Financial Mathematics and Business Statistics - Assignment Example

Financial Mathematics and Business Statistics Question Due to the weakness in the market suffered by most companies in the last few years, the management team at Dreamcatcher, a games’ retailer has been reviewing its stock ordering system and has managed to negotiate a reduction in ordering costs with their main supplier from £280 to £190. Despite selling different games, their size and packaging allows games to be managed as a single product. The expectations regarding next year’s state of the economy are as follows: Scenario Crisis Continues Slow Recovery Medium Recovery Fast Recovery Probability 20% 15% 50% 15% Demand 40,000 65,000 105,000 250,000 Considering that the average carrying cost per unit is £1.45: a) Make a recommendation to the board of Dreamcatcher, as to how often it should order and what level or stock should be ordered each time; b) Assess the savings achieved by the recent renegotiation with the main supplier; and c) Assess the impact of this recommendation at the different levels of demand. Solution: a1) The average level of demand is (0.2*40,000+0.15*65,000+0.5*105,000+0,15*250,000)/100=107,750 The average carrying cost is 107,750*1.45=156,238/- The ordering cost is 190. Total cost=Average carrying cost + holding cost = 156,428/- Reorder level = sqrt(2*107750*1.45/190)=40.55 b) The stock should be ordered for every 3 months ie.0.25 years. b) Savings achieved = 156,238 – 107,705 = 48,533 c) For different levels of demand we formulate the problem as follows: For demand of 40,000 units the impact is very weak. For demand of 65,000 units the impact is weak. For demand of 105,000 units the impact is very strong and For demand of 250,000 units the impact is very weak. Question 2: A furniture manufacturer has recently noticed a reduction in the profit achieved in their Tables’ product line and in order to address the product is conducting a review to understand which model(s) are more profitable and it should concentrate on. It currently makes four different tables using three different materials and labour. Prices, inputs and constraints are summarised in the following table: Table Square Round Rectangular Elliptic Max. Available Price £600.00 £700.00 £900.00 £1,050.00 Wood 11 units 12 units 52,000 Metal 7 units 8 units 36,000 Other Materials 6 units 4 units 9 units 8 units 48,500 Labour 1½ hours 2¾ hours 2 hours 3½ hours 20,000 Max. Demand 3,500 3,000 1,500 2,000 The costs for the inputs are £25 per hour for Labour and £15, £20 and £10 per unit for Wood, Metal and Other Materials, respectively. Formulate this problem as a linear program and use Excel’s Solver to arrive at a solution. Write a short report describing your procedure, justify your formulation and give a recommendation to the firm on the best daily production mix. Solution: The formulation of the LPP is as follows: Maximize profit Z=600x1+700x2+900x3+1050x4 Subject to 11x2+12x4≤52000 7x1+8x3≤36000 6x1+4x2+9x3+8x4≤48500 1.5x1+2.75x2+2x3+3.5x4≤20000 x1≤3500, x2≤3000, x3≤1500 and x4≤2000 x1, x2, x3 and x4≥0 where x1 denotes number of square tables, x2 denotes number of round tables, x3 denotes number of rectangular tables and x4 denotes number of elliptic tables. The solution is as follows: The authorities should make a decision to manufacture 3000 round tables, 1500 rectangular tables and 2000 elliptic tables and avoid square tables. The maximum price earned is £4,650,000/- (Excel sheet is attached herewith for reference) Question 3: The majority of banks, when making decisions on mortgage applications, will look at two indicators: salary and borrowing as a percentage of purchase price. On the first indicator, banks are normally willing to lend 2.5 times one’s salary or 3.25 times joint salary in a joint mortgage application, while currently most banks will lend up to 75% of the property price on their best rate with penalties for higher percentages. Alan and Karen are getting married and decided to buy a flat to move into once they do. You have been given the following data: Alan’s current salary is £52,000 p.a. and Karen’s is £38,500 p.a. plus a bonus likely to be around £5,000 (based on previous 3 years experience); Both have jobs where they partly telecommute, so on average each works from home 2 days a week; Their total savings at the moment are £73,000; Karen owns a flat which he would sell, and has been advised that he should be able to sell it for £189,000. The mortgage outstanding on this flat is £102,000; Alan and Karen are planning to apply for a 20 year mortgage; The average price of flats in the area they would like to move into is as follows: studios £145,000; 1-bedroom £230,000; 2-bedroom £375,000; 3-bedroom £510,000; 4-bedroom £680,000 Having contacted a financial adviser, he has identified the following as the best available mortgage rates: Assess: a) What is the maximum Alan and Karen can borrow while taking advantage of the bank’s best mortgage rate; b) What amount you advise them to borrow, given their financial and professional situation; c) Which is the best mortgage that Alan and Karen to take out (assume they take out the amount you recommended in b); d) Whether that advice would change if interest rates went up or down by up to two percentage points. Solutions: a) Maximum Alan and Karen can borrow is 300,000 b) Amount advised them to borrow is 250,000 c) The best mortgage that Alan and Karen to take out is 265,000 d) Yes that advice would change if interest rates shoots up to 2 points. Question 4: Pierce plc has developed two new products (X and Y), both of which require an investment of £5million, the amount of finance the company can raise, which means they can only launch one of them. Given the different technologies used in the two products, product X’s life cycle is likely to be longer, while product Y is likely to need replacing within five years. Pierce has estimated the cash flows for the two different products, which are identified in the table below. Pierce’s required rate of return on investments is 10.75% and the estimated cash flows for the two options are as follows (in ‘000s): Year Product X Product Y 1 500 1,000 2 750 2,000 3 1,100 3,000 4 1,200 2,000 5 1,500 1,000 6 1,800 7 2,300 8 3,000 9 1,500 10 750 Required: a) Discuss and compare the different types of investment appraisal methods Pierce can use, including a discussion of the advantages and disadvantages of each. b) If Pierce had a rule that all investment projects need to payback within 3 years, what project would be chosen? Comment. c) Make a recommendation as to which project should be undertaken. Solutions: a) The different type of investment appraisal methods Pierce can use are Payback period, Average Rate of Return, Discounting, Net Present Value etc..\ PAYBACK PERIOD Considering the Payback period, the return on product X is realized after 5 years (return £5,050,000) whereas the return on product Y is realized even after 3 years (£6 million, ie 1 million more than the investment). For X: the payback period is 4 years + 1,450,000/1,500,000*100=4+0.97 years=4.97 years For Y: the payback period is 2 years + 2,000,000/3,00,000*100=2+0.67 years=2.67 years So when comparing the above product Y’s payback period is low. So it is preferred. This technique is easy to calculate and understand. Also this technique is helpful for companies with weak cash flow. The disadvantage of this technique is this method focuses on time but ignores profit and also it ignores income after payback time. AVERAGE RATE OF RETURN (ARR): This method calculates Average annual profit on investment as percentage of the sum invested. Average Profit per year ARR = ––––––––––––––––––– x 100 Capital Outlay It provides the company with percentage figure that can be compared with the percentage rates of returns on other uses of its capital. The technique is as follows: AP X (total cash flow-total investment)/No. of years=(14,400,000-5,000,000)/10=940,000 AP Y (total cash flow-total investment)/No. of years=(9,000,000-5,000,000)/5=800,000 ARR(X)=940,000/5,000,000*100=18.8% ARR(Y)=800,000/5,000,000*100=16% The average rate of return of Product X is superior when compared to product Y. For both the products, the average rate of return is better than 10.75% which is the required ARR. Advantages: Unlike payback, ARR actually focuses on profit. ARR can easily be compared to other investments as it is in the form of percentage return. Disadvantages: Timing: Actually it looks at profit, it ignores the timing of that profit to occur. The ARR is based on average profit. The actual profit may dramatically vary during the course of duration of the investment years. Discounting method is ignored as there is no discounting concept in the above said problem. NET PRESENT VALUE (NPV) Comparing the NPVs of both products, The NPV of product X (considering 10% discount, without discount, the NPV can not be computed) is £459,325 whereas the NPV of product Y is £464,530. The NPV of product Y is slightly greater than that of product X. So product Y is slightly better in terms of NPV. Advantages: 1. The NPV takes account of the time value of money. 2. The NPV takes account of all the cash inflows and outflows for the whole duration. Disadvantages: 1. Choosing the discount rate is difficult especially for long term projects. 2. It is complex to calculate and often misunderstood. (Excel sheet is attached herewith for reference) b) If Pierce had a rule that all investment projects need to payback within 3 years project Y should be taken because its payback period is lesser than that of project X. c) To be recommended is project Y, because it is superior in two investment options ie. Payback method and NPV method. Question 5: The table below represents data for the profits, sales, average shop size and number of product lines sold by the 20 branches of a retailing company. You have been asked to analyse the data, using the Data Analysis tool in Excel, and make recommendations, including the following: a) Summarise the distribution of profits of the twenty branches and comment on the results, including identification of any particularly good or poorly performing branches. b) Identify whether there is evidence that the average number of lines stocked per branch is significantly different from 150. c) Identify whether there is a significant difference between the profits of two groups of branches, split by the level of sales, with the threshold being £600,000. Profit (£000s) Sales (£000s) Size (000s sq. ft.) Lines 77.5 613.9 3.2 80 91 217.4 4.3 200 20.7 900.9 3.1 164 40.8 673.4 1.5 150 45.8 424.7 3.2 69 41.1 542.2 1.8 128 47.5 564.6 2.5 75 80.4 662.1 3.1 182 16.5 583.6 4.2 126 22.3 720.2 0.6 164 40.8 881.5 1.8 145 68.1 227.7 0.8 130 17.7 807.4 3.8 154 66.2 656.4 0.3 124 31.3 632.8 2.3 142 15 548.5 5 178 67.8 533.6 1.5 173 55 147.5 1.7 199 8.6 311.4 3.8 98 16.5 450.1 4.6 148 a) The mean profit is 43.53 with SD 25.06, the better performers are those whose profit is above the mean level. So the branch numbers 1, 2, 5, 7, 8, 11,12, 14, 15, 17 and 18 are the better performers and the others are not performing well. b) Since the sample is only 20 ( Read More