BUSINESS MATHEMATICS & QUANTITATIVE METHODS FORMATION 1 EXAMINATION - APRIL 2010 NOTES: You are required to answer 5 questions. (If you provide answers to all questions, you must draw a clearly distinguishable line through the answer not to be marked. Otherwise, only the first 5 answers to hand will be marked). All questions carry equal marks. STATISTICAL FORMULAE TABLES ARE PROVIDED DEPARTMENT OF EDUCATION MATHEMATICS TABLES ARE AVAILABLE ON REQUEST TIME ALLOWED: 3 hours, plus 10 minutes to read the paper. INSTRUCTIONS: During the reading time you may write notes on the examination paper but you may not commence writing in your answer book. Marks for each question are shown. The pass mark required is 50% in total over the whole paper. Start your answer to each question on a new page. You are reminded that candidates are expected to pay particular attention to their communication skills and care must be taken regarding the format and literacy of the solutions. The marking system will take into account the content of the candidates' answers and the extent to which answers are supported with relevant legislation, case law or examples where appropriate. List on the cover of each answer booklet, in the space provided, the number of each question(s) attempted. The Institute of Certified Public Accountants in Ireland, 17 Harcourt Street, Dublin 2.
THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND BUSINESS MATHEMATICS & QUANTITATIVE METHODS FORMATION 1 EXAMINATION - APRIL 2010 Time Allowed: 3 hours, plus 10 minutes to read the paper. You are required to answer 5 questions. (If you provide answers to all questions, you must draw a clearly distinguishable line through the answer not to be marked. Otherwise, only the first 5 answers to hand will be marked). All questions carry equal marks. 1. In an expansion of the Production Department, the Superior Products Company is investing in a specialised machine costing 90,000. The machine will last for 5 years and will be sold for 3,000 at the end of that year. At the end of each year maintenance will be carried out on the machine. The cost at end of year one is 2,000, increasing by 15% for each succeeding year. The revenue produced by the machine will be 25,000 for the first year, increasing by 10% for each succeeding year. To test the financial viability of the machine you, as the Management Accountant, are required to: Set out the net cash flows for the machine. (8 Marks) Derive the Net Present Value using a discount rate of 18%. (6 Marks) (iii) Find the Internal Rate of Return. (6 Marks) 2. The Do-It-Better Manufacturing Company operates a shift loading system whereby 60 employees work a range of hours depending on company demands. The following data was collected: Hours worked No. of employees 16 < 20 1 20 < 24 2 24 < 28 3 28 < 32 11 32 < 36 14 36 < 40 12 40 < 44 9 44 < 48 5 48 < 52 3 You are required to: Calculate the mean and standard deviation of the hours worked. (8 Marks) Present the data on a cumulative frequency curve and derive the median hours worked. (8 Marks) (iii) Derive the skewness of the data from the observations in and and interpret your result. () 1
3. The Impart Trade Union claims that female members of its union are being treated unfairly based on the average earnings and average hours worked by both male and female members. However, the company claims that the situation has improved greatly and that the conditions for both are almost the same. It produced the following data, for the past four years, to support its claim. Year Average Earnings / week Average Hours worked / week Male Female Male Female 2006 600 350 42 35 2007 700 360 50 35 2008 650 380 45 30 2009 670 420 45 32 You, as the Equality Officer, are asked to adjudicate on the claim. In order to do this: Plot graphs showing the average earnings per week and average hours worked for both male and female employees. (8 Marks) Plot a graph showing the average hourly rate for both males and females over the period. (6 Marks) (iii) Interpret the data for the company. (6 Marks) 4. The DIY division of your company has produced the following table of four commodities used in the construction industry. The prices and sales by value are set out below: 2003 2008 Price per Sales by Price per Sales by 1000kgms value m 1000kgms value m Sand 148 444 178 712 Cement 162 648 184 920 Limestone 202 1010 242 1210 Gravel 130 390 135 405 The company wishes to establish an index for the commodities for 2008 taking 2003 as the base year. Using Laspeyres and Paasche index numbers you are required to: Calculate price and quantity index numbers for 2008. (1) Interpret the index numbers that you have calculated and explain the difference between both index numbers. (8 Marks) 2
5. Unsold seats on airplanes represent a loss of potential revenue to the airline operator. LowCostAer wants to estimate the average number of vacant seats per flight over the past year. From its data it examined the records of 125 flights which were randomly selected and the number of vacant seats for each of the flights was recorded. Based on this, the sample mean and standard deviation are 12.1 and 4.2. If the potential revenue loss per vacant seat is 50, estimate the revenue loss of vacant seats per flight with a 90% level of confidence. (8 Marks) Your company has set up a fund to develop a community hall with an initial payment of 12,000. This is compounded at six monthly periods over 4 years at 5% at each six months. Calculate the size of the fund at the end of four years and the effective annual interest rate. (6 Marks) (iii) The Financial Accountant advises you that the machine the company purchased for 90,000 will depreciate at 15% per year. Using reducing balance depreciation, calculate the level of depreciation of the machine at the end of the 5 years. (6 Marks) 6. The annual CPA conference is being held in December on the role of Financial Mathematics in the business and investment environment. In the context of this subject explain the following: (iii) (iv) (v) Simple and Compound Interest. Annual Percentage Rate. Annuities. Mortgages. Sinking funds. END OF PAPER 3
SUGGESTED SOLUTIONS THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND BUSINESS MATHEMATICS & QUANTITATIVE METHODS FORMATION 1 EXAMINATION - APRIL 2010 SOLUTION 1 Net Cash flows for the machine are set out in the following table Year end Cash Outflows Cash Inflows Net Cash Flows (Costs) (Revenues) 0 (80,000) ------ (80,000) 1 (2,000) 25,000 23,000 2 (2,300) 27,500 25,200 3 (2,650) 30,250 27,605 4 (3,042) 33,275 30,230 5 (3,498) 36,603 30,105 (3,000) Net Present Value Discount Present Discount Present Factor @ 18% Value Factor @ 24% Value Year 0 0 (80,000) 0 (80,000) Year 1 0.8475 19,493 0.8065 18,550 Year 2 0.7182 18,099 0.6504 16,390 Year 3 0.6086 16,800 0.5245 14,479 Year 4 0.5158 15,593 0.4230 12,787 Year 5 0.4371 13,159 0.3417 10,287 83,144 72,493 Net Present Value 3,144 (7,507) 6 Marks (iii) Internal Rate of Return Calculating the IRR by means of N1I2 - N2I1, where N1 = 3,144, I1 = 0.18, N2 = (7,507), N1 - N2 I2 = 0.24 3 Marks = 3,144 x 0.24 - (7,507) x 0.18 3,144 - (7,507) = 754.6 + 1351.3 = 2,105.9 = 19.77% 10,651 10,651 3 Marks 5
SOLUTION 2 Cumulative frequency curve and median Hours Mid Value Frequency Cum Worked x frequency fx (x x) (x x)2 f(x x)2 16 < 20 18 1 1 18 (18) 324 324 20 < 24 22 2 3 44 (14) 196 392 24 < 28 26 3 6 78 (10) 100 300 28 < 32 30 11 17 330 (6) 36 396 32 < 36 34 14 31 476 (2) 4 56 36 < 40 38 12 43 456 2 4 48 40 < 44 42 9 52 378 6 36 324 44 < 48 46 5 57 230 10 100 500 48 < 52 50 3 60 150 14 196 588 60 2,160 2,928 Mean = x = fx = 1540 = 30.8 f 50 Standard deviation = σ = f (x - x)2 = 4798.76 f 50 = 95.97 = 9.79 Median; the median is described as the centre of a set of data. Since you are asked to derive the median from the graph, the value is approximately 31 hours worked. The median is the value of the data such that 50% lies above and below this value. Graphical derivation of the median. 60 50 Cum Frequency 40 30 20 10 Median 15 20 25 30 35 40 45 50 55 Hours worked 6
(iii) When data is symmetrically distributed the mean, median and mode are equal. If data is skewed the co-efficient of skewness gives a measure of the degree of skewness in a set of data where the co-efficient of skewness is derived from 3(mean median) = 3(30.8 31) = - 0.06 standard deviation (σ) 9.79 If the co-efficient of skewness is zero the data is perfectly distributed the median equals the mean. If a negative value is obtained this indicates skewness to the left where the mean is less than the median, often due to the presence of an extreme value. Extreme values affect the mean since these values are used explicitly to calculate it. In the present case, both the mean and the median are approximately the same. From a visual examination of the data it is obvious that there are no extreme values which could cause a major distortion. A comparison of the mean and median gives a general method for detecting skewness in data sets. 7
SOLUTION 3 The graphs for average earnings / week v average hours worked are set out below: Year Average Earnings / week Average Hours worked / week Male Female Male Female 2006 600 350 42 35 2007 700 360 50 35 2008 650 380 45 30 2009 670 420 45 32 Comparison of average earnings per week 800 Male 700 600 Average Earnings/week 500 400 Female 300 200 2006 2007 2008 2009 Comparison of average hours worked per week 60 50 Male 40 Hours worked Female 30 20 10 Year 8
Comparison of average hourly rates Average hourly rate Male Female % Difference 2006 14.3 10.0 30.1 2007 14.0 10.3 26.4 2008 14.4 12.7 11.8 2009 14.8 13.1 11.5 3 Marks 16 Male Average hourly rate 14 Female 12 10 8 2006 2007 2008 2009 Year 3 Marks (iii) Analysis. The presentation of data graphically appears to give a very negative picture of the comparison of both groups. Weekly earnings of males are substantially higher than that of females. The average earnings per week for males range from 600 to 700 compared to females of 350 to 400. Females work from 16% to 30% less than males. When this is put into context of the smaller number of weekly hours worked by females it is necessary to derive the average hourly rate to compare the earnings for both categories. Over the period the average hourly rate for males shows a very slight increase while it increases substantially for females particularly over the past two years. The hourly rate difference between both groups reduced significantly from 30% at the beginning of the period to 11% in 2009. The company s statement that the situation has improved is correct but equality has not yet been achieved. However, if the trend continues the average hourly rate shows signs of convergence in the short-term. 6 Marks [Total 20 Marks] 9
SOLUTION 4 The data provided is set out below 2003 2008 Price per Sales by Price per Sales by 1000kgms value m 1000kgms value m Sand 148 444 178 712 Cement 162 648 184 920 Limestone 202 1010 242 1210 Gravel 130 390 135 405 Using Laspeyres and Paasche Price and Quantity indices. To find Laspeyres index, base year sales can be used as weights for the price and quantity relatives. However, to find Paasches index current year s sales cannot be used. It is better to find actual quantities by dividing values by prices. We are using 2003 as the base year. Po PoQo Qo Pn PnQo PnQn Qn PoQn Sand 148 444 3 178 534 712 4 592 Cement 162 648 4 184 736 920 5 810 Lime 202 1010 5 242 1210 1210 5 1010 Gravel 130 390 3 135 405 405 3 390 2492 2885 3247 2802 Price Indices Laspeyres: PnQo x 100 = 2885 x 100 = 1.157 PoQo 2492 Paasche: PnQn x 100 = 3247 x 100 = 1.158 PoQn 2802 5 Marks Quantity Indices Laspeyres: PoQn x 100 = 2802 x 100 = 1.124 PoQo 2492 Paasche: PnQn x 100 = 3247 x 100 = 1.125 PnQo 2885 5 Marks Interpret the index numbers and explain any difference between both These indices are weighted aggregate indices. Normally prices are weighted by quantities and quantities by prices. The Laspeyres price index uses base time period quantities as weights while the Paasche price index uses current time period quantities as weights. When prices are rising the Laspeyres index tends to overestimate price increases and the Paasche index tends to underestimate price increases. The Laspeyres price index is considered a pure price index since it compares like with like from period to period, while, in the case of the Paasche price index, the weights change from period to period and like is not being compared with like. Since the Laspeyres price index uses base period quantities as weights, it can become out of date while the current quantities used as weights by the Paasche index are always up-to-date. 10
The Laspeyres price index needs only base period quantities regardless of the number of periods for which the index is being calculated. This is an advantage over the Paasche index which needs new quantities for each time period. Similar considerations to the above apply to the Laspeyres and the Paasche quantity indices. In general the Laspeyres index is more frequently used, for the above reasons, than the Paasche. The Paasche price index implies that base year quantities do not vary over time. However, in the present case both indices are of the same level. Although there appears to be substantial increases in both prices and sales values over the 5 year period, the quantities over the period are substantially similar. This means that there is little difference between the price and quantity indices. However, this summary should be put into the context of the above analysis. 11
SOUTION 5 The general form of a 90% confidence interval for a population mean is: X ± 1.645(σ/ n) where X = 12.1, s = 4.2 (since we do not know the value of σ we use the sample standard deviation). For 125 records, the confidence interval is 12.1 ± 1.645 (4.2/ 125) 12.1 ± 0.62 or from 11.48 to 12.72 Since the cost of a vacant seat is 50, the loss of revenue pre flight is between 574 and 636. After 4 years (8 six monthly periods) the accumulated value (A) of the investment is A = P(1 + i)n where P is the initial investment of 12,000; i = 5%; n = 8. Therefore, A = 12,000(1 + 0.05)8 = 12,000(1.05)8 = 12,000 x 1.477 = 17,729. The effective annual rate is (1.05)2 1 = 1.1025-1 = 10.25%. (iii) Cost of machine: 90,000; depreciation rate 15% pa. Since reducing balance depreciation is the converse of compound interest with larger amounts being deducted from the original asset value each year, the following formula can be used. Vt = Vo(1 i)t where Ao = 90,000, i = 0.15, t = 5. Value of machine after 5 years: 90,000(1 0.15)5 = 90,000(0.85)5 = 39,933. Therefore the total amount of depreciation is (90,000 39, 933) = 50,067. 6 Marks 12
SOLUTION 6 The financial terms have been discussed in the Student Bulletin and are detailed below. Simple interest is a fixed percentage (i%) of the principal (Po) paid to an investor each year irrespective of the number of years (n) the principal has been left on deposit. However, in modern business this approach is rarely adopted the interest is compounded, that is, At = Po(1 + i%n) Compound interest. Methods are developed for calculating the accumulated value and present value of an investment. If a person deposits Po with a financial institution at a rate of interest of i% per annum and leaves any interest to accumulate within the account the interest is earning interest. After t years the initial investment grows to Po(1 + i)t. After t intervals of time, where t can be a month, quarter year, half year, etc, the accumulated value is At = Po(1 + i)t. Annual Percentage Rate (APR): Rates of interest giving an actual rate of interest over a stated interval of time, are effective rates of interest. Where the effective rate of interest is expressed as a fraction of a year (1/p) it may be converted to an annual rate by multiplying by p. thus, 3% per quarter would be quoted as 12% per annum, converted quarterly. Interest rates quoted in this way are known as nominal rates of interest. Corresponding to a nominal rate of interest, there exists an effective annual rate of interest. A rate of interest expressed as 10%pa, convertible half yearly, is the same as an effective rate of interest of 10.25% or quoted as the APR (annual percentage rate). This process may be generalized as follows: Nominal rate compounded n times per year: At = Po(1 + i/n)nt APR rate compounded annually: At = Po(1 + APR)t Since the yield is the same, Po(1 + i/n)nt = Po(1 + APR)t giving an APR of (1 + i/n)n 1. Annuities. An annuity is a series of equal payments, investments or withdrawals made at regular time periods generally a payment made towards insurance policies, personal loans, hire-purchase payments, investment and pension funds, weekly, monthly or annually. There are various different types of annuity but if the investment is made at the time of compounding it is an ordinary annuity. Annuities may be paid at the end or the beginning of payment intervals. The term of an annuity may begin and end on fixed dates, may be a contingent or perpetual annuity that carries on indefinitely. The means to derive the value of an investment over a period of t years, where Ao is the amount invested at the end of each year, is the series Ao + Ao(1 + i) + Ao(1 + i)2 + Ao(1 + i)3 +.. + Ao(1 + i)t-1. This is a geometric series where a = Ao, r = (1 + i) and the sum, that is, the total amount of the annuity, becomes [Ao(1 + i)t 1]/i the total value of the annuity at the end of t years. This basically is the compound interest for fixed deposits at regular intervals of time. However, if a series of equal payments or withdrawals will be made in the future, it is necessary to get the present value of the annuity. The key question being asked is: how much should be invested now (Vo) for t years at a given rate of interest, i% per annum, to cater for a series of equal annual payments, Ao. The value of a series of regular payments at the end of t years is Vt = [Ao(1 + i)t 1]/i; if the amount invested [Vo(1 + i)t] is adequate to provide for this series of payments, then Vo(1 + i)t = [Ao(1 + i)t 1]/i. Therefore, the value of the investment at the end of t years is equal to the investments compounded annually. This simplifies to Vo = Ao[1 - (1 + i)-t]/i where [1 - (1 + i)-t]/i is called the annuity factor. Mortgages. Many companies and individuals borrow monies and make arrangements that the loan and interest is repaid in equal amounts over equal periods of time. A loan of this type is a repayment mortgage and is said to be amortised. For such a repayment the values of the loan and interest are normally known but the regular amount to be repaid must be calculated. To calculate the value of each repayment, generally in monthly periods, the debt at the end of t years will equal the value of a series of monthly repayments over the period of t years, that is, an annuity. This gives Po(1 + i)t = [Ao(1 + i)t 1]/i where Po is the mortgage and Ao is the annual repayment. If the payment is monthly, t is replaced by 12t. By simplifying this equation the regular annual payment (Ao) is Po[i/{1 (1 + i)-t}]. Sinking Funds. A sinking fund is created by setting aside a fixed sum of money each year with the objective of repaying debts/loans or making provision for the replacement of assets or equipment. This is similar to establishing an annuity to make provision for the loan repayment where a deposit is made at the beginning of each year. If a fixed sum is set aside each year, Ao, the value of the fund will grow annually. At the end of year 1, the value of the fund is Ao(1 + i) + Ao, etc. At the end of t years, the value of the fund is Ao(1 + i)t + Ao(1 + i)t-1 + (1 + i)t-2 + ---------- Ao(1 + i). By using a geometric series, as previously, the value of the fund is derived as Ao(1 + i){[(1 + i)t 1]/i}. If the fund is compounded monthly, i and t become i/12 and 12t. 13