share wealth increase, people are more likely to choose a variable rate mortgage because income and share savings can be used to selfinsure against default risk. In contrast, the statistics homework help negative association statistics assignment help between increased bond savings and the probability of choosing a variable rate mortgage is consistent with home buyers viewing it as likely that if mortgage rates were to increase, then the statistics help tutors value of their bond portfolio would decrease, which leads them to be help with statistics assignment less likely to choose a variable rate mortgage as their bond savings increase. In general, results are not affected too much by omitting observations that are outliers by having high incomes or bond or share savings, although point estimates of the pay for statistics assignment effect of both types of savings increase.
The major shortcomings statistics assignment help of the model include the lack of additional variables that would relate to risk. For example, inclusion of the mortgage rate as of the date of the origination of the mortgage would presumably add explanatory statistics homework help power. For similar reasons, an indicator for whether both partners to a marriage work might pick up an effect through the decreased risk of loss of earnings from having two earners rather than one. Despite these omissions, the model predicts more than 70% of the choices correctly for a sample in which the choices are split nearly evenly.