BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
NOTES:
FORMATION 1 EXAMINATION - AUGUST 2009
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 - AUGUST 2009
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.
As the Management Accountant to a company, you are assessing a purchasing proposal for the Production
Department. The Manager has proposed the purchase of a machine for €50,000. He projects that the cash inflows
and costs shown in the following table will arise:
Year
1
2
3
4
5
6
Sale Revenues €
10,000
15,000
20,000
15,000
10,000
5,000
Maintenance Costs €
500
500
1,000
1,000
1,500
2,000
The machine can be sold after 6 years for €10,000. All cash flows are at the end of the year. The cost of capital
to the company is 10%. To assess the proposal you are required to:
(i)
Calculate the net present value of the project.
(iii)
Provide an explanation of these terms and the most appropriate method of appraisal of a project of this type.
(6 Marks)
(ii)
(8 Marks)
Calculate the discounted payback period.
(6 Marks)
[Total: 20 Marks]
1
2.
The percentage annual growth of 50 small pension funds over the year 2008 was collected by the Pensions Board.
The frequency table of the data is presented below. For a company management meeting you are required to
perform the following:
(i)
(ii)
(iii)
Draw a histogram and cumulative frequency graph of the data.
Estimate the median growth in the funds from either of the graphs and compare it to the
calculated median.
Comment on the value of the graphs for presentation purposes.
% Growth per annum
0 <X < 4
4 <X < 6
6 <X < 8
8 < X < 10
10 < X < 12
12 < X < 14
14 < X < 16
3.
Frequency
6
12
11
9
5
4
3
(6 Marks)
(4 Marks)
[Total: 20 Marks]
As the Manager of DIY Manufacturing Ltd. you use sample statistics to make inferences about population
parameters. You have been assigned the task of confirming that the production of its miniature batteries is
statistically acceptable. The company claims that the average weight of the batteries is 30 grams. The batteries
are filled with powder to an average weight of 29.5 grams with a standard deviation of 2.1 grams. From a random
sample of 36 batteries:
(i)
(ii)
(iii)
Calculate the probability that the average weight is 30 grams or more.
(8 Marks)
If the sample size was smaller than 36 how would this affect the results you derived in
(i) and (ii).
(4 Marks)
Calculate the limits within which 95% of all batteries weigh.
(You are required to use appropriate diagrams to support your calculations)
4.
(10 Marks)
(8 Marks)
[Total: 20 Marks]
Over the past 5 years the % level of unemployment has been tracked by the Central Statistics Office and is set out
in the table below. Political opponents claim that the underlying trend is increasing and is disguised by the seasonal
nature of the data. In order to present an accurate assessment of the data you are required to:
(i)
(ii)
(iii)
Smooth the data by means of a four quarterly moving average and briefly describe the method used.
(10 Marks)
Plot the trend on a graph.
(6 Marks)
Comment on the claim that the “underlying trend is increasing”.
Year
2003
2004
2005
2006
2007
March
2.0
2.6
3.6
2.5
2.1
June
2.1
3.0
3.2
2.7
2.3
2
September
2.4
3.4
3.3
2.5
December
2.4
3.6
2.9
2.1
(4 Marks)
[Total: 20 Marks]
5.
As a Financial Consultant for CPA International Investments you are required to provide advice to the company’s
clients on the following:
(i)
(ii)
(iii)
6.
Client A wishes to invest in a supplementary pensions product to support his retirement income when he
retires in 10 years time. He wishes to invest a lump sum each year of €2,000. He expects to receive an
interest rate of 6% per annum. Advise him on the value of the fund at the end of the 10 year period.
(8 Marks)
Client B has been offered the opportunity to invest in a property development. He believes that the average
rental income for an apartment is at most €10,000 per year. A random sample of 36 apartments gives a
mean rent of €10,200. The sample standard deviation is €1,450. You are asked to calculate a 90%
confidence interval for the mean annual rental income. Do the sample results support Client B’s belief?
(6 Marks)
The Superior Products company will only supply jeans when a price greater than €25 per unit is available.
It will increase output by 2 units for every unit increase in price. Write the equation of the supply function for
the company so that it can estimate the units to produce when the price is €80.
(6 Marks)
[Total: 20 Marks]
“Without doubt an index is an extremely fashionable tool”. In the context of this statement explain the following
terms with regard to index numbers.
●
●
●
●
●
Indices – their use and construction.
An expenditure index.
A price index.
A volume index.
The base period.
[Total: 20 Marks]
END OF PAPER
3
SUGGESTED SOLUTIONS
THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND
BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - AUGUST 2009
SOLUTION 1
(i)
Net Present Value of the project.
The cash inflows, cash outflows and net cash flow is prov9ded in the following table. Applying the discount factor
of 10% provides a Net Present Value of €7,054.5. On this basis the project is viable and should proceed.
(8 Marks)
Year
0
1
2
3
4
5
6
NPV
(ii)
(iii)
Cash
Outflow
€
Cash
Inflow
€
50,000
500
500
1,000
1,000
1,500
2,000
---10,000
15,000
20,000
15,000
10,000
5,000
10,000
(2 Marks)
(2 Marks)
Net
Cash
Flow
Discount
Factor
@ 10%
(50,000)
9,500
14,500
19,000
14,000
8,500
13,000
1.000
0.909
0.826
0.751
0.683
0.621
0.564
Discounted
Cash
Flow
Discounted
Cash
Outflow
Discounted
Cash
Inflow
7,054.5
54,360.0
61415.0
(50,000.0)
8,635.5
11,977.0
14,269.0
9,562.0
5,278.5
7,332.0
(4 Marks)
(50,000)
(454.5)
(413.0)
(751.0)
(683.0)
(931.5)
(1128.0)
(2 Marks)
----9,090.0
12,390.0
15,020.0
10,245.0
6,210.0
8,460.0
(2 Marks)
Discounted payback is the time it takes for the discounted incoming cash flow to recover (or break even with) the
discounted outgoing cash flows. It is regarded as the time it takes for the project to become profitable. In the
present case the breakeven occurs in the first quarter of year 6. This is in agreement with the NPV.
(6 Marks)
Explanation and comparison of the terms.
The NPV measures the added value to the company of undertaking the project. It is calculated by using a rate of
interest equal to the rate of return that the company would expect to pay to finance the project. NPV measures the
overall profit of the project. The risk of the project is taken into account by means of the cost of capital
employed.Cash flows should, therefore, not include interest payments as these are emplicitely taken into account
in the discounting process. If cash inflows occur later than cash outflows, the cash outflows, the cash inflows will
be discounted more in the NPV calculations.
Discounted payback is often useful if capital is scarce. A project which becomes profitable earlier may be
considered better than one which becomes profitable later. It allows capital / loans to be repaid and other
investments to be proceed. A project which takes longer but which repays capital more slowly might have a higher
NPV> However, if the cost of capital used in the NPV calculations is correct, it should not be of concern when the
project becomes profitable. Later cash flows are discounted more. However, it may be better to use a higher
interest rate for discounting cash flows from projects that take a greater length of time. This is a better way to take
time preferences into account than using discounted payback. It is often considered useful to know the discounted
payback period for the process of financial planning.
‘Payback period’ is the same as discounted payback except that cash flows are not discounted. The discounted
payback period for investment appraisal is not often used. It is a method, however, of ensuring the preference for
early cash flows over later cash flows is taken into account but is is better to do so using the NPV approach.
(6 Marks)
5
[Total: 20 Marks]
SOLUTION 2
(i)
A histogram and cumulative frequency graph.
% Growth per annum
0
4
6
8
10
12
14
≤X
≤X
≤X
≤X
≤X
≤X
≤X
≤
≤
≤
≤
≤
≤
≤
Frequency
4
6
8
10
12
14
16
Cumulative
frequency
6
12
11
9
5
4
3
Frequency
density
6
18
29
38
43
47
50
1.5
6.0
5.5
4.5
2.5
2.0
1.5
Histogram
6
Frequency
Density
5
4
3
2
1
0
Cum Frequency
2
4
6
8
10
12
14
Growth (% per annum)
16
(4 Marks)
50
40
30
20
10
0
2
4
Approx Median 7.2%
6
8
10
12
Annual Growth (%)
6
14
16
(4 Marks)
(ii)
(iii)
The median can be calculated or derived from the graph and is approx. 7.2%.
(4 Marks)
Therefore, median is 6 + (7.5/11) x 2 = 7.36%.
(4 Marks)
The calculated value of median is: (N + ½)th observation. Since there are 18 observations up to this interval, the
25½th value is the 7½th value in the interval.
Value of graphs for presentation purposes.
Cumulative frequency tables are suitable for discrete and continuous data and show the number of observations
below the end of the current interval. This allows us to see that 29% of funds (the majority) reported levels of growth
below 8%. A bar chart or histogram is an effective way of displaying discrete or categorical data. In an histogram
frequency is represented by area rather than height. For this reason it is not necessary to have intervals of equal
width. Histograms depict continuous data so the rectangles are drawn touching each other. The vertical scale
represents ‘frequency density’ (frequency / class width) rather than frequency otherwise the first bar would give a
misleading representation of growth if actual frequencies were used rather than relative frequencies.
The histogram gives a better visual representation of the data for presentation.
The ‘ogive’ gives a greater facility for calculating a range of values, eg. How many funds had growth in excess of
12% etc. or what is the interquartile range (measure of dispersion not affected by extreme values) 9.9% - 4.3% =
4.6%.
(4 Marks)
[Total: 20 Marks]
SOLUTION 3
(i)
The question asks for the probability that the mean is greater than or equal to 30; P(x ≥ 30).
Since the sample size is greater than 30, the sample is normally distributed.
Therefore, the mean and standard error of the distribution of mean is
µx = µ = 29.5
σx = σ/√n = 2.1/√36 = 2.1/6 = 0.35
(2 Marks)
Therefore, for x = 30, z = 30 – 29.5 = 1.43
0.35
(2 Marks)
From the Normal tables, the area in the tail of the distribution = 0.0764.
This is the required probability: P(x ≥ 30gm) = 0.0764 = 7.64%
(2 Marks)
The appropriate Normal Distribution is
Area = P(x ≥ 30)
µx = 29.5
30
0
X
Z
7
(2 Marks)
(ii)
Two limits that contain 95% of weights; σ = 2.1
0.025
x = µ - z (2.1)
- 1.96
0.475
0.475
µ = 29.5
0.025
x = µ + z (2.1)
0
+ 1.96
X
Z
Since we are dealing with individual values we use σ.
From the Normal tables, the z value for the tail area 0.025 is + 1.96.
(2 Marks)
The number of units from the mean is z x σ = 1.96 (2.1) = 4.116 gms.
(2 Marks)
That is, 25.384 to 33.616. The weights of 95% of batteries are within these limits.
(2 Marks)
Hence the two limits are µ ± zσ = 29.5 ± 4.116
(iii)
(2 Marks)
If the sample size is smaller than 36, this could involve the distribution for small samples – Student’s t distribution
where it cannot be assumed that the population is normally distributed. However, if the assumption is made that
the smaller population is normally distributed, smaller samples give a smaller standard error of the mean.
In (i) above the larger standard error will result in a smaller value of z giving a larger value of probability.
(2 Marks)
In (ii) since the question does not involve samples but individual batteries, there is no change.
8
(2 Marks)
[Total: 20 Marks]
SOLUTION 4
Year
March
June
September
December
2004
2.6
3.0
3.4
3.6
2003
2.0
2005
3.6
2006
Quarter
M (1)
J (2)
S (3)
D (4)
M (1)
J (2)
S (3)
D (4)
M (1)
J (2)
S (3)
D (4)
M (1)
J (2)
S (3)
D (4)
M (1)
J (2)
2004
2005
2006
2007
(ii)
2.9
2.5
2.3
Unemployment
2.0
2.1
2.4
2.4
2.6
3.0
3.4
3.6
3.6
3.2
3.3
2.9
2.5
2.7
2.5
2.1
2.1
2.3
2.4
3.3
2.7
2.1
Year
2003
2.4
3.2
2.5
2007
(i)
2.1
2.1
Annual Total
Pair Total
Trend
8.9
9.5
10.4
11.4
12.6
13.6
13.8
13.7
13.0
11.9
11.4
10.6
9.8
9.4
9.0
18.4
19.9
21.8
24.0
26.2
27.4
27.5
26.7
24.9
23.3
22.0
20.4
19.2
18.4
2.4250
2.4875
2.7250
3.0000
3.2750
3.4250
3.4375
3.3375
3.1125
2.9125
2.7500
2.5500
2.4000
2.3000
(2 Marks)
(2 Marks)
(2 Marks)
The trend is calculated as the centred moving average and is plotted on the graph below. The four point moving
average is taken as the natural time period of the series and data is quarterly. The centred moving average
therefore gives the trend.
(2 Marks)
Raw data (dashed line)
4
3
Unemployment
(%)
2
Trend (dotted line)
1
1
2
3
4 5
2003
6
7
8 9
2004
10
11 12
2005
13
14
15 16
2006
17
18
19
2007
Quarter
(6 Marks)
The graph shows a seasonal affect for the first two years where it peaks at the end of 2004 and beginning of 2005.
It tends to peak again at the beginning of 2006 but the trend is downwards for 2007. It is difficult to discriminate
between the raw data and the trend since the variation in the percentage data is so small.
(2 Marks)
9
(iii)
Although the raw data shows an increase in the trend over the period, the data for 2007 is reducing. The trend
indicates that the data peaked over the period 2005/2006 and since that time shows a reducing or downward trend.
The claim that “the underlying trend is increasing” cannot be supported by the data.
4 Marks
[Total: 20 Marks]
SOLUTION 5
(i)
This payment is an annuity – a series of equal deposits made at equal intervals of time. The principle (P) of the
investment is
Year 1 = €2,000
Year 2 = €2,000 (1 + 0.06) + €2,000
Year 3 = €2,000 (1 + 0.06)2 + €2,000 (1 + 0.06) + €2,000, etc.
This series is a geometric progression of the form
For 10 years, the value is
€2,000[(1 + 0.06)10 - 1]
0.06
=
=
(ii)
Sn = a(rn - 1)
r - 1
€2,000(1.7908 - 1)
0.06
€26,360
To construct the confidence interval you require:
X, sx = s/√n, z (90%)
The 90% confidence interval is x ± zsx, where x = €10,200, z = 1.6449,
sx = 1450/√36 = 1450/6 = 241.6
(3 Marks)
(2 Marks)
(2 Marks)
Hence, the 90% confidence interval is €10,200 ± 241
(2 Marks)
Therefore since the confidence interval contains €10,000 your client’s belief is confirmed.
(2 Marks)
That is, €9,959 to €10,441. You are 90% confident that the mean lies within this range.
(iii)
(3 Marks)
The general form of a linear equation to represent this supply function is
Y = aX + b where Y = Quantity and X = Price.
Q = aP + b.
From the data given, P = €25 when Q = 0 and the slope of the line is 2
(that is the change in Y per unit change in X).
Therefore,
0 = 2 x 25 + b; b = -50
The supply equation is Q = -50 + 2P
When the price is €80, the quantity produced is 110 pairs.
(3 Marks)
(3 Marks)
[Total: 20 Marks]
10
SOLUTION 6
Indices – Use and Construction
Economists, statisticians and business specialists have constructed a means to measure the magnitude of
economic changes over time. This measure is an index number and is used for international comparisons of
economic and business data. Because they work in a similar way to percentages they make such changes easier
to compare. The Consumer Price Index is probably the most widely quoted index for everyday economic purposes.
It attempts to measure the change in the price of a wide range of goods and services that we purchase regularly.
The cost of living index is probably the most familiar to many people but a wide range of other indices are in use:
Index of Retail Prices, Index of Industrial Production, Dow Jones, Nasdaq, etc. A particular time period – the base
period - is chosen and the variable for that period is given a value of 100. An index is calculated for the remaining
periods on the assumption that the base period has a value of 100. An index gives the change (percentage) that
has taken place since the base period.
(4 Marks)
Simple Aggregate Index.
If we take account of unit prices over current and base periods, this gives the simple aggregate index. It is not very
useful because the quantities may differ so much from each other over the periods and the unit prices may be for
different quantities of products.
An Expenditure Index.
Expenditure is made up of two different elements, prices and quantities bought, any change in either will cause a
change in expenditure. If base year prices and quantities (Po, Qo) and current year prices and quantities (Pn, Qn)
are used an expenditure index can be calculated in the form ∑PnQn/∑PoQo. In this case account has been taken of
the different quantities by multiplying the unit prices by the corresponding quantities. This process is called
weighting.
(4 Marks)
A Price Index.
To measure prices from a base year we can use the quantities purchased in the base year to weigh the unit prices
in both years. If quantities are held constant any change in outlay or expenditure will be entirely due to the change
in price. This is a base weighted price index or Laspeyres Price Index which is given by ∑PnQo/∑PoQo. In practice,
the Laspeyres price index is usually calculate using price relatives. For this method the expenditures in the base
year are used as weights. This method is used because it is easier to obtain data on expenditure than on actual
quantities bought when we are dealing with a large complicated index. The base weighted index has the advantage
that the base year expenditures have to be worked out once. These can then be used in the calculation of the index
in any subsequent period. However this index can be misleading. For example fluctuations in a particular element
might have a considerable impact on an index. The popularity of a particular element could dramatically affect the
quantities sold and an index which uses base year quantities from some time in the past could be misleading. This
particular problem can be avoided by using Paasche’s price index or the end year weighted price index. In this
particular case we have ∑PnQn/∑PoQn.
(4 Marks)
A Volume Index.
However, if the prices remain relatively stable and the quantities of items change, it is more useful to calculate an
index based on quantities, using prices as weights. These are the base weighted or Laspeyres Volume index and
the end weighted or Paasche’s Volume index and are given by ∑PoQn/∑PoQo. and ∑PnQn/∑PnQo.
(4 Marks)
The Fisher price index gives a geometric mean of Laspeyres and Paasche indexes.
Fischer = √Laspeyres x Paasche.
The price indices answer the question: By how much have prices increased?
The Volume indices answer the question: By how much have quantities increased?
While a third question could be asked: How much more money is being spent? This can be answered by the Value
index ∑PnQn/∑PoQo
Base period.
One problem in the development of any index number is selecting a suitable base period. A base period should be
chosen where prices or volumes are not unnaturally high or low. Also a base period should not be too far in the
past. Since tastes and availability can change substantially over time such an index can be seriously misleading.
One means to avoid such problems is to use a chain based system where, in calculating successive index
numbers, the base used is the previous period. A chain-based index number is particularly suited for period by
period comparisons while a fixed-base index number makes it easier to compare movement of prices over time.
(4 Marks)
11
[Total: 20 Marks]