Consider a pool of home mortgages. Prepayments of mortgages in the pool affect t

ID: 3180134 • Letter: C

Question

Consider a pool of home mortgages. Prepayments of mortgages in the pool affect the mortgages cash flow, so mortgage lenders, servicers, and investors all have an interest in predicting mortgage prepayments. Mortgages may be prepaid for a variety of purposes, including selling the home, taking cash out of the property to fund home improvements or other consumer expenditures, or refinancing the mortgage to change the monthly payment schedule. Narrow your focus to mortgage prepayments that are made for the purpose of refinancing. If there were no costs to refinancing, you would refinance to reduce your monthly payments every time the current mortgage rate dropped below the rate on your mortgage. In actuality, however, there are costs to refinancing, such as points and closing fees. Therefore, the spread between the current mortgage rate and your own rate must be big enough to more than make up for the costs, or you wouldn't be interested in refinancing. The economics of refinancing suggest that compared to mortgages that aren't refinanced, refinanced mortgages have higher mortgage rates. Define population 1 as mortgages that are refinanced, and define population 2 as mortgages that are not refinanced. Let mu_1 equal the mean mortgage rate on refinanced mortgages, and let mu_2 equal the mean mortgage rate on mortgages that are not refinanced. similarly, let sigma_1 and sigma_2 equal the standard deviations of mortgage rates for populations 1 and 2. Assume that sigma_1 = 0.55 and sigma_2 = 0.66. In a study, professor Michael LaCour-Little selected independent random samples of mortgages that were refinanced and mortgages that were not refinanced, and he collected data on mortgage rates. For the sample drawn from refinanced mortgages, the sample size n_1 = 31, and the sample mean cap x_2 = 8.62. For the sample drawn from mortgages that were not refinanced, the sample size n_2 = 30, and the sample mean cap x_2 = 8.09. The point estimate of mu_1 - mu_2 is ___. In this study, the sampling distribution of cap x_1 - cap x_2 is approximated by a ___ distribution with ___ and a standard deviation ___. Use the Distributions tool to help you answer the questions that follow. The 95% confidence interval estimate of the difference between mu_1 and mu_2 is ___ to ___. You want to determine whether refinanced mortgages have a higher mean mortgage rate than mortgages that are not refinanced, as the economics of refinancing suggests. You test the hypothesis that refinanced mortgages have a lower mean mortgage rate than mortgages that are not refinanced. The null and alternative hypotheses are formulated as: H_0: mu_1 - mu_2 greaterthanorequalto 0, H_a: mu_1 - mu_2 0 H_0: cap x_1 - cap x_2 lessthanorequalto 0, H_a: cap x_1 - cap x_2 > 0 H_0: mu_1 = mu_2 = 0, H_a: mu_1 - mu_2 notequalto 0 The test statistic for the hypothesis test is ___. The P value is ___. A level of significance of alpha = .05 is specified for the study. The null hypothesis is ___. Therefore, you ___ conclude that refinanced mortgages have a higher mean mortgage rate are not refinanced.

Explanation / Answer

Answer:

Point estimate = 0.53

Sampling distribution is approximated by a z distribution with normal approximation and standard deviation 0.1558

95% confidence interval for mean difference =(0.2246, 0.8354)

Option 2

H_0: mu_1 - mu_2 leq 0 H_1: mu_1- mu_2 > 0

Test statistic = 3.40

P value =0.0003

Null hypothesis is rejected.

You have sufficient evidence to conclude.

Hypothesis Test: Independent Groups (z-test)

8.62

8.09

mean

0.55

0.66

std. dev.

0.53000

difference (g1 - g2)

0.15581

standard error of difference

hypothesized difference

3.4015

.0003

p-value (one-tailed, upper)

0.22461

confidence interval 95.% lower

0.83539

confidence interval 95.% upper

0.30539

margin of error