BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
NOTES:
FORMATION 1 EXAMINATION - APRIL 2010
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:
(i)
Set out the net cash flows for the machine.
(8 Marks)
(iii)
Find the Internal Rate of Return.
(6 Marks)
(ii)
2.
Derive the Net Present Value using a discount rate of 18%.
(6 Marks)
[Total: 20 Marks]
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
16
20
24
28
32
36
40
44
48
<
<
<
<
<
<
<
<
<
No. of employees
20
24
28
32
36
40
44
48
52
1
2
3
11
14
12
9
5
3
You are required to:
(i)
Calculate the mean and standard deviation of the hours worked.
(iii)
Derive the skewness of the data from the observations in (i) and (ii) and interpret your result.
(ii)
Present the data on a cumulative frequency curve and derive the median hours worked.
(8 Marks)
(8 Marks)
(4 Marks)
[Total: 20 Marks]
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
Male
Female
600
350
700
360
650
380
670
420
2006
2007
2008
2009
Average Hours worked / week
Male
Female
42
35
50
35
45
30
45
32
You, as the Equality Officer, are asked to adjudicate on the claim. In order to do this:
(i)
(ii)
(iii)
4.
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.
Interpret the data for the company.
(6 Marks)
(6 Marks)
[Total: 20 Marks]
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:
Sand
Cement
Limestone
Gravel
Price per
1000kgms €
2003
148
162
202
130
Sales by
value €m
444
648
1010
390
2008
Price per
Sales by
1000kgms €
value €m
178
184
242
135
712
920
1210
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:
(i)
(ii)
Calculate price and quantity index numbers for 2008.
(12 Marks)
Interpret the index numbers that you have calculated and explain the difference between both index
numbers.
(8 Marks)
[Total: 20 Marks]
2
5.
(i)
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)
(ii)
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.
[Total: 20 Marks]
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:
(i)
(ii)
(iii)
(iv)
(v)
Simple and Compound Interest.
Annual Percentage Rate.
Annuities.
Mortgages.
Sinking funds.
[Total: 20 Marks]
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
(i)
Net Cash flows for the machine are set out in the following table
Year end
0
1
2
3
4
5
(ii)
Net Present Value
Year
Year
Year
Year
Year
Year
0
1
2
3
4
5
Discount
Factor @ 18%
0
0.8475
0.7182
0.6086
0.5158
0.4371
Net Present Value
(iii)
Internal Rate of Return
Calculating the IRR by means of
Cash Outflows
(Costs)
(80,000)
(2,000)
(2,300)
(2,650)
(3,042)
(3,498)
(3,000)
2 Marks
Cash Inflows
(Revenues)
Net Cash Flows
-----25,000
27,500
30,250
33,275
36,603
(80,000)
23,000
25,200
27,605
30,230
30,105
Discount
Factor @ 24%
Present
Value
2 Marks
Present
Value
(80,000)
19,493
18,099
16,800
15,593
13,159
83,144
3,144
0
0.8065
0.6504
0.5245
0.4230
0.3417
6 Marks
N1I2 - N2I1, where N1 = 3,144, I1 = 0.18, N2 = (7,507),
N1 - N2
I2 = 0.24
= 3,144 x 0.24 - (7,507) x 0.18
3,144 - (7,507)
= 754.6 + 1351.3
10,651
5
=
2,105.9
10,651
= 19.77%
4 Marks
(80,000)
18,550
16,390
14,479
12,787
10,287
72,493
(7,507)
3 Marks
3 Marks
SOLUTION 2
(i)
Cumulative frequency curve and median
Hours
Worked
16 < 20
20 < 24
24 < 28
28 < 32
32 < 36
36 < 40
40 < 44
44 < 48
48 < 52
∑
Mid Value
x
18
22
26
30
34
38
42
46
50
Mean = x = ∑fx
∑f
1
2
3
11
14
12
9
5
3
60
Cum
frequency
1
3
6
17
31
43
52
57
60
fx
18
44
78
330
476
456
378
230
150
2,160
(x – x)
(18)
(14)
(10)
(6)
(2)
2
6
10
14
(x – x)2
324
196
100
36
4
4
36
100
196
=
f(x – x)2
324
392
300
396
56
48
324
500
588
2,928
4 Marks
= 1540 = 30.8
50
Standard deviation =
(ii)
Frequency
σ = √ ∑ f (x - x)2
∑f
√ 95.97
=
√ 4798.76
50
=
4 Marks
€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.
4 Marks
Graphical derivation of the median.
60
50
Cum Frequency
40
30
20
10
Median
15
20
25
30
35
40
Hours worked
6
45
50
55
4 Marks
(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)
standard deviation (σ)
=
3(30.8 – 31)
9.79
=
- 0.06
2 Marks
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.
2 Marks
[Total: 20 Marks]
7
SOLUTION 3
(i)
The graphs for ‘average earnings / week’ v ‘average hours worked’ are set out below:
Year
Average Earnings / week
Male
Female
600
350
700
360
650
380
670
420
2006
2007
2008
2009
Average Hours worked / week
Male
Female
42
35
50
35
45
30
45
32
Comparison of average earnings per week
800
4 Marks
Male
700
600
Average Earnings/week
€
500
Female
400
300
200
2006
2007
2008
Comparison of average hours worked per week
60
2009
4 Marks
Male
50
40
Hours worked
Female
30
20
10
8
Year
(ii)
Comparison of average hourly rates
Average hourly rate €
Female
10.0
10.3
12.7
13.1
Male
14.3
14.0
14.4
14.8
2006
2007
2008
2009
16
Average hourly rate
€
14
% Difference
30.1
26.4
11.8
11.5
3 Marks
Male
Female
12
10
8
2006
(iii)
2007
Year
2008
2009
3 Marks
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
(i)
The data provided is set out below
Price per
1000kgms €
148
162
202
130
Sand
Cement
Limestone
Gravel
2003
Sales by
value €m
444
648
1010
390
Price per
1000kgms €
178
184
242
135
2008
Sales by
value €m
712
920
1210
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.
Sand
Cement
Lime
Gravel
∑
Po
148
162
202
130
Price Indices
Laspeyres:
Paasche:
Quantity Indices
Laspeyres:
Paasche:
(ii)
PoQo
444
648
1010
390
2492
Qo
3
4
5
3
Pn
178
184
242
135
∑PnQo x 100 = 2885 x 100
∑PoQo
2492
PnQo
534
736
1210
405
2885
PnQn
712
920
1210
405
3247
=
1.157
∑PnQn x 100 = 3247 x 100
∑PoQn
2802
=
1.158
∑PoQn x 100 = 2802 x 100
∑PoQo
2492
=
1.124
∑PnQn x 100 =
∑PnQo
=
1.125
3247 x 100
2885
Qn
4
5
5
3
PoQn
592
810
1010
390
2802
5 Marks
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.
2 Marks
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.
2 Marks
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.
2 Marks
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
4 Marks
SOUTION 5
(i)
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).
2 Marks
For 125 records, the confidence interval is
12.1
± 0.62 or from 11.48 to 12.72
12.1 ± 1.645 (4.2/√125)
Since the cost of a vacant seat is €50, the loss of revenue pre flight is between €574 and €636.
(ii)
4 Marks
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)
2 Marks
4 Marks
2 Marks
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.
4 Marks
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.
4 Marks
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’.
4 Marks
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}].
4 Marks
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.
4 Marks
[Total: 20 Marks]
13