Repayment Mortgage - Assignment Example

Quantitative Methods work BSc Business Studies Part Question The two types of mortgages considered during the were the Repayment Mortgage and the Interest Only Mortgage. The first type, the repayment Mortgage can be paid by a series of regular payments or instalments. Its value is equal to the present value of all future instalments. An example of this mortgage is a mortgage taken on a property. If an Interest Only Mortgage is taken out then only the interest is paid over the mortgage period. The sum of the actual loan (capital) is paid in a lump sum at the end of the mortgage period and is normally accumulated by a regular saving arrangement. The monthly cost of this mortgage comes out to be the monthly interest plus the amount to be accumulated. For calculating the monthly expenses for the both types of mortgages first the nominal interest rates should be used to calculate the effective annual interest rates. This is done by using the following formula: I=(1+r/x)^x-1 Nominal interest rate Number of compounding periods per year Effective annual interest rate 5.00% 1 0.05 5.25% 1 0.0525 5.50% 1 0.055 5.75% 1 0.0575 6.00% 1 0.06 6.25% 1 0.0625 6.50% 1 0.065 6.75% 1 0.0675 7.00% 1 0.07 7.25% 1 0.0725 7.50% 1 0.075 7.75% 1 0.0775 8.00% 1 0.08 8.25% 1 0.0825 8.50% 1 0.085 In order to obtain the effective annual rates the Microsoft Excel formula was used: EFFECT(nominal_rate,npery), where nominal_rate is the annual nominal rate and npery is the number of compounding times per year. The monthly expenses for the Repayment Mortgage are listed in the table below: Annual interest rate Number of months of payment Amount of mortgage Monthly expenses 0.05 300 350000 - 2,046.07 0.0525 300 350000 - 2,097.37 0.055 300 350000 - 2,149.31 0.0575 300 350000 - 2,201.87 0.06 300 350000 - 2,255.05 0.0625 300 350000 - 2,308.84 0.065 300 350000 - 2,363.23 0.0675 300 350000 - 2,418.19 0.07 300 350000 - 2,473.73 0.0725 300 350000 - 2,529.82 0.075 300 350000 - 2,586.47 0.0775 300 350000 - 2,643.65 0.08 300 350000 - 2,701.36 0.0825 300 350000 - 2,759.58 0.085 300 350000 - 2,818.29 These values were calculated in Microsoft Excel using the formula: PMT(rate,nper,pv,fv,type). Rate is the interest rate of the mortgage, Nper is the total number of repayments for the loan, in this case 300 months (25*12), Pv is the present value of the total repayments that are to be made, Fv is the future value that one wishes to attain after the last repayment, in this case 0 and lastly Type indicates whether the repayment is made at the beginning of the month (0) or at the end (1), in this case we assume it is made at the start of the month so 0. In order to evaluate the Interest Only Mortgage option we must first calculate the effective annual interest rate on the deposit placed in the sinking fund. This will be done in the same way as for the Repayment Mortgage. The results are presented in the table below: Nominal interest rate on deposit in sinking fund Number of compounding periods per year Effective annual interest rate on deposit in sinking fund 5.50% 1 0.055 5.75% 1 0.0575 6.00% 1 0.06 6.25% 1 0.0625 6.50% 1 0.065 6.75% 1 0.0675 7.00% 1 0.07 7.25% 1 0.0725 7.50% 1 0.075 7.75% 1 0.0775 8.00% 1 0.08 8.25% 1 0.0825 8.50% 1 0.085 8.75% 1 0.0825 9.00% 1 0.09 The monthly cost Interest Only mortgage consists of the interest paid over the mortgage period and the amount accumulated in the sinking fund. In order to calculate the monthly payment into the sinking fund using Excel the PPMT(rate,per,nper,pv,fv,type) formula was used where rate is the interest rate per period, per is the period, nper is the total amount of payments in the sinking fund, pv is the present value of the mortgage, fv is the future value expected in our case 0 and type indicates when the monthly payments are due in this case we assume that they are made at the beginning of the period so we take the value 1. The results are listed below: Annual interest rate Number of years in the loan Amount of loan Payment into investment plan 5.50% 25 350000 545.14 5.75% 25 350000 524.79 6.00% 25 350000 505.05 6.25% 25 350000 485.93 6.50% 25 350000 467.39 6.75% 25 350000 449.44 7.00% 25 350000 432.06 7.25% 25 350000 415.24 7.50% 25 350000 398.97 7.75% 25 350000 383.23 8.00% 25 350000 368.02 8.25% 25 350000 353.33 8.50% 25 350000 339.13 8.75% 25 350000 325.42 9.00% 25 350000 312.19 The second stage is to calculate the monthly interest. This was done by multiplying amount of the mortgage (350000) by the annual interest rates which in turn were divided by 12 to get the monthly rates. The final stage was to add together the monthly interest with the monthly savings. The results for the total monthly expenditure using the Interest Only mortgage are recorded in the table below: Mortgage Interest rates per year Monthly interest Payment into investment plan Total monthly expense 350000 0.05 1458.333333 545.14 -2,003.47 350000 0.0525 1531.25 524.79 -2,056.04 350000 0.055 1604.166667 505.05 -2,109.22 350000 0.0575 1677.083333 485.93 -2,163.01 350000 0.06 1750 467.39 -2,217.39 350000 0.0625 1822.916667 449.44 -2,272.36 350000 0.065 1895.833333 432.06 -2,327.89 350000 0.0675 1968.75 415.24 -2,383.99 350000 0.07 2041.666667 398.97 -2,440.64 350000 0.0725 2114.583333 383.23 -2,497.81 350000 0.075 2187.5 368.02 -2,555.52 350000 0.0775 2260.416667 353.33 -2,613.75 350000 0.08 2333.333333 339.13 -2,672.46 350000 0.0825 2406.25 325.42 -2,731.67 350000 0.085 2479.166667 312.19 -2,791.36 Based on the monthly cost estimates it can be concluded that an Interest only mortgage should be taken out as the monthly expenses for all of the rates are smaller that those of the Repayment mortgage. Question 2 As the introduction of the new cut-resistant golf ball coating is a good opportunity to increase the market share of Par Co., we must test the new golf balls driving distances. To find out if the new golf balls have a driving distance comparable with the current model we must perform a Hypothesis test. One-tail test should be used in this case as we are interested solely in determining whether the new balls have shorter driving distances than currently produced model or not. In this case, the samples are independent. H0 (null hypothesis): new cut-resistant golf balls have a driving distance comparable with the current model. The difference between sample means is less than or equal to zero. H1 (alternative hypothesis): new cut-resistant golf balls have a shorter driving distance than the current model. The difference between sample means is greater than zero. Current Model New Model (with coating) Sample Summaries Sample Size 40 40 Sample Mean 270.275 267.500 Sample Std Dev 8.753 9.897 First, the mean and standard deviation values for each of the samples are calculated, using the formula AVERAGE (number1, number2..) and STDEV(number1, number2..). We can observe that sample mean for the new model is somewhat less than for the model currently produced; however the standard deviation for the new model is greater as well. In order to decide whether to reject the null hypothesis or not, the test statistics and p-value have to be calculated. Test statistics (t-value) for independent samples test of differences between means can be found by using a formula: T-value = ((1- 2) - D0)/(Sp x SQRT (1/n1 +1/n2)), Where 1 = 270.275, and 2 = 267.500 are the two sample means, D0=0 is the hypothesized difference between means, Sp =2.089 is the pooled estimate of the common population standard deviation, n1 = 40, n2 = 40 - sample size. Using the obtained t-test statistic equal to 1.3284, we calculate the p-value for one tailed test with the formula TDIST(x, degrees of freedom, tails). The p-value of 0.0940 will lead to rejection of the null hypothesis if the significance level is set to 10%, however there is not enough statistical evidence in order to reject the null hypothesis at significance levels of 5% and 1%. Such results highlight the possibility of the first or second type error when making a decision based on this sample results. It is recommended to perform an additional test. t-Test Statistic 1.3284 p-Value 0.0940 Null Hypoth. at 10% Significance Reject Null Hypoth. at 5% Significance Don't Reject Null Hypoth. at 1% Significance Don't Reject Question 3 Let X1, X2, X3, and X4 be quantity of glass doors, windows, patio doors, and solid doors produced by the company correspondingly. The objective is to maximize total weekly profit given the constraints. LP problem formulation: MAX: 24 x X1 + 10 x X2 + 30 x X3 + 20 x X4, subject to: 2 x X1 + 3 x X2 + 4 x X3 + 1 x X4 Read More