Financial Mathematics and Business Statistics - Assignment Example

Financial Mathematics and Business Statistics: work Question i) Using the conditional profits as given in problem and the probabilities of states of nature, The expected monetary value can be calculated for each of three states of nature. Table 1 Probability International Expansion Domestic Expansion No change Pessimistic 0.25 7,500 8,600 11,000 Average 0.55 13,800 14,000 12,300 Optimistic 0.20 21,900 19,500 16,700 Table 2 International Expansion (S1) Domestic Expansion (S2) No change (S3) Pessimistic 1,875 2,150 2,750 Average 7,590 7,700 6,765 Optimistic 4,380 3,900 3,340 EMV=pijaj 13,845 13,750 12,855 The total weighted profit $13,845 is the highest due to action S1, it should be chosen, since EMV for International Expansion (S1) is the highest. ii. Conditional Opportunity Loss From table 1, we reformulate table 3 by subtracting minimum element from each row: Table 3 International Expansion (S1) Domestic Expansion (S2) No change (S3) Pessimistic 0 275 875 Average 825 935 0 Optimistic 1,040 560 0 Since we are dealing with conditional profits, higher value for each state of nature shall be considered for calculating opportunity loses. The calculations for expected opportunity loss are shown in table 4. Table 4 Probability International Expansion (S1) Domestic Expansion (S2) No change (S3) Pessimistic 0.25 0 68.75 218.75 Average 0.55 453.75 514.25 0 Optimistic 0.20 208 112 0 EMV= 661.75 695 218.75 EOL*=EOL(S3)=$218.75 iii. The expected profit with perfect information (EPPI) can be determined by selecting optimal course of action for each state of nature, multiplying its conditional values by the corresponding probability and them summing these products. EPPI calculations are shown in the following table. Table 5 Probability Optimal course of action Conditional Loss Weighted Opportunity Loss Pessimistic 0.25 S1 661.75 165.4375 Average 0.55 S2 695 382.25 Optimistic 0.20 S3 218.75 43.75 Total 591.4375 Since we are dealing with expected profit EPPI=591.4375, now expected value of perfect information is given by EVPI=EPPI-EMV*=591.4375-218.5=372.9375 Question 2: Let the decision be to produce x units of cheap, y units of average and z units of premium. Then the objective function is to maximize the profit Z=(75-17.625)x+(150-36.0625)y+(375-96.875)z=57.375x+113.9375y+278.125z Since the total cost constraint to each of the specification is given by: Cheap: 0.5*4.5+1.75*6.25+0.25*7.75+0.25*10=17.625 Average: 0.75*4.5+3.5*6.25+0.75*7.75+0.5*10=36.0625 Premium: 2.25*4.5+9*6.25+2*7.75+1.5*10=96.875 Subject to the constraints x≤3,000, y≤2,400 and z≤1,400 also the total cost constraints are: Forming: 0.5x+0.75y+2.25z≤2,250 Machining: 1.75x+3.5y+9z≤8,750 Assembly: 0.25x+0.75y+2z≤3,250 Testing: 0.25x+0.5y+1.5z≤2,375 x,y,z≥0 Now we solve this problems using MS Excel solver: By introducing the slack variables s1, s2, s3, s4, s5, s6 and s7, the problem becomes standardized one: Maximize Z=57.375x+113.9375y+278.125z +0s1+0s2+0s3+0s4+0s5+0s6+0s7 Subject to x+s1≤3,000, y+s2≤2,400 z+s3≤1,400 0.5x+0.75y+2.25z+s4=2250 1.75x+3.5y+9z+s5=8750 0.25x+0.75y+2z+s6=3250 0.25x+0.5y+1.5z+s7=2375 x,y,z,s1,s2,s3,s4,s5,s6,s7≥0 a. By solving with Excel solver add-in, we get x=3,000, y=1,000 and z=0 and the maximum profit is $286,062.50 which means that the company shall product 3,000 units of cheap amplifiers, 1,000 units of medium amplifiers and no units of luxury amplifiers (refer to file lpp.xlsx sol1 sheet). b1). Maximum available Machining hours were 10,500; Suppose maximum available machining hourse were 10,500, then the strategy should be To produce 2,400 units of medium amplifiers and 200 units of luxury amplifiers and the profit would be $329,075 (refer to file lpp.xlsx sol2 sheet) b2. The company was able to charge £200 for the Average amplifier. Then the profit for average amplifier would be selling price – cost =200-36.0625=163.9375, Now the decision is changed to producing no units of cheap and medium amplifiers (see sol3 sheet of lpp1.xlsx file), but to produce 972.22=972 units of luxury amplifier and the profit would be $270,338 (refer to file lpp.xlsx sol3 sheet). c) Among all the above, the best strategy to adopt is that increase the maximum machining hours to 10.500 and make a profit of $329,075, that is, to produce 2,400 of medium amplifiers and 200 luxury amplifiers. Question 3: a) By consulting the website of a bank in one of the "big four" UK banking groups (clearly indicate in your report the website link and date consulted): http://www.bankofengland.co.uk/publications/Pages/news/2015/003.aspx (Date Consulted: 13 March 2015) a. Explain the term Offset Mortgage; The term offset mortgage means the deduction of savings from the mortgage amount which enables one to offset the interest one pays for the savings they have. So if one has £1 million as mortgage and £20,000 as savings then they can pay interest for only they have to pay for the remaining £80, 0000. b. Identify the lowest rate the bank is currently charging for an Offset mortgage; Bank of England maintains Bank Rate at 0.5% and the size of the Asset Purchase Programme at £375 billion (Bank of England). c. Identify the lowest rate, application fees (if applicable) and respective period the bank is charging for a fixed rate mortgage; Mortgage rates remains very low nowadays, the lowest rate now prevails is 1.5% for a two year and for five year fix it is below 3% The Bank of England has been keeping at 0.5% from March 2009 and they estimate that it would not have risen. b) Calculate the amount you advise Adam and Eve to borrow and which size flat to buy, given their financial and professional situation; Based on Adam’s salary of £49,000 p.a., he would get £122,500 as loan and based on Eve’s salary of £42,500 p.a., she would get £106,250. So the total amount is £228,750. But when they jointly apply for mortgage they would get (£49,000+£42,500)*3.25=£297,375 which is £68,625 more than what they get individually together. This loan of £297,375 is 90% of £330,416, that is, Approx. £330,000. So if they purchase around £330,000, they would avoid the penalty of 1.5% For this situation they can prefer minimum 1 bed room house which costs around £330,000. If Adam sells his present flat @£185,000 and subtracting his mortgage amount of £124,000, he would have £61,000 in his hand. The savings of them are £45,000. If we add savings, the total amount would be £106,000. So £106,000+£330,000=£436,000 they can afford for purchasing 2 bed room house. If they apply for 25 year mortgage, the best deal to buy is 2 bedroom flat since after their marriage, they would have children and they also need separate bed rooms. This would ease the situation when the children are growing. On the other hand, if they prefer 3 bedroom, £575,000 would become more burden for them to repay. The best deal is to buy 2 bedroom house. c) Using the two rate identified in a) calculate which is the best mortgage that Adam and Eve can take out (assume they take out the amount you recommended in b); As already given in section (b), the increase in mortgage rate by 3% would be a burden to Adam and Eve, so they can stick up to 2 bed room flat. d) Whether that advice would change if interest rates went up or down by up to three percentage points. Even with the increase in interest rates, the repayment is affordable for Adam and Eve. Question 4: a) Summarise the distribution of profits of the twenty branches and comment on the results? The distribution of profits is given in the following table. This is done by using histogram of the Excel Data Analysis option. Profit No. of branches 40 1 Total 50 Summary Statistics of Profit Mean 11.684 Standard Error 2.949272 Median 6.8 SD 13.18955 Sample Variance 173.9641 Kurtosis 0.653467 Skewness 1.250159 Range 44.82 Minimum -2.69 Maximum 42.13 Sum 233.68 Count 20 b) Is there evidence that the average number of lines stocked per store is significantly different from 78? For this we set up Null hypothesis H0: The average no. of lines stocked per store is =78 Alternative hypothesis H1: The average no. of lines stocked per store is not 78 ie.#78. Level of significance: 5% level or =0.05. Test statistic: to=(xbar-)/SE(xbar)= (xbar-)n/S follows t distribution with n-1=20-1=19 d.f xbar is the mean of no. of lines=100.5, =78 S=38.76, SE(xbar)=8.67, now to=2.595 whereas the critical value of t is te=2.43 for two tailed test at 5% level of significance for 19 d.f. Since the observed value to is greater than the expected value te (probability 0.018), we have enough evidence to reject null hypothesis H0 and accept alternative hypothesis H1 and conclude that the average number of lines stocked is significantly different from 78 at 5% level of significance. c) If you divide the branches in two groups with, one of branches with sales above £150,000, and the other with sales below that value, is there a significant difference between the profits of the groups? The following table shows the division of groups profit group 41.54 1 50.84 1 62.78 1 90.08 1 91.75 1 91.8 1 96.85 1 109.05 1 140.78 1 141.57 1 166.93 2 190.59 2 198.69 2 231.6 2 263.92 2 265.28 2 377.04 2 548.31 2 702.11 2 748.82 2 t-Test: Two-Sample Assuming Equal Variances Descriptive statistics group1 group2 Mean 91.704 369.329 Variance 1138.664 47754.16 Observations 10 10 Pooled Variance 24446.41 Hypothesized Mean Difference 0 Df 18 t Stat -3.97042 P(T Read More