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## Bank of England lowers main interest rate to 0.25%

Question 1:
On the 4th of August 2016, the Bank of England (BoE) lowered its main interest rate to 0.25%, the lowest on record since the Bank has published rates in 1970, which still remains unchanged. As a consequence of this low level of rates, Virgin Money’s Lifetime Tracker Standard mortgage rate stands at 3.9%. Jeff Green is about to buy a house in the countryside which costs £650,000 and is taking out a 25-year repayment mortgage for 75% of the acquisition value of the house. Jeff has asked you to assess his ability to pay, in case the BoE increases its interest rate, as his maximum monthly payment cannot exceed £3,200. You are required to:
1. Calculate the monthly payment John will need to meet under current conditions.
2. Assess the effect on John’s monthly payments for each quarter percentage point increase in the BoE rate, up to 4% and identify at which rate Jeff would no longer be able to make the monthly payment.
3. Determine the shortest length (to the nearest month) of mortgage Jeff could take out if he wanted his monthly payment to be exactly the maximum he could afford (based on current mortgage assumptions).
Question 2:
A company that manufactures electrical appliances is looking at one of its lines (microwave ovens), where it offers three different levels of specification: Cheap which sells for £50, Average which sells for £100 and Premium which sells for £250. The production of each oven goes through four different stages and you have been provided with the following data table:
ProcessMachine Cheap Average Premium Cost per hour Max. Available
Forming 1 hours 1.5 hours 3 hours £4.50 4,500 hours
Machining 3.5 hours 7 hours 13 hours £6.25 17,500 hours
Assembly 0.5 hours 1.5 hours 3.5 hours £7.75 6,500 hours
Testing 0.5 hours 1 hours 2 ½ hours £10.00 4,750 hours
The marketing department has also done some market research and believes demand for each of the models is limited to 2,000 units of the basic model, 1,200 of the medium and 700 of the luxury. You are required to:
1. Formulate this problem as a linear program and use Excel’s Solver to arrive at a solution, identifying what is the maximum profit the company can achieve in the
microwave ovens product line.
2. How would your answer change if the following happened:
a. Maximum demand for Premium model was 1,000; OR
b. Maximum available Machining hours were 20,000.
3. Write a report with a recommended production and marketing plan for the company.
Question 3:
A statistician is trying to find whether there is a relationship between the number of hours of study and exam results, or whether exam results are random. The first part of his study was to generate random numbers of study hours and exam grades, which will
be compared to the actual results once the exams are written. The data randomly generated is given in the table below:
1 25 80.0 45 96.0
2 5 8.0 3 29.0
3 18 16.0 14 46.0
4 29 87.0 49 32.5
5 17 95.0 11 61.0
6 39 14.0 19 61.0
7 49 9.5 43 12.0
8 25 10.0 44 46.5
9 6 35.0 0 95.0
10 22 58.0 38 66.0
11 37 8.0 38 24.0
12 31 89.0 50 62.0
13 18 57.0 17 60.5
14 45 22.0 33 10.0
15 5 39.5 33 61.0
16 4 90.0 42 36.0
17 17 23.0 45 39.0
18 29 55.0 34 86.0
19 16 74.0 15 87.0
20 22 29.0 29 100.0
21 28 69.5 49 76.0
22 6 27.0 39 50.5
23 21 12.0 31 55.0
24 4 82.0 14 67.5
25 24 26.0 10 94.0
26 12 13.0 17 33.0
27 4 23.0 34 85.0
28 38 24.5 45 33.0
29 31 77.0 3 19.0
30 5 71.0 44 38.0
Required:
1. Summarise the distribution of expected grades for both exams, according to the data given.
2. Construct a 95% confidence interval for the exam marks for each of the subjects. Is there a significant difference between them?

Bank of England lowers main interest rate to 0.25%

1a. In this problem we will compute the monthly payment which is to be made in terms of the given situation. We have considered the rate of interest as 3.9% and loan amount of £ 650,000 which will be paid in total period of 25 years. The computation of the monthly pay is done by dividing the mortgage payments with the total time available to clear the debt amount. This is seen as £ 254636.35/25 = £ 10185.454 (Di Maggio et al. 2017).

1b. In this case the assessment of the monthly payment for John in each quarter is depicted with the percentage increase of 4%. The computation of the monthly pay needed in this scenario is seen to be based on dividing the final monthly amount with the final payment which is to be taken into consideration. The calculation of monthly pay needed in this case is seen to be 10292.82/12 = £ 857.73.

1c. The depiction of the shortest length for the payment of mortgage is seen to be depicted with a total of 332 repayment to clear the loan amount of £ 650,000. The time required for this is computed as 27.2 years (Keys et al. 2014).

2.a The linear program has been formed with the use of excel solver. In order to formulate the linear program, we have set the decision variable to maximize the profit and consider the different types of the ways to apply the same in the microwave ovens products line (Hand and Voroninski 2016).

2b. (a)

If the maximum demand for the premium model is 1000 units, the per unit production for the month will change from 6045 units, 3023 units and 1511units to 2000 units, 1200 units and 1000 units.

2b. (b)

In this case the maximum available hours are set as 1000 hours and the unit of production will change to 2000 units, 1200 units and 700 units.

2c.

As per the given situation the problem formulated has been considered with the overproduction of the materials. The product demand has been seen to be 2000 units for the cheap, 1200 units for the average and 700 units for the premium process. Despite of this, the production of the materials is seen as 6045 units, 3023 units and 1511 units. It shows that there is a significant need for optimization for the monthly production units and maximize the overall profit of the company (Ware, Singh and Banwet 2014).

## Calculation of monthly payment under current conditions

If the premium model’s demand was 1000 units, it would have been adequate for the company for producing 2000 units for the cheaper variant of the product, 1200 for the average variant and 1000 for the premium variant of product. Moreover, the if available machine hours are seen to be 2000 units then the company is required to produce 2000 of the cheap ovens, 1200 of the average ovens and 700 of the premium ovens. Henceforth, it is recommended for the company to maintain a production level of 2000 units of the cheap variant, 1000 units of the premium variant and 1200 units for the average variant.

The marketing of the product needs to be further based on the motto of earning the maximum profit and strategy formulation aimed at promoting the products in the cheap and average category. The rationale for the decision is further taken into consideration with the demand constraint. It seen that the demand for the average and cheap product is highest in nature. This strategy will be able to reduce the costs and increase the profit ( Gallego, Ratliff and Shebalov 2014).

3a. The descriptive statistics evaluation of the grade distribution for both QM Exam and acting exam is listed below as follows:

3b.

The important considerations in the given scenario shows that we cannot reject the null value. In addition to this, the upper limit and the lower limit for the confidence interval can be stated as the 46.53 and -22.11. This signifies that there no significant difference among the results with a 95% confidence interval (Hinton 2014).

3c. The summarization of the various output for the regression analysis of QM study is depicted as 0.35. The comparison of the value of the summary output pertaining to the acting exam is determined as 0.4. Henceforth, it can be stated that the acting exam study hours is having more impact on the grades of the exams. The computed result is also seen to be significant in nature (Meret, Hellström and Hagelund 2015).

3d. The model of QM study hours and QM exam for the students studying for more than 20 Hours is depicted with a significance level of 0.218. It needs to be also noted that the different types of the consideration for the summary output pertaining to the “Acting Study hours and Acting Exam Grade for students studying more than 20 Hours” is depicted with 0.33. It needs to be noted that the Acting Study hours and Acting Exam Grade for the students studying more than 20 hours is having more impact in compared to the QM study hours and QM exam grade who are studying higher than 20 hours. This result is inferred with a similar level of significance with the previous results (Kuppuswamy and Bayus 2018). The use of the second model has been depicted to be based on the “QM study hours and QM exam grade for students studying less than 20 Hours” with a significance level of 0.989. The comparison of this result with the acting study hours shows that the grade for the students studying less than 20 hours for the acting exam the significance level is seen to be 0.867. This interpretation of shows that the that QM hours of study and QM grade of the exam for students studying less than 20 Hours is having a higher impact in compared to the acting study hours. Therefore, it can be stated that the use of various types of the variables pertaining to the finding between the exam grade and study hours is having significant relation. This shows that this model is better in compared to the previous model (Pyrczak 2016).

## Assessment of effect on monthly payments for each quarter percentage point increase in BoE rate

3e.

I have found inconsistencies within the data in many occasions. The most noteworthy occasions relating to this needs to be seen with the fact that despite there is less number of the study hours, the students are able to achieve high grades. However, such a scenario is unrealistic in a real examination. For example, if a student is specialised in the acting exam with zero hours of study then also a grade of 95 is achieved by the student. Similarly, in case of considering the QM exam, a student studying for four hours and five hours has also achieved a grade of 82 and 71. In contrast to this result, student studying for the 49 hours and 45 hours has still achieved a grade of only 9.5 and 22 in the exam. This shows that the replication of the results will not be possible in a real exam (Plesinger et al. 2015).

4a. The EOQ computed then there is no change in the factors is seen as 4264 units..

As per the findings of the EOQ, firstly we see that when there is no change in any factor then the EOQ will be 4264 units for a particular product. In any instance if there is simultaneous increase in all the factors by 7.5%, the EOQ will be 4421 units. On considering the third scenario, there has been significant decrease of 7.5% in all the factors and the overall EOQ is depicted as 4101 units. If there has been an increase of the 7.5% in D, then the company will need to order 4421 units which is similar to simultaneous increase of 7.5% on all the factors. If there is an increase of 7.5% in C’, the EOQ will be 4113 units. The increase in S’ by 7.5% will lead to an EOQ of 4421. In case there is decrease of 7.5% in D, the EOQ needs to be accommodated with 4101 units, which is alike simultaneous decrease of 7.5% on all factors. The decrease of 7.5% in the C’ will lead to an EOQ of 4434 units. This is justified for the company as a decrease in the annual cost for carrying one unit will allow the company to order more units of the same product. Similarly, the decrease of 7.5% in S’ will have an EOQ of 4101 units. The overall observations show that D is seen with the highest effect on the changes pertaining to the EOQ followed with S’ (Taleizadeh, Stojkovska and Pentico 2015).

## Shortest length of mortgage Jeff could take out if he wants his monthly payment to be exactly the maximum he could afford

In the given scenario the recommendation as per the building the factory is made with the use of capital budgeting tools. The important aspects of the capital budgeting tools to be used is seen with use of “NPV, IRR, payback period and profitability index”. If the cost of capital is 11.25% then Allen, plc. Should needs to opt for country A to setup the factory. This decision is seen with an NPV of £ 1023401 in Country A compared to £ 956313.1 for country B and PI of 1.49 for country A to 1.46 in country B. Even for a low payback period and high IRR for setting the factory in country B, Allen, plc needs to choose country A. The rationale for this is seen to be based on a rule that in case there is a situation of conflict then NPV determination is considered as ideal in nature as it is able to reflect better primary goal which focuses on financial growth. Even if the cost of capital decreases to 10% or increases to 12.5% the decision will be consistent in nature (Chittenden and Derregia  2015).

References

Chittenden, F. and Derregia, M., 2015. Uncertainty, irreversibility and the use of ‘rules of thumb’in capital budgeting. The British Accounting Review, 47(3), pp.225-236.

Di Maggio, M., Kermani, A., Keys, B.J., Piskorski, T., Ramcharan, R., Seru, A. and Yao, V., 2017. Interest rate pass-through: Mortgage rates, household consumption, and voluntary deleveraging. American Economic Review, 107(11), pp.3550-88.

Gallego, G., Ratliff, R. and Shebalov, S., 2014. A general attraction model and sales-based linear program for network revenue management under customer choice. Operations Research, 63(1), pp.212-232.

Hand, P. and Voroninski, V., 2016. An elementary proof of convex phase retrieval in the natural parameter space via the linear program phasemax. arXiv preprint arXiv:1611.03935.

Hinton, P.R., 2014. Statistics explained. Routledge.

Keys, B.J., Piskorski, T., Seru, A. and Yao, V., 2014. Mortgage rates, household balance sheets, and the real economy (No. w20561). National Bureau of Economic Research.

Kuppuswamy, V. and Bayus, B.L., 2018. Crowdfunding creative ideas: The dynamics of project backers. In The Economics of Crowdfunding (pp. 151-182). Palgrave Macmillan, Cham.

Meret, S., Hellström, A. and Hagelund, A., 2015. Friend, Foe, or Tolerated Guest? Role and Position of the Sweden Democrats, the Danish People's Party and the Progress Party in Mainstream Press Editorials. Part I Descriptive Statistics. In Trust Us! the Sweden Democrats in Sweden, the Progress Party in Norway and the Danish People´ s Party in Denmark. Berghahn Books.

Plesinger, F., Klimes, P., Halamek, J. and Jurak, P., 2015, September. False alarms in intensive care unit monitors: detection of life-threatening arrhythmias using elementary algebra, descriptive statistics and fuzzy logic. In Computing in Cardiology Conference (CinC), 2015 (pp. 281-284). IEEE.

Pyrczak, F., 2016. Making sense of statistics: A conceptual overview. Routledge.

Taleizadeh, A.A., Stojkovska, I. and Pentico, D.W., 2015. An economic order quantity model with partial backordering and incremental discount. Computers & Industrial Engineering, 82, pp.21-32.

Ware, N.R., Singh, S.P. and Banwet, D.K., 2014. A mixed-integer non-linear program to model dynamic supplier selection problem. Expert Systems with Applications, 41(2), pp.671-678.

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